Chapter 1. How by this Iewel to take the altitude of the Sunne, Moone, or Starres.

THough this matter be too too wel knowne alreadie, euen to the common sort: yet since other authors haue made it their first Chapter, I refuse not to follow the same course. A degree in latin Gradus, is the 360. part of a circle: Altitudo, in English altitude, or height: the same in all Astronomical obseruation, is taken by the degrees of a circle, or a relation vnto them. VVherefore to take the al­titude of the Sunne, let hang the Iewel in your hand at libertie, turning the edge of the Limbe towards the sunne, the rule on the backside lying euen in the line of leuell, which line representeth here your horison: then lift vp that ende of the rule next the sunne (or put downe the other, all is one) vntill you may see the sunne beames to peirce both the sight holes, and looke what degree of the Limbe the rule cutteth, counted from the line leuell, the same is the Sunnes al­titude for that instant. For starres you must with one eye shut, looke through the greater sight holes, made of purpose for stars, till you see the starre thorow them both, and then doth the rule shew your altitude. Also a man may, if his light holes be too little, or the starre dimme, take the starre euen with the vpper edges of the sights, so they be parallel to the holes, which of some is thought the best way. Further if your instrument the heauie, or your hand vncertaine, it is best to hange it by a naile on a tree, staffe, or post, this taking of altitude will be much in vse hereafter.

Note that when you take the altitude of the Sunne, Moone, or any starre being in the Meridian circle, then is the same called the meridian altitude.

Chapter 2. How to get or rather examine the Sunnes greatest declination, and distance of the Tropickes, and thereby also to learne the latitude of any countrie.

THe Eclipticke lieth alwayes byas, or inclining to the equinoctiall, according to the sunnes greatest decli­nation, which in these dayes is 23. degrees 30. minutes, or rather after late writers 23. degrees, 28. minutes, which to proue, take the meridian altitude of the sunne in the highest of sommer, being in the very Solstice, [Page] [Page]

[Page] [Page 25]and againe in the deepest of winter, being in the winter Solstice Deduct the lesser meridian height, out of the greater, the remainder is the distance of the tropicks: halfe that, is the Sunnes greatest declination. For example.

In the yeare 1579. about S. Barnabies day, because the countrie folke account that the longest day in the yeare: for two or three dayes before, and after, I watched the sunne comming to the south, and there I tooke his highest alitude ech day, and I found that the 13. day of Iune the sunne had the highest Meridian altitude, which was 62. degrees that I noted in my book. Then about saint Lucies day in winter, counted the shortest day in the yeare, I watcht the sunnes lowest meridian altitude, and found it the 13. of December to be 15. degrees then of these two I subduct the lesser vz. 15. out of 62. the greater: there resteth 47. and that is the di­stance betweene the tropickes, the halfe of that vz. 23 ½ is the greatest declination of the sunne, and so much doth the zodiacke decline from thequinoctiall, and as much doe the poles of the worlde, which are also the poles of thequinoctial, differ from the poles of the zodiacke.

Heere must you note that this greatest declination of the sunne hath not alwayes beene found one, of se­uerall authors at seueral times: for Ptolomeus & Aristarchus, Samius and Timochares, that were 400. yeares before him almost, did finde by obseruation that this declination was 23. degrees, 51. minutes, and 20. sec. and from their time hitherwards, it is founde to decrease by little and little. For mahometes Aratensis after Pto­lomeus 749. yeares, found it 23 degrees 35. minutes: after him 420. yeares, Prophatius Iudeus 23. deg. 32. minutes. Those of our times, as VVernerus, and Copernicus, haue found it 23. deg. and 28. minutes, and so is it thought to be at this time, of those that haue made obseruations With large instruments.

But now by this declination is had the latitude of any place, verie commodiouslie. For example: because be­fore I found the sunnes lowest meridian altitude 15. deg. adde thereto the sunnes greatest declination nowe knowne vz. 23. degrees. 30. minutes, it maketh 38. deg. 30. minutes, thequinoctials height aboue the horizon, which you shall deduct from the quadrant of a circle vz. 38. 30. out of 90. there resteth 5. 1. deg. 30. minutes, the height of the pole, or latitude at Reading, where I made the foresaid obseruation. But being made by smal instruments, I haue somwhat erred, for it is heere 51. ⅔. The contents of this Chapter serueth most truly as before, to all those that dwell in the temperate zones, that is whose zenithes are betweene the tropicke and pole circle. But to those that haue either tropicke in the zenith, the lowest meridian altitude of the sunne, be­ing in the winter solstice must be taken out of 90. & so resteth the distance of the tropicks. And to those that haue their zenith betweene the two tropickes, the two meridian altitudes taken in the solstices, must be ioy­ned togither, and the sunne amounting must be taken out of the semicircle vz. 180. and so resteth againe the distance of the tropickes, whose halfe vz. the sunnes greatest declination being added to the sunnes lowest meridian altitude, maketh theleuation of thequinoctial aboue any such horizon, which being takē (as before) out of 90. degrees, leaueth the poles eleuation. Likewise whose zenithes are in the colde zones, who haue the sun continually aboue the horizon certaine dayes, there must you take the suns greatest meridian altitude, being in the Solstice, and likewise his lest meridian altitude the same day (for the sun commeth twise visiblie in 24. houres, to the meridian euen as our north stars do) those two meridian altitudes ioyned togither, make the whole distance of the tropickes, those that haue either of the poles their zenith, there the highest meridi­an altitude, is the sunnes greatest declination, and those that haue the pole circles zenith, the greatest meridi­an altitude, in the sommer Solstice is the distance of the tropickes.

Blagraue. All these and many more you may learne by the very beholding of the Iewel, and might haue done if nothing had beene written thereof, for the Iewel being placed from latitude to latitude, as in the 2. Booke and 19. Chapter is shewed: you may reckon euen vpon the limbe all these altitudes aboue the Finitor, seuerally at pleasure, for the Almicantares are had Ipso aspectu, and truely all the circulations of the globe, more readie then if you had the sphere in your hand, and therefore reade well the last part of the second booke, to vnder­stand euery part of your Iewel.

Chapter 3. How by the theoricke on the backe of the Iewel, to get the sunnes place or degree in the zodiacke any day.

SEeke the day proposed in the circle of monethes, on the backside of your Iewel, and thereto lay the rule which in the circle of signes adioyning, sheweth the degree sought for.

Example.

This present day being the 7. of Iune 1582. I would know where the sunne is, vz. in what degree of the zodiacke, I resort to the backe of my Iewel, and seeke in the Limbe of monethes fore Iune: and there the 7. day, as they be figured, where on I lay the rule which I finde to cutte in the Limbe of the signes, adioyning the 25. degree, and almost 40. minutes of Gemini, seruing to the beginning of the said 7. day of Iune, which is the degree of the sunne sought for.

But here by the way you must note, that the circle of the yeare conteineth but 365. dayes, where as in truth it is 6. houres more verie neare, which is a quarter of a day, therefore when you make your theoricke, you must know to what yeare it is made, either the first, second, or third after the leape yeare: that knowne, your theorick serueth exactly ynough for that yere to which it is made, but to the next yere you must adde a quar­ter of the next day following to your day proposed, and then take the degree of the zodiacke seruing that, to the second yeare, take halfe the day following, to the third three quarters, and then the fourth is the yeare of the theoricke againe, and needeth nothing to be added: deuide the yeare of our Lord by 4. and if there remaine nothing it is leape yeare, if one two or three, as neuer it is aboue, it sheweth the yeares past from the leape yeare. Example.

This yere 1582. deuide 82. by 4. (for always the M. & C. yeare are leapyears) there resteth two, wherby I [Page 26] know that it is the second from the leape yeare, so that if my theoricke were made for the first yeare, then this being the second yeare, I adde to euery day, a quarter of the day following through the whole yeare, and so of the rest. These sixe houres foure times gathered make a day, and therefore euerie fourth yeare Februarie hath 29. dayes, and then is called the leape yeare, in latin Bissextus or annus intercularis, because a day is then put in betweene, other minutes wanting are not here to be regarded.

Chapter 4. How to finde the sunnes declination for euery day in the yeare, and also the declination of euery part of theclipticke.

VVHat the sunnes greatest declination is, it is said in the 2. Chapter. In like manner, euerie degree of the zodiacke hath his seuerall declination denominated by the parallel, cutting the same deg. wherefore theclipticke line being drawne on the Mater, and deuided as in the 4. Booke; and 6. Chapter is shewed: Learne by the last Chapt. the sunnes degree in the zodiacke, for the time proposed, seeke the same degree on theclipticke line, and then doth the parallel crossing the same, shew in the Limbe (or other­wise reconed from thequinoctial) the declination sought for. Example.

The first of may this yeare 1582. I woulde knowe the sunnes declination: by the last Chapter I finde the sunne to be in the 20. degree, and almost an halfe of ♉ I seeke that degree on theclipticke of the Mater, and there do I see to crosse the 18. ½. parallel, reckoned from thequinoctiall, and because the same parallel lieth towards the pole articke, or north pole from thequinoctiall, I conclude the sunne declineth 18. ½ degrees northwards, and so may you see the declination of any degree of theclipticke, whither the sunne be there or not: if theclipticke be not on your mater, or be vndeuided, the rule or Finitor may supplie it.

Chapter 5. How to finde the latitude or poles eleuation, any day proposed in any place, or countrie whatsoeuer.

THe latitude or poles eleuation, are in effect all one, and what they are is shewed in the first Booke 11. Chapt. first by the 3. Chapter seeke the degree of the sunne, for the day proposed, and by the 4. Chapt. the sunnes declination, then take the meridian altitude by the first, and reckon the same among the al­micantares, from the Finitor vpwards, turning about the Reete till the same almicantare do touch in the limbe, and that rightwards from the pole articke, the sunnes parallel of declination: and so doth the Finitor stande fixed to the latitude of that place: the degrees in the Limbe betweene the Finitor and the pole articke, sheweth theleuation thereof aboue the horizon of that countrie, or place.

Example. The 10. of Aprill 1582. the sunne being in ♉, by the 3. and his declination 11 ⅔. by the 4. I tooke his meridian altitude 50. deg. which done, I told out the 50. almicantare, and turned about the reete til the said 50. almicantare did meete the 11 ⅔ parallel (reckoned towards the north pole) on the limbe, and there the reete fixed, I reckon on the Limbe the degrees betweene the Finitor and the pole, or betweene the zenith and thequi­noctiall, which are alwayes equall, I finde 51 ⅔ degrees, which is the height of the pole, or latitude at Rea­ding. Note that in all workings for the most part, the zenith point of the reete must be aboue thequinoctiall towards the North pole.

You may do the same another way without the Iewel, thus: if the declination of the sunne be north, subduct the same from the meridian altitude, as here it was 11 ⅔, which taken out of 50. the remaine videl. 38 ⅓ is the equinoctials height aboue the horizon, which alwayes subducted from 90. leaueth the latitude vz 51 ⅔. But if the declination be south, adde the same to the meridian altitude, and so haue you thequinoctials, height which taken from 90. leaueth likewise the latitude.

Chapter 6. How to know what it is a clocke at any time the sunne shining, and in any region in the world.

I Cannot choose but set downe this Chapt. next because it is one of the chiefest that any yong beginner can delight in, and by my Iewel is one of the easiest where, in Gemma F. instrument, it was one of the hardest, [Page 27] and therefore he appointed it the 30. Chapter, because a man should be first well acquainted with his instru­ment. The latitude of your place or abiding, being first had by the 5. or otherwise, as shalbe shewed in other Chapters following, you must get the sunnes declination for the day proposed, by the 4. and then at the in­stant that you would know the houre, take the sunnes altitude by the first: which done, the Finitor being set to the latitude reckon the sunnes altitude so taken amongst the almicantares, and marke where his parallel of declination cutteth the same almicantare, for there the houreline passing by, doth shew the houre and deg. euerie deg. being 4. minutes, thus in briefe always the altitude being taken the houreline passing through the crossing of the Almicantare, and the sunnes parallel is your desire.

Blagraue. As oft as I looke what it is a clocke by my Iewel, and see with what facilitie, certaintie, and redi­nesse for euerie countrie I doe it, ouer that I was wont by Gemma Frisius instrument, with his Cursor and Brachulum, I can not but meruaile that hee and his sonne Cornelius G. coulde bee content with so tro­blesome a practise, which trulie I lothed euen vpon the first sight, which not onely in working this Chapt. but also in most of their whole booke is verie combersome, although in some Chapter more then in other, yet is Gemma Frisius to be excused, because he was the inuentor. Facilius enim est inuentis addere quam de nouo componere.

Example. The sixt day of October 1580. in the morning I went foorth of my studie at our mansion house at Bulmersh by Reading, where I was writing the first copie of this proposition at that present instant. The latitude being there 51 ⅔ degrees, and at the same time the sunne being in the 23. deg. 8. minutes of ♎, his de­clination by the 4. was 9. degrees southwards, and 6. minutes, all these things to him that vseth his Iewel daily will alwayes be in memory, I say, I went forth then, and turning the edge of my Iewel to the sunne, I founde his altitude 12. degrees by the first, which is as much to say, as the sunne was that instant in the 12. almican­tare, wherefore by the second booke 19. Chapter, setting the Finitor to our latitude vz. 51 ⅔ deg. I marked where the 12. almicantare reconed from the finitor, did cut the 9 6/60 parallel, reconed from thequinoctiall line towards the south pole, for that the sunnes declination was south, and there did the houreline that crossed this intersection shew me that it was 3. deg. past 8 of the clocke, that is 8. of the clocke, and 12. minutes past. For euerie degree is 4. minutes in reconing the houre at all times, though euerie degree in euerie other respect hath 60. minutes, and euerie minute 60. seconds, and euery second 6. thirds &c. vnto tenthes. It was 12. mi­nutes past 8. and therefore time to go to breakfast, for him that lay not last a bedde: I made hast into the hall where I found my companions at it, and so sharpe set, that had not my Iewel beene readier then Gemma Fri­sius Astrolabe, I had lost my breakefast with looking what it was a clocke, and therefore I had great cause to commend it at that time.

But what did I see more, euen the azimuth that the sunne was in at that instant also crossing the intersecti­on of the sunnes parallel, and almicantare, which being reckoned from the zenith line, did shewe me that the sunne was in the 32 ½ vertical circle reckoned from the East, the azimuthes indeed do represent, as it were the points of the compasse, and before the compasse found, men sailed thereby, and therfore if you did distinguish them into the points of the compasse, by some notable difference, then might you see at all times on what point the sunne, Moone, or any starre were without helpe of the lodestone, there be 32. points, by which de­uide 360. and the quotient sheweth 11 8/60 degrees, seruing euerie point seuerally, but this serueth to no great vse but onely on the sea.

Chapter 7. The latitude of any place knowne, how to get the declination of the sunne, or any starre, planet, or comet, by obseruation.

THe latitude knowne, subduct that out of 90. there resteth alwayes the height of thequinoctiall in that horizon, for the latitude and thequinoctials height, are one the complement to the other, they both ma­king 90. degrees. This knowne, take the meridian altitude of any starre or planet by the first Chapter, which if it be found greater then thequinoctials height, then to vs that dwell on this side of thequinoctial, the starre shalbe north declining, if lesse, it shalbe south: therefore take the lesser height out of the greater, the remaine, it is your desire.

Example. Heere at Reading the latitude being 51 ⅔ the complement of 90. is 38 ⅓. degrees, thequinoctials height. I tooke the meridian altitude of the great starre called Arcturus, or bootes, and found it 60 ⅓ degrees, out of 60 ⅔. there resteth 22 ⅓ degrees, so much is the declination of Arcturus from thequinoctiall, and that, northwards, because his height was greater then thequinoctials, and so may you do for the declination of any starre, plannet, comet, or the sunne.

But there are a great many starres that to vs neuer come at the south, because they neuer come beyond the zenith, which are those whose distance from the pole, is either lesse, or equall to the distance betweene the ze­nith and the pole. The distance betweene the pole and the zenith, is alwayes equall to thequinoctials height, wherefore the meridian height of such starres may be northwards, twise taken, once aboue the pole and a­gaine vnder the pole. For the declination of these, take the latitude out of the greatest meridian altitude, and the remainer out of 90. so resteth your desire. But if you will worke by his lesser meridian altitude, subduct it out of the latitude, and the remainder out of 90. there resteth then the declination sought for. Againe if you list, subduct the lesser meridian altitude out of the greater, and part the remainder in halfe, and subduct that halfe from 90. the same declination appeareth againe.

There be other starres which come past the zenith, and yet neuer set: their declination may be had both wayes. The Iewel it selfe yeldeth so plaine demonstration thereof, that it needeth no example, for the finitor being at the latitude, the parallels shew what stars may set, or set not: rise or not rise. I take the lesse, vz. 38 ⅓ in that countrie.

Chapter 8. How the latitude or poles eleuation may be had in any region by the starres that neuer set.

TO those onely that dwell vnder thequinoctiall, all the starres of the firmament do both rise and set eue­rie 24. houres: but to those that haue either pole eleuated, by howe much the more it is eleuated, by so much the more number of starres are continually aboue the horizon, and as many continually vnder, in somuch that where the pole is zenith, the one halfe is euer aboue, the other vnder. By those starres that are alwayes aboue any horizon may the latitude be thus found.

Take both the meridian altitudes of the starre vz. the highest and lowest altitudes in the meridian, as in the 7. there will be 12. houres betweene these altitudes, and yet oftentimes it may be done in a winters night, otherwise you must haue certaine monethes space betweene. Adde the two altitudes, the halfe thereof is the poles eleuation.

Take here Gemma Frisius example. He saith at Louane the 13. of December 1547. about sixe of the clock at night, he did diligently obserue the greatest altitude of the starre in the taile of Vrsa minor, called Cynosura which we call the pole starre, to be 53. degrees, 58. minutes, the next day about fixe of the clocke in the mor­ning he obserued his meridian altitude, and found it 47. deg. 43. min. these two added make 101. degrees 41. min. the halfe hereof vz. 50. degrees 50. minutes is the height of the pole at Louanium.

Blagraue. But the cause why I tooke Gemma Frisius example, is to shew you that it appeareth by his obser­uation, and the middle part of the Chapter last before, that the starre which the common sort and vnlearned do take for the pole of the worlde is distant 3. degrees 8. minutes from the pole, the pole it selfe being no starre, but onely a point imagined 90. degrees distant from ech part of thequinoctiall. Farther you must note that this serueth but for starres equall or lesse distant from the pole, then the zenith: but where as the one me­ridian altitude shalbe north from the zenith, and the other south, adde them as before, and halfe the product is the declination of the starre, subduct that out of the greatest altitude there resteth thequinoctials height, which subducted from 90. leaueth the latitude.

Chapter 9. The latitude knowne, how to finde what day of the moneth it is, though it were lost, as they say, by the sunne shining at noone.

IN the 3. Chapter is shewed by the theoricke to get the sunnes place by the day giuen. But here we suppose the day vnknowne: or else admit we would examine the account made by Alphosus tables. First gette the latitude as before, then take the meridian altitude of the sunne, and thereby get his declination by the 7. and whether it be northwards, or southwards: if therefore the declination be north, reckon it on the Iewel a­mong the parallels towards the north pole: but if southwards then among the south parallels, and the same parallel so reckoned from thequinoctiall, shall cut thecliptick in the degree that the sunne is in at that instant.

Example. In Autumne 1580. on a day vnknowne for want of an Alminacke, I tooke the meridian alti­tude [Page 29] of the sun 32. deg. so that by the 7. I found his declination south 9. deg. Therefore among the paral­lels southwards I reckon 9. I finde that the 9. parallel doth cutte the 22 ½ of ♎, or ♓, but because it is in Au­tumne, therefore it must needs be of ♎, if you doubted whether it were Autumne or spring, obserue the al­titude againe a day or two after, and if it be lesse it is Autumne, if greater, the spring. But now hauing the de­gree of the sunne in the zodiack, to finde the day of the moneth it is but Vice versa to the 3. Chapt and 4. for euen as in the 3. hauing the day, you seeke the sunnes deg. so heere hauing the deg, you may in the circle of monethes of the theoricke, see the day answerable accordinglie, thus: seeke in the theoricke 22 ½ deg. of ♎, and there shall you see to answere the 5. day of October your desire. Hoc rudiori minerua.

But this obseruation although it be excellent, by which both the quantitie of the yeare, and sunnes motion hath beene by old practisers obtained, yet doth it require instruments verie large, and for all that about the Solstices it is scarse exquisite, because then the dayly ascense of the sunne cannot easily be discerned by instru­ments, by reason of the transuerse fight of the zodiacke.

Blagraue. I will shewe you another as reddie way, or reddier by my Reete, and the rather because in small instruments theclipticke line may be either left out, or not exactly diuided in the Mater, and is not much diffe­rent from my addition in the 4. Chapt. For hauing taken the altitude, and founde the declination as before, then reckon that declination on the Label from the Limbe inwards, vz. 9. because it is Autumne: let the Label runne ouer the autumnall signes, which are ♎, ♏ and ♐, vntill the same 9. degree doe touch or crosse the zo­diack, & you shal finde that it wil light on the 22 ½ deg. of ♎, euen as before, and if the Label be not deuided, turne the autumnall signes on the nooneline, and you shall finde the 22 ½ of ♎ to touch the 9. deg. reckoned on the noone line from the Limbe: if it were for the spring time, then you should take ♈ ♉ and ♊, if in win­ter ♑ ♒ ♓, in sommer ♋ ♌ ♍, &c.

Further you shall vnderstand that though Gemma Frisius appointeth this to be done, the sunne being in the meridian, yet by my addition in the 7. Chapter, you may do it any time of the day, and as well or better towards night then at noone. For by the Azimuth and altitude of the sunne, you may get his declination, as I there shewed you to get the declination of the starre, and somwhat more easie is it in the sunne: for you neede but hange a plum line in your hand betweene the sunne and the Iewel, if the sights be not long ynough, and then turne the rule about till the shade of the line fall equall with the fiduciall line of the rule.

Chapter 10. To finde the right ascension for any portion of the eclipticke, and what degree of the Equinoctiall doth coascend, or rise togither with euery degree of theclipticke in a right sphere.

THeclipticke by reason of his oblique site to thequinoctiall, doth continuallie change his scituation in e­uery horizon: so that somtimes he is scituate somwhat right, and then goeth he slower, somtime more oblique, and then goeth he swifter. But thequinoctiall because in euerie seuerall horizon, he keepeth one inclination, and forme of his angles in his motion, there doth alwayes rise in equall time equall porti­ons thereof: wherefore Authors haue measured the motion of theclipticke by thequinoctial partes. There­fore the portion of the equinoctiall, which riseth with any part of thecliptike is called Ascensio recta, right ascention: in an oblique a rising, or ascention. In a right sphere, this rising is called Ascensio recta, right ascention: in an oblique sphere, Ascensio obliqua, oblique or byas ascention, and beginneth his number where 0. in ♈ cutteth the Equi­noctiall.

This matter is easie by the Iewel. For hanging it by the handle or equinoctiall line of the Mater, the axtree­line representeth the right horizon. And the meridians cutting euerie degree of theclipticke, if you fol­low them to thequinoctiall line (whereto the rule with his numbers must be set) do expresse what degrees of the one do rise with the degrees of the other. Note also that euery meridian is a right horizon to some place or other.

Example. Take the Iewel in your hande, and set the rule to thequinoctiall line, because of the diuisions, and then if for the right ascention of the 5. degree of ♉, you follow the meridian, cutting the same degree of the eclipticke line drawne on the Mater, till you come to the Equinoctiall: there the rule sheweth you 32 ⅔ deg. and a little better, which is the right ascention of 5. deg. in ♉. And for the 5. degree of ♏. The same meridian sheweth in the rule 212 ⅔ degrees and better, for the vppermost numbers of the rule, serue the vppermost carectars of theclipticke line vz. from ♑ to ♋, and the lowermost the other vz. from ♋ to ♑. And now it is easie to know how many degrees of thequinoctial rise with any whole portion of theclipticke: as for ex­ample, with the whole signe of ♉, the two meridians comprehending it, do shew in the rule 29. degrees 54. minutes of thequinoctiall.

Chapter 11. The same more compendiously by the Reete.

PLace the Reete so, that 0. in ♈ of the zodiacke touch the north pole in the Mater, and 0 in ♎ the south: and now the Limbe heere representing thequinoctiall, as I haue shewed in the 2. Booke 20. Chapt. is de­uided accordingly, the numbers beginning from the said intersection at 0. ♈ in the centre being now the pole or poles of the worlde. It must needs be therefore that euerie line imagined from the centre to the, [Page 30] Limbe representeth a meridian, which cutting the zodiack of the Reete performeth the scope of the last Chap. For lay the Label on any degree of the zodiacke, and it presently sheweth in the Limbe his right ascention: this needeth no example.

Note that opposite signes and also signes equally distant from thequinoctiall points do ascende in a right spheare in equall time as ♈ ♓ ♎ and ♍ likewise ♉ ♌ ♏ and ♒ also ♊, ♋, ♑, ♐.

Note also being that euerie Meridian is a right horison to some place, as in the last cap. I sayd, therefore the like degrees of theclipticke and thequinoctial must needes cut togither the meridian or any place as do cut in any right horison, but then are not called right ascention, but caelimediation or Culmination, whereby you see that the right ascention and mediation or Culmination are all one in substance: but yet to say the trueth, the right ascention is properly the degree of thequinoctiall, the Culmination is the degree of thecliptike coascending the right horizon with any starre.

G. Frisius 14. Contrariwise to any right ascention proposed any man may gather the degrees of thecliptick coascending performed, visa versa.

Chapter 12. What point of theclipticke it is, as well in the first quadrant as in the rest also, that hath the greatest difference betweene the arch of theclipticke, and the arch of thecquinoctiall coascending with him.

THis working requireth heede. Lay the rule vpon the pole circle of the mater, mouing it to and fro there­on, in such wise that the pole circle do cut like degrees in the rule numbred from the centre, as the rule doth cut in the Limbe numbred from thecquinoctiall, and then reckon the degrees in the Limbe, from the pole to the rule, the same is the declination of the part sought for, then by the declination so known, the point of theclipticke is had by the 9. Chapter.

Example. Let the pole circle (to auoide profers for breuities sake) cut 73 ¼ deg. in the rule, the same num­ber you shal finde the rule to cutte in the Limbe. Then betweene the pole & the rule reckoned on the Limb, are 16 ¾ degrees, the declination of the point sought for. Therefore by the 9. to that declination, there answe­reth 16 ¼ degrees of theclipticke numbred from 0 in ♈, or ♎, so that in the first quadrant it lighteth in the 16 ¼ degrees of ♉ in the second, the 13 ¾ of ♌ in the third, 16 ¼ of ♏, in the fourth, 13 ¾ of ♒. now because the right ascension of the 16 ¼ deg. of ♉ is 43 ¾ deg. of thecquinoctiall, and that that part or degree of thecliptick is distant from 0 in ♈ 46 ¼ deg. subduct the lesse from the greater, there resteth 2 ⅓ deg. almost which is the greatest difference betweene the arches of theclipticke and thequinoctial in the assensions of the right sphere which hapeneth in the said 4. places.

Note that if the greatest declination of the sunne vz. 23 28/60 deg. should in time alter to be more or lesse as it hath in time beene found lesse and lesse by Ptolomeus, Vernerus, Copernicus and others, then would the number a fore vz. 73 ¼ deg. alter by reason the pole circle then altereth also, or else not.

Chapter 13. Of the difference of ascensions.

IT appeareth by the last Chap. that the greatest inequalitie of ascension in a right sphere, cannot be aboue 2 ⅓ degrees betweene the rising of the degrees of the quinoctiall and theclipticke: but in an oblique horizon there hapeneth great varietie and difference, because theclipticke and the horizon make for greater muta­tions of angles, and inclinations, then in a right sphere, VVhereby the signe ♈ hath a farre deale lesse ascenti­sion in regions, declining to the North, then in a right sphere: and contrariwise ♎ far greater: So hath ♉ lesse then ♏, and ♊ then ♐, which in a right sphere haue equal ascentions. The cause is for that those rise a­boue the horizon in a more oblique scituation, and these in a more right, insomuch that the sixe signes from the beginning of ♋ vnto thend of ♐ are called right ascending, the rest oblique ascending. And the more the sphere is bending towards the poles, the greater is this diuersitie: this difference of ascentions, either in excesse or decrease, from those of the right sphere, is called Ascensimum differentia (that is) the difference of ascen­sions, yet looke how much wanteth to signes oblique, ascending of their right ascensions, so much againe the opposite signes right ascending do encrease: so that the right ascentions of two opposite signes are equall to the oblique ascensions of the same signes taken togither. But now to the matter.

How to finde the difference of ascensions, place the Finitor to the latitude, then seeke in thecliptick line that degree whose ascensionall difference you require, and followe the parallel, cutting that degree vnto the Finitor, marke what meridian cutteth the Finitor there with the same parallel, looke howe many degrees di­stant that meridian is from the axtree line, so many is the difference of ascention of that degree of the eclip­ticke.

Example. I would know the ascensionall difference of the beginning of ♉, I place the Finitor to the latitude vz. 51 ⅔ here at Reading: then following the parallel, cutting 0. in ♉ vnto the Finitor, I finde betweene the axtree line & the same point of the Finitor 14 5/60 meridians, therefore I conclude that the difference of ascen­sion of 0 in ♉. is 14. degrees, 5. minutes. And because ♉ is a signe oblique ascending (as is said) I say that be riseth so much sooner in our horizon then in a right horizon, as 14. deg. 5. min. commeth to. The like di­uersitie of ascension hath the beginning of ♏, which is opposite to ♉, but that you must take away the diffe­rence [Page 31] from the right ascension in the first, where in this you must adde it, and so is it in all opposite signes and therefore to saue labour, the one being had the opposite is knowne.

Chapter 14. The same very plainely by the Reete.

YOur horizon knowne by the 2. Booke 20. Chapt. If you would know the ascentionall difference of a­ny degree of the North signes, lay the same degree to cut on the northeast quarter of your horizon, and the rule thereto, and then numbring on the Limbe the distance betweene the rule so placed, and theast, or 6. of the clocke line, the same is your desire. If you seeke this difference for south signes, you must lay the degree on the south east part of your horizon, and then doe as before, but you may as is saide, to saue la­bour, know by the one halfe the opposites, yea by one quarter the whole for this matter, if you take them alike from 0. in ♈ and ♎ for ♓ ♈ ♍ and ♎ haue like difference, ♒ ♉ ♏ and ♌, so ♋ ♐ ♑ and ♊, but not in like degrees at the opposites haue.

Chapter 15. To know the oblique ascenscions.

BY the 11. and 12. you learned to finde the right ascension of any part of the zodiacke, and by the 13. the difference of ascentions, these knowne, then to the right ascentions of the south signes, add the difference, and from the right ascentions of north signes, take the difference of ascention, and so haue you your de­sire.

Example. The right ascen. of the 4. deg. of ♊ by the 11. & 12. is 62. degrees, the difference by the 13. is 28. deg. 18. minutes: now because ♊ is a north signe, I subduct the difference vz. 28 18/60 out of 62. the right a­scension, there resteth 33. deg. 42. minutes the oblique ascension. Likewise the right ascension of his opposite deg. vz. the 4. of ♐ is 242. to it I adde the difference vz. 28. 18. there commeth 270. deg. 18. minutes, the ob­lique ascension, and so may you do for euerie eleuation or latitude.

These preceps though they shew barren, yet haue they great vses in dimensions of times, and inquisions of the heauenly motions.

Chapter 16. Of descentions both right and oblique, that is, what degree of thequinoctiall doth descend with theclipticke.

TRuely the Iewel it selfe sheweth it, and therefore I will not stand vppon it, you must worke for them in the west parts of the horizon, as before in the 14. Chapter you did on theast, the ascensions and de­scensions in a right sphere are all one, and the difference in the oblique are all one, onely in this it alte­reth, that if you added the difference for the oblique ascension as in the 15. you must substract it for the de­scention, and so contrariwise.

A briefe rule. Adde 180. vnto the oblique ascension of the opposite degree to the degree proposed, and so haue your desire.

Example. For the oblique descension of the 4. of ♊. adde 180. to the oblique ascension of the 4. of ♐, which is 270. degrees 18. minutes, as before, it maketh 450. 18. minutes, the oblique descension of the 4. in ♊ if it had not exceeded 360. But now you must out of this vz. 450. 18. minutes, cast 360. the remaine vz. 90. deg. 18. minutes is the descension sought for.

Chapter 17. The longitude and latitude of any starres proposed, to know their right or oblique ascention, and how much they decline from thequinoctiall, and how to place starres in the Mater of the Iewel.

HEere Gemma Frisius bursteth out in these words. This instrument, saith he, hath such aboundant and wonderful vse, that I scarse know to let it downe orderly, and therefore am forced oftentimes to reduce many things into one Chapter, How then (gentle reader) shall my poore capacitie being euerie way too much inferior to Gemma Frisius, set downe and performe so great a charge orderly, as my Iewel casteth vpon me farre surmounting the other instrument, and well the worse, being, that this wrangling world yeldeth me, time but by snatches.

But to the matter: set the Finitor euen with theclipticke line of the Mater, and so do the aximuthes represent the longitudes, and the almicantares the latitudes of theclipticke, therefore reckon the longitudes and lati­tudes of your starre, which in diuerse tables you may haue amongst them: looke where they crosse, there is the starres place in the Mater, then marke what meridian and parallel of the mater, doe crosse the same place of the starre so founde, the one sheweth his right ascention, at in the 10. and the other his declination as in the fourth.

Example. The faire red star of the first light of the nature of ♂ called Oculus tauri, the buls eye, rising always to sight about 2. perches vnder the seuen starres: This star I finde by Stadius tables, who is one of the newest writers, his longitude to be 63. degrees, 40. min. or in ♊ 3. deg. 40. min. all is one, and his latitude 5. deg. & 10. min. Southwards, this knowen, I set the Finitor to the Ecliptike lyne, so that the Zenith be towardes the South pole, because the starres latitude is South. Then on the Finitor or Eclipticke, which heere are all one frō ♈ towardes ♋, I reckon the starres longitude, vz. 63. degrees 40. min. or the 3. deg. 40. min. of ♊, and in the azimuth issuing from that pont of the Eclipticke, I number his latitude, vz. 5. degrees 10. min. in the same point, that is to say, where the 63 40/60 azimuth, and the 5 10/60 almicantare do crosse ech other on the mater, is the starres place, and there may you graue him in and write his name: but because in small instruments it would hurt the worke and also for that the starres do remoue in 100. yeares almost a degree, therefore make but a pricke with yncke for the time being, and then turne away the Finitor vnto the latitude vz. 51 ⅔ deg. being our eleuation here at Readinge. Now the rule beeing sette to the equinoctiall may you see the 62 ½ meri­dian, cutting the said prick or star so placed, which sheweth his right asc. to be 62. deg. 30. min and the 15 ⅚ parallel shewing his declination to be 15. deg. 50. min. & that towards the north from thequinoctial though he were south from theclipticke, lastly for his difference of ascension, you may easily know as in the 13. by following the starres parrallel to the Finitor, and there take the number of the hourelines to the axtree line which here are 21. and better, by the which and the 15, Capt. you may gather his oblique ascension. For be­cause the starre is on the north part of thequinoctiall, therefore subtract the difference vz. 21. from the right ascention before found vz. 62. deg. 30. minutes and there resteth 41. degrees, 30. minutes, the oblique ascen­sion: but if you adde the difference vz. 21. vnto the right ascention vz. 61. 30 thereof commeth 92. degrees 30. minutes, the oblique ascension. But if it happen so that the starres parallel touch not the Finitor then being a north starre, he neuer setteth in that region, or being a south starre, he neuer riseth there.

Chapter 18. The declination of fixed starres proposed with their latitude, which neuer altereth, how to finde their longitude in the zodiacke.

THe like Hipothesis did Ptolomeus vse, & before him Hipparchus Samius, whereby they found that the sphere of the fixed starres, had also a motion contrary to the first mouer, and that it is perfourmed ac­cording to the motion of the Zodiacke, and about his poles. For insomuch as the latitude of all the fixed starres from the Eclipticke, hath alwayes beene found one, and neuer altering, both of old writers and also of those which now do obserue the motions out of the heauen it selfe, and that it is most certaine that the starres do continually goe forwards (from the Equinoctiall points, describing circles parallell to the Eclipticke) they cannot therefore performe those motions vpon any other then his poles. And hereby it commeth to passe, that although they be always of like distance from the Eclipticke, yet they come neere or depart further off from the Equinoctiall.

It is shewed in the 7. Chapter how to get their declination from thequinoctiall: their latitude may be ta­ken out of Ptolomeus tables, or any other, because they neuer chaunge. Let vs therefore take an example out of Ptolomeus, who saith that Timochares which liued 432. yeares before him, did obserue the starre cal­led Spica, in his time to be declining north from thequinoctial 1. deg. ⅖ that is 24. minutes, but in the time of Ptolomeus, he was south from thequinoctiall halfe a degree, my desire is to knowe the true places of Spica, that is his longitude & latitude for both those times proposed. The latitude of Spica is 2. degrees south from theclipticke, therefore I place the Finitor to theclipticke, the zenith towards the south pole of the mater, for that the latitude is south: and there do I note where the latitude of Spica being here the second almicantare, and his declination for Timocharis time being 1 ⅖ parallel northwards, do crosse the azimuth passing on the same crossing to theclipticke, doth shew the 22. deg. of ♍, the longitude of Spica for Timocharis time. In like manner working, you may finde his longitude for Ptolomeus time, to be in the 26 ⅓ degrees of ♍.

VVhereby also Ptolomeus did gather that the fixed starres did go forwards one deg. in an hundred yeares according to the sequele of the signes, because betweene him and Timocharis passed 432 yeares. But nowe sinse Ptolomeus obseruations, the fixed stars haue gone forwards 21. deg. almost, for now Spica is in the 17. of ♎ 36. minutes. Therefore to one degree there cannot alwayes be giuen an 100. yeare: since there are but 1406. yeares passed, but making estimation according to this rate, the starres should bee mooued one deg. in 67. yeares, yet we must account of ancient writers, as we would haue ours accounted of, and confesse that there is a certaine Anomalion or inequallitie of motion in the fixed starres, which no man hath hither vnto better coequated then Copernicus.

Chapter 19. The difference of ascension of any part of the zodicke or fixed starre proposed, or the oblique ascension to know what the latitude of the countrie is.

Number the difference of ascension in the parallel of the place, point, or starre, proposed from the ax­tree; line leftwards, for north parts of starres: and rightwards for south, thereto set the Finitor which shall shew in the Limbe the poles eleuation, but if the oblique ascention be proposed, get the difference by the 15. vz. subducting the oblique ascension out of the right ascension, if the part or starre be north: but if [Page 33] it be south, then subduct the right asension out of the oblique, so haue you the difference, and then worke as before.

Example. I would know in what region the beginning of ♋ doth rise with 60. degrees of thequinoctiall In a right sphere it riseth with 90. degrees. Now because it is a north signe, I subtract the oblique ascension out of the right ascension vz. 60. out of 90. there resteth 30. degree, the difference of ascension: then in the parallel drawne by 0. deg. of ♋ which is the verie tropicke, I number 30. deg. reckoned by the hourelines from the axtree line towards ♑, and thereto lay the Finitor, it sheweth in the Limbe the 49. latitude almost: By this Chapter, the distinction of the Geographicall parallels, and climates are performed very easily.

This Chapter is also performed easily by the Reete, numbring in the Limbe the difference of ascensions from 6. of the clocke downewards for north signes and starres: and vpwards for south, and thereto see the rule, then vnder the fiduciall line of the rule, set the deg. or starre proposed, and marke which horizon it toucheth, so haue you your desire.

Chapter 20. To finde the amplitude of the rising and setting of the sunne or starres.

ALl men know that euery horizon hath foure principall points, videl. east, west, north, and south. There is neither the sunne, nor any starre that riseth full east, except he be in thequinoctiall where the sunne is neuer, but being in the beginning of ♈ & ♎, at al other times he is either north or south & riseth north­east or northwest accordingly as in the 2. Booke 2 ⅓. Chapt. is shewed. These points in the sunne rising dif­fer euerie day, and are called Amplitudo ortus, and thus are had.

Place the Finitor to the latitude, and follow the starre or sunnes parallel vnto the Finitor, the deg. betweene that point and the centre is the amplitude.

Example. The sun being 0. deg in ♋, follow the tropicke parallel vnto the Finitor, placed to the 51 ⅔ latitude it cutteth on the Finitor 40. degrees reckoned from the centre which is the sunnes amplitude of rising.

Chapter 21. The amplitude of the rising of any knowne starre giuen, or of the sunne, with his place in the Zodiacke to know the latitude of the countrie agreeable.

ADmit the amplitude of the sunnes rising being 0. degrees in ♋ were giuen to be 50. degres, to know in what latitude that happeneth, reckon 50. deg. on the Finitor from the center, and applie the same to and fro till it cutte the sunnes parallel, which then is the tropicke, and you shall see in the Limbe the pole ele­uated 58 ⅔ degrees almost, the latitude sought for, and so may you do with theamplitude of any starre and his parallel.

Chapter 22. To know the rising and setting of the sunne, and the semidiurnall arch of the sunne and starres.

PLace the Finitor to the latitude, and then marke where the sunnes parallel doth cut him, the houreline lighting there sheweth your desire.

Example. The sunne being in the 10. degree of ♉, I woulde knowe his houre and time of rising and setting, his parallel of declination by the 4. is 15. northwards, therefore I follow the 15. parallel to the Finitor, and the houreline cutting him there, doth shew me 5. degrees before fiue of the clocke, the sunne rising, that is 40 minute, or ⅔ of an houre after 4. and by the afternoone figures the same houreline sheweth 5. deg. after seuen of the clocke the sunne setting, that is 20. minutes, or ⅓ of an houre after 7. for euerie deg. of thequi­noctiall is 4. minutes of an houre.

Now for the semidiurnal arch the portion of the sonnes parallel betweene the Finitor and the Limbe, or me­ridian sheweth it videl. an 100. deg. or 7. houres and 5. deg. then reckon backe againe from the Limbe to the Finitor, or double it, & so haue you the whole dyurnal arch vz. 200 deg. or 14. houres 10. degrees, which ta­ken out of the whole circle, which is 360. deg. or 24. houres, it leaueth 160. deg. or 9. houres, and 5. deg. the length of the night.

The same may be done easilie by the Reete, place the deg. of the sun on theast part of the horizon of your countrie, as in the second Booke 23. Chapter, and thereon also lay the rule, and it sheweth in the Limbe the houre of the sunne rising, from which houre, number the degrees vnto the nooneline, so haue you the semidi­urnall arch.

But for starres it is best to be done the first way bicause the starres parallel in all respectes performeth it as the sonnes parallel did.

Example. The Finitor being set to our latitude vz. 51 ⅔, I follow the parallel of Oculus tauri, which either by the 7. or 17. I find to be the 15 50/60 parallel vnto the Finitor, there do I see to be betweene the Limbe and it 111. degrees, or 7. houres, 6. degrees, his semidiurnall arch, which doubled, maketh 222. deg. or 14. houres and 12. deg. his diurnal arch, which is as much to say, as so many houres and deg. the same starre doth remaine aboue the horizon in our countrie, and the said 222. degree, or 14. houres, 12. deg. subducted out of the circle, lea­ueth his nocturnall arch, 138. deg. or 9. houres 3. degrees.

Heere note that the difference of ascension of the sunne, or any starre learned by the 13. added to 90. ma­keth the semidiurnall arch, for North signes and starres, but for south it must be taken out of 90.

Also the houres of sunne rising, and setting added, alwayes make 12. also if you double the houre of sunne set, thereof commeth the length of the day, that taken from 24. leaueth the length of the night, as if the sunne set at 7. of clocke, double that, it maketh 14. houres the length of the day, the rest of 24. is the night.

Chapter 23. The quantitie of the longest day of any countrie proposed, how to get the eleuation of the pole there, and of the distinction of the climates and parallels.

IT is saide that at Constantinople the longest day is 15. houres, 12. minutes, the question is, what the lati­tude is, there deuide the day in halfe, there commeth 7. houre 36. min. or 9. deg. for euery 4. min. are one degree, the houre of Sun set. Now therfore apply the Finitor to the place in the Sommer tropicke, where it cutteth the 7. a clocke houre line, and 9. degrees past, and so shal it shew in the Limbe 43. deg. the height of the pole at Constantinople.

Chapter 24. To know what houre of the day or night the Moone, or any planet or starre riseth or setteth on the horizon.

TO performe this proposition you must haue the day giuen with the deg. of the sunne, the latitude: and whither he be a North or South starre, then proceede thus: lay the apex of the starre vpon the northeast quarter of the horizon proposed, if it be a North starre, or on the Southeast, quarter, it it be a South starre, then the rule laide to the degree of the sunne, sheweth in the Limbe the houre required.

Example. The first day of Ianuarie 1580. the sunne being by the 3. Chapter in the 21. of ♑, I afraide to know what time the great starre called Canis maior, the great dogge did rise on our horizon at Reading. Now because he is a south declining starre, I laid his apex on the Southeast quarter of the 51 ⅔ horizon, and then turning the rule to the 21. of ♑, it shewed me in the Limbe 6. of the clocke and 5. degrees past and better: whereby I concluded that in our horizon or latitude at Reading, Canis maior did rise the said first of Ianuarie 5. degrees. or 20. minutes and better after sixe of the clocke at night. Likewise for his setting the same day (or rather night I might say) I laide the starres apex on the Southwest part of the same horizon, and the Label to the saide degree of the sun, it sheweth in the Limbe 3. of the clocke and 8. degrees past, that is 32. minutes after 3. of the clocke in the morning, at which time Canis maior did set againe vnder our horizon that day.

But if you would know the rising and setting of any starre that is not made or placed in the Reete, then by the 21. get his semidiurnall arch, and reckon the same on the Limbe from the noone point Eastwards, and thereto lay the rule, then turning about the Reete, place euen with the rule the deg. of mediation, or culminati­on of the star, that is, the deg. of theclipticke, rising with the same starre in a right horizon by the 11. Chap­ter: that done, turne the rule without mouing the Reete vnto the degree of the sunne, & it shall shew as before, the houre sought for.

Admit for example that Canis maior were not in my Reete, therefore I gette his semidiurnall arch, which by the 21. Chapt. I finde to be in 68. deg. ¾ or 4 houres, and 8 ¾ degrees all is one, which being reckoned on the Limbe from noone Eastwards, I lay thereto the rule, then by the 11. finding that the degree of culmination of Canis maior is the 6 ½ deg. of ♋, I lay the same euen with the rule, and there the Reete resting, I turne the rule to the degree of the sunne, as before vz. 21. of ♑, there I finde in the Limbe 5. degrees after 6. of the clocke, as be­fore, the rising of Canis maior: and for his setting I reckon 68 ¾ deg. from noone westwards, and thereto lay his culmination vz. 6 ½ of ♋, and thence turning the rule to the 21. of ♑ it sheweth there in the Limbe 8. deg. past 3. in the morning, his setting, as before. Remember alwayes in your working of starres that Locus solis ostendit boram.

Likewise for the Moone or any planet hauing no latitude from thecliptick, which seldome happeneth but they haue laide the deg. of the zodiacke on the horizon, as you did the starres apex, and then Locus solis ostendit boram: Looke more in the 33. Chapt. of the 2. Booke.

Blagraue. You shal vnderstand, it the moone or any planet be much in latitude, as they cannot be aboue 6. deg. and the moone not aboue 5. at the most, that then there may happen a great diuersitie of their rising and setting, if you should but onely lay their deg. of the zodiacke to the horizon as before, which the expert practi­ser for all the planets will easily remedie: but for the moone you haue more to do, for if you take her deg. out of the ephemeris, or the common Almanacks, the same is but her degree at noone for that day, for at 6. of the clocke at night, she wilbe 3. degrees farther in the zodiacke: the cause is, she walketh the whole zodiacke in a moneth, and euery 2. houres after a sort one deg. and somwhat more: wherefore you must adde for euery two houres past noone, one deg. & before noone subduct, and yet commonly in tables the day endeth at noone, & this day beginneth yesterday at noone.

Chapter 25. To know what starre neuer rise or set to euery region, and which are verticall.

FOr these take these general preceps, that star whose declination is equal to the latit. so that the declination & latit. be both north, or both south, doth still once in 24. houres come to the zen. as appeareth by the zen. point of the Iewel, which toucheth in our latit. the 51 ⅔ parallel: also those stars whose declination southwards is equal to, or greater the thequinoct. height, are neuer seene in that country, as for example set the finitor to our [Page 36] latitude 51 ⅔ you shall see on the Iewel that the 38 ⅓ parallel southwards doth but touch the Finitor, the rest be­yond him neuer are seene: contrariwise those North starres whose declination exceedeth the equinoctials height, do neuer set, as by the Iewel may apparently be seene. Farther if you would know to what countrie any starre or signe is verticall, seeke his declination, and where his parallel lighteth on the Limbe, thereto set the zenith point, and so the Finitor sheweth the latitude.

Chapter 26. Of the Cosmicall Eliacall and Acronicall rising and setting of the starres.

THe Cosmicall rising is when as the starre riseth on the horizon with the sunne neare one instant, and is had if you place the starres apex on the East part of the horizon, the deg. of the zodiacke cutting then the horizon, is your desire, then by the theoricke you may finde the day or the yeare answering thereto. But if the starre desired be not in the Reete, then reckon his semidiurnall arch, from noone Eastwards, and thereto lay his degree of culmination, had by the 11. and marke what deg. of the zodiacke, doth cut the ho­rizon on the East part, the same is also your desire.

The Cosmical setting, is when any starre doth set at one instant when the sunne riseth, or neare it, which is as easily had as the other by laying the starres apex on the VVest part of the horizon, and then the deg, of the zodiacke rising on theast is your desire: if the starre be not in the Reete reckon his semidiurnal arch West­wards, and lay thereto his culmination, and so againe the ascendent is your desire, euen as before.

The Acronicall rising is when a starre riseth the sunne setting, place the starres apex on theast part of the horizon, and then marke the deg. of the zodiacke, descending on the VVest, for when the sunne commeth thither, then doth that starre rise Acronice, and this may you do by his semidiurnall arch also as before.

The Acronicall setting, is when a star setteth with the sunne, and is easie to be vnderstood by the premises.

The Heliacall rising of a starre, is when he getteth out of the sunne beames, and the Heliacall setting, is when he getteth into the sunne beames: but heere you must note that the greater starres may be seene when they are more neare the sunne then the lesser. Starres of the first light are not seene except they be 12. degrees di­stant from the sunne, of the second light 13. of the third 14. of the fourth 15. of the fifte 16. of the sixt 17. Then for the planets, Saturne requireth distance 11. degree ♃ 10. ♂ 11 ⅓ ♀ 5. ☿ 10. but these be not alwayes certaine, for the planets turning in their epicicles, doe now shew greater, and now lesse, also the inclination of the zodiacke to the horizon may somwhat alter them, and the state of the aire also.

Chapter 27. How to know what it is a clocke in the night time by the starres.

THe sunne onely sheweth the houre at all times, yet by the starres or planets in his absence we doe finde him out, and so get the houre.

Take the height of the starre, and marke whither he be East or VVest from the meridian (for always the farther from the meridian the starre is, the surer you worke) then set the Finitor to the latitude, and reckon the starres height so taken among the almicantares, & where that almicant. and the starres parallel do crosse the houreline cutting there, sheweth the houre distance of the starre from the south, which is called Hora stella, the starres houre: but now by this to get the true houre, reckon this houre distance of the starre on the Limbe Eastwards, if the starre were on theast side, or westwards, if on the West: and thereto lay the starres apex by helpe of the rule, or his degree of Culmination, if the starre be not in the Reete, and the Reete there stan­ding turne the rule to the deg. of the sunne, it sheweth the houre sought for on the Limbe. This asketh much businesse by Gemma Frisius Cursor and Brachiolum.

Example. The 6. of October 1580. I went forth and tooke the altitude of a faire great starre called Hircus the Goate 20. deg. being then northeast, whose declination is 45. degrees North, then placing the Finitor to the latitude I marked where the 20. almicantare cut the 45. parallel, the houreline crossing them there in their point of meetings shewed foure houres & 2. degrees, the houre distance of Hircus from the north part of the meridian Eastwards: and that which more is then Gemma Frisius Astrolabe could shewe at one instant, the azimuth crossing there also with them, sheweth in what verticall circle the starre was, but to the matter: Be­cause the starre Hircus was on theast side of the North part of the meridian: therefore in the Northeast quar­er, I reckoned on the Limbe from North Media nox Eastwards, the starres houre vz. 4. houres 2. deg. and there­to by helpe of the rule I laide the starres Apex, the Reete there standing, I turned the rule to the degree of the sunne which then was in the 23 ⅓ deg. of ♎, and found it pointing on the Limbe 7. of the clocke, and halfe an houre past within little the houre sought for.

Gemma Frisius 32. Heere Gemma Frisius maketh a Chapter to reckon the houres from noone vnto 24. ending the next day at noone, as the Astronomers doe, and likewise from sunne rising or sunne set to 24. as diuerse nations do, but seeing it so easie and friuolous for our countrie I omit it: I shall adde much and there­fore had neede be short in some.

Chapter 28. Of the vnequall or planetarie houres.

EVerie artificiall day is accounted betweene sunne rising and sunne set: the night accordingly from sunne set to the rising, the 12. part of euerie such day is called a planetarie houre, so that as the dayes are longer and shorter, so are the houres, and therefore called vnequall. They are called planetary because auncient [Page 37] Astronomers haue held opinion that the seuen planets in order haue dominion, these houres giuing that pla­nette to the first houre of the day, after which hee is called as on sunday, ☉ hath the first houre the next ♀ then ☿ ☽ ♄ ♃ ♂, which endeth in the 7. houre, and then beginne agayne the 8. houre with ☉, and then forwards ♀ ☿, &c. as before with the 12. night houres also, and you shal finde that ☽ will hitte righte to begin the 1. houre of the day following, whereof the day hath his name, which is the cause that the week dayes are not named according to the order of the planets.

Blagraue. Gemma Frisius appoynteth a particuler quadrant on the backe of the Astrolabe for the plane­tary houres, eyther the same, or very like it which Stophler appointed, but I like not to haue any thing parti­cular to one latitude, in so general a Iewell. I would rather doe thus: deuide the semidiurnall arch by 6. and by the quotient, deuyde the arche of the houre sought from Sun rising: as for example, the Sun being in the beginning of ♋ the semidiurnal arch is 124. deuide that by 6. the quotiēt is 20. 4/6 deg. admit at 10. a clocke before noone, I would know the planetary houre, the arch thereof to the Finitor is 94. deuide that by 20. & let the 4/6 alone, thence commeth 4 14/20 that is to say, the 4. planetary houre, and 14/20 past, that is as good as ⅔ partes.

But for those that canne no Arithmeticke, an easier waye is thus, and for that cause chieflye did I distin­guish all my meridians by sixes, with great strokes: beginne from the Finitor with the next fixt meridian that cutteth the paralel of the sun, and thence tell euery 6. on the same parallel with a sticke as they say, vnto the Limbe, and count how many they be, and looke how many fixes you haue on that parallel, so manye degrees reckon to euery planetary houre on the same parallel, as before in the last example: you shall finde 20. sixes, and foure degrees ouer: respect not those 4. but go on with 20. for the first planetary houre from the Finitor, and 20. for the next: and so on till you come to the 10. houre desired, and if you will work exquisitely for e­uery planetary houre reckoned, account 4. and then see how oft you can take out 6. & so many degrees, adde to the end of your reckoning, and so haue you the planetary houre exactly, as in example before, at 10. of the clocke, it was found the fourth planetary houre, almost the 5. therfore I heape 4. together 5. times, it maketh 20. thence, I can take 6. 3. times: which 3. I adde to the planetary houre before found: so that now it is the 4 17/20 houre, that is almost the fift.

Also the Finitor being set to the latitude, he that would deuide the 3. arches of the Equinoctiall, and the two tropicks, into sixe equall partes betweene the Finitor and the Limbe, and so draw arches from tropicke to tropicke, by helpe of eche of these 3 pricks, might so haue the planetary houres for that latitude, but this be­ing particuler, is not conuenient in the Iewell. But if any man did deuide euery 3. parallel betweene the tro­picks in 6. parts, and did but only set great perceiueable pricks at euery part, it would serue and not muche hurt. This matter is not weighty, and therefore let euery man doe his pleasure.

Chapter 29. How by the Iewell to know the Meridian or greatest altitude of the sunne and starres.

THe innermost circle of the Limbe in respect of the Mater, representeth alwayes the Meridian in euery la­titude: wherefore the Finitor being set to the latitude, marke where the parallel of the Sunne or any starre doth touch the Lambe, the almicantare touching therewith him, sheweth streight the Meridian altitude.

Chapter 30. How to finde the degree of Medium Coeli, or Culmination called midde­heauen at any time proposed.

THe degree of the Zodiacke being in the Meridian at any instant, is called in Latine the Gradus mediā Coe­li, of some Culman Coeli, and Cor Coeli. that is the top and heart of the heauen.

By helpe of the Reete. it is easilye had, laye the degree of the Sunne in the Zodiacke vnto the houre and time proposed, by helpe of the rule, and then looke what degree cutteth the noone lyne, the same is your desire.

Example, the Sun being 6. degrees in ♌, I would know the degrees of the mydheauen or Culmination, at 8. a clocke in the forenoone, I set the 6. of ♌ vnto 8. a clocke by the rule, and then in the noone line towards the handle, I see there 9 ½ deg. of ♊ the deg. sought for, and in the same noone line vnderneath I see the 9 ½ of ♐, which is called Gradus Imi Coeli, the deg. of deepe heauen or the Angle of the earth.

To do the same by the Mater without the reete, first to get the right ascention of the degree of the sunne, and then his distance from the South in degrees and minutes, take this distaunce for forenoone howres out of the right ascension of the Sunne, if it may not be, adde 360. to the ascension, & then subduct it, but for afternoone howres, adde the distance to the ascension, & if the whole summe exceede 360. cast away 360. so haue you the right ascention of the culmination, then the degree of the Zodiacke answering, is had by the 10.

Example. the Sun being in the 6. of ♌ as before his right ascention is 128 ⅓. & because the howre before is of the forenoone distance from the meridian 4. houres or 60. degees. I subduct 60. out of 128 ⅓. there re­steth 68 ⅓. degrees, the right ascension of the degrees of culmination: then by the 10. Chap. I finde the de­gree of the Zodiacke, answering thereto, to be 9 ½, degree in ♊ as before.

Chapter 31. To know how much any degree or point of the heauen is distant from the Meridian at any time by degrees of the Equinoctiall.

LAY the degree of the Sunne vnto the houre proposed on the limbe by helpe of the rule, and then turning the rule about to any degree, starre or other point in the reete, it sheweth in the limbe the distance thereof from the meridian, either in degees or houres. This proposion is necessary for directions, and needeth no example. Note that for starres not in the reete, their deg. of culmination serueth the rule applyed thereto.

Chapter 32. How by the Iewell to know the height of the Sun aboue the horrizon euery houre and time of the yeere sitting within dores: and thereby to make tables to any latitude, for making of particuler Dials, as Cylinders, ringes, quandrants, & such like.

Gemma Frisius heere commended the profite of this Chapter, specially for making tables for particuler di­als, saying that it hath a wonderfull facilitie of working by his Astrolabe, but truly by my Iewel the working is an hūdred fold more easie: For my finitor being once placed to y^e latitude, he that vnderstādeth the 6. chap. the 21. & 26. may see his table as readie made vpon his Iewel as G. Frisius, or any man els can make him the other way with 3. or 4. houres paines: and that which more is, you neede but the finitor vnto any other latitude, and so is your Iewel a table readie made also to the same, so that you may heereby peruse tables for many latitudes in lesse time then you could peruse for one latitude after G. Frisius by his cursor and Bra­chiolum.

Example. the 13. of September 1588. the Sun then in the 0. degree of ♎ I woulde know of what height the Sun is euery houre that day, I place the finitor to the latitude vid. 51. ⅔. here at Reading, then do I marke what Almicantares do cut the Sunnes paralell being here the equinoctial, in the crossing of euery houre line: I see at the limbe being 12. of the clock the 38 ⅓ degrees or Almicantare: at 11. whiche also serueth for 1. in the afternoone 37. almost, at 10. & 2. 32 ½. at 9. & 3. 26. at 8. & 4. 18. & better, at 7. & 5. 9. & at 6. & 6. 0. because the Sun then setteth, & is nought aboue the horison. This doth G. Frisius aduise to set in a table. do­ing the like with euery paralel, or euery 3. paralel at the least, of the Sunnes declination, & when you haue done, your table shalbe no redier then my Iewel: for the paralel of declination knowne, I can goe vpon hym from houre line to houre line, either for quarter, half houre, or other minute, & there see presently what Al­micantare doth limit the height, which your table cannot do, except you make him almost infinit to euerie minute, for which purpose only my Iewel were worthie to be accounted of.

Chapter 33. How to know the height of any starre aboue the horizon sitting close within dores, and thereby to learne to know the starres in the skie.

THis chapter bringeth any man to the knowledge of euerie seuerall star & their names, and for other vses is commodious: if the star be in the reete whose altitude yee seek, then lay the degree of the Sun to the houre, & part at which you desire, the height by help of the rule, & then turne about the rule to the starres Apex, which sheweth in the limbe, the starres houre, or distance from the South, by the 30. and if you reckon on the rule the degrees, from the starres Apex to the limbe: you haue the starres declination also, and whether his declination be North or South I shewed in the 2. booke 21. chap. This done, set the finitor to the latitude, & then on the starres paralel reckon from the limbe his houre distance, before found, & mark what almicantare doth crosse there, the same sheweth the starres height for that instante, & the Azimuth also cut­ting there, sheweth the verticall circle, in which the star is, which is more then was proposed.

But if the starre be not in the reete, then must you get his declination by tables, & likewise his culmination and vse his degree of culmination in steed of the starres Apex, as in the 24. 25. 26. &c. and then proceede as before in all respectes.

Blagraue Here G. Frisius setteth no example, but truly I think it good, because this capter brought me too knowe the starres as well in the skie as I did on my reete. This fifteenth of October 1576. about eyght [Page 39] of the clocke at night, whilest I was reading of Copernicus reuolutions, and Stadius, conferring them toge­ther, I found there that the auncient Astronomers appointed the first deg. of the Equinoctiall to begin from a starre of the second light, neere the Eclipticke ca led Cornis arietis, whose longitude was in olde time in the very vernal intersection of thecliptick with the Equinoctiall, which starre by reason of the motion of the starres Apogea, or more plainly of 8. and 6. speeres called Motus precessionis Equinoctiorum, is gone forwards 27. deg. 40. min. so that now he is in the 27. deg. 40. min. of ♈. For this cause I was desirous to know that starre in the sky as well as on my Reete. The Sunne then being in the 27 ⅙ deg. of ♎, I layde that deg. to 8. a clocke in the Limbe, thence turning the rule to the starres apex, I found in the Limbe, the starres houre distance from the South 3. houres and 13. degrees, almost towards the East, & on the rule his declination 17 ½ deg. VVher­fore I set the Finitor to the latitude, and on the 17 ½ parallel Northwards, I reckoned 3. houres and 13. deg. inward by the houre lines, and I found there crossing the 33 ½ almicantare the heighte of the starre: & there presently I saw likewise the 15. azimuth reckoned from the East or zenith line, to cut the same place of the starre. Then with speede I set the rule with his sights on the backside, ready vnto so many deg. vz. 33 ½ aboue the line of leuell, and going abroade I turned my face towards the 15. azimuth as neere as I could gesse from the East Southwards, and there might I see the starre sought for through the sight holes, and so by noting his place and the starres about him. I knew him euer since.

Gemma Frisius 39. To fynde in what verticall circle or azimuth the Sunne, or any planet or starre is, anye time.

Blagraue. Here Gemma Frisius maketh a special chap. for that which my Iewell yeeldeth vnasked, as by the 6. 31. 38. and diuers chap. appeareth. Note that before the lodestone, found, men sayled by the verticall circles of the Sun and the starres.

Chapter 34. How to finde the Meridian line, and the foure quarters of the worlde called cardines, diuers wayes, and specially by the Astro­labe.

BY the Sun you may doe it any houre, first getting the azimuth by the 6. and then lay the rule to the lyke degree Eastwards, or VVestwards on the Limbe, as the Sunne shall then be, and the rule there fixed, lay the Iewell flat on a stoole, or by some deuise leuell with the horizon, and turne him about till the Sunne shine through both the sight holes, and then will the noone line of the Iewell, lye euen in the Meridian, by which you may draw your line, but here had you neede of a wyre set vp plumme in the furthermost sight hole, and so lette his shade couer the other, because the Sunne beyng aloft cannot pearce the sightes, as in the seuenth is said.

Another way. Erect a stile or wyre plum on any horizontall plaine in the forenoone, and hang the Iewell that the Sunne pearce the sight holes, and marke both what deg. the rule cutteth in the Limbe, and where the shade of the top of the style endeth on the plaine, and there set a pricke, let the Iewell hang still, till the Sun in the after noone pearce agayne the sights, then set another pricke where the stiles shade endeth, then frō the stile draw a line iust betweene those prickes, the same is the Meridian, and crosse the same square with a­nother, and that sheweth East and VVest.

Another way. Make a conuenient circle on a plain in the centre, stick a stile plum, marke both in the fore­noone and after noone, where the stiles shade toucheth that circle, then draw a lyne from the centre to the middest of that arche betweene the touches, the same is the Meridian.

Another. The Astrolabe placed on an horizontall plaine, as before, and the Sunne shining, at Sunne ry­sing, and likewise at sunne setting through the sight holes, the degree betweene both, sheweth the Meridian line.

Another. Get the amplitude of the Sun by the 19. chap. number it on the Limbe from the noone line, and therto lay the rule and the Iewel leuel with the horizon, marke at Sun rising, that the beames pearce the sights and then is the noone line your desire.

Blagraue, Though Gemma Frisius hath set downe wayes inough to find the Meridian, yet all of them de­pend vpon the shining of the Sun, and in the day time. Therefore I wil shew two other wayes in the night, by the starrs, nothing inferior to the other. For, in like maner as you did the first way by the Sun, may you do by any starre hauing gotten his azimuth by his houre distance from the South by the 26. or 32. the deg. of the azimuth, reckoned on the Limbe, and the rule fixed therto, laying the Iewel flatte, as before in the Sun, turne him til you may see the starre, euen with the sightes which here had neede be wyres both.

The second way is that which I vsed, before I knew any other almost to set dyalles by, and in my opinion is nothing inferior to anye of those before. Thus woulde I do whē I had made my horizontal dial, & sodered on the cocke or Gnomon, I wold choose a bright starry night, & then bring forth my dial, & set him on the post where he should stand, then had I a playne square little boorde, or rather then fayle, a fayre square trencher, this square boord would I fasten or hold perpendiculer, close to the cocke, and so turne about cocke, dyall, boorde and all together, till I might see the last starre of the tayle of Vrsa minor, which they commonly cal the pole starre, euen with the boord, and then was I sure the cocke stood North and South, and so would I naile my dyal on.

But in this way you must not neglect two things, one is that the boord stand plumme vp with the cocke, at the time of view. The other which truely in those dayes I tooke no regard vnto, because I knew it not, for I could make dyalles before I studied Astronomy: and that is this: you must by helpe of your Iewel, choose your time, so that the beginning of ♈, or ♎, be in, or neere the meridian either aboue or vnder the earth. For [Page 40] when ♋ & ♑ are in the meridian, the pole starre declineth 3. degrees from the meridian once euery night that time may be had, halfe an houres misse will not alter much, or els watch at any time: when the first star of the tayle of vrsa maior, called allioth, and commonly the thill horse of charles waine, doth stand directly o­uer or vnder the pole starre, as it were both euen with a plumme line, then be sure the pole starre is directlie North to strike your true meridian by.

Chapter 35. The meridian founde, to finde the Azimuth of the Sunne or starres by sight without the houre.

THE noone line of the Iewel placed in the meridian on any horizontall playne by the 33. you may turne about the rule to the Sunne or any star by helpe of wier sights, as in the 33., and it sheweth in the limbe your desire.

Chapter. 36. The verticall circle of the Sunne or any starre, the degree of the Sunne, with their al­titude aboue the horizon, giuen howe to knowe the latitude and what it is a clocke.

TAke the Azimuth by the 34. and the altitude by the 1. reckon the same among your almicantares and azimuthes on your Reete, & mark where those two crosse one another, thē turn about the reete til the same point of crossing light on the paralel of the Sunne or Starres declination, & the houre line light­ing there with them, sheweth the houre of the Sunne of Starre. The Finitor likewise sheweth the latitude in the 7. This chapter is set downe è conuerso almost: if you vnderstand the 6. the 26. Chapter and others, this is very easie.

Chapter 37. The latitude, the Sunnes Azimuth and almicantare giuen to finde his place in the Zodiacke.

ABout the Tropickes this rule is vncertaine, as in the 9. chapter is shewed, where in the end of my addition this Chapter is perfourmed.

Chapter 38. To know what houre the Sunne or any Starre commeth to any azimuth proposed.

PLace the Finitor to the latitude, and marke what houre line cutteth the azimuthe proposed on the Sunne or Starres parallel, the same is your desire, sauing for Starres remember, locus solis ostendit horam, as in the 26.

Blagraue. I haue made short of these last chapters without examples, because my reete with the azimuthes and almicantares doth readily expresse them: but in deede by Gemma Frisius, working in his Cursor and Bra­chiolum, examples had been needfull, and specially in this chapter, which by G. Frisius requireth great paines.

G. Frisius 45. & 46. These chapters by reason of my new haue no matter.

Chapter 39. A Comet, a Planet, or an vnknowen starre being seen in the skie, how to get his longitude and latitude in the Zodiacke, and how by the right ascension and declination of any stare to get his longitude and latitude.

VVHen you see any Comet, Planet, or Starre vnknowne: First take his altitude, and the same In­stant the houre and minute by the 26. or otherwise very certainely, and his azimuth by the 34. Then set the finitor to the latitude, and marke what parallel cutteth the crossing of the Co­mets, almicantare and azimuth, so found, the same is his parallel of declination, and the houre line cutting the same crossing, sheweth his distance from the meridian by the 26. This done, place the degree of the Sun to the houre in the Limbe before found: and then in the noone line may you see the degree of culminatiō whose right ascension seeke by the tenth. Nowe if the Comet bee on the East part adde to this right ascension his houre distaunce, if on the West, substract it, and so shall you gette the right ascention of the Comet, but if in adding, the number exceede 360. cast away 360. also if the houre distance may not be subducted, then add 360. and so the product is the Comets right ascension. And nowe hauing the right ascension & declination [Page 41] of the Comet, marke where they crosse on another, and make a prick with ynke, for there is the Comets ve­ry place in the heauens then: to know his longitude and latitude, set the finitor to the eclipticke, and so the a­zimuth and almicantare cutting the pricke or Comets place before made, sheweth your desire, as in the 17. This is a notable practise and hath wonderous vse in Astronomie, and therfore worthie of an example at large.

Blagraue. VVherefore I will shew an example, whiche I my selfe for mine owne delight and experience did obserue. The nienth of October 1580 walking foorth at night, as I commonly dyd, to behold the con­stellations, I perceiued Southwardes a shimering or dimme light, in bignesse to sight, about an hand breadth, but because the skyes were cloudie, I coulde not well iudge wheather it were a Comet or some starre, with his light dissipated by the clowdes. The tenth of of October, being a cleere starry night, I perceiued wel that it was a Comette, or rather of some called a crinitall starre: it was then placed vnder the three starres of the [...] light, called [...] to fight about two yardes, so that it made an equicrurall trian­gle, with the two vttermost of those three starres. The eleuenth of October, the nyghte was cloudie and no starres seene: the twelfth of October, a cleere starrie night againe, and then I sawe the Comet was gotten aboue the saide starres about an ell to sight, and there made an equicrurall triangle, with the midlemost of the same three starres and the vtter starre VVestwardes, wherefore seeynge I was too wryte this Chapter: shortlye after seeynge the skyes so deere, I thought good too take a little paynes for that cause.

But hauinge on that sodaine no meridian readie founde, neither my Astrolabe furnished wyth wyer sights, nor placed as in the 33. and 34. to get the Azumuth, I presently deuised by help of two long poles, set somewhat a slope, vnderteeled wyth two forkes, too hang from the toppes of eache of them a corde with a plummet pitching them too and fro, tyll at the last standing South from them a pretie way off, I might see the pole star euen with both the cordes, which deuise here serued me in steed of a meridian line, for then I went on the North side of the cordes, and watched tyll the Comet came euen with them: whiche hap­pened presently after I was readie for him: then was I sure that he was in the meridian or South azimuth, which done, I presently tooke the height of the Comet 38 ½. deg. also of the great star called Aldebura [...], the bulles eye, by whiche according to the 26. I founde it eyght of the clocke the sunne then beeing in the 29. degree, 10. minutes of ♎.

Nowe because the Comets height, videlicet, 38 ½. was equall in manner to the Equinoctials height by the 7. I concluded his declination was 0. videlicet, in the Equinoctiall it selfe, and by the 29. his degree of culmination 24 ½. degree of ♒, and by the 10. his right ascension 327. And then hauing his right ascensi­on and declination, I seeke them out on the Mater by the 17. and marked where they crosse, and there make a pricke with ynke, for it was the Comets place at the time of my obseruation. Lastly, I turned the finitor to the Eclipticke, and there by the seuenteenth chapter, the azimuth cutting the pricke or place of the Comet, shewed his longitude 329. and better, whiche more plainelier is the 29 degree of ♒, and the Al­micantare cutting the same pricke shewed me 12 ⅔. degree, the latitude of the Comet Northwardes from the Eclipticke. By this chapter and the 17. a man might make a globe, with his constellations truer then some are that are solde.

Note that I sawe the same Comet or crinitall starre, about the East circle, and about 12 degrees high as neere as I coulde gesse vpon the sixe and twentie day of Nouember next following, for in truth, his proper motion was a mayne towardes the Sunne, though wandering, and then the Sunnes motion being towardes it, they had met and passed by eche other.

Chapter. 40. Howe to knowe the quantitie of the angles which the Eclipticke maketh with the meridian at any moment.

IT hath been shewed before that theclipticke doth alwayes diuersly or vnequally passe ouer as well a ryght horrizon or the meridian of any place, as the oblique horrizon, yet wyth a more diuersitie and vnlikenesse in the last, then in the first. The cause (as it hath been saide) is the deformitie of the angles or inclinations of the Eclipticke to the meridian. The determination of this chapter therefore is to teache how too knowe this inclination, which is very commodious in many respectes as shall bee shewed in their place.

First you must vnderstande that as oft as two great circles deuide, or crosse one another, they make either 4. right angles, or els 4. angles equall vnto foure right angles. Further the 2. opposite angles, at eyther point of the intersection are alwayes equal which in the Theoremes before my spericall triangle is manifested. VVherefore one of them beeynge knowen, all the reste are knowen lykewise, for doubling the angle, known, & then taking that out of 360, degrees, which is the quantitie of foure right angles, there remaineth the other two angles, the halfe thereof is one of them: as too the purpose, when ♋ is in the meridian, all the foure angles are right angle when ♈ or ♎ is there, then the twoo lesser angles are alwayes equall to the complement of the Sunnes greatest declination: the other twoo, then are easily knowen as before. Know also that the degrees of the Eclipticke alike distante from eyther of the Equinoctial points, do make like angles of inclination, but in sundry partes of the heauen. In this Chapter shalbe shewed to find the two lesser angles.

First therefore by the 29. learne the degree of culmination (that is the degree of the Ecliptick) in which you desire to learne the angles that he maketh with the meridian, then marke howe farre this degree is distant from the next Equinoctiall point: the like distance number on the artike circle from the poynt [Page 42] where it toucheth the meridian or limbe: for at the ende of this reckoning must needes bee the pole of the Zodiacke when ♈ or ♎ is in the meridian. Then applying the rule to the same poynt in the pole circle, it sheweth on the limbe the lesser angle sought for, if you reckon from the Equinoctiall to the rule.

For example, I woulde knowe the quantitie of the foure angles whiche the ecliptike maketh with the meridian in the beginning of the tenth degree of ♉. The distance of this poynt from the next equinoctium, videlicet, from ♈ is 39. degrees, then numbryng 39. in the pole circle from the limbe inwards, & therto laying the rule, I finde in the Limbe 71. degrees ⅓ the quantitie of one of the lesser angles, whiche doubled, maketh 142 ⅔. this taken out of 360. leaueth 217 ⅓. the quantitie of both the greater angles, each of them being 108 ⅔. or thus 71 ⅓. taken out of halfe 360. videlicet, 180. leaueth 108. ⅔. your desire: Thus haue you all the foure angels.

Chapter 41. To doe the same another way.

LEarne the distaunce as before from the next Equinoctiall poynt, number the same in the rule from the center outwards, seek also the right ascentiō of the distance, now learned as thogh it were distant frō ♈ and reckon the same among the parallels: then moue the rule vp and downe, vntyll suche time that the point of the role before noted do light exactly on the saide parallell, reckoned from the Equinoctiall, and so doth the rule shewe in the limbe, the lesser angle sought for as before.

For example, I would knowe the angles which the 6 ⅓. of ♑ maketh with the meridian, the distaunce thereof from ♈ is 82 ⅔. the right ascention of that distaunce from ♈, by the 10. chapter, is 81 ⅔. Therefore numbryng on the rule from the center 82 ⅔. and among the parallels 81 ⅔. I turn about the rule tyll the 82-⅔. degrees thereof doe cut the 81 ⅔. parallell, and then doth the rule in the Limbe shew 87. deg. the quantitie of the lesser angle.

A thirde waye is thus, gette the declination of the point proposed, by the fourth Chapter, number the same on the rule from the Limbe inwardes, and laye it on the pole circle, it sheweth on the limbe your desire.

As in the first example, the declinatiō of the tenth degree of ♉ is 14. degrees ½. that numbred on the rule from the limbe inwards, and laide on the pole circle, sheweth 71 ⅓. as before, I coulde shew another way but it needeth not.

Chapter. 42. How to finde the horoscope, called the Ascendent at anytime appointed.

THE powers of the planets and starres, as thy differ in respect of their nature, so are they either increa­sed or decreased according to their cituation in the heauens. For we see dayly that tempestes are chief­ly eyther raised or stayed by the Sunnes rising, setting or comming to the meridian, & that the moone by commyng to those foure place, doth rule the ebbyng and flowing of the Seas, and sometimes rayse tem­pests. Besides that, he that wyll marke it well shall finde, that the rising and settyng, and comming to the meridian, of the great starres, and specially with the Sunne, as in the 25. woorketh great effects in altering the ayre.

And therefore olde writers called these 4. places, the Cardines, and other newer call the angles or points, of the 4. Cardines or angles. Ptolomeus appointeth the midheauen to be chiefe and signifieth glory and honour. The next too bee the ascendent, and signifieth life. Then the descendent, and signifieth deathe. Lastly, the Imum coeli called angulus terra, the angle of the earth, and signifieth contrarye to the mydde heauen.

But nowe to the matter againe: the horoscope or ascendent is the degree of the eclipticke rysyng vppon the horrizon at any time proposed, or in the beginning of any thing, and chiefly in the natiuitie of a man, and is very easily founde by helpe of the Reete thus.

Place the degree of the Sunne in the Zodiack, vnto the houre proposed by helpe of the rule applied to the limbe, and streight looke what degre of the Zodiacke, cutteth your horrizon on the East, the same is the as­cendent, and the degree cutting on the VVest, the descendent.

Example.

Admit a childe were borne the 12. of October 1580. at eight of the clocke at night in our latitude, and that you woulde knowe the ascendent of this natiuitie. The Sun was then in the 29. degree 10. minutes of ♎, whiche degree, lay vnto eight. of the clocke among the afternoone houres in the limbe, and then looke vnto the 51 ⅔. horrizon what degree cutteth him on the East part, I finde the 1. degree of ♋, whiche is the ascendent. In this example the North halfe of the Zodiacke crosseth both the North and the South parts of the horrizon, but because it is not the North halfe of the Zodiacke that riseth on the horrizon, therfore you must take the degree which lyeth on the Northeast part of the horrizon, which is as I saye, 1. degree of ♋. For on the South part of the horrizon, there lieth the 10. degree of ♉. whiche is not too bee accounted of. But to proceede further in the example you may also (the degree of the Sunne standing at eight of the clocke as before) see the first degree of ♑, in the VVest parte of the horrizon descendente, where also the South halfe or the Zodiacke lyeth crosse both halfes of the horrison, so that the tenth degree of ♏. cutteth the North parte of the horrison, whiche muste bee refused, remembrynge alwayes that the North halfe [Page 43] of the Zodiacke must cut the North halfe of the horrizons, and rise and set on them, and no other, & so like­wise, the South, on the South, as is shewed in the second booke & 23. Chapter. Further hauing thus the as­cendent and descendent, you may also see the degree of midde heauen or culmination in the vpper part of the noone line 24 ⅔. in ♒, and in the lower part, the degree of Imum cali, or the angle of the earth, the 24 ⅓. of ♌, and so haue you the foure angles or principall houses of your figure for that natiuitie vz. the 1. 4. 7. & 10. houses.

Chapter 43. How to finde the horoscope or ascendent another way by the Mater.

FIrst get the degree of culmination for the houre and time proposed by the 29. and his meridian altitude by the 31. Next the quantitie of the angle, which the eclipticke in that degree maketh with the meridian by the 39. and 40. These had, number in the Limbe, the alt. of the deg. of culmin. from the Pole towards the Equinoctiall, and thereto lay the rule. Then in the Equinoctiall line from the Limbe in wardes reckon the quantitie of the saide angle, founde by the 39. and 40. and from that place following the houre line there, vnto the rule, and from the rule vnto the pole: note the degrees contained on the same houre line, betweene the pole and the rule, for those are the degrees of the Zodiacke contained betweene the degrees of culmina­tion, and the neerest part of the horrizon. For you must vnderstande that notwithstanding the horrizon and the eclipticke, beeing both great circles of the sphere, doe alwayes deuide one another into equall partes, yet is not the degree of medium coeli or culmination, iust in the middest of the halfe Zodiacke aboue the horri­zon, but onely when ♋ or ♑. are in the meridian by reason of the obliquitie of the Zodiacke: but is diuided by the meridian into vnequall portions, whereof the lesse this chapter findeth out. But sometime the lesser portion is on the East part, and sometime on the VVest, and is thus knowne. For when ♋ is on the East part the lesser portion is on the VVest, and then must those degrees bee subducted out of the degree of cul­mination, and when ♋ is on the VVest, the lesser portion is on the East, & then those degrees must be added and thereof commeth the horoscope.

Example.

In our latitude of 51 ⅔. admit the Sunne to bee in the 20. of ♉, and the degree of culmination at the houre proposed to bee 20. in ♋, whose meridian height by the 31. is 60 ½. degrees: the angle of inclination of the Eclipticke to the meridian in that degree is 82. by the 40. These knowen number in the limbe from the pole downewardes, the meridian altitude vid. 60 ½. and thereto lay the rule, then reckon on the equinoctiall from the limbe inwardes the quantitie of the angle of inclination, videlicet, 82. which lighteth on the 82. houre line reckoned from the limbe, wherefore reckon the degrees vpon the same 82. houre line betweene the rule and the pole, and you shall finde 85 ⅓. which is the distaunce betweene the medium coeli or culmination, & the parte of the horrizon next it. And now because ♋ was on the VVest part, you must adde this 85 ⅓. to the degree of culminatiō, vs. the 20. of, ♋, & so proceeding according to the sequele of the signes, the numeratiō falleth on the 15 ⅓ ♎, which is the ascendent sought for. Thus haue you two wayes to come by the ascendent, the first easier, but this more artificiall by the reasons of spericall triangles, and by my fift booke may be perfor­med two or three wayes.

Chapter 44. Of the twelue houses, and what a circle of position is.

ANtiquitie which ought to be imbraced, hath appointed 12. houses or places, wherein the starres & pla­nets haue great powers, of which 12. the 4. angles or cardines are chiefest. For euen as the Suns recourse doth not only alter the yeere beyng in the two equinoctiall points and solstices, but also in euery of the 12. signes, which causeth euery moneth to differ from other: euen so reason and experience teacheth in di­stinguishyng the 12. houses. But now all writers agree in setting out the 4. cardines, and greatly disagree in the other 8. For some beginning from the ascendent, deuide the Zodiacke into 12. equall partes by circles meetyng together at the poles of the Zodiacke, and to these doth Ptolomeus agree, Other some deuide the equinoctiall by circles from the poles of the world into 12. equall partes, as Alkabitius, & Iohannes de Sax­onia. Othersome (which in deed is the best) according to Regiomontanus, deuide the equinoctiall into 12. equall parts, by circles meeting together at the two intersections of the meridian with the horrizon, and this way sayth he, standeth with most reason, since they are distinguished by circles, like vnto those which appoint the 4. cardines, and meeting together with them at the same points. A fourth way did Campanus a famous mathematician appointe, deuiding the verticall circle of the East into twelue equall partes from the said in­tersection of the meridian with the horrizon. VVel, howsoeuer the houses bee deuided, those circles which distinguish them, or rather semicircles (for twelue semicircles do perfourme it) are called circles of position and the like semicircles may bee drawen by anye starre, degree or poynt of the heauen, and are called also circles of position. Therefore in short, a circle of position according too Io. Regiomontanus, whome wee meane to followe, for that all menne nowe doe so, is a seemicircle passing from the twoo intersections of [Page 44] the horrizon with the meridian by any starre, degree or point in the heauens. Notwithstanding by the Ie­well we shall teach to perfourme them to euery mans minde.

But nowe for the order of the 12. houses, therein they all agree. The first house beginneth in the ascen­dent, and so proceedeth vnder the earth, and the rest in order by the Imum coeli vntoo the descendent, and thence by the medium coeli vnto the ascendent againe, where the 12. houses end. Euerye house hath 25. de­grees forwardes, and fiue degrees of the house that went before. Also the 12. houses are set downe in a Geometricall figure, as you see here to gether with their properties.

Chapter 45. To finde the circle of position according to Io. Regiomontanus and Campanus, of any point proposed, and howe much the pole is eleuated aboue any such circle.

IT is shewed in the last chapter what a circle of position is. They are also as it were certaine Horrizons, vpon the which the point or starre proposed doth arise. And in truth euery of them doth represent some one horrison in the worlde, whose latitude, or poles eleuation wee seeke, and so by sphericall reason wee finde, what degree of the eclipticke is in euery of them. And therefore generally thus doe:

Get the declination of the point proposed by the 4. or 7. and his distance from the meridian by the 26. and others. This done, set the East Azimuth or zeneth line to the latitude in steed of the Finitor, and so doe the azimuthes represent all the circles of position. The zenith line beeing placed thus to the latitude, as I say, then in the parallel of declination of the point proposed, reckon his houre distaunce, from the meridian before founde and the azumuth there cutting, sheweth the circle of position, if you count him from the Ze­nith line: which being had, then to knowe the poles eleuation aboue it, doe thus, reckon amongest the Almi­cantares from the Zenith point, the latitude of your countrie vpon the circle of the position found: and ther­too lay the rule, then reckon the space between the same point of the circle of position, and the Limbe by the degrees of the rule, and so haue you the eleuation sought for. By this chapter any man may easily make tables of positions for any countrie.

For example, in the 41. chapter I supposed a natiuitie the 12. of October 1580. at 8. of the clocke at night: I woulde now knowe in what circle of position the Sunne then was, I lay the Zenith line to our latitude, vz. 51 ⅔ insteede of the finitor, then for 8. of the clocke. I reckon 8. houres from the South in the Sunnes parallel being 11 ½ by the 4. and there I finde to crosse the 28 ½. azimuth vnder the Zenith line, whereby I conclude that the Sunne was then in the 28 ½. circle of position vnder the earth, in the Northwest quarter. This done, reckoning our latit. vi. 51 ⅔. on this 28 ½. azimuth, & y^e from the zenith point found, therto lay the rule & you shall see betweene the Limbe and the same point vpon the rule vpon the rule 43. degrees. ½. the poles eleuation aboue the same 28 ½. circle of position, some were wont to name the circles of position according to their latitude as that this should be the 43. ½. circle of position.

Blagraue, Heere you may see the wonderfull diuersitie in vse of all the circles in my Reete. Insomuche that at any time when ♋ is in the South meridian, as he is on the Iewell, then placing the Zenith line, too the latitude, the azimuthes doe shew the circles of position of euery starre placed in the Iewell, and point in the ecliptick or other part of the heauens without any more ado, whereas by G. Frisius Cursor and Brachiolum [Page 45] you must stand setting for euery particular place with much trouble. Heere note that where the Azimuthes do not reach any point, you may but turne about the Reete so that the zenithes nadir be to the latitude, and then he supplieth the whole.

Chapter 46. Of the other eight houses according to Regiomontanus, and how to set a figure.

THe foure angles haue beene exactly taught before in the 41. and others, and the other 8. houses also a­ny man of capacitie might picke out by the last Chapter, but this working hath a pretie methode, which I will not hide: for you shall vnderstand that all the 12. houses are diuided with 6. whole circles of po­sition, which diuide thequinoctial into 12. equal partes. Two of those sixe are the horizon and the meridian which limit out the foure angles, the other 4. are two of ech side alike distant from the meridian, bending to­wards the horizon alike, and haue like equations ech to his match, and ech of them boundeth two houses. So that euen as the meridian doth bound the midde heauen, and angle of the earth, so doth the circle of position next the meridian on theast part, appoint the 11. and 3. houses, the second next the horizon, the 12. & 2. likewise two circles on the west part, sheweth the other foure houses. Further knowe that opposite houses are numbred by like degrees of opposite signes, so that hauing founde sixe houses, they are had all 12. Therefore 4. houses knowne with the Horoscope and midheauen, there needeth no more for this turne: & those 4. are knowne if the two circles of position of the 12. houses, which also is of the 2. & of the 11. which also is of the 3. be knowne by their depressions vnder the pole in this manner.

Place the zenith line to the latitude as in the last Chapt. and then marke what azimuth cutteth euery 30. deg. of thequinoctiall line, the same is the circle of position of euery such house, to which get the eleuation of the pole by the last Chap. Thus reckon on thequinoctial from the Limbe inwards 30. deg. and there shal you finde to cut the 47. azimuth or circle of position, seruing to the 11. and 3. houses, whose depression vnder the pole by the last Chapter is 32 ⅓ deg. Then reckon on thequinoctiall 30 deg. farther in, videl. 60. from the me­ridian, and there shall you finde to cut the 19 ¾ circle of position seruing to the 12. and 2. houses, whose de­pression vnder the pole by the last Chapt. is 47 ⅓ deg. These two eleuations vz. 32 ⅓ and 47 ⅓, keepe well in minde: for they serue you to get your 12. houses for euer in our latitude: you shal do wel to choose them out among the horizons in the Iewel, and with prickes or otherwise disseuer them from the rest that you may redi­lie finde them, writing the number 12. and 2. to the 47 ⅓ horizon on ech side, and 11. and 3. to the 32 ⅓ ho­rizon, for so I alwayes did.

These things done, you shall thus haue your 12. houses and the degrees of theclipticke, answering them, lay the degree of culmination of the time proposed on 6. of the clocke before noone in the Limbe, where he lyeth in a right line or right horizon, thense for the 11. house moue forwards the degree of culmination 30. deg. in the Limbe, that is vnto 8. of the clocke by helpe of the rule, then in the horizon of the 11. house videl. the 32 ½ horizon, you shall finde the degree of theclipticke, beginning the 11. house, as in the natiuitie mentioned in the 41. Chapt. lay the degree of culmination, being 24 ⅔ of ♒, vnto 8. of the clocke before noone, you shal [Page 46] see to cut the Horizon of the 11. house videl. the 32 ⅓ the 25 ½ of ♓, which is the beginning of the 11. house: Now for the 12. house I moue the deg. of Culmin. 60. deg. that is to 10. of the clock, & there do I see in the horizon of the 12. house videl. the 47 ⅓ the 20. deg. of ♉, which is the beginning of the 12. house, but then mouing the Culmin. to 12. of the clocke, there appeareth in our horizon videl. 51 ⅔ the Horoscope or ascend. videl, the 1. deg. of ♋. Now for the 2. house I moue the deg. of Culmin. 30. deg. forwards vz. vnto 2. of the clocke at the afternoone, then in the horizon of the 2. house, which is all one with the horizon of the 12. vz. the 47 ⅓ horizon, I see the 21 ½ of ♋. Then remouing the Culminat. 30. deg. farther videl. to 4. of the clock, there I finde in the horizon of 3. and 11. house videl. the 32 ⅓ the 7. deg. 10. minutes of ♌, the beginning of the thirde house, and nowe haue 16. of the 12. houses. For the rest, I neede but set downe in the oppo­site houses, opposite signes & deg. and so is the figure set of that natiuitie, as you may here see. By this meanes you may make table of the 12. houses to euery latitude, perhaps truer then some that are printed.

Blagraue. Note that you must in remoouing the degree of Culmin. alwayes take the degree rising in theast, on the said horizons, and further that you be sure to remember, when the south signes rise, to take the deg. cutting the Southeast part of that horizon, and when North signes rise, the deg. cutting the Northeast part as in the 2. Booke 23. Chapt. and in the 41. is shewed.

Chapter 47. Of the other eight houses, according to Companus and Gazulus.

HE that hath well vnderstood the last 3. Chapt. neede no instructions more as I thinke, for Campanus, as is before said, deuideth the circle of theast into 12. equall parts as Regiomontanus doth thequinoct. and that also from the same opposite points of the meridian, crossing the horizon, wherefore laying the zenith line as before, to the latitude, then doth the Finitor represent the circle of the East, or the East azi­muth being there redily diuided into all the circles of position, which the azimuthes do now represent, and euery 30. azimuth beginneth one of the 12. houses in order from the ascendent vnder the earth, and so for­wards. Getting also their depressions vnder the poles in al respects as before, onely herein is the greatest diffe­rence, that you must also here get the home distance of euery house from the meridian, which the hourelines on the mater shew vnasked, for the zenith line standing at the latitude, you may see the 38 ½ houreline recko­ned from the Limbe, to cut the 30. circle of position reckoned on the Finitor from the Limbe, and likewise the 66 ½ houreline to cut the 60. circle of position. The rest are distant in like deg. from the meridian, though in seuerall quarters. And where in the 45. you laid the deg. of Culmination to 30. deg. or 8. of the clocke from theast, for the 11. house, you must insteede thereof lay it on the first houre distance from the meridian found vz. 38 ½. And in steede of remouing the Culmination to 60. deg. or 10. of the clocke, you must lay it to the second distance videl. 66 ½ for the 12. house, then for the 2. house reckon 66 ½ deg. from sixe a clocke at night towards noone, and for the 3. 38 ½ from the west vpwards also, for you must note that the axtree line here representeth the meridian and the nooneline or horizon after a sort, in working.

Blagraue. I leaue out here an example because it differeth not from the vnderstanding of the last Chap. but onely in reckening the houre distance, of the which I presume, my Iewel being so furnished with all needeful circles, supplieth the meaning primafacie without serching about the bush as Gemma Fris. doth. And as for the other kind of houses after Firmicus and Alkabitius and others, I vtterly leaue out because my Iewel being full fraught with al necessarie circles, a I say, yeldeth greater matters with small instruments, & also for that they are vtterly reiected, and none now in vse, but the first way after Iohn Regiomontanus.

Chapter 48. To know in what house any starre is at any instant and to place the chiefe starres in the 12. houses.

Some haue had a delight to place diuerse of the great stars in the figure being set, which must thus be done. Get the stars houre distance from the merid. at the instant proposed by the 30. & number the same in his parallel of declin. then set the zenith line to the latitude, as in the 2. last Chap. and looke what azimuth or rather circle of position cutteth the same point in the said parallel, as in the 44. then mark in thequinoct. line what deg that circle of position lacketh of the meridian (for euery house is 30. deg. of the quin. after Io. Regi. whom we follow). The meridian is alwayes the beginning of the 10. house, wherfore so many times 30. as the stars circle of position is distant from the meridian on thequinoct. line, so many houses it lacketh, or is beyond the 10. house: so that by knowing in which quarter of the skie the star is, the house, & part is easily reckoned.

Blagraue. He that vnderstandeth al before, needeth no example here, but thus much farther, I wil shew you as in the 45. you may plainly see that 4. circles of position with the Finitor, & the meridian, do shew the whole 12. houses, which are as there is said, the 47. and 19 ¾ azimuthes or circles of position, reckoned on ech side of the zenith line as he is there placed: wherefore if you did disseuer out with some distinction by prickes or o­therwise those 4. azimuthes, and set their figures to them, namely at the zenith line, 1. because hee being the horizon beginneth the 1. house at the 19 ¾ azimuth vnder him: or leftwards two, for the 2. house: at the 47. azi­muth 3. at the 90. which is the Limb. 4. and so round backe againe by the same circles ending at 12. in the 19 ¾ azimuth aboue the zenith line. These circles being thus distinct & numbered, are in truth your 12. houses, the zenith line being at the latitude. So that now the star or sunnes distance from the meridian reckoned on their [Page 47] parallel as aboue, make there a pricke with incke, for there is the sonne or starres place for the time being, and now in what house he is, the said azimuthes distinguished and figured as before, shewe without asking. This could neuer be done by G. Frisius Astrolabe in this manner.

A short example: the sunne being in the tropicke of ♋, would know in what house he is at 7. of the clock before noone. I set the zenith line in steede of the Finitor to the latitude, and then I finde that the 27 ½ circle of position cutteth the tropicke with the 7. a clocke houreline, which accordingly the numbers euen now placed sheweth me that the sunne is farre within the 11. house about ¾. neere towards the 12. Or thus more plainly, follow the said 27 ½ circle of position vnto thequinoctiall, and there doth he shew that the sunne is in the 20. deg. numbred on thequin. within the 11. house towards the 12. that is ⅔. Likewise I shewed in the 26. chap. that the great starre Hircus was 4. houres and 2. deg. past the north meridian eastwards in his parallel being 45. there make a prick, & mark in what house he then was, you shal finde him in the 11. house somwhat more then halfe in.

Chapter 49. Of directions, what direction is, and by what meanes it is performed.

TO direct saith Io. Regiomontanus, is no other thing else then to turne about the sphere vntil the second place be brought to the place of the first, that is to say, vntill the second place light on the semicircle of position of the first, some haue called this prorogation, some inambulation, some Aphesin. I thinke it ra­ther saith G. Frisius Dimission, or Emission, because one place of the heauens by turning of the sphere is emised or sent forth vnto the site of another place. Whereby Ptolomeus doth call those places which hee termeth chiefe, that is to say, those which may receiue, the chiefe signifiers of our life, and adde power to those signifiers Topous Apheticous: and also calleth those signifiers Aphetae, as it were Dimissaries, or Emissaries. The Arabians call them Hylech Plilippus, prorogators. To be short, Apheta called commonly both in latin & english, Significator, or signifier, is either a star, or some notable place in the heauens, hauing the chiefest dominion ouer the life of man, and Ptolomeus hath 4. almost, the Sun, Moone, the pars Fortune, & the Horoscope. There be other pla­ces also accepted of authors, as the midheauen, and any planet hauing gotten some notable dominion: and are called Apheta and signifiers, vnto whom other place of the planets, or places filled with the beames of planets are dimised or sent, that is to say, dyrected.

For declaration hereof, take G. F. own example of the natiuity of King Philip, somtime husband to Queene Marie. He was born 1527. at a quart. past 4. of the cloke in the afternoone, the sun being 9. deg. in ♊, & the lat. of the country being 40. The figure thereof being set there, is found 2 ½ deg. of ♏ in the ascendent, and in the midheauen 8 ½ deg. of ♌. And now to the purpose, the Horoscope is alwayes a notable place. For hence do they take their iudgement of the helth of the body, of life and of iourneying, wherefore it is called Apheta, or a signifier. And Mars which then was in the 2. house, is called the promiser, & the 2. place, which by the mo­tion of the sphere is brought vnto the first place, that is to say, vnto the horizon which is the circle of position of the horoscope, & this is called direction or emission. The question is chiefly how many deg. of thequinoct. shall passe ouer the circle of the horizon or the meridian, in the meane time, while the place of ♂ be brought vnto the place of the Apheta. For it is signified that there are so many yeres to come before the effect of the pro­miser shalbe performed on the Apheta. This matter is easily done by the Iewel for any region or point of the heauens. Seeke the circle of position of the first place or signifier, and his horizon by the 44. and 45. or rather 47. and vpon the same horizon lay the deg. of the Apheta in the zodiacke, and marke then what deg. is in the midheauen. And then turne about the Reete vntil the 2. place of the promiser come to the same horizon of the signifier, by and by moue the rule vnto the deg. which before was in the midheauen, it sheweth on the Limbe the degrees passed of thequinoctiall, which signifie the yeares of the direction.

As in the said example, place the Horoscope being the signifier of life vnto his circle of position, that is, vnto the 40 horizon, you shal see on the line of midheauen ♌ 8 ½ deg. then turning the Reete vntill the place of ♂ which then after Alphonsus, was in the 29. deg. 17. min. of ♏ be brought vnto the same 40 horizon, and then setting the rule vnto the 8. deg. 30. min of ♌, you may see that the Reete hath gon forwards on the Limbe 34. deg. ⅙ almost, and so many are the deg. of emission or direction of ♂ vnto the horizon. But heere must you diligently note that for Aphetaes placed in theast halfe of the heauens, that the horizons be taken on theast part of the Iewel, & for those of the west halfe, on the the west part, the rest may be performed as before. Nei­ther must you here neglect the latitudes and declinations of the planets, for which I taught you remedie in my addition to the 23. Chapt. And here another way to remedie the latitude of any planet, is thus: Get the deg. of right ascen. and thereto lay the rule, whereon reckon his declination from the Limbe inwards & there is the true place of the planet: this rule also serueth for all stars not being in the Reete, and also for starres in the Reete in workings for time long passed or to come, because fixed starres go forwards in long. a deg. at the least, in an. 100. yeares.

Chapter 50. Of conuerse or euerse direction, or dimission.

WHen as the signifier is on theast part Ptolomeus, always emiseth or bringeth the 2. place vnto the first circle or place, or to the Aphetaes place, & reckoneth the deg. past in the meane time. And this he cal­led) proiection of beames, because that either the planets places or their beames are brought vnto the Aphetaes place, and this is called direct, because the second place differeth from the first according to the se­queale of the signes. But when the Apheta differeth from the midheauen towards the West, the seconde place shall be the VVest it selfe, and then shall the Apheta bee brought vnto the VVest part of the horizon [Page 48] to know the direction of the Apheta vnto the Anareta, called Intersector, and is saide to be contrarie to the order of the signes, because the second place, that is to say, the descendent or point of the west, which then is the Anareta, is distant from the first, contrarie to the order of the signes. Of the first manner is spoken in the last chap. Of the latter there needeth not many words, for it is euen like. For the deg. of midheauen be­ing placed in the nooneline, turne the Reete till the Apheta be brought to the descendent, and then the rule laid to the deg. which was in midheauen, you may see howe many deg. the sphere hath now gon forwards which are the degrees of the direction.

Chapter 51. How farre, or into what part of the zodiacke, Dymission or direction shall come any yeare.

IT hath beene plainely and exquisitely shewed in what time the Dymission or direction of any place is per­formed. But oftentimes the question is before al the Dymission be performed, into what deg. of the zodiack the direction shall come, and he that hath wel vnderstood the premises, shal performe this Chapt. with lit­tle adoe: for the question it selfe hath a certaine difficultie by the manner of pronuntiation, which I thinke best first to expound, for many either pronounce ill, or else vnderstand not the matter wel: for whē as we say, how farre commeth the direction of the Horiscope this yeare. It seemeth to be vnderstoode that the Horo­scope (which now we cal the Apheta) should be moued in the heauens: but the matter is otherwise when the Apheta is in theast part. For then we do not emit or send forth the Horoscope, or any Apheta, but do mooue the places following, or promisers vnto him. Therefore when it is demaunded how farre the emission of the Apheta placed on theast part, hath gone forwards: the question is, what deg. of the zodiacke shal come vnto the semicircle of the Apheta in this or that yeare. But the Apheta standing on the West part: It is more right­ly said that his Dymission doth occupie this or that deg. of theclipticke, because the Apheta is knowne to be moued towards the West. Now therefore togither with an example will we explaine the matter.

First in direct direction. Therefore in the natiuitie of king Philip, let vs appoint the Apheta to be the Horo­scope it selfe, let vs see then vnto what degree of the zodiacke the Dymission, or direction shall come that yeare 1554, that is to say, what part of the zodiack shal then come vnto the Horizon circle by Dymission vid. vnto the Aphetaes place. First consider how many yeares be passed from the natiuitie it selfe, which are 27. for he was born, anno 1527. Secondly, place the Apheta vnto his place, which is the Horizon, & note the deg. of culmination, then remoue the same deg. of culmination 27. degrees forwards in the Limbe, and there shall appeare in the 40. horizon ♏ 23. deg. 40. minutes almost, therefore this deg. of the zodiacke then commeth vnto the Horoscope.

But now let the Apheta be in the west part of the heauens, as the part of Fortune is in the said natiuitie, let vs seeke his Dimission for the same yeare 1554. place the deg. of Culmination in the midheauen videl. ♌, 8. deg. 30. minutes: and numbering thence 27. deg. in the Limbe, and thereto laying the rule, place there the degree of Culmin, videl. the 8 ½ of ♌. Now then marke diligently the distance of the part of Fortune from the south, numbering the degree of the Limb betweene the nooneline and the rule placed on the part of For­tune. In our example they are 45. deg. 45. minutes almost, with the distance therefore, and the declination of the Apheta found before videl. 22. deg. 25. minutes north, seeke the circle of position of the same Apheta for that time, and theleuation to it, and how many degrees of thequinoctiall the same semicircle is distant from the meridian. Therefore by the 44. or 47. Chapt. the circle of position is found distant from the meridian in theast circle 41. deg. 48. minut. Theleuation of the pole is 25. 40. The same semicircle doth cutte off from the west quadrant of thequinoctiall 34 deg. 30. minutes as neere as may be gathered by a small instrument. Now therefore we haue the circle of position of the part of Fortune for 27. yeares passed, the question now is, what deg. of theclipticke was in this circle of position in the beginning of this natiuitie. For it may right­ly be said that the Dymission or direction of the part of Fortune is come vnto it. But seeing this is a generall rule, it is better to shew it in a Chap. following.

Chapter 52. To finde what degree of theclipticke occupieth any circle of position at any time.

THis Chapter is very commodious, yet combersome by tables, & therefore the more pleasant by the Iewel, it is but in effect the 45. and 46. euersed. Get the circle of positions eleuation by the 45 and his distance from the meridian, as in the 47. These had if the circle of position, shall be oriental and nocturnall (that is) vnder the earth, reckon on the Limbe his distance from the meridian found, from theast point towards mid­night, and thereto lay the deg. of Imum celi, or the angle of the arch, if orientall and diurnall then towards noone, and there lay the Culmination, if occidentall and noct. then from the VVest towards midnight and there lay the Imum caeli, if there diurnal, towards noone, and there lay the degree of Culmination as in the 46. Chapt. And so the deg. of theclipticke cutting the circle of positions horizon is your desire.

Example. Admit the sunne be in 8 ½ deg. of ♏ the Medium caeli, or Culmination, then 6 ½ of ♈. the Imum caeli or angle of the earth 6 ½ of ♎, theleuation of the 48. position vnder the arth eastwards, being by the 44. 31 ⅔ his distance from the meridian 29 ½, which I reckon from theast towards midnight on the Limbe, and thereto lay the deg. of Imum caeli videl. 6 ½ of ♎ then looking on theast part of the 31 ⅔ horizon, I find there the 10 ½ of ♍, the deg. of thecliptick, occupying the 48. circle of position nocturnal. Likewise for the 48. position diur­nall Eastwards the deg. of colmin. videl. 6 ½ of ♈, laid 29 ½ deg. towards noone, you shall finde in the same 31 ½ horizon 18 ⅔ of ♉, remembring alwayes to looke on the North Horizon for north signes, & so contra­riwise, as in the 2. Booke and 23. Chapt. Looke farther the 4 Booke.

Chapter 53. To know the quantity of the angle of the inclination of the Elipticke to the ho­rizon at any degree of the Eclipticke.

THis chap. is very profitable about the occultations and appearing of starres, and also for the deforma­tion of Eclipses. First therefore place the deg. proposed vnto the horizon on the East, then reckon the de­grees of the Eclipticke betweene the ascendent and midheauen, if they exceede 90. then take the deg. betweene the midheauen, and the descendent, or take the first out of 180. all is one: next get the meridian al­titude of the degrees of midheauen. These had, reckon on the rule that distaunce betweene the horizon and mydheauen outwards from the centre, and moue the rule vp and downe till the same poynt do cut the pa­rallel answerable to the height of midheauen, and so doth the rule shew in the Limbe, reckoning from the E­quinoctiall the quantity of the angle sought for: which is also the lesser angle, as in the 39. and therfore sub­ducted out of 180. leaueth the greater, and this lesser angle is alwaies the height of the Ecliptickes 90. deg. or nonagess. gradus from the horizon.

Example. In the yere of our Lord 1590. the 20. of Iuly the 19. houre 30. min. which is at halfe an houre past 7. the next morning (for Astronomers begin the day at noone, and reckon agayne til noone vnto 24. in their tables) there shall be a great Eclipse of the Sun, the Sun and moone both being in coniunction in the 7 ½. deg. of ♌: wherefore placing the deg. of the sun vnto 7. ½. of the clocke 30. min. I see 10. ⅔. of ♍ in the ascendent, and in the descendent 10 ⅔. of ♓, and in the mydheauen 4. ♊, I would now know the quan­tytie of the lesser angle made betweene the Eclipticke and the horizon. From the ascendent to the mydhea­uen, are 96 ⅔. deg. take those out of 180. there resteth 83. ⅓. the meridian altitude of the degree of culminati­on, vz. of 4. of ♊, by the 28. is 59. ½. deg. This done, reckon the degrees betweene the midheauen and the des­cendent, vz. 83 ⅓. on the rule from the centre, and lay the same degree on the 59 ½. parallel, & so doeth the rule shew in the Limbe, reckoning from the Equinoctiall 60. deg. the quantity of the angle sought for, which is the height of the nonages. grad. or 90. deg. of the Eclipticke aboue the horizon.

By this angle many things are brought about, as shalbe touched hereafter.

Blagraue. It may be that Gemma Frisius tooke the course before about this chap. for better vnderstanding of the chap. following, or els I must say hee ouerclipt another more easie way, as hee himselfe saieth, learned men do many times, which I wil here shew you.

Place the deg. at which you require the angle, to be ascendent: thence reckon vpwards 20. deg. on the Zo­diacke, therto lay the rule, which sheweth in the Limbe the distance of that 90. deg. from the South by the 30. and betweene the Limbe and this 90. deg. of the Zodiacke, rckoning on the rule, is the declination thereof by my addition to the 4. chapter. These had, seeke on the Mater where this declination and distaunced do crosse by the 26. and the Almicantare (the Finitor beyng at the latitude) cutting there, sheweth the angle sought for.

As in example before, the 10 ⅔. of ♍, being ascendent, I reckon on the Zodiacke vpwards 90. deg. it ligh­teth on the 10 ⅔ of ♊, thereto I lay the rule, where I finde the distance of the same 10 ⅔. of ♊, from the south to be 7. deg. ¼. and his declination by reckoning the deg. on the rule betweene it and the Limbe, to be 22. and better, North. These two being had, I set the Finitor to the latitude, and then marke where the 7 ¼. houre line reckoned from the Limbe, and the 22. parallel do crosse, and there I see cutteth the 60. almicantare, as before, the angle sought for, with lesse trouble.

Chapter 54. how to finde the altitude of the Sun or any point of the Ecliptick aboue the horizon at any time otherwise then hath yet bin shewed.

THe houre giuen, with the Sunnes place: the ascendent is knowen by the 41. and by the 52. the altitude of the nonag. grad. or 90. degrees from the horizon, which I reckon of the Limbe on the Reete, and thereto lay the labell: this done, note how many degrees the Sun, or point giuen, is from the ascendent or descē­dent, so many deg. reckon on the rule from the centre, where the Almicantare cutting, sheweth the altitude sought for.

As in the example of the last chap. the houre being 7. and 30. min. the ascendent thereby 10 ⅔. of ♍, and so the altitude of the 90. degree 60. VVherefore these being had, I would also know the altitude of the Sunne at that instant, being then as is saide in the 7 ½. of to ♑, to perform this, I reckon in the Limbe or vttermost azimuth of my Reete, the altitude of the 90. deg. of the Eclipticke, vz. 60. deg. and thereto lay the Labell, then I reckon on the Label, from the centre the space between the ascendent, videl. 10 ⅔. ♍, & the suns place, vz. ♌ 7 ½. whi­che is 33 deg. ¼. And there do I see the 28 ⅓. almicantare to cut the Label, which is the height of the sunne for the time of that Eclipse. In like maner may you come to the height for that instant, of any deg. of the eclip­ticke by their distance from the ascendent or descendent, the Labell standing still there which in calculating the Eclipses standeth in great steede.

Chapter 55. To find what angles the circle of altitude maketh with the Eclipticke at any point thereof assigned.

THis chap, serueth chiefly to delay the sunnes Eclipses, wherein is almost the greatest labour and diffy­culty of Astronomie and Arithmeticke but shal be easily performed by helpe of this Iewel. First note that Gemma Frisius meaneth by the circle of altitude, the vertical circle or azimuth, wherein the sun or any other deg. proposed, is. Then to proceede, gette the distaunce of the point proposed from the zenith, by sub­ducting his height learned by the last chap. out of 90. Likewise learne by the 52. the distance of the nonagest. grad. or 90. deg. the Eclipt. from the zenith: these 2. being known, reckon the greater of these on the rule from the centre, & then moue that note on the rule vnto the paral. of the lesser distance, and so doth the rule shew in the Limbe the quantity of the angle, reckoning from the Equinoc. As if in the last question of the E­clipse, I wold know of the angle which the circle of altitude passing by the Sun, doth make with the Eclipt. I seeke the distaunce of the Sun from the zenith, by subducting 28 ⅓. from 90. 61 ⅔. Likewise subducting the height of the 90. deg. vz. 60. from 90. there resteth 30. his distaunce from the zenith also. Now doe I reckon on the rule 61 ⅔. deg. and there make a note which I lay vnto the 30. parallell, and so is the angle sought for found to be 34 ½. shewed on the Limbe by the rule.

This is alwaies one of the lesser angles, as in the 39. chapter, and is towardes the East, when the poynt of the Eclipticke assigned, is on the East part of his 90. degree, and on the VVest if otherwise. Wherefore this angle found, vz. 34. ½. subducted out 180. there resteth 145 ½. the qūatity of the gteater angle as in the 39.

Furthermore know this for a certain truth, that the angle found in this chap. is that which they cal the an­gle of latitude, because the parallaxis or diuersity of aspect, which happeneth in latitude is subtended or ly­eth vnder it, and this angle if we subduct him out of 90. there shall bee lest the angle which they call the an­gle of longitude: but whereto these belong, and to what vse they are sought, shall bee shewed here following.

Blagraue. In this chap. the Equinoctiall representeth the Ecliptick, and the rule, the vertical circle of the sun, the poynt noted in the rule the zenith, or rather the like distaunce from the Sunne: the other way or down­wards the centre of the Iewell representeth the point of intersection with the sunne: the line between the note in the rule (which as I saide is the zenith) and the Equinoctial, is the distance betweene the zenith, and the 90. deg. but rather between the like deg. on the azimuth beneath the sun. The reasons hereof I will shew some­what more large in my 5. booke, and omit them here, both because too much tedious matter together might cloy a young student, and also for that I meane to shew, how to find al spherical triangles on the Iewel, which Gemma Frisius instrument could neuer doe.

Chapter 56 How to finde the parallaxis or diuersity of the sight of the moone in the circle of altitude.

IT is by demonstration affirmed in Ptolomeus works & other famous authors, that the globe of the earth is but a point in respect of the heauen it selfe: that is to say an vnsensible bignesse in comparison to the hea­uens. And that whether a man be in the centre of the earth, or on the circumference, there commeth ther­by no diuersity of any apparents in the heauens: heereof it commeth that all dialles are made, as though we dwelt in the centre of the worlde, neyther hath any man found diuersity therein, were he neuer so ingeni­ous or practick. But as these things are most true, and stand vpon good demonstration, yet in some of the pla­nets it is not so. For although the fixed starres, and the highermost planets, because of their vnmeasurable di­stance from vs, make no diuersity of sight to vs, from that which is seene in the centre of the worlde: yet in the lowermost planets how much the neerer they be vnto vs, so much the more is perceiued the euariation of their place appearing from that which thy haue vnto the centre of the world. For you must vnderstand that all calculations of the motions of all the starres and planets perfourmed by tables or Ephimerides, are done & made to the centre of the world. Now because the speere of the moone is neerest vnto vs, there is perceyued a manifest diuersity betweene her place, which we see dwelling here on the circumference of the earth, & her place which a line drawn from the centre of the world through the moon, doth shew: because the semidiame­ter of the earth, according to which quantity we dwel from the centre, hath a sensible bignesse vnto the dy­staunce of the moones speere from vs. This diuersitie is perceyued not only in the moone, but also in Mercury and Venus, and somewhat in the sunne, though verye little, and scarce perceiuable: this diuersity of sight called of Ptolomeus, Parallaxis, is greatest in the moone: in the rest it is almost neglected, neither is it alwaies a­like in the moone: which is a sure argument that the moone is not carried in Homocentre speeres, as some woulde haue auouched. But how much the more the moone departed from the centre of the world, and be­commeth. Apogea, so much the lesse happeneth this parallexis, and how much the neerer she commeth to the earth being Hypogea. so much the greater is this parall. so that the greatest parallaxis of all, is one degr. 6. min. And the least which happeneth, the moon being in perig. is but 50. min. omitting a few seconds: in the sunn this parall. is 3. min. onely. This parallaxis, when it is simply spoken of, it is meant the moone or planet beyng in the Finitor or horizon, in the zenith they can haue no parallaxis: for there the beame proceeding from our sight, is all one with that which commeth from the centre of the world, but the more neere to the horizon, the greater. These things are plaine to those that vnderstand the theorickes, and are excellently demonstrated by Ptolomeus.

The scope therefore of this chap. is by the greatest horizontall parallaxis, giuen how to get the parallaxis of the moone for the altitude she is in.

Gette therefore out of some tables of Eclipses, or out of Copernicus, the moones greatest or hory­zontall [Page 51] parallaxis for the fight, she is in her speeres, and likewise the moones height aboue the horizon by the first or 32. though commonly in an Eclipse, she differeth not from the sunnes height. Then number the horizontall parallaxis on the Limbe from the Equinoctiall vpward, and therto lay the rule: which done, rec­kon on the rule from the centre, the moones distāce frō the zenith, or inwards from the Limbe the moones alti­tude all is one, the parallel cutting the rule there sheweth your desire.

This institution bicause it is most pleasant, by the which not without admiration of most men, Eclipses are prognosticated long time before they happen, deserueth an example.

As in the Eclipse like to be 1590. mentioned in the 50. chap. the sun & the moone shall he in the 7 ½. de. of ♌, this I say is their place calculated to the centre of the world: in the which luminaries shalbe coniun. Now therefore the question is, what the sun & moones visible place shalbe, thereby to gather the euariation of the Eclipse. For by reason of the euariation of the sight or parallax. they shall not be seene here coniunct, as they are found to be in respect of the centre of the world: the altitude of the ☽ is found 28 ⅓. as well her latitude being 20. min. North, as her longitude accounted of, the greatest parallaxis I found 52. min. Therfore in the Limbe I number not 52. min. as I should doe, but I reckon 5. deg. ¼. almost: so that ech deg. supplieth 10. min. the 5. deg. being here for the 50. min. and the other ¼. almost for 2. min, which is done because els a huge in­strument would be too little to perform this matter exactly. The rule being so set al 5 ¼. deg. almost, I rec­kon inwards thereon the height of the moone. vz. 28 ⅓ and there I see to crosse the rule the 4 ½. parallel, now taking for ech of the 4. parallel 10. min. and for the half 5. min. the moones parallax­is according to that height and place: that is to say, the moone shall be seene here 45. min. lower then in truth she is, in respect of the centre of the world in that Eclipse.

Likewise appointing 5. deg. for 3. min. the greatest parallax. of the sun, euery deg. is 40. secondes: there place the rule, & rec­kon the said 28 ⅓. deg. inwards it sheweth the 4 ¼. parallel, whiche commeth to 8. score and ten, or 170. seconds, deuide that by 60. there commeth 2. min. & 50. seconds, the parallaxis of the sun. Here you must take heed that you seeke not to work grea­ter angles in this manner: and in these lesser, that you exceede not 10. deg. on the Limbe: for this ease serueth but in small an­gles, Where the question is of minutes, which in greater wil not holde proportion, but if a third or 4. part of a minute be wan­ting the errour is not much, but so much of a degree is not al­lowable.

Blagraue. Gemma Frisius saith before in this chap. that the moones greatest horizontall parallaxis is 1. deg. 6. min. beeyng in her Perigeon, and that the least horizontall parallaxis is the moone being in her Apogeon 50. min. For the rest he referreth you to tables. But now because I woulde nor haue any desirous hereof, to be stalledr of lacke of rabies or sight in the Theorickes: I will shew you after a blunt sort how to gette the horizontall paral. of the moone according to her sight between her apogeon and perigeon, by helpe of an Almanacke.

First I would haue you vnderstand that at euery change and full of the moone, she is in, or very neere her apogeon, and at euery quarter, in, or neere her perigion, therefore when as if it be true as Gemma Frisius sai­eth, that the greatest paral. in the perigeon is 1. degree 6. min. and the least in the apogeon 50. min. subdu­cting the lesse from the greater, there resteth 16. min. the difference. The space of time betweene the quarter and full, or change, is 7. dayes, 9. houres or thereabouts, which maketh 177. houres, but account it here (for it hurteth not greatly) 180. houres. Now therefore this knowen, reckon the difference of these horizontall pa­ral. vz. 16. min. on the Limbe from the Equinoctiall, or rather as you were wont 8. deg. so euery deg. represen­teth 2. min. thereto lay the rule: and by helpe of an Almanacke learne howe to any houres the moone is di­staunt from the next ful or change, and reckon on the rule for euery two of those houres one deg. on the rule from the Limbe inwards, and there marke what parallel cutteth: for it sheweth the minutes to be added to the moones least horizontall paralaxis, videlicet 50. I would needes knowe her horizontall parallaxis, I looked in the almnacke, and there I sawe she was at the full the 22. of October, at 6. of the clocke at nighte. I caste howe many houres were passed and found 50. houres, then laying the rule on 8. deg. in the Limbe, which here represent ech 2. minutes. I reckoned thereon from the Limbe inwardes, for those 50. houres 25. deg. there I saw the 7 ¼. parallel to cut, which here representeth but 14 ½. min. to be added to 50. it maketh 64 ½. min. or 1. deg. 4 ½. min. the horizontal parallaxis for the time sought for. VVhich being had, I might therby as before get the moones parallaxis for the height, she was at sed haec rudiori Minerua, by reason of the moones Epicircle, which causeth some errour.

Chapter 57. How to finde the parallaxis of the moone or any other planet, or starre in longi­tude and latitude of the Eclipticke.

THE parallaxis beyng now found in the circle of altitude, in whiche circle his must needes happen: You [Page 52] shall vnderstand that sometime it happeneth in longitude, whereby the starres seem to haue another longi­tude then is found in tables calculated to the centre of the world, insomuch that if the starre shalbe from the ascendent, lesse then 90. deg. he shal seeme to haue a greater longit. thē in truth he hath, but being situate VVest wardes from the 90. degree he appeareth to sight in lesse longitude: but in the middest or 90. deg. of the E­clipticke, he maketh no parallaxis in longitude, but all in latitude. For then the circle of altitude is all one wyth the cyrcle of Longitude of the starre yssuyng from the pole of the Zodiacke, by the Zenyth. But when the parallaxis falleth wholly in Latitude then the Eclipticke muste of necessitye cutte the Ze­nith and bee made one circle with the circle of altitude, this happeneth only to them that dwel either vnder one of the Tropickes, or betweene both. In other places the parallaxis doeth partly alter the longitude, and partly the latitude, so that the neerer the moone shal be to the 90. deg. of the Eclipticke, so much the greater is the parallaxis of latitude and of the longitude lesse: and contrariwise the further she is from the 90. degree, the greater is the parallax of longitude, and the lesse of latitude. And how much either of them is, may thus be had.

Seeke by the 54. the angle which the circle of altitude maketh with the Ecliptick in the place of the moon or of the sun: and by the last chapt. the parallaxis in the circle of altitude, reckon the quantity of the angle on the Limbe from the Equinoctial, and thereto lay the rule, then on the rule from the centre outwards, reckon the parallaxis found by the last chap. reckoning for euery 5. minutes therof one deg. on the rule, as you there did on the Limbe, the parallel cutting there, sheweth the parallaxis of latitude, accounting accordingly euery paral­lel for 5. min. Likewise following the houre line cutting the point of the rule with the same parallel vnto the Equinoctial line: it sheweth the parallaxis of long reckoning from the centre, taking euery houre line for 5. min. also. As in the 54. chap. the angle of the Eclipticke with the circle of altitude was found 34 ½, the paral­laxis in the same circle by the last chap. was then 45. min. I reckon on the Limbe 34 ½ deg. and thereto lay the rule: then on the rule I reckon from the centre 45. min. or rather 9. degrees, so that euery deg may heere re­present; 5. min. then doe I finde to cut the rule in that 9. degrees, the 5. parallel which here representeth but 5. times 5. min. videlicet 25. so many min. is the parallaxis in latitude. Then following the houre line, cutting the said 9. deg. of the rule, with the said 5. parallel he sheweth on the Equinoctial line 7 ⅔ degr. reckoned from the centre, which taking euery deg. for 5. min. are 39. min. almost, which is the parallaxis in longitude of the moone at the time of the sayde Eclipse. And because she is then Eastward, from the 90. deg. therefore the longit. of the moone seen, is greater then the true longytude. And therby it is euident that the visible coniun­ction shal goe before the true, so long time as the moone by ouergoing the Sun may get 39. min. which is the euariation of sight or parallaxis in longitude.

Chapter 58. Howe to finde when the Eclipse of the Sun shall be.

FIrst search diligently when the time of the Suns Eclipse shall be, which then in truth is, when as the Sun and the Moon are seene in one longitude. It hath byn already saide in generall if the coniunction of the lights happen between the ascendent and the 90. degr. of the Eclipticke. that the visible coniunction go­eth before the true, as much as the parallexis in longit. commeth to: but if the place of coniunction be VVest from the 90. deg. then the visible coniunctions (that is the midst of the eclipse) shal follow the true coniuction according to the proportiō of the parallaxis in longitude. Therfore the time of the Eclipse is gathered by the parallaxis in longitude. At in the example before, the parallaxis is found 39. min. in longitude, so much the moone doth exceede the Sun almost at the houre of the true coniunction. The question is now, how muche time it wil come too, while the moone is ouergoing the Sun this little space. Therefore we must gather how muche the Sunne and how much the Moone doe goe in an houres space, and howe much the motion of the Moone is greater then the Sunnes in an houre, and that according to sight, if a man wil handle this mat­ter curiously. The houre motion of the moone is thus had. The time of the true coniunction of the ☉ and ☽ knowen by tables for that purpose, and the parallexis in long found by the last chap. if the coniunction be to come, adde one hour vnto the time of the true ☌: and againe for that houre seek the altitudes & parallaxis of [Page 53] the ☉ and ☽ as before. Let the parallaxis of longitude be added to the true place, if the moone be Eastwards from the 90. deg. of the Eclipticke, but subduct it out of the true place, if the ☽ be VVastwards, and so shall you haue theyr loca visa, places seene. But if the true coniūction be past which happeneth when the place of the true ☌, is in the oriental quarter: then subduct one houre, & seek the loca vera the true places of the ☉ & ☽, for that time, and their parallaxes in longitude and latitude. And againe, adde or substract the parallaxes of long to or from the loca vera the true places of the light, and so haue you their loca visa their places seen. Thē subduct the seene place of the time going before out of the seen place of the time following, so haue you the morus visua the motion seen: that is, how much the planet goeth forwards in the Eclipticke according to sight. But if you delight curious calculating, you must do this, as well in the Sun as in the moon. And the you must take the Sunnes motion seene, out of the moones motion seene. So shall remaine the superation or exceeding of the moone aboue the Sunne in one houre according to sight. Now therefore by the proportion of the pa­rallaxis in longitude vnto this houre superation you may gather the time between the true and seene con­iunction or Eclipse: appointing the first number of the proportion, the houres outgoing or superation of the moone. The second one houre or 60. min. The third, the parallaxis in longitude. So shall come forth the min. of the time sought for, which being added too or subtracted from the time of the true coniunction, sheweth the time of the Eclipse. These min. are added when as the place of the coniunction happeneth in the West quadrant of the Eclipticke, and are subducted in the East quarter. But this profitable institution must needes haue an example.

VVherfore, in the Eclipse hitherto intreated of the parallaxis of the moone in longitude, is found 39. min. These I adde because they are in the East quarter, vnto the true place of the moone, which is 7 ½. or ♌, and so the place of the moone according to sight, which I call the place seene, or locus visus, shall be 8. deg. 9. mi. &, 0. seconds in ♌. The parallaxis in longit. of the ☽ I will here omit for breuities sake, whereby the place of the Sun shall still remayne 7. deg. 30. in ♌, being both the true place and here allowed for the seene place. Nowe because it is most certaine that the middle of the Eclipse which as before sayd, is the visible coniunction, doth goe before the true, we are againe to get the true places of the ☉ and ☽, with their parallaxes for one houre before the true coniunction, that is, for 6. of the clocke 30. min. And then is the Sunnes true place 7. deg. 28. min. of ♌, the true place of the moone 6. deg. 56. min. of ♌: because the houre motion of the moone is 34. min. The Horoscope or ascendent for this tyme is 0. deg. 20. min. in ♍ the deg. of culmination 19. degr 30. min. of ♉, whose meridian altitude is 56. Further, the distance of the deg. of culmination from the horoscope is 100. deg. 50. min. of the Eclipticke, whereby the angle made betweene the horizon and the Eclipticke, is found to be 57. degrees 10. min. by the 51. Likewise the Sunnes altitude againe to bee 18. deg. ½. and the moons altitude to be 18. deg. 20. mi. The angle of the Ecliptick with the circle of altitude is 53. almost by the 53. & this is the angle of latitude, therfore subducting frō 90. there resteth 55. the angle of longit. of the moone. The parallaxis of the moone in her circle of altitude is found 52. min almost by the 54. and in the sun 2. min. 30. seconds: therefore by the 55. the Sunnes parallaxis in longitude is almost 2. min. which beeing ad­ded to the true place of the Sunne, maketh for this houre the Sunnes place seene 7. deg. 30. min. of ♌. Lastly, the moones parallaxis in longitude is 42. min. and better, by the 55. Therefore her place seene, is by addying those min. found to be 7. deg. 38. min. of ♌. Now therefore let vs take the place seene of the Sun going be­fore, vz. 7. deg. 30. min. of ♌, out of the place following, vz. 7. 30. ♌, so there remaineth 0. This nowe is the Sunnes houre motion appearing or seene, which for want of paines commeth to nothing. In the like manner subduct the former motion of the moone appearing. vz. 6. deg. 56. of ♌ out of the later which were gathe­red to be 7. deg. 38. min. of ♌, and so is the houre motion of the moone founde to bee 42. min. For so muche shall the moone goe forwardes in one houre according to her motion appearing or seene. And because the Sunne maketh 0. his houre motion seene, and the moone 42. min. taking 0. out fo 42. the moone shal ouer­goe the Sunne 42. min. in an houre. The parallaxis of the moone in longitude at the time of the true coniun­ction was found 39. min. and of the sunne (if I had respected it) about 1. min. 20 seconds, but you must take this but as an example, and worke it exactly your selfe, therefore the moone had ouergone the Sunne then 38. min. 40. seconds according to fight. Last of all, because the moon doth ouergo the Sunne in an houre 42. min. she shall make 1. deg. 20. seconds (which is the difference of the Sunne and moones place seene at the time of the true coniunction) in 2. min. 6/2. of an houre, with 6/42. of a min. ouer. This time therefore be­cause the coniunction seene, goeth before the true, we must take out of the time of the true coniunction, so shall remayne the time of the coniunction or Eclipse, appearing vnto vs. vz. 7. of the clocke and almost 28. min.

Chapter 59. Another manner to finde out the parallaxis in longitude and latitude, as wel in the ☉ as the ☽.

I Haue often times in this my booke spoken of Gemma Frisius Cursor and brachiolum, but perhappes euery man knoweth not what they are, the Cursor therefore is a rule, in length a semidiameter of the Instrumēt, and is so wrought with a rabbott or a done tayle, as workemen call it, at one ende: that hee alwayes stan­deth square vpon the rule, and so slippeth vp and downe from the one ende to another, his one side is deuided as halfe the rule, the other into an 100. or rather a 1000. equall parts, the brachiolum is a little arme at the ende of the cursor, &c. This Chapter was to be performed by Sinicall working, and therefore impertinent to the Iewel.

Chapter 60. To finde the latitude of the moone.

THe Sun neuer digresseth from the Eclipticke, but the way of the moone lyeth by, as 5. degrees from the E­clipticke [Page 54] crossing him alwayes in 2. opposite points, of which the point Northwards is called Caput draconis, the other Southwards Cauda draconis. In these points only is the moone in the Ecliptick, and in or very neere them, the moone is at all Eclipses, they haue a continuall motion contrary to signes sequele, and perfourme it in 28. yeeres, and almost 5. monethes once round.

Therfore by tables or the Ephimeris, you must get the place of the moone for the time proposed, and like­wise the place of Caput draconis, if now the moone be departed from the Caput towards Cauda, the latitude shall be north: but from the tayle towardes the head, South. This knowen, reckon 5. deg. on the Limbe from the e­quinoctiall line, thereto lay the rule, then by subduction get the distaunce of the moone from one of knottes vz. Caput drac. or Cauda dracon, to which soeuer she shall be neerest, reckon this distance on the rule from the cen­tre, and so the parallel cutting there, sheweth the latitude sought: this needeth no example.

Chapter 61. How much of the Sunne shall be darkened.

VVHen as the Synode or coniunction of the lights happeneth neere one of the knots called Caput and Cauda draconis, then is there greate likelihood of an Eclipse of the Sun to be. But it is a more certayne token if the place of the middle coniunction called Synodus media, happen lesse then 20. deg. 40. min. before the Cauda, or after the Cauda lesse then 11. deg. 20. min. Likewise if it happen before the Caput within the space of a 11. deg. 22. min. or after it within 20. deg. 40. min: then is it possible that the Sun may be eclypsed, without these boundes there can be no Eclipse. But the most certaine token of all is, if the latitude seene of the moone doe exceede the quantity of both the semidiametres of the sunne and moone seene, at the time of the Eclipse seene, then shall the Sun suffer no Eclipse. Of these things the whole inquisition of the greatenesse of the Eclipse doth depend, vz. of the semidiametres of the lights, and the latitude seene at the time of the con­iunction visa, as well of the ☉ as the ☾. But as much as belongeth to the inuention of the semidiametres ap­pertayneth not properly to this instrumēt, but is gotten by their proper tables. And here generally it is inough to shew, that the semidiameter visus, of the Sun neere his Apogeon, that is in ♋, is seene to be 15. min. & 49. seconds almost, for which you may take 16. min. in sleight calculating, the sunnes Semidiam. visus, in the pe­rigeon is 17. min. Therefore when as the greatest differeth from the least but one min. there is no cause to seek ouer curiously for the other places, althogh the mutatio occentrotetis, maketh a litle diuersity in this behalf, but being almost insensible, is here by me neglected. Now for the mnones Semidiametri apparentes, the least is 15. min. and the greatest that may happen in the coniunctions 17. min. 49. seconds, which differ little from 18. min. These therefore make ynough for our purpose.

For this take Gemma Frisius owne example, first mentioned in his 62. he recyteth of an Eclipse of the Sun to be anno 1560. the 21. day of August, at one of the clock 22. mynutes after noon, by Stophers tables, the Sun and Moone then being 7. deg. 45. min. in ♍, the latytude at Louan beeing 51. degrees almost, where Gemma Frisius remayned. Nowe because the ☉ in this Eclipse is more neere his apogeon beyng ♋, then his perigion beeyng ♑, take for his Semyademetre sixteene mynutes. The Moone is from hir Apogeon 5. dodecatemoria, that is 5. common signes and three degrees, and by that account beyng neere her perygeon, hath for her Semidiameter apparens 17. min. These Semidiametres therefore gathered into one summe, make 33. mynutes: and if now the Latitudo visa of the moone were greater then this Sunne, then as I sayde before, there coulde be no Eclypse. The Latitudo visa of the moone is thus gotten. You must by the 55. gette the pa­rallaxis in latitude for the time of the Coniunctio visa: and likewise of the sunne, if there be any. Also the la­titude of the Moone for the same instaunt, calculated as in the last Chapter by her true place and distaunce from the neerest knotte. If therefore this Latitude vera be North, and the parallaxis of latitude lesse then it, deduct this from the true latitude, and so remayneth her Latitudo visa Northwardes: but if the parallaxis of Latitude, be greater then the North Latitude, subducting the North latitude out of the parallaxis of latitude, there shall be left the Latitudo visa Southwardes. But if the Latitude vera shall be South, adde the parallaxis of Latitude to the Latitudo vera, and so shall you haue the Lititudo visa Southwardes. This therefore beyng got­ten, you must take regarde also to the Sunnes paralaxis in Latitude, that is alwayes Southwards, because the Sunne is alwayes in the Eclipticke, and all parallaxis happen downewardes. This therefore is all one wyth the parallaxis of Latitude. And it is alwayes added, when as the moones Latitudo visa is Northwardes, and is subtracted in her South Latitudo apparens, vnlesse the same be lesse. For then it is taken from the Latitudo visa of the Sunne, and so remayneth the moones latitude from the Sunne appearing to our fight: which if you subduct from the number gathered of the Semidiametres of the sun & moone, there shalbe left the part of the sunnes diametre darkened: and multiplying that by 12- and deuyding the product by the sunnes diametre, you shal haue the poynts or dygites (as they cal them) of the Sunnes diametre darkened, as in this example the Coniunctio apparens or visa, by the 56. is at one of the clocke and 16, minutes, the horoscope 3. degrees in ♐, the angle of the horizon: and the Eclipticke 44. deg. and 45. mynutes by the 51. the angle of latitude 88. degr. almost by the 53. The altitude of the sunne 44. degrees 42. minutes almost by the 31. whereby the paralax­is in latitude of the moone is 43. mynutes almost by the 55. So also the paralaxis of the ☉ in latitude is two minutes, the Latitudo vera of the moone is 23. mynutes Northwardes, Nowe because the parallaxis of laty­tude is 43. minutes, and the Latitudo vera 23. minutes North, subduct it out of 43. there remayneth 20. minut. the moones Latitudo visa Southwardes: but the Sunne also is seene distaunt from the Eclipticke 2. minutes per aspectum. Therefore the Moone is seene more Southwardes then the Sunne eighteene mynutes: thys is the Latitudo apparens of the Moone from the Sunne, which subducted from the number gathered of the semy­diametres gotten before to be 33. minutes, there are left 15. minutes for the part of the sunnes diametre dar­kened.

Lastly, these 15. being multiplied by 12. digites of the diametre of the sun, make 180. which deuided by the whole diametre of the sunne, or by 30. there are produced 6. points or digites of the suns diametre, which is one halfe: some call them points or digites of the Eclipticke.

Chapter 62. How long the Eclipse of the sunne shal endure.

MAke therefore a circle about the centre A, according to the bignesse of the Sunne, so that AB, bee 15. min. and the whole diameter BC, 30. min. and AE. The Latitudo apparens of the Moon from the sun being in this last example 18. min. Therefore E is the centrum apparens of the moone, AD, the whole diametre of the moone, vpon the point E let there bee EF perpendicular: and then open the com­passe according to the number gathered of the Sunne and moones diametres, which in the example before was 33. min. And so placing the one poynt of the compasse in the centre A, let the other touch the line EF in the poynt F. Now it is most euident that the Moone being in F, can couer no more of the sunne, and that there is the end or beginning of the suns eclipse. And when the moone shalbe in E, then is the middle of the eclipse, which you may the easier beleeue, seeing GF is the Semidiameter of the ☽, the Sun and Moone both touching in G.

Therefore EF is halfe the space which the moone doth run in the whole Eclipse of the Sunne: how much that is, you may get by our compasses onely on the scale or els multiplying AF into it selfe, and AE likewise, and then ta­king the square of AE, out of the square of AF, there shall be left by the last propositiō saue one of Euclides 1. book, the square of EF, whose roote is the quantity of the line EF. As in this example, AF is 33. min. the square of this is, 1089. the line AE, being the Latitudo apparens of the Moone from the Sunne is 18. min. the square hereof is 324. this taken out of the firste square, vz. 1089. There resteth 765. the square of the line EF, whose square roote is about 27. min. ½. the quantity of the lyne EF. This space of the motus apparens of the moon is reduced vn­to time by the motus horarius apparens of the moone founde, as in the 56. which in this example wilbe 28. min. Now say by the rule of proportion 28. min. of the moones motus visa, make one houre or 60. min. How much then shall 27. min. ½. and so fol­lowing the rule, you shall finde 59. minutes, whiche are commonly called minuta casus or minuta incidentia, that is to saye, howe muche it is from the beginning of the Eclipse vnto the middest, and thence vnto the end whereby doubling these min. of incidence, you haue almost the time of the whole eclipse, which here is two houres lacking 2. min. But yet I know well that these 2. times (that is from the beginning to the midst, and thence to the end of the Eclipse) are not alwaies equall vnlesse the Eclipse happen in the 90. deg. of the Eclip-e ticke from the ascendent, and that because of the vnequality of the moones motus apparens, for the which in th orientall part the parallaxis of long it. is added to her true place, and is deducted in the occidentall.

He therefore that woulde performe this matter curiously shoulde as in the 56. get the motus apparens of the moone for one houre before the Eclipse, and also for one houre after the middest of the Eclipse, and thē mea­sure the space of the beginning of the eclipse vnto the middle called spacium incidentiae, by the motus horarius, go­ing before, and the space between the middest, and the end of the Eclipse called redditus luminis by the houre motion of the houre following. And thus haue you all the reason hereof, although perhaps I haue not beene curious in the examples, since I haue set downe the instruction so plaine.

Blagraue. Note that I haue set downe these tearmes motus visa, Latit. visa, and such like in Latine words, least any man should in English sense ouerslip them as words of course, being indeed words of art, and such as this matter dependeth on,

Chapter 63. Of the Eclipse of the moone, the greatnesse and duration of the same.

THe Eclipse of the moone hath an easie calculation, the cause whereof is, that it dependeth not vppon the difference of seeing it in any respect (as before in the sunne.) For as oft as there happeneth an E­clipse of the moone, because that she beyng opposite to the Sunne, lighteth into the shadow of the earth, which extendeth it selfe farre beyond the speere of the moone, and taketh away the light which she had of the Sunne: she by this meanes eclipseth in like bignesse and manner to all places of the world: and continueth like time, and is seene euery where at one moment, although that it be accounted at diuers houres in dyuers places, according to the distaunce and difference of the meridians, as hereafter I will shew in the longitude of places.

Blagraue. It appeareth here that Gemma Frisius had a meaning to intreate of the longitude of places whi­che Oh would God he had liued to haue performed, that it might haue beene my good happe to haue seen his vttermost, in that behalfe, by which meanes I should haue eyther spared altogether the greate endeuour which I of necessitie must now for the credites sake of my Iewell employ in that behalfe, or at the least haue receyued such light from his doynges, that I might with ease haue gone through, with it. But now Gemma Frisius is gone, lamenting will not helpe: he left his sonne Cornel. Gemma behynde, who published his Fa­thers woorke of the Catholicon, and stryued to augment it by his owne industrye: but God knowes, against the streame. For a man may easily knowe by the very matter and manner of it, his doyngs (of whi­che I haue omitted the most part) from his fathers.

But now to come againe to the matter, as the Eclipse of the moone is vniuersall at one instaunt, it is farre otherwise in the Sunne: for one selfe same Eclipse of the Sunne to some appeareth great, and lasting: to other some little and gone in a moment: to some the North part, to some the South part of the Sunne is seene to be darkened: and that in infinit diuersities.

The cause of so great variety, is the diuersity of the places whence the sun is beholdē: bicause that in the eclipse of the Sunne he looseth not his light, as the common sort do imagine: but only that the moone commeth betweene vs and the sun. VVhereby all countries see not the Eclipse in one maner. Insomuch that some doe see the sunne in his ful light, when that others see him almost wholly eclipsed. For the see­ing of it, according to the scituation in sun­drye places, is the greatest cause of the diuersity of the Eclipse of the sunne. But in the Eclipse of the moone, the view euery where doth vary it nothing, neyther in lō­gitude nor latitude, for the moone is darke­ned indeede, where there the sun is not (as is already said:) and therfore are we forced to seek so many variations of aspect in the sunne which in the moone needeth not. It shalbe sufficient here to take the time of the true opposition of the sun and moone for your meridian, either out of their proper tables, or out of Ephimerides, rightly calcula­ted: & for that time to get the true place of the sun in the eclipticke, whose opposite is the iust place of the moon: & then seek the latit. of the mooue very diligently, as is shewed in the 58. chapter. Lastly you must gette the diameter or se­midiameter of the moone: and likewise the semidiameter of the shadowe of the earth, of what quantity it is in the place of the moones passage. The same is somtime greater, somtime lesse, for two causes: for when the ☽ is neere her perig. she findeth the shade of the earth greater then in other places: the shadow of the earth runneth out taperwise vnto a point at the last, & becommeth lesser by so much the further as it extendeth frō the earth. Secondly, the shadow of the earth may be greater in the selfsame place of the moons passage, by rea­son of the distance of the sun from the earth. For the neerer the sun commeth toward the centre of the earth, so much the shadow of the earth groweth the narrower & shorter, contrarywise, by the sunnes bearing backe from the Earth, it is extended and enlarged in the same place of the speere of the moone, in which it was nar­rower before, as by these figures you may see.

Therfore to our purpose you must know by tables the time of the true opposition of the sun & the moone in the Eclipse: and the place of the sun & the moon for the same time, together with the latitude of the moon. And then the semidiameter of the moon, & of the shadow, through which the moons passage is. And here wil I shew in brief as much as shal suffice the learner to get the semidiamet. of the sun, the moon, & of the shade by the motus horarii, or diurnall motions of the Sunne and the moone. It is found out by the industry of ar­tificial men, that what proportion 20. beareth to 11. the same doth the suns diurnal motion vnto his diamet, appering. Therfore multiply the suns diurnal motiō gottē out of any tables into 11. & deuide the product by 20. so haue you the diam. of the sun. The cause is, for that both the greatnes of the diamet. apparens of the ☉, & y^e swiftnes of the suns motiō are increased & diminished in like proportion, according to y^e site of the ☉ in his excentricke.

The diametres also of the Moone do likewise keepe a proportion with her motion, so that the diametre Apparens of the Moone is almost equall to her Motus Horarius: Puruacchius saith, what proportion 48. beareth vnto 47. the same hath the mot. horarius of the moone vnto her Diameter apparens: therefore if a man take the motus horarius, or houre motion of y^e ☽ for her diametre, he shal not misse a minute. But to haue it ex­actly, multiply the motus horarius of the Moone into 47. & deuide the product by 48. & therof shal come the moones diametre. Lastly the diametre of the shadow of the earth you shal thus get. Multiplie the diametre of the moone founde as before into 13. the product deuide by 5. and so shall you haue the greatest quantitie of the shadowe in that passage of the Moone, that is to say, the sunne being in the Apogeon: but the sunne being in other places, the shadow abateth in the same constitu­tion of the ☽, as is said, by so much the more as the sun commeth neare to the earth: and how much the diametre of the shade is in another place, that is found by the motus horarius of the sunne, for by how much greater it waxeth in other places, the shade of the earth is made lesse, by so much ten times told: These rules are both generall, exact, & most excellent, which by a familiar ex­ample I will manifest.

In the yeare 1555. the 4. of Iune the 24. houre, and 42. min. there is like to be a great eclipse of the Moone, which is to be knowne in that the place of the ☽, at the time of the true opposition is not farre from Cauda Draconis. For the place of the sunne is according to com­mon tables ♓ 22. deg. 46. minutes, whereby the Moones place is ♐ 22. 46. also the Caput Draconis is in ♓ 23. deg. 8. minutes, therefore Caeuda is in ♐ 23. 8. so that the true place of the Moone is but 21. minutes from Cau­da Draconis, wherby there is not onely knowne that an Eclipse shall be, but also a verie great one: for as oft as the Moone is aboue one deg. 8. min. in latitude any way, then shall the Moone auoide the shadow in her race. But heere the latitude of the Moone is scarce two min. which is a token that the Moone shalbe verie neere the centre of the shadow of the earth in this Eclipse. Now to know the quantitie & continuance of the same, we must know the semidiametre of the Moone, & of the shade of the earth by the rules before shewed. The Motus Diurnus of the ☽ in that place is 13. deg. 5. therefore her Motus horarius shalbe 32. min. 40. sec. and somwhat lesse: then this shalbe the diametre of the moone: but if you wil know it exquisitly, multiply 32. min. 40. sec. by 47. & deuide the product by 48. & so shal you haue 32. min. 20. sec. almost, & the same is the iust diametre of the Moone: halfe that vz. 16. min. 10. sec. is her semidiameter: which multiplie by 13. and the product deuide by fiue, you haue the diametre of the shade of the earth in the place of the Moones passage in her sphere, the sun being in his apogeon, the multiplication of the diametre maketh 25 220. sec. the diuision produceth 5044. sec. that is 84. min. 4. sec. so that the semid. of the shade is 42. min. 2 sec. And because the sun is neare his apoge­on, the shadow of the earth cannot be diminished any quantity, to be accounted of by reason of the suns com­ming neare vnto the earth, otherwise you should heere seeke the motus horarius of the ☉ being in his apogeon and also for the place where he now is, and the difference of these motions decuplied, that is multiplied by 10. should be taken from the greatnesse of the shade here found, & so shall you haue the exquisite bignesse of the shade of the earth, but here for methodes sake we thinke not good to prosecute euery least matter, but leaue such curiositie to those that haue more leisure, or who are whollie addicted to these studies. By these things found, you may onely by drawing of a figure knowe the greatnes and duration of theclipse, draw therefore the streight line AB, which shall be the 22. deg. of ♐ deuided into 60. min. Then in the 46. min. in which shall be the centre of the shadow of the earth, that is to say, opposite to the ☉ at the time of the true opposition, place the one foote of your compasse, and there according to the quantitie of the semidiametre of the earth now founde videlizet 42. minutes, make a circle for the shadow of the earth, the centre being C, whereon draw the line DE perpendiculer to AB, which is the circle of the latitude of the Moone, and because the latitude of the ☽ is found 2. min. northwards, take in your compas 2. of the 60. parts of AB, & place the same from C. towards D in the point E. & that is the centre of the Moone: in this centre according to the semidia­metre of the ☽ vz. 16. min. 10. sec. make the circle of the moone, which being done you shal streight see before [Page 58]

your eyes that the whole moone is drenched in the shadow of the whole earth without any calculation. And if you deuide the whole diametre of the moone into 12. digits, you shall apparantly see how many digits or points theclipse of the moone shall be. For looke how many digits of the moones diametre DF. shall con­taine (with the distance from the vttermost shade of the earth vnto the edge of the moone farthest drenched within the darknesse) so many digits or points as they now call them shall theclipse be said to be. For D is the point of the shadow of the earth, most bending from thecliptick towards the which the moone departeth, but by numbers we shall thus proceed. Because EF is 16. min. & 10. sec. and the latit. of the Moone EC. 2. min. therfore CF shalbe 14. min. 10. sec. to which there being added CD, being 42. min. 2. sec. the whole DF shal be 56. min. 12. sec. Now by the rule of proportion, if 16. min. 10. sec. be worth 6. digits, what shall 56. min. 12. sec. be worth: & so following on the rule, you shal get 19. digits & 41. min. of a digit such as the whole dia­metre of the moone maketh 12. To be short the whole moone containing 12. dig. shalbe wholly drenched in the shadow: & that so deepely that the shadow shal exceed the moone 8. dig. almost, which is as much to say, as that theclipse of the moone shalbe almost 20. digits. The moone her selfe conteineth not aboue 12. points, but the depth of the moone within the shadow of the earth shalbe almost 20. dig. wherby you shal vnderstād that when as the greatnes of theclipse exceedeth 12. digits, then shal there be an eclipse with tarriance, as they call it, that is, the moone shal tary a certaine space in the darknes before she shal receiue her light againe of the sun: & how many the more digits there shalbe found, by so much the longer shall the tariance of the moone be in the darknesse, & the whole eclipse endure: both which exactly to descrie, thus must you do. Open your com­passe to the quantitie of the semidiam. of the earth and the ☽ ioyned togither as in this example, vnto 58. min. 12. sec. and on the centre of the shadow of the earth vz. C. describe a hidden or blind circle, and note wel the touching of this circle with the way of the Moone. I cal the way of the moone a line drawne by the centre of the moone vz. E, parallel if you list to the line AB. or if you be disposed to prosecute these matters more nar­rowly: let this line make with DF the blunt angle DEM. of 95. deg. as heere the line LEHM. is, for this is the true way of the Moone whereby H is the place of the next knot. This line is cut of the blind circle in the points L & M, in which points the Moone being, maketh the beginning and end of theclipse. Lastly, from the 2. intersections of the shadow of the earth, and the way of the moone, set the semid. of the moone by helpe of your compasse in the way of the Moone vz. in N and O. For in the one of these places the Moone is altoge­ther entred into the shadow of the earth and in the other, beginneth to get out thereof. Therefore you haue 5. moones if you list, the first on the centre L. where the ☽ begins to eclipse: the next on the centre o, when she is wholly darkned: the third on the centre E, when she is in the middle of theclipse, the fourth on the centre N. when she beginneth to get light againe: the last on M. where theclipse endeth, so that LO is the space, that the moone goeth from the beginning vntill she be all darkned, and ON of her whole darkning, which they call the tariance in Latin Mora: therefore OE is halfe her tariance called (Dimidia mora.) All which parts you may measure with your compasse by the parts of theclipticke before made: and deuiding ech of them by the Supe­ratio horaria of the moone you shall haue all the times sought for, or the whole taken togither vz. LM. And by numbers may thus be performed. For that EC is 2. min. in latitude, and standeth almost perpendiculer on LM. the way of the ☽, therfore the square of LC shal be equal to the squares of LE, and EC. therfore taking the square of EC out of the square of LC, there shal remaine the square of LE by the last proposition sauing one, of the first booke of Euclide: For LC is the sum gathered of the semidiametres of the moone, and the shade vz. 5. 8. minutes, 12. seconds, but letting passe the seconds, the square of LC shall be equall to 3364- herehence taking the square of CE, vz. 4. there is left 3360. the square of LE, whose side is somwhat lesse then 58. and that is the line LE, so much also is EM. Let no man carpe heere because we saide that CE was perpendicular to the Moones way, when as before we drew him perpendiculer to theclipticke, the best learned men will neglect so much, because it bringeth no perceiueable error. Therefore the line LE be­ing knowne to bee fiue 58. minutes almost, take out of that the semidiametre of the Moone vz. 16. minutes 12. seconds, the line LO shalbe 41. min. 48. almost, which they call minuta incidentiae, and NM is almost equal vnto it, called minuta repletionis. Lastly subduct LO being the minuts of theclipse waxing out of LE, there shal be left OE halfe the tarriance vz. 17. min. 12. seconds almost. Now because the Motus horarius of the Moone [Page 59] was 32. min. 40. seconds, and the sunnes Motus horarius 2. min. 22. sec. take the one out of the other there re­steth 30. min. 18. sec. the superation of the moones gate aboue the sunne in one houre. By this deuide all the parts of the Moones way before founde and you shall haue the tymes correspondent vnto them. As be­cause the Mynuts of the waxing of the Eclips were 41. mynuts 48. so deuide those by 30. mynuts 18. seconds by reducing each number vnto secondes, so shal amount one howre 22. mynuts, the time of the waxing and likewise of the wearing of theclips. Likewise deuide the minuts of Dimedia mora vz. 17. my. 12. by 30. min. 18. seconds there commeth 33. my. of an howre, wherby the whole darckning is 1. howre 6. my. the whole time of the Eclips 3. howres 50. mynutes.

Chapter 64. How to get the parallaxis of the Moone out of the heauens at any houre proposed, the latitude knowne with the longitude and latitude of the ☽.

THe Moones place giuen at any instant proposed, calculate hir altitude on the Iewel by the 32. take the same instant the altitude of the Moone, by obseruation, if these two agree there is no paralaxis, which can neuer bee except the Moone be neere the zenith, the difference of these two is the paralaxis, but this requireth exact calculation and obseruation.

Chapter 65 To know the greatest, or horizontall parallaxis of the Moone when she is to be seene.

GEt the paralaxis of the ☽ for the altitude she is in at the instant proposed, then reckon the same altitude on the rule of the Iewell inwardes, and then moue to and fro till that poynt of the Rule doe cut the paralell answering the paralaxis of altitude, and so doth the Finitor shew in the Limbe the horizontall pa­ralaxis sought for. In this dooing you may accompt euery degree of the paralels for 10. mynuts. as in the 54. and 55. Cap. The greatest horozontall paralaxis is the ☽ being most in her perigeon 63. mynutes almost. Hereof shall I treat at large in my worke of Longitudes.

Chapter 66. The Longitude and Latitude of two Starres giuen, how to know the distance betweene them.

BY this Chap. and others following is shewed to find the distance of stars, & also of townes, cities, & pla­ces as easily, and redily as on the globe it selfe, which no doubt must needs make the instrument very de­lectable. The longit. and latit. of any two stars had, subtract the greater long. out of the lesse, so haue you the difference, which being had, number in the Limbe from thequinoctial the greatest latitude of the two stars southwards, if the latit. be south, or northwatds, if north: and thereto lay the Zenith point of the Reete, then in the meridian reckoned from the Limbe answerable to the difference of longit. number the lat. of the other star and that from thequinoctiall accordingly, and the azimuth cutting there, sheweth the distance sought for, if you reckon thereon thence to the zenith point.

Example. I would know how many degrees Canis maior is from Oculus tauri I finde by Stophler (for any ta­bles will serue the turne, as well old as new for that their difference of long. and their latit. neuer alter) that the longit. of Oculus tauri was then in ♊ 2 18/60 deg. that is from 0. in ♈ 62 18/60 deg. and his latitude, 5. deg. 10/60 southwards. Likewise Canis maior in ♋. 7 18/60 which commeth to 97 18/60 deg. from ♈, and his latit. to be 36 10/60 southwards also. Now subducting the lesse longit. vz. 62 18/60 out of the greater vz. 97 18/60 there resteth 35. the difference of longitude. First therefore I reckon in the Limbe from thequinoct. line, the greatest latit. of these vz. 39 10/60 towards the south pole, because the latitude is south, and thereto I set the zenith of my Reete then in the 35. meridian counted from the Limbe, I reckon the lesser latitude from thequin. line vz. 5 10/60 deg. south­wards, also because that lat. was likewise south: this done, I see the 52 ¼ azimuth to cut the place of the second starre whose deg. betweene the same place and the zenith by helpe of the almicantaries, I finde to bee 46 ¼ therfore I boldly pronounce that the distance betweene Oculus tauri and Canus maior is 46 ¼ degrees.

Chapter 67. How the distance of any two starres vnknowne may be had.

THis may you do by the 38. Chap. getting their longit. and latit. out of the heauens, & then proceeding by the last Chap. But yet an easier way without al that circumstance is thus: take the altitude and azi­muthes of both starres at one instant, or at the least the distance of their azimuthes without respect of the meridian: these had the working is all one with the last Chapt. but onely chaunging the names, the dif­ference of their azimuthes, for the difference of longitude: and their altitudes for their latitudes, and where as the equinoctiall did in the last Chapter represent the Eclipticke, hee must heere represent the horizon and the parallels, circles of altitude, &c. This needeth no example, for if you will suppose the altitudes [Page 61] of two starres equall with the latitudes in the last Chapt. and the distance of their azimuthes equall to the difference of longitude there, the matter is already performed.

Chapter 68. To know how long the taile of a Comet is.

VVOrke altogither as in the last Chapt. with the head and last point of the taile of the Comet, as though they were two seuerall starres and the distance found is your desire.

Chapter 69. Another way to serch the distances of starres one from another.

GEt their declination by the 7. Chapt. and their difference of right ascension by the 17. or 38. or by obser­uation, these had, worke altogither as in the 64. Chapt. onely chaunging the names, as is said in the 65. And thus haue you diuerse and excellent wayes to examine the distance of starres.

Blagraue. Truely these are very prettie conclusions & pleasant, but yet not sufficient to correct and amend the places of fixed starres and planets, and the whole globe, as G.F. affirmeth, except they be performed by a very huge instrument at the least of 10. foote diametre. But the onely way instrumentally to performe these things in my opinion (Saluo meliori iudicio) is by their crosse staffe to take their distances, and thereby to rectifie their places of longitude and latitude, vpon a great quadrant, as I will shewe you hereafter in my booke of logitudes if God giue me leaue and leisure once to come to it, to the which no doubt these Chapters are good preambles and introductions, notwithstanding for those that take delight in varieties, I will somwhat helpe to fit their fancies. If you would know the distance of any two starres placed in my Reete, turne it about till some one meridian touch the apex or point of either star, and the degrees of the same meridian included be­tweene them, numbred by helpe of the parallels sheweth your desire, but then you must be sure that these starres be both North, or both South, declining from thequinoctiall. In my Reete I appointed all the North starres to point outwards from the centre, and the South inwards.

Example. Oculus tauri, & Canis minor are north stars, and in my Reete their apex pointeth outward, turne the Reete about & at last you shal find the 17 ⅓ meridian to touch both their apex & the distance betweene them on the same meridian 46 ½ deg. But if one star be North & the other South, yet wil I wish you wel: then turne about the Reete vntill the apex or points of either starre do touch ech a seueral meridian, but equally taken on ech side of the axtreeline, and then must you reckon from the one of them to the pole, and from the pole backe againe to the other, and so haue you your desire.

Example. Of Oculus tauri & Caris maior, the one in my Reete pointeth outwards, the other inwards, wherfore I turne the Reete till at the last I finde Oculus tauri to cut the 46. meridian on the left side of the axtreeline, and Canis maior the 46. meridian on the right side of the axtree, where I finde from Oculus tauri to the pole 23 ¼ deg. and from the pole to Canis maior 23 ½ deg. these put togither make 46 ¾ deg. the distance of these two starres.

Chapter 70. The longitude and latitude of two starres giuen, to get the angle of station of one from another.

THe angle of station is the distance of the azimuth, wherein any starre is from the meridian of some cal­led the angle of position. Place therefore the two starres giuen according to their difference, longitude and latitude giuen in all respectes, as in the 64. Chapt. and the very same azimuth that there sheweth their distance, sheweth also the angle of station, of one of them from another, being counted from the Limbe, and therefore I neede set no other example, but looke in my booke following of Cosmographie, and you shal perceiue it plaine, the like may you performe by their difference of ascenscion and declination, or their alti­tude and difference of azimuthes.

Chapter 71. To know whither three starres or cities be in one great circle.

THis Chapter is verie pleasant and no lesse profitable in amending the places of the fixed starres in the globe celestiall. First therefore gette the angles of station by the last Chapt. of one of the starres from ech of the other, and if these angles be equall then are they in one great circle, else not.

Blagraue. This Chapt. of Gemma Frisius is true, so that you get the angles of position of the middlemost starre, and of one of the other from the third, but if you get the angles of position of the other two from the middlemost, then if the angles of position being equall do make one streight line, that is to say, if the one be Northeast, then if the other be Southwest or so forth, you haue your desire, euen like as in the compasse any two opposite points make one streight line.

Chapter 72. That the taile of a Comet extendeth it selfe directly from the Sunne, and howe you may try it.

THE Philosophers affrme that a Comet is ingendered of a soft clammie slime, or viscous substance, apt to nourishe fire drawen vp into the higher moste region of the ayre, and there is set on fire, by the firy element, to whiche he is very neer: & that his taile cōmeth of the flowing or dragging of y^e matter, like as in starres that shoote, where the matter of exhalation is, by his shutting drawen in length. But as concer­ning the tayle, that cannot bee so, both because his motion is verye slowe, and also for that it hath alwayes been obserued of learned men, as namely, Apian and others, that the tayle of euery Comet extendeth it selfe directly from the Sunne, so that the Comet his taile, and the Sun are in one great circle, as at all times by the last chapter, you may proue when any Comet happeneth.

Cornelius Gemma. It hath pleased God, as it should seeme, to take away Gemma Frisius, when he hadde perfourmed thus farre, and before he had furnished this chapter, whiche his sonne Cornelius G. took in hand to accōplish & prosecute the same very tediously, to little purpose, as hee doth all the propositions & chap­ters afterwards, which I mean to abreuiate in as much as shalbe necessary, & the rather because I shall stuffe out this worke with such store of matter, that there will be no place for superfluities. Hee taketh occasion here vtterly to condemne all the Philosophers, who say that there is a firie region or clement next aboue the ayre, affirming that all fire and heate remayneth in the Sun. And further, that Comets are set on fire by the Sunne only: his greatest reason is, for that if the comet were set on fire by force of the firy element, then must of necessitie his taile bee rounde about alike, but because the tayle is alwayes found directly from the Sunne, therefore it is manifest, that the Sunne is the only causer of his flamme, appoint the Sunne, the Comet and his taile, insteede of the three starres in the last chapter, and the working is all one, if you list to make expe­rience thereof.

Chapter 73. To knowe what point of the heauen is in the Meridian of any place at any in­staunt proposed, and what it is a clocke in any other countrie.

PLace the degree of the Sunne to the houre proposed in the limbe by the help of the rule then number in the Limbe, from the noone line, the difference of longitude of that countrie from yours Eastwardes, if it lye Eastwardes, or VVestwardes, if otherwise: and thereto lay the rule, and it sheweth the degree of the Zodiacke, or any starre lying in the same meridian. Nowe if you turne about the reete til the degree of culmi­nation of the instant proposed lye vnder the rule, and then remoue againe the rule too the houre proposed, marking what degree of the Zodiacke he there cutteth, lastly, remouing the rule and the reete together, be­ing still on this last degree, vntill the degree of culmination come againe to the noone line, it sheweth then in the Lmibe the houre of the other countrie sought for.

Chapter 74. How to direct a ship by the starres in any voyage.

VVHen you set foorth from the shore, set the degree of the Sun vnto the same houre or instaunt, and thence turne the rule vnto the difference of longitude, as in the last chapter, of the place you meane to sayle vnto: then numbering on the rule from the limbe inwardes, the latitude of the same place you goe to, there is the Zenith of the same. Nowe if there chaunce any starre to be in this Ze­neth, direct your ship full vpon him, & you shall goe as certainly, as if Mercurius were your guide. But if the starre be in the same meridian but not in the zenith, reckon howe many degrees on the rule hee is distaunt

Chapter 1. An aduertisement of the Authour concerning necessarie Instrumentes.

HAuing in my thirde booke finished all the Propositions and conclusions, whiche haue been written by any authour heretofore vpon their seuerall Astrolabes lately: and lastly set downe, gathered togeather, and wonder­fully encreased and augmented by that worthy and learned man Gem­ma Frisius, on his Cetholicon Astrolabe, and applyed all those too my Ie­wel, together with diuers of my owne additions, whiche performeth them an hundred folde more playnely, sensibly, and compendiously, then hys Catholicō, as he that wyl trie both shall testifie: It were much for me now to come after so many learned men, and to bring in place any thing more worth the liking. Some wil think I go about to finish Apelles picture, that no mā yet durst vndertake, & perhaps say, who is bolder then blind bay­ard? or Asinus ad Lyras: knowing my rusticall education, and simple capa­citie. VVell, bee as be may is no banning, such a ones words are not wor­thie the scanning, hee that feareth euery grasse must not pisse in euery medow. Babling tongues be hanging rype, I wey not theyr glauering, I acknowledge my learning little, and truly my leasure lesse, notwithstan­ding as much as lyeth in my lot I will shewe my good intent in these volumes following. The rest I leaue to those that haue better wherewithall: to whom I yeelde my labours but as a preparatiue. The shoema­ker can goe no further then the shoe, neither can the tayler make his coate bigger then the cloath wyll yeeld. Such as it is, yet I doubt not but it will bee liked of some at the least. Therefore too the matter, I woulde wish any man that woulde more effectually go to worke in these matters, ouer and besides his little portable Iewel of mettall, which I taught to make in the first booke that must be his companion in all companies, and his page in all places, to haue neuerthelesse at home in his chamber or studie, a great Iewell of two foote dia­metre, if not of mettall, yet made of a round boorde, couered with a fine paste boord, whereon hee may di­stinguish the necessary lines with coloured ynkes very beautifully. The reet he may make of like paste boord, which hadde neede too bee made of good strong peper royal, foure papers in thicknesse, for neither is your bought paste boord broad enough, not good enough: the cēter of this paste boord Reete wold be lined with a thin plate of brasse for wearing: I made my selfe one of them that serued me to great purpose, and there­fore I am sure I tell you true. Let your spaces of your meridianes and your paralelles in the mater, be degrees a peece at the most, and so likewise your azimuthes and almicantares in your reete, yet it is better that those almicantares, that you cut out be eche a degree & the other that you leaue in like barres or the spiders web, to be ech but halfe a degree: for if you will haue your worke sufficiently transparent, two partes at the least must be cut out, and a thirde left in. Also bee sure that you deuise to leaue strings enough, in placing your starres, and cutting out the lower part of your reete, that the paste boord may beare out it selfe, and drawing them like braunches handsomely, the reete will shew the fayrer though they be not necessarie. I did also at euery 15. azimuth, leaue a string, a degree in breadth from the finitor to the Zenith, to keepe my almicantares a part at their true width, take heede you wishe not you had done so too when it is too late. In euerye of these stringes I did write the numbers of my Almicantares from 5. to 5. &c. tyl 90. And then if you had two other, the one made for a globe terrestriall, and the other celestiall, the meridianes and parallels, 5. degrees wide a peece, as hereafter in my Cosmographie shall be shewed, and beyng all three of equall bignesse, one reete may serue them all, they will be verie pleasaunt. These had, you are in heauen alreadie, except you chance to be in loue with a quadrant, and the crosse staffe, which in or before my booke of longitudes of regions, I wyll teach to make, without which you cannot be, if you will pearce vnto the exactnesse of euery thing.

Chapter. 2. How to make an instrument of three streight rules better then any compasses to describe any arch with, be his center neuer so farre off.

IN making my Iewell on a boord of two foote diametre, I was not a little troubled to drawe the arches of the meridians and paralels, which come neerest too the center, and therefore I should doe you wrong if I should leaue you to seeke in that behalfe, truly I hadde deuised to make a beame compasse of a Iauelin or [Page 65] speare of 18. foote long, and yet neither would that length reach all my centers nor draw any arche with­out sagging: insomuch that I was driuen to finde to make them without any kinde of compasses by shorte streight lines drawen from 10. degrees to 10. degrees, &c. the which I had thought wyth a long circum­staunce heere to haue set foorth for theyr better helpe, that should attempt the like. But in the meane time came a little booke to my handes of one Guydo Vbalbus de Theorica Astrolabii, wherein I founde a very sleight in­strument, made of three rules, to perfourme these great arches easily, which I sought by great ambages. VVhē I sawe it, I was almost out of conceite with my selfe for the time to thinke that my grosse witte coulde not deuise it before: for I no soner sawe a glimse of it, but streight I perceiued the reason without reading of anye woorde in the booke: and more then that I coulde streight goe too the tree where Vbalbus gathered that frute though he himselfe concealed it: namely, the 20. proposition of the thirde booke of Euclides elements which sayth thus, If in one portion of a circle there be made diuers angles touching the circūference of the arch (the corde of the same arche, beeing base vnto them all, for that is his meaning as appeareth in the Cō­ment) all those angles are equall one to another.

Also in the 30. proposition of the same booke hee sheweth that all angles in a semicircle touching, the arch be right angles, as in my second conclusion appeareth, which proueth also y^e former in respect. A semicircle is but an arche. For your better vnderstanding beholde, these two figures in the Arche, YX the three an­gles marked wyth A, hauing all of them the corde of the arche to their base, are euery one equall eche to other: likewise in the semicircle CD, the three angles marked with E, are all right angles, this is mani­festly prooued in Euclide by demonstration, & you your self may experimentally examine the equalitie of these angles if you list, by the 10. cōclusiō in my first book. But let this passe, I wil shew you the making of y^e Instrument somewhat more commodiously then Guydo Vbalbus hath set foorth, and lesse combersome, which by this bare figure following will appeare better then by manifolde woordes, and therefore beholde it well,

AB, and AC are twoo streight rules ioynted together like a payre of compasses, at A, but so that the ioint A, must bee as small as may be possible, the riuet being hollowe, and as bigge as may containe a crowes quil: then betweene AD are broade cheekes to helpe strengthen the smalnesse of the riuet. HG shoulde bee very neere one thirde part in length of AB or AC, hauing neare the ende GA a rounde pin erected wher­on the thirde rule EF is put, and then the screwed boxe M keepeth hym on the rule. EF hath three or foure holes towardes the ende E, to keepe him backe from shutting ouer the rule AB. Also KI is equall in length to HG & is double, or rather slit, so that the lower iaw is equal in thicknes to HG, & the vpper­most thinner: whiche the skrew pinne L doth force together, to holde fast the rule EF, when neede is. NO is not much vnlike to KI: yet for fashions sake it is made somewhat like the cocke of a qualifier or gunne, and standeth an inche aboue the rule AB with a foote, his office is to hold a penne or a steele poin­ter fast, being put in the hollowe riuet A, that it stirre not in grauing, to which purpose one checke must bee thicke, and the other thinner, as KI was. Q is the steele pointer or grauer, whiche beeing put through the chappes of NO into the hollwe A, is there holden fast by the skrewe pinne P. Thus muche for your in­strument: But to helpe him to worke, he must haue the assistaunce of two dogges as I call them, such as you see R and S, whiche haue each a waight of leade fastened on their backes, their height must not exceede 3. quarters of an inche, that NO may ryde ouer them, the formost foote, whiche I call the nose of the dogge must bee made like the blade of a knife, and the edge a little blunted, and set forwardes, but verye streight downe, and the backe befiled away vnto a point.

Chapter 3 The vse of the former instrument in drawing of any Arche.

THE vse of this trianguler compasse, is by three prickes giuen too describe an arche, cutting them all three: whiche too doe you must sette the twoo dogges their noses in the two vttermost prickes, whereto applie the edges of the two rules containing the angle A, so that the pointer or grauer that is in the hollowe riuet A, bee in the midlemost pricke, and the three rules there fastened by helpe of the [Page 66]

skrewe pinnes, leade them about close by the dogges noses, and so the poynter shall describe the arche desired. Example, in this figure let XYZ bee three prickes assigned: first I set the two dogges noses in the two vt­termost prickes XY, the skrewe pinnes L and M, being first at libertie, I place the pointer Q which stan­deth in the riuet A, in the midlemost pricke Z, and there holding hym downe, I open the rules vntyll the side AC doe touch the dogge standing in Y: and the side AB touch X, which done, I wring hard the skrew pinnes L and M, and so is your instrument set fast in his angle. Nowe neede you but like a mower with hys Sythe leade along these three rules, so that the side AB doe still touche the dogge standing in X, and AC the dogge MY, and so shall the pointer in A, describe an arche as true as any compasses whatsoeuer. For the dogs shal still keep you so within your bounds that you shall not choose, but mowe your swathe true compasse. In this figure the lines AB, and AC, do represent the sides of the rules, which because they include one angle whose base is XY, therefore in respect that XY doth alwayes keepe the Angle A, at one base the angle A must needes finde out the circumference of the arche as he goeth, as appeareth by the figures of the last chapter, by this you may gather also, that the two rules AB and AC, must bee at the least as long as XY, if they be longer, it is superfluous for this arche, but perhaps may serue another the better, and there­by for my purpose you may cōclude that your two rules need be no longer then the diametre of your Iewell because the two prickes farthest distaunte are the extreames of the diametre, so that one of these made of 3. rules, eache sixe inches long, wil make a Iewel of sixe ynches diametre: the best way is to haue two, one of 9. inches, another of twentie or foure and twentie.

Chapter 4, Of a certaine secrete in diuiding the meridians of great Instruments.

THough this inuention with three rules bee so excellent to drawe great arches in great instruments that all other deuises be almost friuolous, yet wyll I shewe you a great secret or two, whiche I vsed for my helpe before the same was knowen, not because I woulde haue you vse it in that behalfe: but for that I haue founde great pleasure, & necessitie of it in working other inuentions and proiections of the spheare, For Astronomicall dials, mappes, and other matters more then I will speake of till time serue: in the meane season make account of it. If you remember, in the seconde booke and eight chapter, I shewed you to deuide the two halues of the Zodiacke from their seuerall poles: assure your selfe in very lyke manner, prooue it when you will, may you deuide any meridian of the Mater into the same diuisions that any of them is deui­ded by the parallels, and that without drawing the parallels, if in the noone line you seeke the center or ra­ther pole of any suche meridian as farre backwardes within the center of the mater, as the meridian that shoulde bee deuided lyeth within the limbe. For example, if you woulde deuide the 30. meridian, counted from the Limbe on the South halfe of the mater, seeke then where the 30. meridian, from the center on the North halfe of the mater, doth crosse the noone line: and there is your pole or center most assuredly, wher­on keeping the one end of a rule, and laying the other ende on euery of the deg. of the South halfe of the limbe, eache after other: you may thereby deuide the saide 30. meridian, euen as eyther halfe of the Zodiacke was deuided from their seuerall poles in the said second book and eight chapter. And when this 30. meridi­an is thus deuided, you shall finde those diuisions equall to those, which are made by the crossing of the pa­rallels, which you may sufficiently gather by the last way, which I taught in the seconde booke and nienth chapter, to deuide your Zodiack by. I say in this maner may you denide any of the meridians if you seek his pole in the noone line, as farre beyonde the center as the meridian is within the limbe, and thereon as from a center deuide him by the opposite halfe of the limbe. For in very truth, as I say, that point is the very pole of euery such meridian, as in my next booke of spherical triangles will appeare. And thus may you not onely deuide the meridians but generally euery arche whatsoeuer, representing a great circle or semicircle of the spheare, which is specially to bee accounted of, and wil, beyng ioyned to the 4. and 6. chapter of the seconde booke, yeeld exceeding fruite and pleasure in proiecting of the globe. Before the inuention of the said three rules I was determined in making of great instruments to deuide euery 5. or 10. meridian in this manner and then by short streight lines to drawe the parallels.

Chapter 5. Another secrete founde by the Authour indeuiding the parallelles of great Instrumentes.

FOrasmuche as the time hath beene with mee, and may bee againe with you in making great quadrants of 7. or 10. foot side, or great astrolobes, that you wold be cōtent to know how to diuide som one parallel or [Page 67] other into the very same diuisions, whiche hee is, or at least should be deuided into by the meridians, though you thinke perhaps you shall not greatly vse it, and I thynke I shall neuer haue the neede therof, I haue had, yet will I not omit to shew it, least at a time you woulde wish you knewe it. Admit therefore for exam­ple that you had drawen the fortie & fiue parallel, how coulde you now know where euery meridian bee­ing truly drawen shoulde crosse, you cannot tell: then say I, thus doe. Let the point of the 45. degree of the limbe, that is to say, where the 45. parallell toucheth the limbe, be vnto you insteede of a center, on which point laying the one ende of a rule, mooue the other ende from degree to degree of the further half of the noone line beyond the center, and thereby deuide the one half of the saide 45. parallel, which deuisions (if your rule be streight and you worke truly) shall without all doubt be equall to those diuisions which the meridians do make. And so may you deuide any of the parallels especially those betweene the 45. parallel, and the pole exactly, but those that lie towardes the Equinoctiall, cannot so exactly bee deuided in this manner, by rea­son that the rule wyll cut them so byas, or oblique that the pointes of intersections can scant bee founde. I feare mee, I shall haue occasion to vse these in making my great quadrant, and therefore I leaue out fur­ther demonstration till then. The time was when I had writtē more of these matters then I meane nowe to publishe. For I haue been driuen by helpe of those diuisions too drawe the meridianes partly by ame.

Chapter. 6. Of drawing and dyuiding of the eclipticke line on the Mater.

THE eclipticke is, or ought to bee drawen crossing the center of the mater to touch the one ende of ey­ther Tropicke at the Limbe, videlicet, leing byas to the Equinoctiall according to the Sunnes greatest declination 23 28/60. degrees, as in the second booke, and fifth chapter, and ought to be deuided into the very like partes or degrees as the Equinoctiall is. VVherefore if you crosse this eclipticke line square on the center, with a blinde line it shall touch in the Limbe the one ende of eyther pole circle, where doubt you not, but the poles of the eclipticke are. Therefore from eyther of those points you may deuide him, by lay­ing the rule on euery degree of the halfe Limbe farthest off from the same point, as in the second booke, and fourth chapter, the Equinoctiall line was deuided from one of his poles: then set too the Caracters of the Signes, the beginning of ♈ and ♎, to be at the center: & of ♋ at the Limbe or end of the Eclipticke on the right hand, ♑ at the left: the rest in order as I hope you knowe by this time. You see here that the reason of this differeth not from the deuiding of the meridians mentioned in the fourth Chapter. For the ecliptick being the great circle of the spheare, and lying 90. degrees within the limbe, towardes the center, euen cros­sing the center: therefore hys gole must needes be 90. degrees from the center, which is at the limbe.

Chapter 7. By the verticall circle or azimuth of the Sunne or Starres, taken to knowe what it is a clocke, very certainely and exactly.

HAuing now instructed you for the making of a great Iewell of two foote diameter, it is conuenient that I shewe some vses needing hys capacitie, notwithstanding that all the thirde booke wyll bee gladde of his acquayntance. I would haue no man presume too much of a small instrument be he neuer so ex­quisitely made: in matters requiring great certaintie, there can no man worke so truly with a small as with a great. For although that small instrumentes in mathematicke speculation be sufficiently certaine to lead a manne to knowledge of thinges, yet there is no doubt but Maiora semper sunt certiora in mathematica practica, nei­ther is it possible to vse any so great as in some cases woulde be required. It is shewed in the sixth chap­ter and third booke, how to knowe what it is a clocke by the altitude of the Sunne, but because the Sun, and likewise the starres in an houre before and after noone, euery day rise and fall very little: therefore a small mysse in the altitude taking, may make a quarter or halfe an houres errour: besides that, the intersection of the Almicantare, with the parallel of the Sunne, or any starre neere the South, lyeth very byas, and speci­ally in South signes and starres, whiche may make more errour for want of the point of the true interse­ction then may well be allowed in some cases. VVherefore the truest and most certaine way to come by the knowledge of the houre, and specially ad minutum, which in the rectifiyng of the starres, planets, &c. and getting the longitude of regions is very needefull, is by azimuth, because he alwaies crosseth the parallel ve­ry distinctly, and is (the meridian line first had) to be found very certainely. It is shewed in the third booke 36. chapter, by the azimuth and altitude taken, to knowe both the latitude and the houre: but here by the a­zimuth, and latitude giuen, is had both the houre and the altitude of the Sunne or starre, prouided that the degree of the Sunne in the Zodiacke, with the starres declination be also knowen. Take either by helpe of a Topographicall instrument or els by the thirde booke and 35. chapter, the azimuth of the Sunne or any knowen starre, which had, set the Finitor to the latitude, and marke where the like azimuth of the reete dooth cut the parallel of the Sunne or Starre, the meridian crossing there, sheweth the houre and the almican­tare crossing there, sheweth the altitude, yet in the starres this houre is but Horastellae, the starres houre, by which you may get the true houre, as is shewed in the thirde booke and 26. Chapter, remembring alwayes that Locus Solis oftendit horam, this needeth no example, it differeth not much from the saide 36. chapter. Fur­ther all the difference from the sixth chapter of the thirde booke is, that here you woorke by the azimuth as there you did by the almicantare.

Chapter 8. By the latitude of any planet and his distaunce from any knowen fixed starre had: how to get his exact longitude.

BEcause in the first chapter of this booke I counsailed you to adde vnto your great Iewell the crosse staffe, I thought good to shew you, partly by this chapter, that he will serue you to necessary purpose. VVhere­fore according to Gemma Frisius instruction in his booke de radio Astronomico, you shal by the crosse staffe take the distance of any planet you shal see in the heauens, whose longitude you desire, frō some one knowē fixed starre, that is to say, such a stare whose longitude and latitude is knowen: which had, you shall then re­sort to the Ephemeris or some other like tables, and get the latitude of the planet for that day, nowe place the Zenith poynt of your reete so many degrees of the limbe aboue the equinoctiall, as the latitude of the fixed star commeth to: the reete there fixed, marke where the Almicantare counted from the zenith point, equall to the distance taken by the crosse staffe, doth cut the parallel counted from the equinoctiall, equall to the planets latitude had in the Ephemeris, and the meridian crossing there with them, followed too the equinoctiall line shall shewe you theron the deg. and min. of the difference of longitude betweene the starre and the planet, which if the planet be Eastwardes of the starre, you shall adde to the starres longitude, if VVestwardes, sub­duct it, and so shall you haue the longitude of the planet desired: remembring this by the way, that if the la­titudes both of the starre and planet be one way, that is to say both North or both South, then must you af­ter you haue placed the zeneth point at the Limbe fixed as before, take the said parallel on the same side of the equinoctiall, as the zenith point is: but if the one bee North, the other South, then take the said parallell on the further side of the Equinoctial.

For example. The 14. day of October 1582. about 6. of the clocke in the morning, I tooke the distāce of the firie redde planet, ♂, from the great starre of the light called humerus dexter orionis, Oriones right shoul­der, by a crosse staffe of my fashion for want of G.F. and founde the same 31. degrees 37. minutes. and 30. secondes, for my staffe was large. The latitude of the said starre of Orion, is, and euer hath been, and there­fore this working is the more certain, 17. degrees southwardes from the eclipticke, which by my tables of fixed starres, you may finde: the latitude of ♂, by Leouitius Ephemeris was then one degree verye neere North­wardes, these had, I reckon on the Limbe from the equinoctiall towardes the South pole of my Iewel, the said 17. degrees, being the latitude of Orion: thereto I place the zenith point of the reete fixed: then among the parallels towardes the North pole, I reckon one degree beeyng the latitude of Mars, found in the Ephemeris.

Nowe then where the 31. 37/60. almicantare counted from the Zenith doth cut the saide first parallel (for I leaue out the secondes in small iostruments) there doe I see the 26 ¼. meridian counted from the Limbe to crosse, the same was the difference of longitude betweene the said starre of Orion and Mars at that instant. The longitude of that starre after Stadius is 23. degrees in ♊, and at that time I sawe Mars Eastwardes of this star. Therefore I adde this difference here found, vz. 26 ¼. to this longitude of the saide starre being 23. in ♊, and counting it according to the sequele of the signes, it falleth out too bee the 19. degree & 15. min. of ♋. which I conclude to be the place of Mars, his longitude at that instant. But some will here say to what purpose is this if wee goe to the Ephemeris for the latitude of the planet, then were it as good to take the lon­gitude there also. True it is in some respect, but yet shall you see commonly that Ephimerides will differ in the longitude of the planets one from another, besides, when Ephemerides are made by diuers men to diuers lon­gitudes, then may they differ, and yet be both true. VVherefore by this chapter you may with greater speede then Gemma Frisius or any other hath before taught, for ought that I euer read, both examine the truth of your Ephemeris, and of diuers of them, which is the truest & most neer to the longitude of your countrie. And though in their Ephemerides they shoulde differ muche in the longitudes of the planets, yet they cannot erre greatly in the latitude, if they doe, a little cannot hinder your purpose in working after this chapter, so you take your distaunce truely, as for the purpose you shall finde there that ♂ goeth but 2 minutes in latitude e­uery day, so that where before you tooke one degree or 16. minutes: if you had taken 60. minutes, which is the latitude of ♂ two dayes before, and wrought as before, you shoulde haue committed no sensible errour in the longitude. Though this experience be most pleasant to examine the places of the planets, yet med­dle not with the Moone: for her tickle swift course found in tables, and her vntowarde parallaxe in takyng her distance, will disappoint you. But if it please God to lend mee life, and that this wrangling worlde alow mee leasure, wherein these sixe or seuen yeeres I haue fared like a tennise ball tost vp and downe, I wyll in my boke of longitude of the places, so sufficiently prouide you for euery shuttle nycitie of the Mone, that he shall haue no starting hole to deceiue you. You shall be fully instructed to meete her at euery turne, yea e­uen to leade her in a string, as they say, and that at pleasure. Then shall her defuse motion in her busie speeres with her euerdaring parallaxe (which hath heretofore stalled the most, and incumbred the best learned, yea and deceyued some of them, as I shall there manifest) be so broade and playnlie opened by my simple indu­strie, that all sortes shall wonder at the facilitie thereof. And therefore let all Cosmographers, Nauigators, Shipmasters, Trauailers, and such like, clap their handes for ioy of this Iewell.

Chapter 9. How at all times you may beholde on the Mater of your Iewel the whole circum­ference of the ecliptike, according to his cituation, aboue the horizon, which bringeth with it pleasant vses as followeth,

FIrst place the deg. of the Sunne in the Zodiacke, to the houre proposed, reckoned on the limb, & there pre­sently shall you see the asc. on the East part of your horrizō by the 3. book 42. cha. & his amplitude by the [Page 69] first part of my addition to the 20. Chapt. of the third booke: The degree of culmination or midheauen, you haue there also by the 3. Booke 30. Chapt. and his meridian altitude by my addition to the 3. Booke 29. chap. there present all these are had at one inspection, by applying the degree of the sunne to the houre proposed. Then to proceed, number the meridian altitude of the degree of culmination on the Limbe, from the pole ar­ticke towards the right hand, if the ascendent be a north signe, but leftwards if he be a south signe: to the end of this numeration place the Finitor, which done, reckon on the Finitor from the centre outwards, the ascen­dents amplitude and that alwayes leftwards or towards Media nox, and mark what meridian of the Mater doth cut on the Finitor the said degree of amplitude, the portion of the same meridian from the Finitor to the pole, representeth the arch of theclipticke from the ascendent to midheauen, & the meridian of like distance frō the axtree line on the other side which also cutteth the like amplitude on the Finitor his portion betweene the pole and the Finitor representeth the arch of thecliptike from midheauen vnto the descendent.

Example. Our most towardly and vertuous young Prince and king, Edward the sixt was borne 1537. the 11. of October the 13. houre 51. min. the sunne then being in ♎ 28. deg. wherefore I would see the constitu­tion of theclipticke to the horizon on my Iewel that thereby I might set the figure of his natiuitie, as I wil shew in the 13. and 14. Chapt. following. First therefore I set the deg. of the sunne vz. ♎ 28. vnto the 13. houre 51. min. which is one of the clocke after midnight, and 12. ⅔ degrees past, where I see in our horizon being 51. ⅔ the ascendent ♍ 5. deg. almost, whose amplitude is 16. deg. ½, and likewise the degree of culmination ♉ 26. deg. whose meridian altitude is 57. ⅓ deg. These had, I reckon the saide meridian altitude vz. 57. ⅓ deg. on the Limbe from the pole articke rightwards, that is towards noone, because the ascendent is a north signe, thereto I set the Finitor, whereon from the centre northwards or leftwards, I reckon the ascendents ampli­tude vz. 16. ½ deg. and there do I finde to cut the Finitor, the 14. meridian counted from the axtree line, where­by I conclude that the arch of this 14 meridian between the pole and the Finitor is now the arch of thecliptick betweene the midheauen and the ascendent, so that the pole being now the 26. of ♉, rekon thense from the same 14. meridian degree after degree as they follow in order on theclipticke, which by helpe of the parallels you may easily do, and you shall finde that the last number at the Finitor will end in the 5. deg. of ♍ which is the ascendent. Lastly the arch of the 14. meridian on the other side of the axtree line between the pole & the Finitor, is here the arch of thecliptick between midheauen and the descendet, cutting on the Finitor like deg. of amplitude, and being accompted from the pole, that is from 26. of ♉ backwards, contrarie to the sequele of the signes, it wil end in 5. of ♓, which is the descendent. This 14. meridian or any like, I must for distinctions sake hereafter call theclipticke circle.

Chapter 10. By the Eclipticke circle constituted as in the last Chapt. how to get the altitude and azimuth of the sunne or any point of theclipticke, and thereby also of the other planets so they haue no latitude.

THecliptick circle being scituate as in the last Chap. to the Finitor for any instant proposed, reckon there­on the deg. of longitude of the sunne or any other planet, not hauing latitude either from the midhea­uen if he be on the East part, or from the descendent, if on the west, because so may you reckon accor­ding to the sequele of the signes, better then from the ascendent vpwards, or from midheauen towards the descendent: where your reckoning lighteth out on this ecliptick circle, the azimuth and almicantare crossing there, sheweth your desire.

Chapter 11. By the same constitution of theclipticke circle how to finde his Nonagessimus gradus with the altitude thereof, being the quantitie of the angle made betweene theclipticke and the horizon.

IF from the ascendent you reckon on theclipticke circle vnto 90. deg. there is your Nonagessimus gradus, & the almicantare cutting, or rather touching there, is the altitude.

Chapter 12. How by the same constitution of theclipticke to see all the circles of position liuely on the Iewel, and what degree of the zodiacke is in any of them.

THis is performed onely by setting the zenith line to the constitution insteede of the Finitor, as for exam­ple: If in the 9. Chapt you had set the zenith point of the Reete 57 ⅓ deg. vnder the pole, there do the a­zimuthes represent all the circles of position crossing theclipticke or 14. meridian whereon you may reckon what degree euery of them cutteth which is your desire.

Chapter 64. How by the same constitution of theclipticke circle to set a figure verie sensible and apparantly.

FOr this purpose you must haue your foure domifying position circles distinguished on the Reete, as in my addition to the 3. Booke 49. Chapt. is shewed, and you shall finde those 4. with the zenith line in this [Page 70] constitution, and the Limbe to cut in the said eclipticke circle, the degrees beginning euery of the 12. houses whereby you may easily set the figure, remembring that by the one halfe, the whole 12. are had.

Chapter 14. How to set a figure with more pleasure and ease, the last Chapter well vnderstood, so that you may beholde your whole 12. houses in their being altogether on your Iewel.

HEere my appointment was that the azimuthes and meridians should chaunge their offices, so that the 14. azimuth should represent the eclipticke circle of the 9. Chapter: and the meridians should repre­sent the circles of position by placing the zenith point of the Reete 57 ⅓ deg. aboue the pole, and (the do­mifying position circles, being first distinguished among the meridians) there shall they cut the 14. azimuth in manner of the last Chapt. and thense following the almicantares cutting there with them, vnto the Limbe, make prickes or notes on the Limbe of the Reete for euerie of them, and so are your 12. houses had without a­ny numbring at those prickes, of which the zenith point of the Reete is alwayes one, yea and the beginning of the 10. house: then if about the Limbe of the Mater the 12. signes bee set and deuided, and amongst them you choose out the Gradus medii cali for the instant proposed, and thereto set the zenith point of the Reete fixed, there do the rest of the prickes euery one choose out the deg. beginning the house he standeth for, which are easily counted from the 10. house, here you see there is no numbring at all.

Chapter 15. How to get the altitude of the Nonagessimus gradus more readily then in the 11. Chapter is expressed.

THe houre and minute giuen, get the ascendent and his amplitude, and also the degree of culmination with his altitude, and thereby place the constitution of theclipticke circle, as in the 9. Chapter is expres­sed aboue the Finitor, and then marke what almicantare doth but onely touch theclipticke circle, the same reckoned from the Finitor is the altitude desired.

I haue giuen ouer these 6. Chapters last past being indeede some of them very tedious and long, as one ve­rie loth to trouble my booke with superfluous matter, for in truth they tended chiefly to the setting of a fi­gure, which when all is done, will (though perhaps not so liuely, yet more speedily) be performed by the third Booke 46. Chap. but I haue let the titles stand, to the end that such as list may the easier boult out the matter of themselues.

Chapter 16. How to place theclipticke aboue the Finitor after another sort then is shewed in the 9. Chapter, and in the same constitution to finde out euery needfull circle, whereby most excellent conclusions are to be performed.

THis constitution did Gemma Frisius finde out by his Astrolabe, as in the 2. Booke 54. Chap. appeareth, yet seruing there onely to get the bare altitude of the ☉ or planets, not hauing lat. which by the 6. Cap. may also be done, but my Iewel by the explanation of this Chapt. and the helpe of his almicantares and azimuthes, performeth the same to great and wonderfull purposes as will more appeare in my Booke of the longitudes of places then in this 4. Booke. Wherefore either by the saide 54. Chapt. place the Finitor as there is shewed, or else by the houre giuen, get the Nonagessimus gradus either by my addition to the 3. Booke 53. Chap. of the 3. Booke, or by the 11. or 15. of this booke, and reckon his altitude on the Limbe from thequinoct. line vpwards, thereto laying the Finitor, but on this condition that if the matter you purpose be in latitude North of thecliptick, then let the almicantares stand vpwards towards the north pole, if south then downwards to­wards thequin. & there is your Iewel in a singuler constitution seruing many pleasant & profitable conclusions as shall follow, wherfore I think it most necessarie to shew or explane the reason therof, thequin. line here re­presenteth the Finitor, and the parallels & meridians represent almicant. & azimuthes, the Finitor representeth theclipticke, his point at the Limbe the Nonagessimus gradus, and at the centre the ascendent and descendent. The almicantares and azimuthes are circles of longitude, and latitude of the zodiacke. The Limbe is the circle of longitude of the Nonagessimus gradus, which issuing from the pole or poles of theclipticke alwayes passeth by the zenith, the centre of the Iewel is the ascendent and descendent, and not the East as it is in the 9. Chap. and all other vses of the Iewel most commonly. But here the East and west points are distant from the centre as much as the amplitude of the ascendent or descendent commeth to, and likewise the miridian as much distant from the Limbe, so that in briefe the Iewel here representeth the globe wrested from the meridian vnto that a­zimuth, which cutteth the Nonagessimus gradus, it is most manifest that all great circles in the sphere are concen­tricke to the sphere, and deuideth other in halfe, howsoeuer they be drawne or crosse ech other as I haue of­ten said, and as in the definitions & theoremes before my spherical triangles following shal appeare, where­fore when as the Nonagessimus gradus is apointed in the Limbe, the ascendent and descendent being alwayes 90. deg. distant equally on ech side from it, must needes be in the centre, which of necessitie must be the place of crossing of the Finitor with theclipticke being both of them great circles, and twise 90. being a semicir­cle. But although that theclipticke do alwayes deuide the horizon in equall parts, yet he doth it continuallie in chaungeable places by reason that hee lieth byas to thequinoctiall, hauing for that cause no vniformitie [Page 71] in his motion, by reason whereof the altitude of the Nonagessimus gradus, is continually variable euerie mo­ment, and so also is the altitude of the degree of culmination, or any other part of the zodiacke. Likewise the ascendent and descendent cutteth euerie moment in sundrie pointes of the horizon: whereas the Equi­noctiall, and all the other parallels cutte the horizon continually in one point and their meridian altitudes are alwayes one, insomuch that the Iewel being set at any time to that constitution, which this Chapter tea­cheth, the East and VVest point cannot be in the centre of the Iewel, but onely when the beginning of ♈ or ♎ is ascendent for then the Nonagessimus gradus being ♋ or ♑ must needes be in the meridian: at all other times the ascendent cutteth the horizon either North or south of the East, which is called the amplitude all in the thirde booke 20. Chapter appeareth, and according to this amplitude shall the East point on the Iewel be remoued.

Chapter 17. How by the constitution of the 16. Chapter to get both the altitude and azimuth of the sunne or any point of theclipticke, or of any other planet or starre how much soeuer they shall be in latitude.

HOw to get the altitude of the sunne, or any point of theclipticke, by this constitution hath beene taught by G.F. as in my 3. book 54. Chap. appeareth, but for any planet, star or point of the heauens hauing lat. you are there neuer the neare, nor yet for the azimuth although it be had in the 11. Chapt. after another constitution for any point of theclipticke. Now therefore by the houre and minute proposed, hauing set your Iewel to the constitution of this last Chapt. you shall by the 3. booke 20. Chapt. learne the amplitude of the ascendent for that instant, which if it be of a North signe, you shall reckon on thequinoctiall line, which heere representeth the Finitor from the centre rightwards towards noone, and there is your East point, and the me­ridian or houreline cutting there, representeth the East azimuth, but if your ascendent be a south signe, then count the amplitude leftwards towards Medianox, and there shalbe your East point and your East azimuth, which point and azimuth being found, you may easily reckon from thense your azimuthes, and that alwayes rightwards towards the Limbe, and thense backwards againe, and so round as occasion shall serue, remem­bring that the 90. azimuth from this East point so counted must needs be your meridian.

As for example, in the natiuity of our vertuous king Edward, the altitude of the Nonagessimus gradus was 58 2/4 deg. by the 11. or 15. wherefore that reckoned on the Limbe from thequinoctiall and the Finitor thereto laid, then doth it represent theclipticke circle placed to this said constitution, the ascendent and descendent being in the centre of the Iewel by the 16. but because the ascendent is of a north signe in this natiuitie vz. the 5. of ♍ and the amplitude thereof by the 3. booke 20. Chapt. 16 ½ deg. I reckon on thequinoct. which now represen­teth the Finitor, the same 16 ½ deg. and that from the centre rightwards towards noone, and there I conclude to be the east point and that the 16 ½ meridian is theast azimuth, and then if from this 16 ½ meridian you rec­kon to the Limbe, and thense backe againe vntill your reckoning come to 90. the houreline there ending representeth the meridian or south azimuth, being here the 16 ½ counted from the Limbe backe againe, so that hereby you may ground this generall rule, looke howe much the amplitude or east point goeth rightwards from the centre, so much the meridian commeth backe from the Limb leftwards, and how much the amplitude goeth leftwardes so much the meridian commeth short of the Limbe in reckoning, wherefore when you haue set theclipticke circle to the constitution of the 16. Chapter, and haue any planet or starre proposed, whose altitude and azimuth you desire to knowe thereby, get his longitude and latitude in the zodi­acke by an Ephimeris or other tables: reckon his longitude on this ecliptick circle being here the Finitor, and that from the centre or Nonagessimus gradus at the Limbe, as in the said 54 Chapt. is shewed, and then on the azimuth there being, reckon his latit. by helpe of the almicantares, where you shal see the parallel reckoned from the e­quinoctiall shewing his altitude, and the houreline cutting there, reckoned from the point or houreline re­presenting theast azimuth found by the 16. Chapt. sheweth the azimuth desired.

Example. I shewed in my addition to the 33. Chapter of my 3. booke, the glorious planet Venus being the 10. of October 1580. in longitude 5. deg. 27. min. of ♎, in latitude 5. degrees southwards, the sunne then also being in ♎ 27. degrees 8. minutes that she rose about 6. minutes after 5. of the clocke in the morning. Nowe woulde I knowe what height shee was at 10. of the clocke the same morning and that by the rule of the 16. Chap. to see how it agreeth with the 3. booke 32. Chapter, first by the houre giuen videlicet 10. of the clock, and my addition to the thirde Booke 53. Chapter or the 11. of this booke, I get the Nonagessimus gradus to be 29 ½ degrees of Leo and his altitude 46. degrees, therefore by the 16. Chapter I reckon 46. deg. on the Limbe from thequin. towards the north pole, thereto I place the Finitor with his almicant. downewards towards the south pole, because the latit. of ♀ was south: the ascendent then by the 3. booke 42. Chapter was ♏ 29 ½ degrees, and his amplitude by the 3. booke 20. Chapt. 33 ½ deg. which I reckon on the equinoctiall line from the centre leftwards, because ♏ is a south signe, and there is my east point, and the 33 ½ houreline cutting there, is my East azimuth. Well then because in succession of the signes after ♌ followeth ♍ ♎ &c. there­fore ♌ being in the 90. degree, I know that Venus was then also in theast quarter, that is to say eastwards of the 90. deg. For which cause I reckō on the Finitor from the Limb inwards, that is to say from the 90. deg. being ♌ 29 ½ vntil I come vnto ♎, 5. deg. 27. min. & it endeth in the 37. deg. almost of the Finitor counted from the Limb, & on the azimuth there standing, I reckon the lat. of ♀ vz. 5. where I see to cut the 31 ⅓ parallel, whereby I conclude that ♀ was then so high, wheras if you had'not respected the lat. but performed it by the long. after the 2. booke 54. Chap. that is to say, to haue taken the parallel cutting the Finitor it selfe in the 37. deg. from the Limbe, you should haue founde ♀ 36. degrees heigh, which is no small error if any man would thinke to [Page 72] know the houre not respecting the latitude. But nowe to know in what azimuth she was then in, from my East azimuth found as before, which is the 33 ½ houreline leftwards of the centre, and there I number right­wards towards noone vntill I come to the houreline cutting the place or latitude of ♀ with the said 31 ⅓ pa­rallel and my reckoning commeth to 79 ½ the azimuth desired, wherein Venus was that instant. And thus may you do for any fixed starre or other planet.

Chapter 18. How by the constitution of theclipticke as in the 16. Chapter to get the longitude and latitude of any starre or planet, that is to be seene in the skie, and thereby also his right ascension and declination.

GEmma Frisius as appeareth in my 3. Booke 39. Chapt. sheweth to get the longitude, and latitude of any starre or comet out of the heauens, as this Chapter doth purport, after another manner then heere shal­be shewed, but in my opinion not so exactly, yet with farre more trouble though both wayes do well to examine a truth one by another. Wherefore thus shal you do when you see any planet, starre, or comet, whose longitude and latitude you desire: gette his altitude and his azimuth by some instrument, and also the houre and minute by the 4. Chapter if it may be: these three must be had all at one instant, and that verie exactlie which done, by the houre had, set the Finitor to the constitution of the 16. Chapter, and then by the East point founde by the saide 16. or 17. Chapter, seeke out the houreline representing the azimuth as there also is shewed, whereon reckon so many degrees from thequinoctiall line as the planet or starres altitude taken come to, and the azimuth and almicantare cutting there, sheweth the longitude and latitude desired accoun­ting them as in the 16. and 17. Chap. is shewed.

Example. Of ♀ in the last Chapt. Admitte I had taken her azimuth 79 ½ and her altitude 31 ⅓ at tenne of the clocke in the morning the seconde of Ianuarie 1579. I being nowe in Gloster shire in the Christ­masse time with the good knight and my deare vncle Syr Iohn Hungerforde, who departed this worlde the 19. of March, Anno 1582. there chaunced a great snow a day or two before, and therewithall followed ri­mie dayes, by which meanes although there were no cloudes to be seene in the whole skie, yet by reason of the rimie thinne vapours that did hang in the lower region of the aire that much brake the force of the sunne beames, I did perceiue the sunne, the moone, and Venus who then was betweene the ☉ and ☽, all three of them at two of the clocke in the afternoone the same day: or rather somwhat before, and so all the day af­ter, which I shewed there to diuerse of my friends and kinsmen which were there as I was, to passe the time with that worshipfull knight, whose hospitalitie was most bountifull. Wherefore admitte the tenth of October 1580. were some such like day, if you will graunt no absurditie, and that then I had taken the alti­tude of ♀ 31 ⅓ deg. and her azimuth 79 ½ at 10. of the clocke shee being to sight Eastwards of the Nonagessimus gradus and woulde thereby knowe her longitude and latitude, which to doe, I lay the degree of ☉ videlicet ♎ 27. degrees 8. minutes vnto 10. of the clocke in the Limbe, where I finde the ascendent ♏ 29 ½, his ampli­tude 33 ½, the Nonagessimus gradus ♌ 29 ½, his altitude 46. by which I set the Finitor to the constitution of the 16. Chapter in all respectes as is done in the last Chapter, then by reckoning the amplitude of the ascendent vi­delicet 33 ½ on the Equinoctiall line leftwards from the centre, because the ascendent was a south signe: there I finde the East point, from which towards the south I reckon among the hourelines, the azimuth of ♀ videli­cet 79 ½ which lighteth out in the 46. meridian counted from the Limbe, in which 46. houreline, I reckon from the Equinoctiall line the altitude of ♀ vid. 39 ⅓ which if your almicantares chance to be turned towards the North pole, you shall perceiue not to reach theclipticke or Finitor: and therefore turne about the Reete that the almicantares may stande downewards towards the south pole, and then doth the said 31 ⅓ deg. of the 46 meridian cutte the 5. almicantare, which sheweth that ♁ was then 5. degrees in latitude, and also it cutteth the 37. azimuth which followed to the Finitor, shewing her longitude if you account from the Limbe or 90. degree thither, that is to say, because ♀ was in the west quarter: therefore the rediest way is according to the sequele of the signes count it from the end of the Finitor at the Limbe being ♌ 29 ½, thence numbring downe towardes the ascendent at the centre vnto the said 37. azimuth, it will fall out to be the 5 ½ of ♎, the longitude of Ve­nus. And thus may you doe with any fixed starre or other planet, the Moone excepted, whose parallexis ma­keth many alterations in this behalfe.

Chapter 19. The altitude and azimuth of the sunne, or any planet or starre giuen, how by the constitution of the 16. Chapter, to get their exact longitude.

THe sunne neuer hath latitude, and therefore we shall soone dispatch with him, the latitude of any other planet or starres, you must get by tables, as in the eight Chapter you did, which had, get at any instant the azimuth of the planet or starre exactly by some large instrument, either by the third booke 35. Chap. or by this next Chapter following, and the houre and minute at the same instant, by which set the Finitor in the constitution of the 16. Chapter with his almicantares to the North pole, if the latitude of the same planet [Page 73] or starre be North or to the South pole, if South, then by the 16. or 17. finde out the houre line representing the azimuth so taken, and mark where the same houre line cutteth the almicantare representing the planet or starres latitude, and there shal you see the azimuth crossing, pointing on the Finitor the deg. of the longitude sought for, if you reckon the same from the centre or descendent, the planet or starre being on the VVest quarter, or from the Limbe or nonages. grad. if on the East as in the sixteene. and 17. chapter, is shewed: as in my former example of Venus, admit that the said 10. of October 1580. I found by the Ephemeris her latitude 5. deg. Southwards, and that the same day at 10. a clocke I took her azimuth 79 ½. deg. from the East by a large instrumēt, then by the 16. chap. I set my Iewell to the constitution of the Eclipticke with the almicantares to­wardes the South pole, because the latitude was South: which done, according as is shewed in the same 17. chap. I found the 46. houre line counted from the Limbe to represent the 33 ½. azimuth: and now whereas this 46. houre line cutteth the 5. almicantare, there doe I see the 73. azimuth reckoned from the Limbe, to crosse, whose degree reckoned on the Finitor from the Limbe or 90. degree, bicause that Venus was on the East quarter, falleth out to be the 5. deg. ½. of ♎, which I conclude to be the longitude of ♀ at that instaunt. Hereby maye any man rectify the places of the fyxed starres very easily, because their latitudes neuer alter, and their azy­muthes are easily taken at all times.

Chapter 20. How to get the true and exact azimuth of any planet, comet, or starre vnknowne.

YOu shall at one instaunt take the altitude as well of the planet or comet, whose azimuth you seek, as al­so of some one knowne starre, not farre distaunt from him, which noted, you shall by helpe of your Iewell, or els of the crosse staffe, take the distaunce between this knowne starre and the same planet: whi­che noted, also you shal then reckon on the Limbe of your Iewell from the Equinoctial line towardes the pole, the greatest altitude before taken, whereto set the Zenith point of the Reete fyxed, and among the parallelles counted from the Equinoctiall towardes the same pole, count the lesser altitude: then among the almican­tares counted from the Zenith poynt, reckon the saide distaunce taken: now where this almicantare and the saide parallel do crosse, the meridian of the Mater cutting there, counted from the Limbe, sheweth the horizon­tall distance betweene the sayde knowne starres azimuth, and the azimuth wherein the said comet or planet was at the taking of the altitudes, which azimuthal distance or difference, you may adde or subtract too or from the starres saide azimuth according to the coast, which you shall see one of them to stande from other in the skye. I trust you can not be to seeke of the starres azimuth, except you forgette the third booke 27. and 33. chap.

This working differeth little from the 8. chap. and therefore needeth no example, for here by the two al­titudes and distaunce you get the difference of the azimuthes as there by the two latitudes and distance you got the difference of longitudes. But here also in the moone her parallaxis will make you misse certaine min. which in my booke of longitudes, I wil teach you to remedy.

Chapter 21. The moone shining on any sun diall to know in the night, what it is a clocke, very speedily by helpe of the Iewell.

MArke what houre and part the shade of the moone giueth in the diall, then goe to an ordinary Al­manacke, & looke in what signe and deg. the moon then is, lay the same deg. of the Zodiack by help of the Label to the like houre, & part on the Limbe of the Iewel, which the shade of the diall gaue you: and thence remouing the Labell to the deg. of the sun, it sheweth you the houre & time sought for: and that very easily without taking the moones altitude, or other troublesome working, as in the 3. booke 27. chap. is required. But by the way you must not forget first to rectify the moones place by adding or subducting one deg. for euery houre, that you imagine the time proposed to be distant from noone, as I haue often warned you.

Chapter 22 A speedy way to know what it is a clocke by the starres without taking their altitude.

VVHen you go forth any night, first behold the North pole, & therby directly back opposite therto, you shal easily fynd the south: then mark whether you see any notable starre there placed at that instāt: if any such so chaunce, and that the same starre be on your Reete, laye his apex on the noone lyne of your Iewell towards Meridies, and placing then the Label on the deg. of the sunne, it sheweth in the Limbe the houre desired.

The like may you doe with any starre, you shall chaunce to see on the North part of the meridian by lay­ing his apex on the noone line towards Medianox.

Also you may with a little exercise iudge by sight very neere how many houres anye starre that you see in the skie lacketh of, or is past the South, if the scantlet be not ouer great, whereby laying his apex by helpe of the Labell, to so many houres of the Limbe, wanting or past the noone line, then locus solis ostendit horam. Truely this Chap. besides the ease and speede that it hath, giueth a great deale more credit among the vnlearned, then the other way, for they murmure if one be long hunting after the houre, and thinke the deuise of smal reputation [Page 74] not respecting art. It needeth no example.

Chapter 23. The vse of the three tables comprehended in the 2. figure of the second booke 13. Chap.

I Thinke it not amisse here to shew the vse of the saide 3. tables to fyt the fancy of some men which may de­light therein. First therfore you must remember that the whole rundle containeth 6. spaces, one compassing in another: of which 6. spaces, in the three vttermost are contained 2. seueral tables: the first seruing to get the dominical letter, the 2. the prime and Epact for euer. And in the 3. innermost of the same 6. spaces is one­ly contained the third table seruing to get Easter day for euer.

Chapter 1. Of the Exposition of a Sinicall quadrant, and al thinges thereto appertay­ning.

A Quadrant is in briefe the quarter of a circle, encluded with two semidiame­ters, as you see here this figure ABC. The arch thereof, vz. BC, I call the Limbe which is deuided as the order is, into 90. degr. The semidiameter erected, vz. BA is the side of the quadrant, the other semidiameter, vz. AC is the groundline or base of the quadrant, from which the Limbe beginneth to be numbred, vz. from C towards B.

An arch in Latine Arcus, is any porcion of the circumference or rynge of a circle, as here, BO, OP, PD, BC, &c.

A Chorde in Latine Chorda, is as it were the diameter to the arche, or the string to the bow, as here the streight line BGP, is the chorde of the arch BOP, and the line DC of the arche DC.

The Complement of an arch in Latine Complementum arcus, is that portion of a circle, which any arch lacketh of a quadrant or 90. degrees, as the com­plement of the arch BD is the arch DC, and the complemēt of DC, is BD. So that alwayes the arche and his complement make a quadrant.

A Syne in Latine Sinus, is a streight line commensurable ey­ther in length or power vnto the diameter of a circle, and to be measured by his parts.

The Syne totall in Latine Sinus totus, is the side or base of the quadrant, as AB, or AC, deuided into an 100. equall parts at the least, but rather into a 1000. 10000. or 100000. parts for the more the better.

A right Syne in Latine Sinus rectus, is anye perpendicular fal­ling from the arch or Limbe of the quadrant on his base, as here you see DF is the right Syne of the arch DC, and OM of OC, and is denominated of those parts of the Syne total, whi­che he contayneth or more briefly a right Syne is halfe the cord of any arch doubled, as the right Syne of BO is the line BG.

A backe Syne in Latine Sinus versus, is any perpendicular fal­ling from the arch or Limbe of the quadrant on his side, as you see DE, which is called the backe sine of the arch DC. By thi you may see and learne if you marke it, that the backe Syne of any arch, is the right Syne of the same arches complement, for DE is the backe Syne of the arch DC, but it is the right Syne of BD. VVherefore in briefe, a backe Syne is halfe the chorde of the arches complement doubled.

The Supplement of a Syne is that which any Syne lacketh of the quadrantes side or Syne totall, as BE is the supplement of the right Syne FD, or AE, which is equall vnto it: likewise FC is the supplement of the backe Syne ED, for ED and FC make the Syne totall.

Note that some Authors make but three distinctions of Synes, as I haue here done, vz. totus rectus & versus, O­ther some as namely Apian and Munster make foure, vz. Sinus totus, then Sinus rectus primus, which here I call the right Syne, and Sinus rectus secundus, which I call the backe Syne, and then Sinus versus to be that which I call the Supplement of the backe Syne, videl. FC, being as the arrow to the bow: but truely I like not thereof, and therefore I haue thought good to call both FC and EB Supplementes as before, for if so be that FC be Sinus versus, then may EB be as wel Sinus versus secundus. The matter is not great for the names, so you vnderstand the [Page 76] meaning, for I shall neede none of them all in this my woorke of tryangles. Therefore I will not striue, yet I thought good here to set these downe, least happily their names come in question vnwares to the amazemēt of some young beginner, as sometime it happened to me: for Ignoratis terminis, ignoratur & ars.

Note further, that in all the Figures of this woorke of tryangles all streight lynes cutting the Centre of any circle, and al arches howsoeuer they be drawne, are to be taken for great circles or portions of them, that is to say of circles of one selfesame quantity, and of one selfesame quantity, and of one selfesame spheare.

Chapter 2 Of certayne definitions needfull to the vnderstanding of this woorke.

VVHat a spheare or globe is with hys axetree and poles, is shewed in the 1. booke 3. chap.

1 Great circles of the spheare or of any sphericall body, are those whose diameters are equall to the spheares diameter concentricke to the spheare, and are ech deuyded by supposition into 390. e­quall and lyke partes or degrees, suche are the Equinoctiall, the Eclipticke, and all the Meridians or any o­ther circle imagyned on the spheare, equall to any of them. As for example, in the spheare AB, CD, all the cyrcles AEB, DEC, FEG, and AL, BM, are equall, E beyng centre to them all, and to the spheare it selfe.

Though the spheare hath properly but one axetree and two poles in respect of his dayly motion: yet must it be here graunted that euerye great circle of the speare whatsoeuer, hath his peculiar axetree and poles, though no motion on them performed: the axetree of a great circle is a di­metient or diametre of the spheare, drawen perpendiculer to the flat of the same great circle through hys centre, as the line AEB, is the axetree of the circle DEC: and so of the rest. But this kinde of axetree line seldom com­meth in question only his extreames, which are called the poles of the same circle, are in this treatise altogether vsed and are thus defyned.

2 I. Regio, lib. 4. cap. 2. The poles of euery great circle are 2. opposite points eche of them distaunt from euery part of his circumference, nienety de­grees or a quarter of a circle, as the pointes A and B are the poles of the circle DEC, and are distant 90. deg. from euery of the poynts DLHEMC, of the same circle. Likewise the poyntes D and C are the poles of the circle AEB, and the poles of the circle ABCD, doe lye both couched in the point or centre E, which of necessitie is 90. degr. distaunt from euery part of the circle ABCD: briefly the pole of any circle is the poynt of the spheare, from whence as a centre, the same circle is described.

3 Io, Regio. lib. 4. cap. 1. A perpendicular arch to any great circle is a portion of a great circle passing by the poles of the great circle whereon it falleth, or at the least such a portion as will passe by, or cut his poles, if he be drawne forth, as in the spheare ABCD, the arch KH, is perpendicular to the circle DEC, because if it were drawn foorth, it wil cut his poles A and B. Likewise BD, BH, BE, & BC, are perpendiculer arches vn­to DEC, because they all yssue from the pole of DEC, vz. B, and so are ED, EA, EB, EC, EF, and EG. perpendiculer arches to the circle AB, CD, by the same reason: and so of the rest.

4 A sphericall angle, is an angle included by the touching or crossing of two great circles or their arches and is denominated by the base of the angle at 90. deg. from the crossing as in the spheare ABG, the 2. ar­ches AB, & AD, do include a spherical angle denominated by the arch BC, which arch BC is in euery 90. degr. from A. So that if the arch BC, containe 50. degr. then is the angle BAD so much: likewise the arches BD, and BC, doe include an angle, whose denomination is EF. But to define it more artificially, a sphericall angle is an angle contained between two great circles, and denominated by that portion of the greate circle, de­scribed on the point of their crossing, which is included betweene them.

5 A Sphericall triangle is a sphericall flat, included by three arches of great circles, as ABD, is a sphericall tryangle included by the three arches AB, BD, and DA: likewise ABD, and BCD, and DAF, are spherical triangles, and so of the rest.

6 A right angled sphericall triangle is, that hath one right angle, or more, as ABC, is a right angled tryangle, whose angles B and C, are both right angles by the thirde definition, because AB and AC, stande per­pendicular on BC. Also ABE, is a triangle of three right angles, and BCD, is a tryangle of one right angle vz. C. because the arch DC, standeth perpendicular on BC, by the 3. diff.

7 Equall sphericall angles are those whose denominations are equall, being in equal spheres.

8 In euery spherical triangle, the side lying against any angle, is called the side, subtending the same angle, the other two sides are called the sides including the same angle.

Chap. 3. Of cereaine Theoremes needeful to be premised for the better vnderstanding of this my present worke of Sphericall trian­gles.

Io. Reg. lib. 3. cap. 19.ALl great circles of the sphere, howsoeuer they be drawne, deuide eche other in halfe which is thus proued. Any two great circles being drawne on a sphere howesoeuer shall both bee concentricke to the sphere by the first definition, and therefore of necessitye concentricke one to another: (that is to say) made both vpon one centre, euen the centre of the sphere: wherefore they must needes crosse one another on the extreames of a diametre of the sphere. Now since by the 9. Geome­trical definition of the fyrst booke, a diameter cutteth alwaies any cyrcle in halfe, therefore of necessity anye two circles crossing on another on the extreames of a lyne being diameter to them both, must needes deuyde eche other in halfe, since the same diameter deuideth them both in halfe. So is it with al the great circles of a­ny spheare, for a line drawne betweene both crossing of anye twoo circles, cutteth alwaies the centre of the Sphere, and is diameter to them both most manifestly as in this sphere, ALBC, the great circles ACB, & AGB, deuide eche other in halfe on the diameter AB. Marke this Theoreme well, for hee often commeth in question.

2 If two greate cyrcles or arches of great circles crosse eche other, the two poyntes of theyr circumfe­rences furthest distaunt, shall bee in eche circle at nien­tie degrees from the crossing, this is manifest by the fourth diff. as in this sphere AL, BC, the twoo arches AD, and AH, crossing in A, I say that they shal be far­thest insunder in N and G, both which are 90. deg. di­stant from A by the said 4. diff.

3 Io. Regi. lib. 3. cap. 17. 18. 20. If 2. great circles cros­sing one another, doe cut ech others poles, they make at eche crossing 4. right angles. This is most manifest by the 3. diff. because they then must needs stand perpendiculer one to another: AEB, and CEL, doe so de­uide themselues.

4 Any two great circles of the sphere, howsoeuer they crosse, make at ech point of crossing foure angles equall vnto foure right angles, the reason is this: looke what the one angle lacketh of a right angle, the o­ther hath it euen as it is in streight lynes crossing eche other by the first booke 15. propo. of Euclide, as at the two pointes of crossing AB of the two great circles KA, BM, and CA, LB: at the one, videl. A, the angles CAG, GAL, CAK, and KAL, are equall to foure right angles, for as CAG, and KAL, are eche lesse then a right angle, so CAK, and GAL, are ech greater by so much, as one of the other want: likewise is it of the 4. angles CBG, GBL, LBM, and MBC, at the crossing B.

5 Of euery two great circles crossing ech other, the two opposite angles at ech crossing, are equall, as the two great circles ACB, and KAGB, crossing MA and B, I say that at the crossing A, the angle CAB is e­quall to the opposite angle KAL: and CAK, equall to BAL. The proofe heereof differeth not from the sayde first booke fyfteene propos. of Euclide, and therefore any one of the foure angles knowne, take him out of the semicircle, videl. 180. degrees, he leaueth the other knowne. As for example, if the angle CAB, be 50. deg. take that out of 180. it leaueth 130. deg. the quantity of the angle BAL, and so much is the angle CAK also by this Theoreme.

6 All great circles or arches of great circles drawne forth, being perpendiculer to anye one greate circle, must needs meet and crosse at his poles: this is manifest inough by the 3. defynition, for otherwise they could not bee perpendiculer.

7 Two semicircles, or equal arches can include no triangle: for then were it repugnant to the defynition of a triangle, which must haue 3. sides.

8 I. Regiom. lib 3. cap. 37. 38. In euery tryangle any two sides must needes be longer then the third, for if two sides should be equall to the third, then ioyned together, they make an arch equall to the thirde side, and then can they not include a tryangle by the 7. theoreme: if shorter, then could they neuer reach the third side, thys is demonstrated in the 1. booke 20. propos. of Euclide.

9 Coper. 2. trian. sph. Euery of the sides of a sphericall triangle, must needs be lesse thē a Semicircle: for if you admit one side of a triangle to be a Semicircle, as ALB, then cannot the other two be lesse then ech of them a quadrant, because they cannot els reach home, if so as AG, and BG, or AC, and BC, you see that at theyr meeting G or C, they become a semicircle, as AGB, or ACB, and then by the seuenth theoreme can in­clude no triangle. If you would imagine the other two sides greater then 2. quadrants, then would they meet without the spheare, and there can make no spherical triangle by the 5. definition.

10 Ioh. Regiomont. lib. 3. cap. 39. The three sides of euery spherical tryangle must needes bee lesse then twoo semicircles, this is sufficiently proued by the nienth theoreme. For otherwise they must needes meete wyth­out the spheare in an irreguler forme, or within the spheare like the three sides AL, BC, with CD, and AD: which 3. you see make no trianguler forme.

11 Ioh. Regiomont. lib. 3. cap. 49. It is demonstrated in the 1. book 32. propos. of Euclide, that in euery right li­ned tryangle, the three angles are alwayes equall vnto twoo right angles. But in all sphericall tryangles, the three angles are alwayes greater then twoo righte angles, as by the sixte definition partely appea­reth

12 How many great circles soeuer crosse in one point of the spheare, euery of their poles are to be found in that great circle, vnto which that poynt of crossing is pole: for example of the circles, CANAGAEA & LA. I say that euery of their poles lyeth in the circle CEL, vnto which the point A is pole by the seconde diff. This is most manifest, for by the 3. diff. the pole or poles of any circle must needes be found in another circle, standing perpendiculer thereon: and seeing by the 3. and 6. theoreme, all those circles are perpendicu­ler to CEL, and CEL perpendiculer to them, therefore by the 2. and 3. diff. all their poles must needes lye in CEL.

These definitions and Theoremes premised, shalbe inough for my purpose.

Chapter 4. If of a right angled spherical triangle, two sides shalbe knowne how to get the quantity of the third side, and the other two angles.

A Right angle as well sphericall as playne, contayneth 90 degrees: wherefore he is alwayes knowen. Lette there be proposed the right angled triangle ABC, whose two sides AB and CB contayning the right angle B, let them be giuen, vz. AB, 50. deg. and BC 40. deg. Now wil I shew you by the Iewel to get the quantity of the third side CA, and of the angles A and C beeyng yet vn­knowne.

Number on the Equinoctial line of your Iewell from the centre outwards the quantity of the side AB geuen, videl. 50. deg. then on the meridiā mee­ting there, being the 50. meridian counted from the centre, reckon from the Equinoctiall towards the pole, the side BC giuen, videlicet 40. degrees, therto lay the rule, and note the degrees thereof cut, counted from the cen­tre, by the saide 40. degree of the said 50. meridian, which you shall finde 60 ⅓. deg. the same is the quantity of the third side sought for, videl. AC, and that which more is, there doe you see your whole triangle on the Iew­ell, as plainly as on any spheare, included by those three arches, videl. 60 ⅓. of the rule, 50 deg. of the Equinoctiall, and 40. deg. of the said 50. meridi­an, which being well marked and vnderstood of you, then following the Labell to the Limbe, it cutteth there also 47 ⅔. deg. which must needes bee the quantity of the angle. A by the 4. difference, being that the side BC videli­cet 40. degrees subtendeth it, and thus haue we one of the angles sought for, the other you shall thus easilye finde.

Reckon the side BC videlicet 40. degrees on the Equinoctial from the centre, then on the 40. meridian there, reckon the side AB, subtending the angle wanting, videl. 50. deg. thereto lay the Labell, and you shal see 60 ⅓. deg. thereof cut as before: the Label cutting in the Limbe 62. degr. whiche is the quantity of C, the other angle sought for. Hereby you may perceiue of your selfe, that it is no matter with which of the two sides gy­uen, you begin in performing this chap.

But you will say what if the side AC, videlicet 60 ⅓. deg. subtending the right angle B with one of his inclu­ding sides as the side BC, be giuen, AB being vnknowen. Here must you begin on the Labell, and reckon ther­on from the centre, the subtending side AC, videl. 60 ⅓. then mouing vp and down the rule, til the same 60 ⅓. degree, doe cut among the parallels the degree of the side BC geuen, vz. the 40. parallell counted from the Equinoctiall, where you shall finde to meete you the 50. meridian as before is sayde to make vp your tryan­gle.

Chapter 5. The declination of the Sunne, Moone, or any other planet or starre from the Equi­noctial, giuen together with the Latitude of the countrey, how by the 4. chap. to get the amplitude, the difference of ascension, the ob­lique ascension, and the diurnal arch of any of them.

Here shall you partly perceiue a tast of the great fruite that commeth since of one chapt. of sphericall try­angles, which extendeth it selfe to infinite questions, therefore it is no maruel, that so many greate volumes haue bin written of them.

I wil take now for example the great starre of the first light, called Oculus tauri, the bulles eye: his declyna­tion I finde after Stadius in the table of starres in my 2. booke, to be 15 49/ [...]. degr. Northwardes, which in thys [Page 79] present figure is the space, GD or M whiche is equal to it, the height of the North pole aboue the horizon being the space EA, which is 51 ⅔. deg. here at Reading: these two giuen, I wil, I say, perform all the premisses: and thus I goe to worke. The circle HG is the 15 49/60. parallel of declination Northwards, in which the starre is fixed, I say fyxed, not exempting his diurnal motion, mentioned in the 1. booke and 9. chap. ex raptu primi mobilis, wherefore at his rising hee must needes cut the point M of the hori­zon EF, the space MO being his amplitude of rising in this countrey, which to fynd, I subduct the declination of the starre 15 49/60. deg. videl. MV out of 90. vid. AV, there resteth his complement of declination 74 11/60. deg. videl. AM. And so haue you two sides of the right angled spheri­call triangle AEM, giuen vz. AM 74 11/60. and AE 51 ⅔. deg. VVherefore by the 4. chap. you now gette the thirde side, EM, 63 ½. deg. the complement whereof, vz. MO, being 26 ½. deg. is the amplitude sought for. That AEM is a right angled triangle is manifest, for that the arch AE standeth perpendiculer on the horizon EMF, by the third def. in respect that being drawne foorth, it cutteth the pole of the horizon EMF, videl. the Zenith point 1.

Then for the difference of ascension and the oblique ascension learne by the fourth chap. likewise the quā­tity of the angle EAM being an angle of the same triangle, which by the 4. def. is denominated by the arch of the Equinoctial CV, videl. 69. deg. whose complement videl. VO is 21. degr. which by the 3. booke 13. chap. must needes be his difference of ascension, and is the quantity of the angle MAO. This arch VO, vi­del. 21. subduct from his right ascension, which by the 3. booke 17, chap. is 62 ½. deg. there resteth 41 ½. degr. his oblique ascension being a North starre, but if it had bin a South starre, you should haue added this arch by the third booke 15. and 17. chap.

Lastly, for his diurnall arche, adde the arch VO, videl. 21. vnto the quadrant OD, vz. 90. thereof commeth 111. the arch VD, or rather MG, his semidiurnall arch: that doubled, maketh 222. his diurnall arch, which 222. deg. deuided by 15. deg. leaueth in the quotient 14 11/15. houres: or more playne 14. houres 3. quarters and more: so long doth Oculus tauri bide aboue our Horizon betweene his rising and setting. Note that in the sun this Semidiurnall arche deuided by 15. sheweth the houres of his rising and setting counted from noone.

Chapter 6. To perfourme the 4. chap. on the Iewell by another manner of woor­king.

FOr this purpose let the right angled triangle in the 4. chap. be proposed againe, whose side AC subten­ding the right angle B, let bee giuen: vz. 60 ⅓. deg. and the side CB, vz. 40. deg. being one of the inclu­ding sides. Now to find the other side and angles after another manner, then in the 4. chap. is shewed, doe thus.

The Zenith line of your Recte, first placed euen in the Equinoctial line, and there fixed: seeke the length of the subtending side giuen, AB, videl. 63 ⅓. deg. among the almicantares counted from the zenith poynt, and the length of the other side giuen, CB videl. 40. deg. amongst the paral­lels of the Mater. counted from the Equinoctiall: in the crossing of thys 60 ⅓. almicantare, and 40. parallel, you shall finde the 50. meridian to cut and also the 42 ⅓. azimuth: both counted from the Limbe, which two with the portion of the Equinoctiall or zenith line lying betweene them, doe shew the three sides of the triangle sought for in their right constitution. The saide 50. meridian cutteth in the zenith line which now lyeth euen in the Equinoctiall 50. deg. for the side AB, and in the saide 42 ⅓. azim. 60 ⅓. deg. for the side AC, both counted from the zenith poynt: and the same 42 ⅓. azimuth, cutteth 40. deg. of the same 50. meridian counted frō the Equinoctial for the side BC. Besides all this, if you follow the 42 ⅓. azimuth to the Finitor line, he there cutteth off 47 ⅔. degrees, counted from the centre, the angle subtended by the side BC, videlicet, the angle A, by the fourth def. euen as it was founde by the 4. chapt. But now for the thirde angle videl. C, you must shift the sides thus. The Zenith line lying stil in the Equinoctiall, reckon there­on from the Zenith point, the side BC, videl. 40. deg. there cutteth the 40. meridian, vpon which reckon the quantity of the side subtending the angle you seek, videl. AB, 50. degrees, and there shall you see to cutte the 28. azimuth counted from the Limbe, which followed to the Finitor, sheweth 62. deg. counted from the centre, euen as in the 4. chap. the quantity of the angle C by the 4. def.

But here you will say: what if the two sides including the right angle be giuen, I aunsweare, there is no difference in the working to finde the subtending side from that which I did last to fynd the third angle: for the one including side reckoned on the Zenith, the other on the meridian, there standing, you shall see the a­zimuth cutting there that meridian to make vp the third side.

Chapter 7. To perfourme the same by the Iewell after a thirde man­ner.

LEt the former triangle ABC remaine, whose side subtending the right angle, videl. AC, 63 ⅓. degr. and the side AB videl. 50. deg. one of the including sides let be knowne, the thyrde BC vnknowne, wyth the angles A and C, which three after another manner then yet hath beene shewed, I will teache you to fynde.

Reckon the including side giuen, videl. AB, being 50. deg. on the Limbe, from the pole towardes the Equi­noctiall, thereto set the rule fixed, then reckon among the parallels from the pole, the quantity of the subten­ding side giuen, videl. AC, 63 ⅓. deg. and where this 63 ⅓. parallel doth cutte the rule, there crosseth the 47 ⅔. meridian, which cutteth off in the rule 40. degrees, the third side sought for, videl. BC: the same meridian fol­lowed to the equinoctiall by the fourth def. sheweth there 47 ⅔. degrees, one of the angles sought, videl. the angle A, subtended by BC. And now for the angle C, you neede but shift the sides: that is to saye, reckoning the side subtending it, videlicet AB, 50. degrees on the rule inwardes, and the other including side BC, videl. 40. deg. on the Limbe from the pole: the rule thereto applied, you shall see the 62. meridian to cut the said 50. deg. of the rule, which followed to the Equinoctiall by the 4. def. sheweth there 62. degrees the angle sought for.

If the two sides including the right angle, had been giuen vz. 40. and 50. you see by this last woorking for the angle C, that the one reckoned on the Limbe from the pole, the other on the rule inwards, and applied to­gether, the third side leaps into them being alwaies a portion of some one of the meridians.

I could yet shew you 2. wayes more to worke with a right angled triangle, but more combersome then commodious, yet in the 24. chap. I shalbe driuen to one of them, & for your farther liking of these past, I wil frame another proposition to apply them to vse as followeth.

Chapter 8. The quantity of the longest day in any countrey giuen, to know the latitude, and in what climate the same is.

FOr example: heere at Reading our longest day is 16. houres 24. min. whereby the semidiurnall arche must needs be 123. deg. as in my 3. booke 22. chap. is shew­ed, admit the latitude vnknowne, which semidiurnal arch is in this figure, XN, or rather YD: whiche if you take out of 180. videl. DYC, there resteth 57. deg. the arch XP, or rather by the 4. def. YC. VVherefore you haue here the right angled spherical triangle, AEX, the angle E being a right angle by the 3 def. whose subtending side AX is knowne. If you subduct the declination of the tro­picke, vz. YX, 23 ⅓. deg. out of 90. vz. YA to be 66 ½. deg. and one of the angles not right: vz. E, AX being 57. deg. as is saide: wherefore by the 9. chap. the side AE shall be found 51 ⅔. deg. the latitude sought for: but this proposi­tion serueth hitherto vnto this next chapt. following, be­ing the speedier course in this question, but if you woulde haue done it by the 4.6. or 7. chapt. which require 2. sides giuen, then should you out of the semidiurnall arch XN or YD, take 90. vz. DO, there resteth OY 33. deg. and then haue you a right angled spherical tryangle, OXY, Y being a right angle by the 3. def. and 3. theoreme, whose two sides including the right angle, videl. YO, 33. degree, and XY 23 ½. deg. are knowne: wherefore, by the first part of the 4. chap. or the last part of the 6. or 7. the side XO is had, which is the Sunnes amplitude in that country, he beyng in ♋, then take XO, now known out of 90. vz. OE, there resteth XE, wherfore now you haue the former triangle AEX of 2. known sides, videl. XE, one of the sides including the right angle E, and the complement of XY vz. XA, subtending E. Therefore by the first part of the 6. or 7, or the last of the 4. you maye easilye knowe the side AE beeyng the eleuation of the pole or latitude sought for, then for the climate you may learne by the thirde booke 22. chap.

I might haue shewed you an easier waye by the tryangle YOX, and neuer driuen you to seeke fur­ther for the angle XO, Y beeyng the arche, EC by the fourth def. is the depression of the Equynoctiall vnder the Horizon EF, whose complement EA, is the eleuation sought for, or more easilye [Page 81] the angle DOF: being equall to the angle XOY, by the 5. theoreme, is thequinoctials height aboue the ho­rizon his complement being DI, by the 1. Chapt. is the latitude sought for by the first booke 12. Chap. Thus you see how aboundantly the sphericall triangles are like to serue your turne in euerie question.

Ioh. Regimon. lib. 4. Cap. 27. Copernic. 4. triangle spher. Chapter 9. If one side of a right angled sphericall tryangle, with one of the two angles not right be giuen, how to finde the other two sides and angle.

LEt there be proposed the right angled sphericall triangle ABC, whose sides AB and AC doe include the right angle A of which AB is knowne to be 35. degrees, and the angle C, subtended by the side AB, knowne likewise to be 44 ½ degrees: the third angle B with the sides AC, and CB knowne.

If you will worke after the 4. Chapter, reckon on the Limbe from the equinoctiall vpwards, the angle knowne vz. 44 ½ deg. thereto fixe the rule, then reckon the subtending side giuen vz. AB 35. degrees among the parallels, and you shall finde the 35. pa­rallel to cutte the rule in the 54 ½ degrees counted from the cen­tre, the same is the quantitie of the side B, subtending the right angle, in which cutting you shal also see the 45. meridian to crosse which followed to thequinoctial, as in the 4. Chap. giueth on the equinoctiall line 45. degrees, for the third side AC, thus hauing al three sides, the third angle at B, you may easily haue by the 4. 6. or 7. Chapt.

You will say, what if the said angle C vz. 44 ½ degrees, and the side AC 45. degrees, had beene giuen, I will shew you to get the other two sides and angle by the doctrine of the 7. Chapter, be­cause I will vse you to them all. Reckon this side giuen vz. 45. on the Limbe from the pole, thereto fixe the rule: then on thequino­ctiall from the Limbe inwards, reckon the angle giuen vz. 44 ½ degrees, and follow the meridian there to the rule, and you shal finde on the same 44 ½ meridian betweene his crossing of the rule, and the pole 54 ½ deg. for the side BC, and the degrees of the rule betweene the same crossing and the Limbe 35. deg the side AB.

Yet me thinkes I heare you say what if the subtending side BC were knowne 54 ¼ deg. with the angle B 60 ½ degrees, and the rest vnknowne, I will then shew you to performe this by the 6. Chapt. because we wil set them all on worke.

Place the zenith line euen with thequinoctiall line fixed, as there is shewed: and then on the Finitor line from the centre vpwards, reckon the angle giuen vz. 60 ¼ degrees, and vpon the 60 ¼ azimuth bounding there, reckon the side BC giuen, vz. 54 ½ from the zenith point, and there shall you see to crosse the 35. meridian counted from the Limbe, shewing in the zenith line 35. deg. for the side AB, and from that crossing you shall finde on the same 35. meridian downe to the zenith line 45. degrees for the side AC, and thus haue I serued euery turne.

Chapter 10. If you would make an horizontall or verticall Dyall to any countrie, how by this last Chapt. to finde what number of degrees euery houres space containeth.

YOu must not here be ignorant that those meridians of the sphere which deuide thequinoctiall into 24. equall parts being euerie 15. counted from the meridian of any countrie, are called specially the houre­lines, these cut any oblique horizon into 24. vnequall parts, as in this figure you may see DE, D, F, DGDH &c. rounde, where the circle BLCK representeth the horizon BC the meridian, and D the North pole eleuated 51 ⅔ degrees aboue B. the North point of our horizon heere at Reading: to which intersections BEFGHI &c. if there be streight lines drawne or imagined from the centre of the horizon vz. A, your di­all is made, and with a circle concentricke to the horizon, you may cut him off to any quantitie, as here you see the little circle, but hereof I must needs more aboundantly speake in the beginning of my 6. booke, and there­fore I referre you thither.

In the meane time to delight you with the singuler profite of these sphericall triangles, I will shew you by them to finde the quantitie in degrees of the horizon betweene euery of these intersections, and because euery quarter of an horizon diall hath like spaces, so that by one quarter the whole may be made, I wil for your ease and mine, beginne in one of the north quarters, though we commonly leaue them halfe out: where, for the 1. houre you may see there is the right angled spherical triangle EBD (for that DB lighteth perpendiculer on the arch EB by the third diff.) whose side DB is knowne to be 51 ⅔ deg. the poles eleuation, and the angle EDB must needes be 15. degrees by the 4. diff. because 15. degrees of thequinoctiall are taken to one houre, wherefore working by the seconde part of this last Chapter, hauing now one side and one angle giuen, you shall finde the side EB 11 ½ deg. Then for the second houre you haue the right angled triangle FBD, whose [Page 82] one side BD is knowne, & one angle vz. FDB, the quantitie of 2. houres which must needs be twise 15. that is 30. degrees, these had by this last Chapter, as before you shall finde the side BF 24 ⅙ degrees, then for the third houre you haue the right angled triangle GBD, whose angle GDB is thrise 15. vz. 45. degrees, by which and the side BD, you shall get the side GB 38. degrees, and so for the rest. Note that BD, vz. 51 ⅔ deg. taken out of BA vz. 90. leaueth DA 38 ⅓ degrees, the complement of BD, which added to AC vz. 90. al­so maketh 128 ⅓ the arch DAC, by helpe whereof I might haue begunne with one of the south quarters, as for the first houre you haue the right angled triangle DCQ. whose side DC is 128 ⅓ deg. and his angle CDQ 15. deg. knowne, as in the former way, and therefore by the 9. Chapter you shall get the side CQ. 11 ½ deg. euen equall to EB, for as I told you in euerie quarter, the match spaces are equall, that is to say, if they be of like distance from the nooneline, also commonly in the sphere Oppositorum eadem est ratio, and so I leaue you for this diall.

Now for the verticall Diall which we commonly call the south wall Diall. The circle LAK, being the verticall circle of theast crossing the meridian BAC, square in the zenith A, is in trueth the horizon of this wall, aboue whose flatte southwards the south pole is eleuated with our antipodes, but Northwards, the north pole D is eleuated, and that visiblie in our countrie according to the quantitie of the arch DA being the complement of BD, as I saide, which flat LAK the 24. hourelines doe vnequally deuide in LXVTS &c. but one halfe of the intersections are quite vnder our horizon, and therefore this Dial hath in our coun­trie visilbie but 12. houres: but to come to the matter, you haue for the first houre of the quarter AL, the right angled sphericall triangle DAR, whose side DA is 38 ⅓ degrees, and his angle ADR 15. degrees, where­fore the side RA cannot be vnknowne by the said second part of this last Chapter, which shalbe the quanti­tie of the first houre. Likewise for the second houre you haue the triangle DAS, his side DA knowne, and the angle ADS, being 30. deg. and so forth with all the rest.

Chapter 11. If the three sides of any sphericall triangle whatsoeuer shall be giuen to finde the quantitie of euerie of his three angles.Ioh. Reg. lib. 4. ca. 34. & lib. 5. cap. 3. & 4. Cop. 3. triang.

OF this matter Ioannes Regiomontanus maketh three long discourses in three seuerall Chapters, with verie tedious businesse by his Sines, Supplements &c. but my Iewel you shall see will make short worke. [Page 83] For if the three sides giuen bee of a right angled triangle, then I perswade my selfe that you are sufficientlie instructed by the 4. 6. and 7. Chapt. to finde all the angles, but yet whether the triangle giuen be right angled or not right angled, (for it is hard to know that by the bare sides giuen) you shall and may worke in manner following to come by the three angles.

Let there be proposed the sphericall triangle ABC, whose three sides are knowne vz. AB 25. degrees AC 36. and BC 30. but the angles AB and C, all vnknowne, which to know you shall reckon any one side (as for example) the side AB vz. 25. degrees on the Limbe from the pole any way, thereto lay the zenith point of the Reete fixed, then number one of the two sides remaining as BC vz. 30. among the almicantares, counted from the zenith point of the Reete, and the third side AC vz. 36. deg. among the parallels, counted from the pole, now where this 30. almicantare, and 36. parallel do crosse, there shall you find to crosse the 58 ½ meridian, and the 82 ½ azimuth both counted from the Limbe, whose portions lying betweene the pole and the zenith, if you marke them, doe liuely represent your triangle sought: for the portion of this 58 ½ meridian betweene this crossing and the pole be­ing as you may there see 36. degrees, numbred by helpe of the parallels, is the side AC: and the portion of the 82 ½ azimuth lying betweene this selfe same crossing, and the zenith being there 30. degrees, is the side BC, and the portion of the Limbe, betweene the zenith and the pole be­ing 25. degrees is the side AB, I say these three arches liuely represent vnto you the forme of the tryangle, whose sides were giuen, and that as expresly as on the Sphere it selfe. VVherefore if you aske counsell of my fourth diff. you will followe this 58 ½ meridian downe to the Equi­noctiall, where it sheweth you 58 ½ degrees counted from the Limbe, the angle subtended by the side BC, vz. the angle A. In like manner the 82 ½ azimuth followed to the Finitor, sheweth 82 ½ degrees, the angle subten­ded by AC, vz. the angle B. And yet doe we lacke the angle C, which will be the easier had, now the rest are knowne: wherefore place one of the sides not subtending the vnknowne angle C, videlicet, either the side AC, or BC betweene the zenith and the pole, as before you did the side AB, and heere for example AC: then among the meridians, reckon from the Limbe the quantitie of the angle subtended by the second side BC, vz. 58 ½ degrees vpon this 58 ½ meridian, reckon from the pole the third side subtending the angle you seeke, vz. AB, 25. degrees, and there shall you finde to cut the 44 ½ azimuth, which followed to the Fi­nitor line sheweth 44 ½ degrees, the quantitie of the angle C: and heere againe doe the three portions of the Limbe, the 58 ½ meridian, and the 44 ½ azimuth represent your triangle proposed.

In briefe: Of the three sides giuen, place the one betweene the pole and the zenith: of the other two rec­kon the one among the parallels counted from the pole, the other among the almicantares counted from the zenith: where these two crosse, there shall meete an azimuth and meridian, whose portions from their meeting to the pole & the zenith, with the portion of the Limbe lying betweene the pole and the zenith, shall represent your triangle, and the two distances betweene the Limbe, and the saide azimuth and meridian are two of the angles. For the third angle you are to shift the sides, that is to say, to place one of the other sides betweene the pole and the zenith, and then worke as before.

Chapter 12 The declination of the sunne giuen at any time togither with her altitude aboue the horizon, how by meane of the 11. Chapt. to get what houre and minute it is, and in what azimuth the sunne then is.

FOr example admit that the 12. day of Iune when as the sunne is in or verie neare the tropicke of Cancer being in this figure PN, and therefore his declinati­on 23 ½ deg. vz. ND, I had then taken his altitude 53 ½ deg. take the complement of his declination, which is 66 ½ and also of his altitude which is 36 ½ degrees: the first is the distance of the sunne from the pole, the se­conde from the zenith at the time of the altitude ta­ken videlicet AAA, and IAA, which with the arch AI, maketh vp the triangle AI, AA of the three knowne sides vz. AAA, 66 ½ IAA, 36 ½ and AI 38 ⅓, be­ing the complement of the latitude, wherefore by the eleuenth Chapter the angles shall be all knowne, of which the angle AI, AA is the distance of the sunne from the south so founde 30. degrees, that is two houres: wherefore if it were morning, then was it tenne [Page 84] of the clocke, if afternoone, two of the clocke. Also AI, AA, must needes bee the distance of the azimuth wherein the sunne then was from the North, found by the 11. 130. deg. if out of that you take 90. there re­steth 40. deg. vz. o. BB, the distance of the same azimuth, from the East or west, and that towards the south. I am the briefer because I meane onely to giue you a tast of the vse of these triangles, knowing this al­so to be verie easie to conceiue, to them that vnderstand the 4. bookes past euen by the bare inspection of this figure.

Ioh. Regiom. lib. 4. cap. 28. Copern. 11. Chapter 13. If of aspherical triangle two sides containing any one knowne angle shalbe giuen, how to finde the other side and angles.

LEt there be proposed the sphericall triangle ABC, whose angle A, let it be knowne 58 ½ degrees togi­ther with the two sides, containing it vz. AC 36. degrees, and AB 25.

Reckon the quantitie of the one side giuen, as of AB, 25. deg. on the Limbe of your Iewel, thereto lay the Reetes zenith fixed: then on the meridian distant from the Limbe, as much as the angle A proposed commeth to videl. on the 58 ½ meridian, reckon from the pole the quantitie of the other side proposed, videl. of AC 36. degrees, and looke what azimuth crosseth the same 58 ½ meridian in the saide 36. deg. the same followed to the Finitor, sheweth the degrees of one of the angles sought for, videlicet of the angle B subtended by the side 36. which you shall finde 82 ½ degrees counted from the Limbe: then are the degrees of the same 82 ½ azimuth betweene the saide crossing and the zenith, the quantitie of the third side sought for, which you shall finde to be 30. deg. Thus doe you see according to your accustomed manner your whole triangle in his forme, his three sides, & two angles knowne, wherefore by the 11. Chapter the third angle is easily had: in the last part of which Chapter, this Chapter was performed ere I was ware, such is the dexteritie of my Iewel, neither woulde I haue repeted it here againe, had it not beene to imitate the course of Regiomontanus and Coper­nicus.

In briefe, lay the one side giuen betweene the pole and the zenith, the other reckon on the meridian as much distant from the Limb, as the angle giuen commeth to, there shal meete you the azimuth making vp your tri­angle.

Chapter 14. How by helpe of the 13. Chapt. to know the height of the sunne at any houre, and minute proposed in the whole yeare, and also in what azimuth he is.

IN my thirde booke 31. Chapter is shewed the same by ordinary vse of the Iewel, and that most exact and redily without any trouble, yet shall it not be amisse to know to do the same by the Art of sphericall triangles, which performeth it as reddily as Gem. Frisius did on his Ca­tholicon, though nothing so easie as my Iewel doth it in the same 31. Chapt.

For example the 12. of Iune, the sun then being in the 1. degree of ♋ his declination there 23 ½ deg. I woulde know of what height, and in what azimuth the sun shall then be. The angle made betweene the 10. of the clocke houreline, and the meridian being in this figure, the angle AA, AI, must needs be 30. deg. for that euery houre is 15. degrees, then taking the sunnes declination for that day vz. DN, or rather CCAA 23 ½ deg. out of 90. deg. videl. CCA, there resteth the arch of the 10. of clocke houre-line betweene the tropicke or sunnes declination, and the pole vz. AA, A being 66 ½ degr. Then the portion of the meridian betweene the pole and the zenith, being alwayes the complement of the latit. vz. AI, is wel knowne 38 ⅓ deg. Here then haue you a sphericall triangle AA, AI, whose two sides AI, and AAA, are knowne, in­cluding the knowne angle AA, AI vz. 30. deg. wherefore by the 13. Chapt. the 3. side IAA shalbe knowne 36 ½ deg. and the angles AI, AA & AA, AI, shal also be knowne: Now then subduct the said third side IAA. vz. 36 ½ deg. being then the distance of the sunne from the zenith out of IBB vz. 90. deg. so resteth the arch AA, BB 53 ½ deg. the altitude of the sunne sought for, and the angle AI, AA, sheweth the azimuth of the sun in the very same manner, as in the 12. Chapt. hath beene said.

Chapter 15. Ioannes Regiomontanus, Lib. 1. Cap. 51. and lib. 4. cap. 29. proposeth and sayth that the knowing of two sides of a triangle not right angled, with one angle subtended by one of those sides, cannot serue to get the other side and angles.

HEere is a wonderfull case, that by their whole art of spearicall triangles, with their signes, coardes, arches complements, proportions, calculations, tables, and such like, they cannot doe that which my poore Ie­wel will doe immediatly without any difficultie. Nay all the sporte is, Regiomontanus spendeth muche speache in the saide 51. and 29. chapter, to prooue that it is vnpossible to bee done, without letting downe a perpendiculer, respecting or subtending the angle proposed, which in the thirtieth chapter following, hee teacheth to doe. But we will doe it, add that without any perpendiculers, a little readier then he doth it.

The figure of the 13. Chap. serueth for this place.

Admit that of the triangle ABC, proposed in the thirteenth chapter, the two sides AB, and AC, wer knowen as there they were, together with one of the angles B or C, subtended by one of the sides giuen: for example, the angle B 82 ½. degrees, subtended by AC, & that the angles A and C were vnknowen, with the side BC: Is it not possible thinke you by these three giuen, to reache to rest, I hope it be, els I loose my labour.

Reckō on the limbe of the Iewell frō the pole y^e side giuen, not subtendyng the angle giuen, vz. y^e side AB, being 25. degrees, thereto lay the zeneth point of the reete fixed, reckon also on the Finitor from the Limbe in­wards, the angle B giuē, vz. 82 ½. de. there haue you the 82 ½. azimuth, then reckon the other side giuen, vz. AC. 36. degrees among the parallels, counting from the pole, mark where this 36. parallel, and the said 82-½. azimuth doe crosse. For the portion of the meridian cutting there, counted too the pole, is the thirde side, which you shall finde 30. degrees, of the 58 ½. meridian counted from the limbe. The same 58 ½. meridian fo­lowed also to the equinoct. sheweth 58 ½. deg. one of the angles sought for, vz. A. The third angle C cannot be vnknowen by the 11. or 13. Cha. Thus you see this vnpossble matter now possible, then iudge whether my Iewel bee a Iewell or no.

Yet thus much will I foretell you, least happily you shoulde be amazed at any time. In some triangles you shall finde that the azimuth representing the angle giuen, will crosse the parallel that worketh this feate in two places, and then shall you not tell which place of crossing to take, yet commonly the crossing farthest within the Limbe is hit: in all the triangles that I haue had experience of, but you may trie by shifting of the sides, as you are wont to doe in finding the thirde angle which is it.

Chapter. 16. The azimuth of the Sunne giuen or taken at any time by Instrument, with his declination, how by helpe of the fifteenth Chapter, to knowe what height the Sunne is of, and what it is a clocke.

IT is taught in the 12-chapter, spherically to get the houre & azimuth of the Sunne by his altitude, contra­riwise, heere by the azimuth shall bee hadde the houre and the altitude.

The figure of the 5. chap. serueth for this place.

For example, the 12. of Iune, admit that before noone I had takē by some Topographicall Instrumēt, or otherwise the azimuth or horrizontall distance of the Sunne 40. degrees from the East towardes the South, which in this figure is OBB, thereto I adde 90. degrees, videlicet, EO: so haue I the arche EBB 130. deg. the declination of the Sunne then is 23 ½. degrees, by the 3. booke and 4. chapter. Nowe it is most mani­fest, and apparantly to bee seene on the Iewel, that any parallell of the spheare doth cut off the meridians, or houre lines, all of like equall distance from the pole, as heere you shall see the arches of the houre lines AX. AZ. AAA, AN, are all complements of the Sunnes declination: each of them, and namely AAA, 66 ½. beeing 66 ½. degrees, the arche betweene the pole and the Zenith, vz. AI, is certaine 38 ⅓. degrees. Here haue you nowe a sphericall triangle AI, AA, whose two sides AI, and AAA, are knowen with the angle AI, AA, which angle is subtended by A, AA, one of the knowen sides, euen the like too that whiche the fifteene chapter proposeth. The third I, AA, and the angles AA, AI, and A, AAI, are vnknowen. VVherfore if the worlde were so harde, and the matter so vnpossible, as Regiomentanus proposeth it, we shoulde bee neuer the neere of our purpose: but working by the 15. chapter, notwithstanding his proofes too the contrary, you shall finde the angle IA, AA, 39. degrees, the Sunnes houre distaunce from the South, which must needes be two houres euen ten of the clocke: also you shall finde the thirde side, I, AA. 36 ½, degrees, whiche taken out of 90. vz. I, BB, leaueth the arch AABB, 53 ½. degrees, the height of the Sun at that instant. As for the thirde angle, you shall finde him 32. degrees, but wee haue no vse of him here.

Ioh. Regio. lib. 4. cap. 31. & 32. Copern. 12. Chapter 17. If of any sphearical triangle, not right angled, two angles shalbe giuen with the side lying betweene those angles or els the side subtending one of them, how to get the other two sides and thirde angle.

IF the side lying betweene those angles shall bee giuen, it is most easie, for placing that side on the limbe be­tweene the pole and the Zenith as you are wont, & reckoning the one angle giuen among the azimuthes, the other among the meridians, where these crosse, ther is your triangle made vp, in your wonted maner. For example, admit that of the triāgle ABC, the two angles A, 58 ½. deg. & B 82 ½. degrees were giuē with the side AB, 25. deg. lying betweene them: reckon AB, vz. 25. deg. on the Limbe from the pole, thereto lay the Zenith of the Reete fixed: that done, reckon the one angle, vz. A 58 ½. deg. amongst the meridianes, the other angle vz. B 44 ½. deg. amongst the Azimuthes both from the Limbe, and where this 58 ½. meridian, and 44 ½. azimuth, do crosse, there shall you finde your triangle made vp after your accustomed maner, for on the 58-½. meridian between the crossing, and the pole, you shal find 20. deg. for the side AB, & on the 44 ½. azi­muth to the Zenith 30, deg. for the side BC, these had, the third angle is to be had by the 11. or 13.

The figure of the 11. Chap. serueth for this place.

But admit, that one of the sides AB, or BC, subtending one of the angles A or C, had beene giuen with the saide angles A and C: as for example, the side BC, subtending the angle A. Heere reckon the one angle giuen, vz. 58 ½. de. subtended by the side giuē as before among the meridians, the other, vz. 44 ½. deg. among the azimuthes, that done, number on the said 44 ½. azimuth counting from the Zenith, the side BC, giuen, vid. 30. deg. and turne the reet til the same 30. deg. of the said 44 ¼. azimuth do cut the said 58 ½ meridiane, and there is your triangle made vp as before for you shall finde the Zenith too cut in the limbe 36. deg. from the pole, as before he did, and the very same crossing as before.

Chapter 18. Howe by the helpe of the 17. chap. to make any declining diall, that is to say, a dial to any wall shouldring or bending from the ful beholding of the South.

ALl walles are commonly made perpendiculer to the horrizon wheron they stand, and point directly to the Zenith. VVherefore of necessitie they must lie euen in the flat of some one of the azimuthes, and therefore euery one, some one horrizon in the worlde by the preambles to my 6. booke, where this mat­ter is handled more at large.

But now to the conclusion of this chapter, where as I taught you in the 10. chap. sphearically too make a diall to the North flat of the wall, lying directlie East and VVest, which wee call a North or South wall, vz.

KL, according to the plage that hee beholdeth. Heere in like maner wil I shew to make a diall to the North flat of a wall, declining from the former Eastwards, howe much soeuer, and for ezample, 40. degrees repre­presented in this figure by the azimuth YZ, where by the angle KA, Y, shalbe 40. degrees, and as much is the [Page 87] angle ZAL, by the 5. Theoreme, further the angle KAY, vz. 40. taken out of KAC, videlicet, 90. leaueth his complement the angle YAC, 50. degrees, to the which the angle BAZ, must needes bee equall by the said 5. Theoreme. This circle ZAY, since it representeth here an oblique horrizon, therfore the 24. meridi­ans, or rather houre lines issuing from the pole D, shall cut the same, ZAY, into 24. vnequall partes, where you see 1. 2. 3. 4. &c seruing for the houres euen in the very like maner, as to the horrizon KAL, they did in the tenth chapter, and how much euery of these houre spaces containeth: I will nowe also teache to finde by helpe of the 17. chap. The first houre line from our meridian VVestwardes towardes L, videlicet, DQ doth cut this horrizon: ZAY, in the point I. VVherefore the quantitie of the arche, IA, shalbe the degrees of the first houre from our meridians VVestwardes, which in this diall is 11. of clock, as by the 6. booke will more at large appeare. Heere haue you nowe a sphearicall triangle DAI, whose side DA, is knowen, as hath often been saide to bee 38 ⅓. degrees. The angle ADI, beeing the quantitie of one houres space, must needes bee 15. deg. the angle DAI, is as before is saide knowen 50. deg. these had, videlicet, the angles A, DI and D, AI, with the side AD, lying betweene them: you shall most easily by the first part of the 17. chapter finde the side IA, to bee 12 ½. degrees, so much is the space of the first houre VVestwardes of the plum line in this diall, then the second houre line, vz. DP, hee cutteth the horrizon ZAY, at the note 2, there haue you another triangle DA, 2. whose angle AD, 2. is knowen to be 30. because it is 2. houres space, & the an­gle DA, 2, 50. as before with the side DA, lying betweene them, vz. 37 ⅓. VVherefore by the 17. you easilye obtaine the arche D, 2. being the distance of the seconde houre line of your diall, westwards of the plumme line. For the third houre you haue the triangle DA, 3. and so cōsequently for all the rest on that side of the meridian BC, where still the side AD, and the angle DAZ, are common too them all. But nowe for the houres, one the East side betweene A and Y, the angle BAY, must bee the common angle, whiche is easily had by adding the angle KAY, videlicet, 40. to bee AK, vz. 90. thereof commeth the angle DAY. 130. deg. so that for the first houre on the East side you haue the triangle DA 11. whose side DA, & the angles DA, 11. and AD, 11. are knowen: thereby get the quantitie of the side A, 11. for your first houres space, & so for the rest.

Heere note that as this figure sheweth the maner of the cutting of the houre circles, to the North side or flat of this wall ZAY, issuing from the North pole D, so if you would haue made the diall to the South flat of ZAY, you must haue placed the South pole in your figure 38 ⅓. deg. Southwardes in the line AC, vid. in AA, from whence the houre lines issuyng will cut the South flat of ZAY, in his right forme: but truly into the very selfe same diuisions as before: yet in contrary quarters, for then the diuisions, which are now in the quarter AZ, will be there in the quarter AY, & è conuerso: then also is the point A, no more the zenith but his Nadir. and AAA, the arch between the Nadir, and the South pole. But what needs all this, when as one mould turned vpside down, or in & out, will serue to diall both the flats, and two more of their cousins as in my 6. booke and 20. chap. is shewed, in which booke very many more things necessary to dials are shewed which to bring in here, wold bring me out of my sphericall triangles, & that must not be. But thus much as­sure your self, you hauing read & well vnderstood my 6. booke of diallyng, you shall euer after by the pre­cepts of this, the 10. & 22. chap. make dials to any wall or flat whatsoeuer, for those three include the whole art. Much more, and many wayes I could write, but for breuities sake, which I wholy intende, these shal suf­fice.

Chapter 19. Ioh. Regio. Lib. 4. Cap. 8. & 30.How to let downe a perpendiculer arche from any angle of a knowen sphea­ricall triangle vnto his base or subtending side, and to know the quan­titie of the same perpendiculer, and of the parts whereinto hee deuideth the base.

LET there be proposed the triangle ABC, of the 17. chapter, of knowen sides and angles, from whose angle C, I woulde let fall a perpendiculer arch to his base, or subtending side AB. First you know that euery perpendiculer arche must haue his originall from on of the poles of the side or circle, whereon it must light by the 3. diff. wherefore you shal alwayes place that side. vz. AB. 25. degrees on the Limbe, betweene the pole and the zenith as you are wont, then is the center of your Iewell the pole or poles of the circle, wherein the same side AB, consisteth by the 2. diff.

Therefore make vp the other two sides of your triangle as you are accustomed, either as in the eleuenth chapter, by the crossing of the almicantare and parallel, e­quall to the other two sides, or els of the meridian and azimuth, equall too the other two angles, as in the 17. at whiche crossing, the angle C, must needs be included. VVell, then vpon the crossing of the 82 ½. azi­muth, and the 58 ½. meridian (if you worke by the 17.) of the saide al­micantare and parallel: if by the 11. laie the fiduciall or center line of the labell, which shall there deuide the 25. lying on the limbe between the pole & y^e zenith into 3 ½. degrees, vid. the arche BD, and 21 ½. degrees, the arch DA, the partes of the base sought for. And the degrees of the labell bee­tweene this crossing, and the Limbe shall bee 29 ⅔. degrees, the perpendiculer sought for, vz. CD.

Chapter 20. To doe the same after another manner of working, yea though one of the other two angles, and one of the other two sides, of the triangle were vnknowen.

ADmit, that frō the angle C, of the triangle ABC of this last chapter, I woulde know the perpendicu­ler, and the partes he diuideth the base into, by another maner of working: to doe this, you shall rec­kon one of the other two angles B or A. For example A. 58 ½. deg. on the limbe from the Equinoctiall vpwardes, thereto lay the rule fixed, whereon you shall from the center reckon the side, which with the base includeth this angle A, vz. AC, 36. deg. & there shall crosse you the 21 ½. meridian counted from the axtree, shewing on the Equinoctiall line 21 ½. degrees, the portion of the base cut of the next angle, A, videli­cet. AD: and the portion of the same 21 ½. meridian betweene the crossing of the rule and the Fquinōctial, you shall finde, 29 ⅔. deg. being the perpendiculer CD. The other portion of the base, is had by subducting this 21 ½. out of the whole base giuen, videlicet, AB, 25. deg. and all this is done without knowing the an­gle B, or the side BC. This working differeth nothing from the 9 chapter, if you mark it: but only it bringeth you acquainted with perpendiculers: you might also haue done it in this maner by the 19. chapter, or partly by the 7. the skill is not great, so you vnderstand well whereabout you goe.

Chapter 21. If two great circles of the spheare crosse one another, making a knowen angle howe to get the perpendiculer arche, falling from any poynt assigned of the one circle vnto the circumference of the other, together with the arche of the same circle, betweene the perpendiculer, and the crossing.

LET there bee proposed the two great circles AB, and CD, cros­sing one another at E, making the knowen angle, DEB, 30. deg. Admit that from the point F, of the circle CD, beeing 20. deg. distaunt from the crossing E, I woulde learne the perpendi­culer arche falling on AB, videlicet, FG, together with the space EG. This matter differeth nothing from the 20. cha. For if on the rule layed at the Limbe, to 30. deg. being the angle at the crossing E, you count the distance EF. vid. 20. deg. frō the center, ther shal meet you the 17 ⅓. meridian whose portion frō the rule to the Equinoct. you shall finde 9 ¾. deg. your perpendiculer FG, and the saide 17 ⅓. is the arche EG. Or thus after the 19. chap. if on the 20. deg. of the 30. meridian counted from the pole, you lay the rule, his part there cut of next the limb, you shall finde 9 ¾. and on the limbe betweene the rule and the 17 ⅓. as before.

Chapter 22. How by helpe of the 21. chapter to knowe both the height of the cocke, to any decli­ning dial, and how much he must be placed byas from the perpendiculer which I call his deflection.

THese rules are general in all vpright walles, the plum line is our meridian, the line where the cock standeth, is the meridian line of the horrison represented by that wail. The meridian circle of euery horri­son, standeth perpendiculer thereon, by the 1. booke, & 4. chap. Lastly the eleuation of the pole aboue any horrizon, which is alwayes the height of the cocke, is the arche of the meridian, betweene the pole and the same horrizon, which arch must needes be perpendiculer to the horrizon, aswell as the whole meridian is. Then to conclude in the figure of the 18. chap. you haue the two great circles BAC, and ZAY, crossing at A, making the knowen angle BAZ. 50. deg. then if from the point or pole D. of the circle BAC. being as in the 18. chap. is shewed 38 ⅓. deg. from the crossing at A, you can deuise to know the perpendiculer arch falling to the horrizon ZAY, vz. D. +, which by y^e 21 you shall find 28. de. the same is y^e height of the cock to that diall, and the space + a, is the quantitie of the angle of deflection, vz. 27. by the 21. chap. It were great folly for me to stand longer of this matter, since the selfe same is done in the 6. booke, 16. 17. & 18. chapters, meaning here but a little to touch the vse of the sphearicall triangles for the encouragement of the young Reader.

Chapter 23. In a right angled sphearicall triangle where one of the angles shal chaunce too bee but certaine min. as it often hapneth, how to get the side subtending that angle presisely notwithstāding.

FOr example, admit that of the right angled triangle ABC, the angle A, were but one deg. 10. minutes, & the side AB, subtending the right angle C were knowen to be 15. degrees, you might now by the 9. [Page 89]

chapter, get the side BC, yet without any exact certaintie, because the angle A is so lit­tle, and that the side BC, wyll fall out to bee but certaine minutes, whiche except your Iewell be monstrous cannot be discerned any thing neere. To helpe this you shall insteed of one degree tenne minutes, whiche are 70. minutes in the whole, reckon on the Limbe of your Iewel from the Equinoctiall vpwardes 7. degrees, and thereto lay the rule as in the nienth chapter, for so doth euery degree of the limb represent 10. minutes. In this case then, as in the 9. reckon on the rule from the center, the side AB, vz. 15, degrees, and there shall you finde the 14 6/7. meridian to cut, whose degrees betweene the crossng of the rule, and the Equinoctiall are 2 ⅕. degrees, the side BE, after the 9. chapter. But here you must allowe each degree, but 10. minutes, as you did at the limbe, so that this 2 ⅕. deg. represent but 22. minutes, so much is the side BC. And in this working you must in no wise exceede 10 or 15. deg. in the Limbe at the most, yea and that where the angle proposed exceedeth not two degrees, for in these small or minute angles, as I may terme them, there is kept a Geometricall propor­tion betwene the parts of the triangle, whiche in greater, the nature of the spheare alloweth not: neither yet in these, but that the error is vnsensible, this helpe did Gemma Frisius inuent, as in my 3. booke 56. chapter appeareth, from whom I woulde be loth to derogate: but in my next edition I will supply it farre better, & more exact, by another deuise euen vnto 30. degrees, which heere you can scant extend to three degrees.

Ioh. Regio. lib. 4. ca. 26. Coperni. 5. triang. sphe. Chapter 24. If the three angles of a right angled sphearicall triangle shall bee giuen, howe to get the quantitie of the three sides vnknowen.

THis proposition had I once half giuē ouer as vnpossible on my Iewell, which made me greatly to mislike him: but in respect hee made mee amendes in perfourming another that Regiomont. left impossible, as by the fifteen chapter appeareth, I thought it a lesse fault, yet still wheresoeuer I went or sate alone, I coulde not choose but be deuising vppon it: and at last while I was abroad, in my eueninges walke, whiche I commonly vsed, it came into my head on the sodaine, but then was it no neede too bidde me trudge a pace home to my Iewel to see it in experience. All the skill thereof is too make the right angle too lye among the crossinges of the meridianes and azimuthes within the limbe in maner as the one angle of all triangles, not right angled doth, so that when the constitution of the triangle is founde on the Ie­well his side, which lyeth on the Limbe betweene the Zenith, and the pole may be the side subtending the right angle giuen: which you shall thus do.

Let there be pro­posed the right angled sphearicall triangle ABC, whose three angles are giuen vz. C. 69. degrees, B 55. and A 90. I reckon one of the angles not right, vz. 69. among the azimuthes of my Iewell, and the other vz. 55. among the meridianes alwayes counted from the limbe: this done, all the misterie of this matter is, how in turning a­bout the Reete to know when this 69. azimuth, and 55. meridian doe crosse another square, which can neuer be by the 3. def & 3. theoreme except they cut one anothers poles, which poles I may easily know where to finde by the 12. theoreme. By which I say that the poles of euerie meridian of my Iewell are to be found in the Equinocti­all line, and not els where. For as it is apparaunt by the third def. and the third theoreme, that the pole or poles of the vttermost meridian, beeing the innermost cir­cle of the limbe, lyeth in the Equinoctiall line, in the center of the Iewel: So looke how muche euery other meridian lyeth within the limbe on the one hand, euen as much doth one of his poles lye distaunte from the center on the other hande. As for ex­ample, the pole of the fifth meridian, next the limbe on the South part, is the 5. degree of the Equinoctial line from the center towardes the North parte, and so of all the rest, as in my fourth book and fourth chap­ter doth also appeare, yet thus much more plainely chose out any meridian whose pole you require, then reckon from him on the Equinoctiall line towardes the center, and beyonde till you come to 90. degrees, and there shall bee his pole: as for example, if from the fift meridian on the South part next the Limbe, you count on the Equino­ctiall Northwardes 90. degrees, the same shal end at 5. de. beyond y^e center, this matter now I hope is plain enough for you to gather that in the very like maner the poles of all the azimuthes are to bee founde in the Finitor, and not els where, wherefore to goe forwards with our triangle you shall by the rules heere last mentioned, reckon on the Finitor from the 69.

Azimuth, whiche as before was ap­pointed too containe the angle C, tyl you come too 90. degrees, and there shall bee his pole: wherfore if you turn about the reete tyll the same 90. degree of the Finitor doe touche the 55. meridian appointed before to contain the angle B, then wheresoeuer you shall finde the same 69. azimuth, and 55. meridian to crosse, there must needes be foure right angles by the thirde theoreme, and therefore [Page 90] no doubt but at the same crossing is made vp your triangle desired in his right constitution after the won­ted maner, as in very deede it is: for from this crossing to the Zenith point of the Reete, you shall find on the 69. azimuth 51 ⅓. deg. for the side AC, and thence to the pole on the 55. meridian 63 ¾. deg. the side AB, & on the limb between the pole and the Zenith 74 ⅙. degrees, the side BC, subtending the right angle A. And now that you haue all, doe, if you lift to followe the 69. azimuth vnto the Equinoctiall line, hee wyll there cutte the 55. deg counted from the center being in truth the pole of the saide 55. meridian by the for­said computation, which is a great confirmation of this matter, els coulde not there haue been a right angle at the said crossing by the third theoreme, if each did not cut others poles. Thus haue you a most pleasaunt and profitable handling of the chapter, yeelding great light to many questions about the same sphear, which I thought good, the better to manifest by this figure.

• BCH, the limbe of your Iewell, K the center.
• B, the North pole of the Iewel: NO, the Equinoctiall line.
• C, the Zenith of the reete placed, ML, the Finitor.
• CD, the 69. azimuth.
• E, the pole of the 69. azmuth, KE, being 69. degrees, or DE, 90.
• BF, the 55. meridian, G, his pole, GK, beeing 55. degrees equall to FO, or FG, being 90.
• P, the point of the 55. meridian, where E, the pole of the 69. azimuth doth touch in turning the reete.
• A, the point of their crossing making vp the triangle ABC, desired.

In briefe, of the two angles not right, reckon the one amongest the azimuthes, the other among the me­ridianes both counted from the Limbe, then turne about the reete vntyll the 90. degree of the Finitor counted towardes the center, and beyonde from the same azimuth doe touch the same meridian, and then whereso­euer you finde that azimuth and meridian, to crosse, there is your triangle made vp in the wonted maner.

Chapter 25. If by the crosse staffe or other Instrument, you shall take the true distance of any planet, co­met, or starre vnknowen from twoo knowen starres, how by helpe of this my woorke of sphericall triangles, to get the longitude and latitude in the zodiacke of the same planet, comet, or starre, and also his declination and right as­cension: and what you had need to do in al sphearical questions.

I Haue hytherto for the pleasure & delight of the learner, set downe diuers Astronomicall questions, and conclusions to be resolued by these triangles in sundrie chapters before: least otherwise, if I should haue gi­uen the bare preceptes, as Regiomontanus and others haue done, this work beyng of all the rest most pro­fitble might happily haue been neglected, & thought on the first blush of small account: knowing that no­thing draweth on mens minds more to any studie thē that they may at the first entrance perceiue their rules to serue to good purpose. And as I haue hitherto touched a fewe sleight questions: so might I haue witten a number, and therewith filled great volumes, the matter beeing in manner endles, were it not that I both hast the end of my trauaile too too long alreadie looked for of diuers of my familiars, and presume that by those, any man may performe all other that be of no greater waight. But now there are in like maner an in­finite number of intricate questions more harder far then any yet propounded, which by Regiomont. Co­pernicus, and others doctrine, grow to great toile with their Synes, calculations, & proportions, wherin first they hunt about for one Syne, which they call inuentum primum, then for another, which they call inuentum se­cundum, & cōmonly inuentum tertium, & perhaps quartum, & so foorth. After all these are found, then they multi­plie & deuide them, & compare their proportions: & when al is done, that they haue founde the Syne sought for: they yet are faine to goe to their tables for the arch correspondent. Those inuenta are not ouer easilye done by my Iewel: but yet an hundred fold easier then as they performe them: notwithstanding you muste haue all your wits about you, & after a sort seek a primum, secundum, & tertium inuentum, as they do, though with no trouble at all: when you haue once the conceite of it in your head. For in all these questions, it is more hard to conceiue how they should be resolued then to resolue them. VVherefore the best way is for a­ny man that hath not well setled the circumstance of the spheare in his head: first by the helpe of his rule & compasse, or els grosly by ame to drawe an Idea, or shape of such circles as hee shall haue to deale withall in his purpose, as also before in my other slight questions I haue done: which done, hee shal by inspection therof very readily spie out which angles or sides be knowen, & thereby get y^e rest, or els see how to adde, inuent, or cast out other circles, arches, & triangles, which may help him to his intention, euen as the lande meater doth commonly resolue a cragged deformed ground into sundry squares & triangles, too compasse the contentes thereof, which to do, I hope by this one example following you will be sufficiently instructed for all.

The 14. day of October 1582. about 6. a clock in the morning, & 30. min. past, seeing the morning cleer & ♂, ♁, & the Moone all 3. aboue our horrizon, according to my accustomed maner, I could not choose but haue a cast at them for my recreation, wherefore with a large crosse staffe that I had made after my fashion, as in my next edition I will shew, I tooke the distance between Mars & the great Star in the right shoulder of Orion, & found it 31. deg. 37 ½. min. Likewise the same instant I took the distance between ♂, & y^e Nor­thermost of the 2. stars of ♊, called Apollo, & foūd that 10, de. 25. mi. ♂ being orient. of thē both. Hereof I thē made a note in my book where at this time I mean it shal serue for an exāple to this chap. and by this di­stance of ♂, taken from those two knowen starres, I will shewe you by sphearicall triangles to get in what longitude, latitude, declination, and right ascēs. hee then was, by a knowen starre, I meane a star whose lon­gitude and latitude is knowen eyther by tables or your Astrolabe. But now the better to conceaue howe to [Page 91] bring this matter to passe, my best way is first to make a slight pattern of the thing I pretende in this maner: I make a circle at aduenture, vz. AB, CD, whose diametre AB, shall represent the Ecliptike, the point C, beeing his North pole, and D the South: this done, I goe to Stadius Tables, and there I finde the longitude of the said star of Orion, 23. deg. 0. min. in ♊, his latitude, 17. deg. 0. min. South of the Ecliptick, and the lon­gitude of Apollo 14. deg. 20. min. in ♋, his latitude 9. degrees 40. min. Northwards: this knowen, I appoint the arche CAD, to bee the circle of Orions longitude: that is to say, the great circle of the spheare, cutting the 23 0. of ♊ issuing from the Ecliptickes poles, then taking 23. 0 of ♊ out of 14.20. in ♋. there resteth 21, deg. 20. min. for whiche I ame out a portion of the Eclipticke AB, containing by estimation 21 ⅓. such partes as AF, containeth 90. vz. AE, and then opening my cōpasse I draw the arch or semicircle CED, which is the circle of Apolloes longitude, vz. of the 14. 20 of ♋.

After this from A towardes D, I ame 17. suche partes as AE, containethe 21 ⅓. or AF, 90. and there I set M, or make a starre, for there is the place of Apollo, according to his lon­gitude and latitude, then in the arch CED, frō E towards the North pole D, I set by ame 9 ⅔. of such like partes being the latitude of Orion, vz. in N: and so are these knowen starres placed: which done, I open my compasse as neere as I can gesse to 31. degrees 37 ½. min. of such like parts, vz. the distance taken by the crosse staffe betweene Orion and ♂, and set the one foote in Orion, vz. M and with the other make a blinde arche cutting the Eclipticke AB, towardes the center: that is to say, with the sequele of the signes, for that ♂ was orientall. Againe I take 10 25/60. lyke partes by gesse, also in my cōpasse: that is to say, the distance taken be­twene Apollo & ♂, & setting the one foot in Apollo, vz. N, with the other I make a little blinde arche crossing the first towards the center F also, because ♂ was orien­tall of them both, vz. in O, where must needes haue been the place of ♂, if my figure had beene truly made which for this purpose needeth not but euen slight for a shew. Then with my compasse at the width of AF at the least, I draw the arches MN, MO, and NO, but with no great curiositie: lastly I drawe the arche COD, which is the circle of the longitude of ♂, and then is my platforme laid, whereupon I descant in this manner.

First to begin with CA, the same is 90. de. and AM, 17. the latitude of Orion: therefore CAM is 107. also CE is 90. degrees by the second def. and NE, but 9. degrees 40. min, the latitude of Apollo: therefore taking NE, out of CE, there resteth CN, 79. deg. 20. minutes. Also the angle MCN, being the difference of Orion and Apolloes longitude is knowen to bee 21. degrees 20. minutes by the 4. def. Therefore nowe hauing two sides of the sphearicall triangle MNC, vz. MC, 107. and NC, 79. 20/60. encluding the knowen angle MCN, 21 20/60. degrees, you shal easily by the 13. chap. find both the 3. side MN, to be 34 ⅔. deg. and also the angle CMN, to be 39 ⅓. Then haue you the triangle MNO, of all three sides knowen. vz. MN. 34 ⅔. NO, 10 25/60. and OM 31. degrees. 37 ½. min. VVherefore by the the 11. chapter you shall easily finde the angle NMO to be 17 ½. deg. which added to the angle CMN, vz. 39 ⅓. maketh 57. almost the quantitie of the angle CMO, and yet you will say, what am I the neere for all this. Soft a while, not too fast for falling: nowe are you past your Inuentum primum & secundum, which in Synicall workyng, perhaps would haue myred you twise ere this, & yet you scant half way ouer. VVel then to the inuentum tertium, you haue a third triangle vz. CMO, whose two sides CM, 107. and MO, 31. deg. 37 ½. min. are knowen, containing the knowen angle CMO, 57. degrees: wherefore by the 13. chapter you shall easily finde the side CO, 88 ¾. degrees, which takē out of CP, vz. 90. leueth OP. 1 ¼. degrees, the latitude of ♂. You shall likewise by the 13. find the quantitie of the angle MCO, 26. degrees, which is the difference of longitude of ♂ from the said great star of Orion, and in reckoning forwardes according to the sequele of the signes: that is say, from the 23. de. of ♊ being the longitude of Orion 26. degrees forwardes on the Eclipticke, falleth out too bee the 19, degrees of ♋, the longitude of ♂ desired: and so is this busie matter at the last brought too good passe by the sin­guler vse & help of sphearicall triangles, & thus shal you be able to performe any other matter of the sphere whatsoeuer, for you shall not lightly finde any question so harde, but if you once drawe your platfourme you shall deuise how to adde some arches or angles, yeelding sufficient helpe too accomplish your pretence, and many times after you are a little exercised, you shall set your platfourme more expresly by the lineamēts of your Iewel, then with your pen, Vsus promptos facit, foemina ludificantur viros: but tarry, I had almost forgot the de­clination and right ascension of ♂, for which you had neede make another platfourme, as heere you see ABCD, is the great circle or colure cutting the ecliptikes poles GH, and the poles of the worlde AB, CD: the Equinoctiall KL, the ecliptick RS, the circle of Mars his latitude, and GOPH, of his longitude: GA. is the distāce of the clip. poles frō y^e poles of y^e world vz. 23 ½. deg. AOB the circle of ♂, right ascension: TO of his declination. Now haue you the sphearical triangle GAO, whose side GA, 23 ½. and GO, 79 ⅔. are knowen containing the angle AGO, being the difference of longitude of ♂ from the beginning of ♋, vz. 19. deg. wherefore by the 13. you may easily get the side AO, and the angle GAO, which had, take AO, out of AX, being 90. leaueth OX, the declination of ♂, then take the angle GAO, out of CAD, vz. 180. it leaueth the angle DAX, the difference of the right ascension between ♂, and 0. in ♋, so that the right ascension of ♋, being alwayes 90. of Mars, it was more at that time by the arch DX.

Chapter 1. Of certaine prenmbles to bee noted in vnderstanding this worke following.

HOw profitable, how pleasant, & how necessary dials are, there is none so simple a soule but seeth it, neither any I thinke so sottish to say against it, since no order could be kept in things, if distinctiō of times were not. For what is the cōseruation, grace, & vpholding of all good ordinances, but euery thing to be done in his due houres. The trauailer by land is not a little cōforted, whē he may with a small toy see how the day goeth away before him: and also may see to what place he may reach before nyght o­uertake him. The Seamā without dials, or such like instrumēts to know their houres and times, coulde not trauaile at all. Chirste himselfe made often mention of houres and times. Therefore since their commodities are so manifest, no doubt but the inuentours were much to be accounted of, and the inuention highly to bee esteemed. Some will say what then? VVhy should wee nowe bee troubled with more bookes thereof, since so many are alreadie extant, a man may haue too much of his mothers bles­sing. But yet sir, if you will follow mee, I may happily leade you a pleasant path, where others are stumbled with many a stile, and were it not that my worke doth for ease speede, and plainesse excell all the authours that euer wrote heereof, yet woulde I not leaue it out, because it wonderfully setteth foorth the singuler ver­tue and efficacie of my Iewell, which as I haue often sayd, & manifestly shewed, will needes supply all vses of Astronomie. For as hee hath in the last booke made smooth all the rockie intricacie of sphearicall trian­gles: so heere with no lesse singularitie he expresseth the making of those Dials, which all other Authours haue left lyke a Labyrinth to the learner, as if you reade Munster Orōtius and others, and especially Andre­as Sconerus, of declining or reclining Dials, you will confesse. VVherefore I doubt no but this woorke of myne shall bee both acceptable too the best, & longed for of the rest, of which I am too prepose these pream­bles following.

1 The whole Globe of the earth in comparison to the circuit of the speere, wherein the sunne is carried [Page 93] eareth a very small bignesse, but a centre pricke to speake of, and in respect of the other heigher, as the 8. 9 and 10. speeres, it carieth no acount at all. For which cause all Dials are made, euen as though we dwelt in the centre of the earth. And betweene a flat heere on the circuit of the earth, and a flat cutting the centre of the earth, so they be parallel, there is no difference in this behalfe.

2 All plaine flats whatsoeuer and howsoeuer they decline, recline, or incline, doe represent some one ho­rizon in the world.

3 Euerie Horizon whatsoeuer, representeth a great circle of the sphere by his definition.

4 Euerie great circle hath his two poles, euen as thequinoctiall hath by the 2. definition.

5 The poles of any horizon are his zenith, and his Nadir.

6 But to auoyd confusion of vnderstanding henceforth in this worke of Dyals, when I shall speake of the pole or poles, I will meane onely the pole or poles of the world.

7 And when I speake of the Zenith or Nadir, I will meane it simplie for the Horizon of the place wher­of I speake, as the common order of speech is.

8 And for that the poles of any Horizon represented by any wall, bancke or flatte whatsoeuer, are in truth the Zenith and Nadir of the same horizon, therefore I will call the vppermost the pole Zenith, and the vn­dermost the pole Nadir of any such horizon, wall, bancke or flatte.

9 Euery Flat whatsoeuer imagined hath two faces, as I may call them, on ech side one, as in a Counter, (which though it be a solide and sorie simile) the one face is crosse, the other pile.

10 In euery great circle there is a flat to be imagined of two faces, or two flats I might say, respecting se­uerall parts of the heauens: as for example, If you imagine a flat in thequinoctiall circle, the same shall haue two faces, the one respecting the North Pole, the other the South Pole.

11 You shall finde no great circle of the Sphere howsoeuer imagined, but that looke how much the north pole is eleuated aboue his one flat, so much is the south pole eleuated aboue his other flat, or rather depressed, to speake it more aptly to my purpose, and so contrariwise, which on your Iewel you may most euidently see. For placing the Finitor to our latitude vz. 51 ⅔ degrees fixed vnder the north pole, you shall streight see the south pole depressed 51 ⅔ deg. vnder the Finitor, which is the eleuation of the south pole to those people which be Antypodes to vs, an so forth in any other latitude.

12 Sythens euery flat whatsoeuer representeth some one horizon in the worlde, and euerie horizon some one great circle of the sphere, as is before saide: therefore euerie flat not being a right horizon, lieth in some one great circle, aboue which one of the poles is eleuated, hauing also an other Antypodall flatte imagined on the backside, vnto which the other pole is as much depressed, or to speake it Antypodally I may say ele­uated: for you must note that euery right horizon hath neither pole eleuated as shalbe shewed.

13 In euerie latitude you shall finde both the poles to be eleuated aboue the flats of the Verticall circle, or East Azimuth (which representeth alwayes the North and South wals) as much as the latitude lacketh of 90. As for example, set the Finitor to our latitude videlicet 51 ⅔: you shall see that the zenith line which alwayes representeth the said verticall circle hath the North pole eleuated 38 ½ deg. aboue his one flatte, and the south pole as much aboue his other flat.

14 Note that when you finde this word (Latitude) simplie in this 6. Booke, I meane the Latitude of the place where you remaine, and that to all other walles and flats, I will vse this word Eleuation, or poles Eleua­tion for distinctions sake.

15 Also I wil vse this word Horizon simplie but to the place of your biding, the rest I wil cal wals or flats when as I meane the Horizons or great circles represented by them.

Chapter 2. Of the distinction on Dials into two kinds, Instrumentall and Sphericall.

THe shapes, formes, fashions, and deuises for Dials are very many in generall: yet in particular they are but of two sorts, for either they are made to descrie the houre by the altitude of the sunne, of which sort are Quadrants, Rings, Circles, Shippes, Cilinders, and such like: and therefore rather to be called instruments then Dials: Or else they are to shew the houre by the shade of a cocke, stile, or Gnomon without respect of sunnes altitude, of which sort are your common Horizontall and Murall Dials of all sorts: your portable Dials with needles: your Equinoctiall, your conuex, concaue, and such like.

Of the first sort the deuises are so manifolde, which euerie learned Astronomer may and hath deuised to his fancie, that a great volume would not contane them. VVherefore since they be troblesome and busie to make, and being made seldome serue but to one latitude, and withall cannot be vsed but of the learned sort: I do not meane to entreate any whit of them, but will set ouer those that are desirous of such fancies vnto Munster, & Orontius de horologiis: to Apian, who hath made a busie Horoscope: to Sconer, not inferiour to any of them: to Stophler, and such ancient writers, where they shall see deuise ynough to werie them. And the rather I omitte these instrumentall Dials, because all their vses are farre more reddily performed by my Iewel: Yet knowne that the vnderstanding of my Iewel, shall cause any man almost at the first inspection to make any of them: for the onely and chiefest matter in their fabrication dependeth on the knowing of the Sunnes true altitude for euery houre through the yeare: which by my 3. Booke 31. Chap. is had for the looking on. And so I leaue all Dials of that sort, to them that like better of them then euer I did. For I neuer loued to ca­rie a Diall about me, whereon I must bee driuen to hunt after the houre when I had neede of him: which minde of mine, was the onely cause that made me to deuise the furniture of the Iewel as you see: For that as I haue often said by Gemma Frisius, Regula, Cursor, and Brachiolum, I could get nothing without hunting after it, and somtimes the Cursor led me such a course that I haue beene faine to giue ouer the chase, or not fully pursued it to the very point. Thus much of instrumentall Dials. The second sort which are more aptlie called Dials, I meane at large to prosecute in this volume, because they depende altogither Ex protractione Spe­rica, [Page 94] as in the next Chapter appeareth, and make not a little to the liuely vnderstanding and wonderful vse of my Iewel. In latin they are called Solaria or Instrumenta Scioterica, also Horologia of the greeke word.

Chapter 3. Of the definition and deriuation of Dyals and the grounde of them all.

A Dyall in verie trueth, might be called a plaine sphere: but onely that it wanteth parallels: for the lines drawne thereon to distinguish the houres, are the verie meridians or hourelines of the sphere prospe­ctiuely proiected on a plaine Superficies, by the Sunnes accesse vnto them: as here following I wil as much as I can, make manifest, bicause the most part will hardly beleeue it. But first to define an oblique Dyall which chiefely commeth in question more artifically: An oblique Dyall is most commonly a rounde Superficies or flatte, lying equiballance to some one horizon, concentricke to his circumference, from whose centre, issue out certaine streight lines, vnto the intersections of euerie 15. meridian or houreline of the sphere with the Horizon: cutting also the circuite of the same Dyall flatte, into like and proportionable parts vnto the saide intersections of the Horizon. For the plainer demonstration of this definition, take my Iewel in your hand and set the Finitor to thequinoctiall, which representeth an Horizon where one of the poles is Zenith: now if in this horizon you imagine a rounde plate to make a Dyall on: the same must needs be equiballance, and con­centricke, both to thequinoctiall and Finitor, because they are in this constitution of the sphere all one circle: and for as much as the parallels grow lesse and lesse, euen vnto nothing at the pole, and that the pole is here the zenith, therefore it must needes be that some one of these parallels is equall and equidistant, to this round plate for your Dyall, so that a perpendiculer line let downe from the zenith or pole, shall cut the centres both of the Horizon, parallel, and plate. Then who knoweth not that thequinoctiall, and likewise all the parallels are deuided into 360. equall parts by the meridians of the sphere, of which 360. the 24. part is 15. therefore euery 15. meridian are commonly called hourelines. These hourelines (or rather spherically called houre circles) doe nowe deuide the Finitor in this constitution into 24. equall and euen parts: wherefore if you imagine 24. lines issuing from the centre of the Finitor, (which is also the centre of the Dyall plate, and represented here by the centre of the Iewel) vnto the 24. intersections of the said houre circles with the Finitor: they shall deuide the Dyall plate into 24. equall parts, because the intersections in this horizon are equall, e­uen as the parallel that standeth directly ouer the plate is deuided: so that by this definition you may con­clude that a Dyall, made to that Horizon, where the pole is zenith, is onely a circle deuided into 24. equall parts which is called an equinoctiall Dyall.

As this demonstration is most manifest in the equinoctiall Horizon, where the saide 15. meridians cut the Horizon into 24. equall parts: so falleth it out plaine ynough in any other horizon, where one of the poles is eleuated aboue the same otherwise howsoeuer. For euery such Horizon, is also deuided by those selfesame 24. meridians, into 24. partes, but vnequall: and by how much lesse the pole is eleuated, the more vnequall they are. For demonstration whereof, set the Finitor of the Iewel, vnto our latitude here at Reading, vz. 51 ⅔ deg. vnder the pole.

Now to see the proportion of the spaces for the hourelines of a Dyall to the same Hori­zon, marke how the 15. prickt meridians or houre circles of the Iewel, doe cut the Finitor, for if you tell the degrees on the Finitor, be­tweene ech cutting you shall haue thereon the degrees seruing to the seuerall distances of the hourelines of your Dyall, so that if as before, you situate in this horizon, a rounde plate for your Dyall, the same shall be con­centricke thereto, and equall to one of the Almicantares, which grow lesse and lesse to nothing at the zenith, euen as the parallels do at the pole, wherefore if from the centre of your plate you imagine 24. lines issuing vnto the intersections at the worlds end, as they say, of the 24. houre circles with the Finitor, they shal cut the round plate into 24. like vnequall parts vnto those, made on the horizon or Finitor in all respects proportio­nably an Symitrially agreeing, so that whi­ther your Dyall plate be great or little, his diuisions are proportionable.

For better explanation heereof vnto the young learners: admit that you had 24. wyers, of equal length, made iust semicircle­wise, and all of them ioyned at two points or poles: these 24. wyers being set ech of like distance from o­ther, would make 12. whole circles: whereof you might appoint any one of them for the meridian circle, then if you did prepare a round plate, or thinne boorde, with a hole in the centre, of equall circuit to one of these 12. wyer circles, and did put him within these wyers, like a bird in a kage, but so that the one pole where these wyers meete, might be iust 51 ⅔ deg. reckoned on the meridian wyer aboue this plate, and the other pole as much vnder: and lastly thrust a streight wyer through both the same poles, & the centre of the [Page 95] plate, as in this figure appeareth: you haue a plaine ocular demonstration of the grounde and reason of all Dyals, for if now you fasten a threede at the centre of the plate, to the axtree line: or else a ruler, and lay the same from place to place, where the 24. wyers do touch the round plate, you may thereby drawe the houre lines for your Dyall, as many as shall serue your turne, as I haue done in this figure, which hath but 12. houre circles, or wyers here represented, because the whole 24. could not wel be drawne, but the other 12. you must imagine to lie so opposite and directly one to another, that euery one hideth his match. Now if you place this plate within the wyers, neere or farther off from the pole, according to the latitude of any countrie whatsoeuer: you may thereby make a Dyall thereunto. So that a man might make a pretie instrument of 24. wyers, to make all manner of Dyals by, if there were no other shift, but you shall haue my Iewel about you when the wyers will be to seeke, neither can the wyers be exactly bent true compasse, nor being bent, keepe their iust places long.

Chapter 4. What the cocke or Gnomon of euery Dyall is, and by what reason they giue the true shade to the Dyall.

FIrst you shall vnderstande that the cocke, Stile, or Gnomon of euery Dyall, I meane the line thereof that yeldeth the shade to know the houres by, lyeth directly in the axtreeline of the world, or at the least, pa­rallel vnto him, which worketh the same effect.

Secondly I shewed in the last Chapt. that the sphere is set about with 360. houre circles, meeting at the poles: which maketh 24. times 15. whereby the space of time, wherein the ☉ commeth vnto euery of these 15. by the motion of the Primum mobile, or first mouer about the axtree of the worlde, maketh one houre: So that an houre is the 24. part of this reuolution once about. These 24. houre circles of the sphere are repre­sented in the last Chapter, by 24. wyers, of which, euery one is so directly opposite to his match against him, that they both make one whole circle. Insomuch that of the 24. houre circles, or rather semicircles, (as they are in deede) there are made 12. whole circles, meeting and crossing ech other in halfe at the poles: wherefore of necessitie the axtree must needes be dyametre to them all by the first definition, and first Theoreme of the fifte Booke. This knowne, I thinke there is no man so ignorant, but may conceiue without the 22. Theoreme of Euclides prospectiue: that if a circle with his diametre or axtree, be turned edgelong towards a mans eye, that then it appeareth to sight no circle, but a streight line like vnto AB in this figure.

By this onely prospe­ctiue reason it commeth to passe, that the ax­tree line giueth shade vnto the Dyal. For when the Sunne is in any houre circle, the shadowe of the axtreeline is extended thereby directlie on the houre circle opposite to the same, and by that meanes sheweth the houre. As for ex­ample, if you place the instrument of wyers mentioned in the last Chapt. so that the round plate encluded within them, may stande equi­ballance to the horizon, & the meridian wier, directly North and South, the pole aboue the plate Northwards, in order spherically: you shall then see that the Sunne being in the me­ridian circle Southwards, shall cast the shadow of the axtree line directly on the same meridian Northwards, which is the reason in all horizon Dyals that haue trianguler cockes, they giue no shade at 12. of the clocke. Likewise the sunne being in the 9. of clocke houreline, casteth the shade of the axtree line directly on his opposite, being the 3. of clocke circle: in so much that in your instrument of wyers, you shall perceiue that at euerie compleate houre, the same houre wyer with his opposite, and the axtree, will all three giue but one streight line for shade, which shall be directly on the houreline, drawne one the rounde plate for the same houre. Insomuch, that by one Dyall exactly made, a verie rude and vnskilfull man may make any like Dyall, if the cocke were set vp to his hande, for at euerie houre his shade will most truely expresse the houreline, and so for a shift he might make a Dyall in a day: if the Sunne chaunce not into a cloude: then he must tarie the Lords leisure. This matter is expressed in the fi­gure of the last Chapter.

Chapter 5. Of the diuision of Horizons, and especially into three sortes.

ALthough that euerie least declining of anie wall, flatte, bancke, or buttresse, is particulerly a diuerse horizon, by the first preamble: and hath a sundrie eleuation of one of the poles, or none at all, so that in this sense the horizons be infinite: yet by two other meanes they are brought to a farre lesse number: whereof the greatest is 90. and the lesser but two. For as in all Astronomicall workings 90. horizons serue through the world, euen as I shewed in the 2. Booke and 21. Chapter, the reason is, as I there said, because he that coulde go continually vpon one of the poles, shall finde that when the pole is once 90. deg. eleuated, hee then is at the heighest, euen in the zenith: then let him go on streight he shall see him sinck againe on the other side of him. So also in another sense, they make but two differences videl. Horizon rectus, and horizon obliquus, yet some make three, which I for distinction of my Dyals thinke best of, vz. Horizon rectus or Pola­ris, Aequinoctialis, and obliquus. Horizon rectus or Polaris a right or polare Horizon, they will to be where the Aequinoctiall is zenith, and the poles leuell with the Horizon, neither of them appearing aboue: [Page 96] and for as much as all the meridians meete at the poles, therefore euerie right Horizon must needes be, and lie in one of the meridians, for that there are no circles of the Speere, which haue their pole zenith, and Na­dir in thequinoctial, but onely the meridians as I partly shewed in my 24. Chapter of triang. Wherefore to conclude, euery East or VVest wall, are right horizons because they lie directly with the meridian, and so are any other reclining or inclining wals or flats, cutting either poles, that is to say aboue which neither pole is eleuated.

Horizon Equinoctialis the Equinoctiall horizon, is where the pole is in the zenith, and thequinoctiall become all one with the horizon, some write he is Horizon obliquissimus, because the signes and the starres in this hori­zon do go all a slope as though they went round about the side of an hill. But if my opinion might serue for authoritie, I would call this Horizon Polaris, for that the pole is zenith, and the other Equinoctialis, for that thequinoctiall is there zenith: my reason is because the denomination of any thing ought to proceede from the worthiest part of it, which must needes be the highermost, for the starres and planets haue most power and force when they are in the heighest part of the heauen, also the Zones, Clymats, and Parallels take their names of those parts of the Sphere, which are in the Zenith, for it were absurde to say that a man were in the hot zone, when thequinoctiall is the Horizon, where he should finde himselfe in the coldest.

Notwitstanding I will yeelde in this behalfe to the authoritie of all the learned men that haue written, ra­ther then presume to offer to stande in armes against them, knowing my validitie too weake to encounter them: yet hereafter in this worke, I must craue leaue of them to terme euerie wall and flatte, by the place where his pole zenith lieth as a south wall, to be a flatte, whose pole zenith lieth in the south point of the ho­rizon, and so of the rest.

Horizon Obliquus, an oblique Horizon is where one of the poles is eleuated how much or little soeuer, till it come to 90. degrees.

Chapter 6. Of the diuision of all Dials into three sorts, and whie they make shew of greater diuersitie then there is.

IN the last Chapt. I distinguished all horizons into three sorts, and that for two causes: the one, for that the formes of making all manner of Dials is but of three sorts, as shalbe shewed in the next three Chapt. therefore marke them well, make those and make all the other: for that the dials to euery of those Hori­zons require their cocke or Gnomon of a seuerall forme. For the dyals of all flats lying in any right horizon, haue their hourelines parallel one to another, and their cocke a long square, although some vse a stile or wyer, which is not so good nor so agreeable to the nature of the diall: because the stile must be of a iust de­terminate length: and then can yelde shade to the Dyall but from the very toppe: whereas the long square cocke yeldeth a long line for shade, and the same alway parallel to the hourelines, and like vnto them. The Dyals of all flats lying in thequin. are but a circle deuided into 24. equall parts, the hourelines meeting at the centre, whose cocke or Gnomon. most commonly is a stile or pin erected: neither is it materiall how long or short hee bee. The Dyals to all flattes lying in any oblique or byas horizon, are a circle deuided into 24. vnequall parts, the hourelines concurring at the centre, whose cocke or Gnomon must needes be a triangle, his side yelding the shade to the Dyall, eleuated in height according to the poles eleuation: though some barbo­rously, as I may terme it, vse a wyer bent a slope to the poles height, which euery bird almost with sitting on it, may way from his pitch, and then all is marde till some skilfull man come to mende it, because euery man knoweth not what the height of the pole meaneth. For I remember well I asked a cunning gardiner once, that had set vp a post aslope in a curious Garden in the middest of a circle, what he would make: a Dyall (said he) why? said I, how heigh is the pole here, saide he sixe foote heigh, pointing to the post that himselfe had set vp: whereat I laughed hartely seeing so cunning a peice of worke towards: but now againe to my matter. As there are but three diuersities of Dyals and horizons, so are they easily to be made in euery country, wher­vnto they serue without any difficultie, if a man were in euery such place seuerally, by occasion of trauaile or otherwise. But yet to make the Dyals, which of right ought to serue in other countries, to shew the houres and serue your turne in your owne countrie, maketh a great shew of skill, and of diuerse sorts of Dyals, to those that vnderstand not the causes of them. For the murall declining, reclining, inclyning orientall and occiden­tall Dyals and all others, the plaine horizontall excepted, are dyals which of right belong to other countries, and horizons, and are but wrested as it were to serue altogether to one place: neither is there any of them but is one of the three sortes before mentioned. There is none of them but of right serueth to a countrie, dif­fering from yours, either in longitude, or latitude, or both: but especially the difference of longitude maketh this great diuersitie and difficultie, for that onely driueth vs to alter the true course of the hourelines, and their numbers to make them serue in our countrie. As for example, in an East Dyall, in the place where the cocke standeth he sheweth no shade, and then is it 6. of the clocke in our countrie, but to that countrie whose difference of longitude is 90. deg, Eastwards, or more plainely whose pole zenith is in theast point of our horizon, it is then 12. of the clocke, and so would it giue also in our countrie, if you laide the same Dyall so that his pole zenith may be in our meridian, his flat cutting the poles.

Likewise in the declining Dyals here in our countrie, the perpendiculer is alwayes 12. of the clocke: but in the horizon or countrie whereto the Dyall should serue, the 12. of clocke is iust where the cocke standeth, so that it is a general rule if you see any Dyal, his cocke yeelding no shade at 12. of clocke: that the same is made to the longitude of the same place: wherefore to conclude my purpose, there are but three diuersities of Dy­als, onely the situation and change of the place maketh the alteration.

Chapter 7. How to make the first kinde of dial to the Equinoctial horizon, that is, where one of the poles is zenith.

THe matter is short and already sufficiently to be had by the 3. chap. deuide a circle into 24. equall parts frō the centre, drawing the houre lines vnto them, then set to the figures frō 1. to 12, & thence from 1. to 12. again, to make vp 24. it is no matter where you begin to number, so you make an end wel: if you would needs make this dial by the Iewel in the 3. chap. you haue instructions sufficient, the cocke is but a stile or wyre erected as plum as may be from the flat, so that the top be no neerer to the one part of the circle then the other. This diall though he be sleight can least be spared, as the shipmen can witnesse, which with a qua­drant annexed behinde him, serueth in euerye latitude where the ship goeth, a thing moste common, or els I would spend sometime about it, if you know it not, look Orontius & Munster de horologits, or in English the Art of nauigation and the regiment of the Sea.

Chapter 8. How to make the second kinde of dial to a right horizon, that is, where the Equi­noctiall cutteth the zenith, together with the ground and reason of al dials to any right horizon.

I shewed in the 3. and 4. chap. the reason of making all dialles to any horizon, where any one of the poles is eleuated to come of the intersection of the houre lines, with the horizō. And by example of wyres the same conclusion, & withal how that the axtree did after a sort delineate out the lines of all dialles, by his shade. But now in a right horizon where neither of the poles are eleuated, because the Zenith is in the Equinoct. one of the meridians must needs be the horizon: & of the rest of the meridians, the one halfe be alwayes wholly a­boue the horizon, & the other half vnder, they al cutting the horizō together at 2. points, which are the poles: which is the cause that they distinguish not the houre lines of the diall by their intersection of the horizon, as in the Equinoctial, and oblique horizon, but yet by helpe of the axetree, they wil do it here as wel as in them, and that in this manner. Take your 24. wyres spherically conioyned, as in the 3. chap. and erect the meridi­an wyre perpendiculer on some plaine flat, so that the axetree wyre may be parallel to the flat as in this figure you see. Now if you were vnder the Equinoctiall, or if you erect one part of this flat, so that the poles & axe­tree of the wyres thus placed, may lye leuel with the poles of the world: that is to saye, in your meridian ac­cording to the poles eleuation, you should then see the Sunne at his rising, to cast the shade of the 6. a clocke wyre, and the axtree wyre directly on the other 6. a clocke wyre: that is to say, paral­lel to the flatte: so that the dialles vnder the Equinoctiall haue no houre line for 6. a clocke. But after that the Sun rising to 7. a clocke, doth cast the 7. a clocke wyre, the axetree wyre and the other against him, al three in one shade, making one streight line equal to the axetree wyre, & may serue for the 7. a clock houre line, if he be not before drawne. Likewise at 8. a clocke, the 8. a clocke houre wyre, the axetree and the other against him, yeelde al three but one streight line for shade: which sheweth 8. a clock, and so at 9. 10. 11. and then at 12. the Sun is directly ouerhead: which would cause the Gnomon or axetree if hee were a longe square, to yeeld no shade: then the Sun by his comming down describeth the after noon houres, as by this figure I hope you plain­ly conceiue.

This wel vnderstood, you may make the dial easily thus: Take a long square stone or flat: and crosse him in the middest both wayes with two lines square ech to other: let the longer line bee East and VVest, videl. AB: the other North and South, vz. CD. Then erecte your Iewell perpendiculer on this East line AB, and directly with it: but so that the plum line of the Iewell maye stand perpendiculer thereon, in the crossing of these two lynes: and there deuise by some meanes to fixe him: that line of your Iewel so fyxed, set your rule with sightes to the line of Leuel, which is then parallel to the stone or flat, and representeth 6. a clocke in any countrey where the Equinoctial is Ze­nith, then lift vp the end of the ruler, vnto the next 15. deg. of the Limbe, and looking through the sights, mark with a pen (as you may easily doe) where your eye beame doth point on the said East line, there draw a lyne square ouer the stone for the 7. a clock houre line, that done, lift vp the ruler other 15. deg. and there marke with a pen where your eye pearcing the sightes, doeth pointe on the sayde lyne, drawing there also another crosse line, for the 8. a clocke houre line, and so doe with euery 15. degree of the Limbe of your Iewel, in the vp­permost halfe, vntill you haue done, as in this Figure you may beholde. Also you may if you list lay the Iewell flat vpon the stone, so that his plumbe line lye direct with the crosse line CD, and his Limbe touching the East line AB and then fyxing a threed in the centre of your Iewel, leading about the same from 15. degr. to 15. deg. of the Limbe with your hande, and at euery place where this threed doth cut the East line, there drawe lines crosse the stone ech parallel to other, and square to the East line: then set numbers thereto as in this Fi­gure [Page 98] you may see. The cock or gno­mon to this dial must be a lōg square like vnto G, erected in the 12. a clock line, whose length is commonly vsed equall to the line CD, or it may bee lōger or shorter at your pleasure, but the height must needs bee equal vnto the space betweene the centre of the Iewel, & the line AB. Thus maye you note that whether your Iewel stande perpendiculer on the flat, or lie down theron the crossings of the long line AB, which some call the touch lyne are al one: so that the leuell line of your Iewel be parall. to the touch line.

Chapter 9. How to make the third kind of diall to an oblique horizon, which they commonly cal an horizon diall, and that very easily.

VVHen you come into any countrey, where you would make a dial: first you must get the poles eleua­tion or latitude, thereby the 3. book 5. & 8. chap. & if you fynd the pole eleuated any whit, then you may be sure it is an oblique horizon: to which if you wil make a dial, set the finitor of your iewel vnto the same latit. found where you shal see the Limbe or vttermost meridian of your iewel and the axetree or East houre line, to cut the finitor 90. deg. or one quadrant apart: which is the cause that in an horizon diall, the first point is to deuide the circle into 4. quarters. Furthermore, if you behold your iewel wel in that scituation, you shall see, that the houre circles, (I meane euery 15. meridian) on the one side of the axetree line, doe cut like e­quall degrees on the finitor, vnto the houre circles on the other side, eche to his matche, beeing both recko­ned eyther from the seueral vttermost partes of the Limbe or on ech side from the centre: which is the cause, that if the deg. of an oblique diall be knowne for one quarter, all the other 3. quarters are to bee drawne by that: so that by this meanes, the two diameters seruing for the noone line and Eastline being drawne, 5. spa­ces more knowne shall serue to make the whole dial by: which in my opinion can be no great toile.

Example: here at Reading the latitude is 51 2/3. deg. and therefore is an oblique horizon. To make a diall thereto, I get a flat stone or plate, and thereon make a circle, or rather a Limbe to the diall: crossing the same square vppon the centre, with 2. lines into 4. quarters, of which lines I appoint one to be the noone line, & at his North end write 12. the other the East line, & ech end write 6. Then for my further ease, because I wold not deuide the Limbe of my dial into 360. deg. as Cor. Gemma teacheth, I make on the centre of this dial a blind circle, equall to the graduated circle on the Limbe of my iewel, al which done, I set the finitor of my iewell to the 51 2/3. latitude, and there doe I marke what deg. of the Finitor, the 1. 2. 3. 4. & 5. a clocke houre circles doe cut: I finde at 1. a clocke 11 ½. deg. at 2. 24 1/6. at 3. 38. deg. almost, at 4. 53 ¼. at 5. 71 ¼. all of them reckoned on the Finitor from the Limbe of my iewell inwardes: all these seuerall spaces of degrees, I take with a payre of compas­ses orderly one after other from the Limbe of my iewell, and set them, or notes for them on ech side of the noon line in the saide blynd circle, beginning from the noone line: and then doe I drawe lines from the centre of the diall, through all the notes in this blind circle vnto the Limb of the dial: seruing to the houre lines, as by this figure you may more plainly perceiue, then by many words.

To these lines I set these numbers, 1. 2. 3. 4. and 5. on the Northeast side, & on the North­west quarter: to them these numbers: 11. 10. 9. 8. 7. as you see: then if you draw forth the 4. and 5. houre lines vnto the Southwest quarter: you haue the 4. and 5. houre lines whiche the Sun riseth before 6. in the Som­mer time. Likewise drawing foorth the 7. & 8. your dial is made: & for the cocke or gno­mon to this diall, if you take the degrees of the latitude, vz. 51 2/3. deg. in your cōpasse & sett the same in your blinde circle, drawing thereto a line from the centre, and erecting at the North end of the noon line a perpen­diculer to meete him there, you shal haue the iust patterne for your cocke, as in this figure you may behold: for it is a generall rule that the cocke of a diall to an oblique Horizon, whether it decline, recline, incline, or howso­euer, must be eleuated from the horizon, equal to the poles eleuation. Note that as I haue performed this by one quarter of the houre lines of the iewel, you may by the iewel perform the whole round: but al commeth to one effect, since as I said the spaces of euery quarter are like, and containe like number of degrees, being taken equally on ech side of the noone line.

Chapter 10. How to make a diall to a perpendicular wall beholding the South fully, called of some Authors Horologium verticale.

THat wall which I call a South wall, is an horizon, or great circle, whose pole Zenith lyeth in the South point of the horizon of the countrey where it standeth, and lieth euē with the East azimuth of the same place, being all one circle with him. Insomuch that he is on my Iewell in euery latitude, represented by the line, which I call the Zenith line, or East azimuth. To this wall the South pole is eleuated alwaies as muche as the latitude of the countrey lacketh of 90. by the 13. preamble, which here at Reading is 38 ⅔. degrees, and therefore is it an oblique horizon by the 5. chap. wherto you may easily make a dial by the last cha. only here in it altereth a little, you need make but bare 12. houres: for that no perpendiculer wall in these partes canne serue aboue that compasse, and that you must turne the dial and the cock downwards, that he may behold the South pole, prouiding that the 12. of clocke lyne be drawne very perpen­diculer down the wal.

For your bet­ter vnderstanding, sette the Finitor to the latitude of Reading, videl. 51 2/3. vnder the pole, then if you behold the Zenith line, which as I said, represen­teth the South wall, you shall finde at the Nadir of the zenith line, the south pole to be eleuated 38 1/3. degr. if you reckon towardes the South parte of your Iewell, wherefore in respect the Zenith line is graduated as wel as the Finitor, you may thereuppon see what deg. euery houre requireth as wel as in the last cha. by the Finitor, & so you shall find the 1. a clocke houre line to cutte 9. deg. the 2. 19 3/7. the 3.31 ⅖. the 4.46 ½. y^e 5.66 ⅔. & the 6. alwaies 90. by which make your diall, as in the last chap. and as in this figure appeareth. Note that the eleuation of the wall knowne, you may set the Fi­nitor to the same latitude vnder either of the poles, without respect, and make your diall therby, which indeed ie the best course, because the degrees are farre more easily to be counted on the Finitor, then on the Zenith line.

Chapter 11. To make a diall to a ful North wall.

I Haue shewed in the 9. and 10. preamble, that euery per­pendiculer wall or flat hath two faces: (that is to say) if a wal be in length East and VVest, then on one side the South beholdeth him full, on the other side the North, in­somuch that looke how muche the South pole is eleuated to the South side, so muche is the North pole eleuated to the North side by the 12. preāble, both which the zen. line reperesenteth. VVherefore you may conceyue that one di­all serueth them both, onely they alter in this: that this di­all to the North wall must be turned vpside downe, that the cocke may beholde the North pole, and all the houres sauing 4. 5. and 6. in the morning, and 6. 7. and 8. at Eue­ning left quite out, because the Sun commeth to a North wall, but before 6. in the morning, and after 6. at euening, as by this Figure you may perceiue.

Chapter 12 How to make a diall to an East wall.

AN East wall is an horizon, whose pole Zenith lyeth in the East poynt of the horizon, whereon it stan­deth, or more plainly an horizon lying euen with the meridian of the horizon whereon it standeth, so that an East wall pointeth directly North and South, whereby you may gather by the 5. chap. that it is a right horizon: and therefore you may easily make a diall thereto, by the 8. chap. only the alteration is in the placing of the diall, and numbering the houres, which you shall thus doe. First drawe a perpendiculer on the wall, videl. AB, then pitching your compasse in some one point thereof most conuenient, vz. C draw a blind [Page 100] arch Southwards of the plumme line videl. DE, whereon number by deg. the eleuation of the South pole a­boue the perpendiculer or Zenith line, vi­del. DG, which as in the 10. here at Rea­ding is 38 ⅔. deg. thē drawing a line from that deg. to the center of the arche, videl. GC, and another perpendiculer to the plum line, or parallell to the horizon, all is one, videl. GF, so haue you GFC the verye patterne of the cocke to the South wall diall, and that in his direct scituation to eyther pole, his Gnomon line issuing from the centre, videlicet GC, represen­ting the axetree of the worlde, and dire­ctly beholding both the poles: wherefore drawe another line square vnto him longe inough for your diall, videl. HK, and the same shall represent the Equinoctiall flatt, to whiche drawe another line parallel as farre dystant as your dial shalbe in width, videl. CL, this done draw a line between the Equinoctial and his parallel, square to them both, euen where you thinke moste conuenient for the cocke to stand, videl. MN, which shall also be parallel to the axe­tree GC.

In this line MN place, the plum line of your Iewell, his Limbe touching the Equinoctiall line HK or CL, all is one, and there your Iewell fastened with waxe, or staied with ones hand, laying a threed to e­uery 15. deg. of the Limbe, extended to the touch line CL, draw your houre lines at ech crossing, so that eue­ry of them be parallel to the axtree line GC, by the 8. chap. But now in an East diall you must number the line where the cock standeth with 6. the reason is for that the meridian of the horizon represented by an East wall is 90. deg. or 6. houres distant from ours: beeing in truth the 6. a clocke houre circle, in our countrey, so that when it is high noone to that countrey whereto our East wall is horizon, it is here but 6. a clock in the morning, the difference of longitude being 90. The rest of the numbers for the houres, sette in as in this figure you may see, because the Sunne riseth to vs neere at 4. a clocke in the morning: therefore I haue set no houres aboue that, and bicause the Sunne departeth from an East wall at 12. therefore past a 11. I cannot goe.

Chapter 13. How to make a Diall to a West wall.

VVOorke in all respectes as you did in the East wall, sauinge that because the Sunne begin­neth to come to the VVest wall but at 12. a clocke: therefore you must chaunge the numbers of the houres, as in this Fygure you may better see, then by many woordes vnderstand.

Chapter 14. Of declining walles, and what circles of the spheare they represent.

IT is a Maxim in philosophie, (as I haue already said) that Omne graue tendit adcentrum, insomuch that if you tra­uel round about the world with a threed and plummet in your hand, the plummet shal still point vnto the centre of the earth: all vpright walles are made perpendicular by a line and plummet: therefore the great circle of the spheare lying euen with any wall, must needes cut the centre of the earth, and be concentricke, & a great circle by the 1. def. And because euery wall perpendiculerly erected, cutteth or pointeth to the zenith, therefore eche of them must needes lye in one or other of the azimuthes: and do crosse, or at the least woulde crosse one another vpon the centre of the earth, & in the zenith, if they were bigge inough to fill out a great [Page 101] circle of the spheare, euen as you see the lines of this present figure, euery of which representeth a perpendicu­ler wall, the circle AD, BC, is here the horizon, his centre is the Zenith: AB is the meridian or South azi­muth, in which circle, the East and VVest wall lyeth, by the 12. and 13. chap. CD is the East azimuth, called of many authors simply, Circulus verticalis, the vertical circle, whereof the South diall is called of diuers, Horologium ver­ticale: notwithstanding al the azimuthes are vertical cir­cles, so called, quia transeunt per verticem, I say in this East a­zimuth CD lieth the South and North wall by the 10. and 11. chap. the line EF representeth a South wall, de­clining East, and a North wall declining VVest, lying in the 45. and 125. azimuth, for the azimuthes are counted from the East vnto 360.

The line GH representeth a South wall declining VVest, and a North wall declining East, lying in the 135. & 315. azimuth, and so of all the rest, which you see in this figure drawn within the lesser circle, representing the globe of the earth, & are to be ex­tended, or as many other as you can imagine. It is moste certaine and manifest by the 2. def. that the poles zenith and Nadir of euery azimuth lie in the horizō, as the poles of euery meridian lie in the Equinoct. VVherfore as the pole zenith of a south wall, must needes lie by the sayde definitiō in the south point of the horizō: euē so the pole zenith of any declining wall respecting the South, de­parteth so much frō the South point, as the declinatiō cō­meth to: & so likewise of any wal, respecting the north: his pole Zenith departeth from the North point, according to the declination, insomuch if you will aske mee what the declination of a wall is: I say it is the distance between his pole zenith, & the south or north points by deg. of the horizon: & looke which way the pole zenith departeth, thence is the declination denominated or called, as if the pole Zenith of a wall lie betweene the South & East point, then is that wall called a South wal declining East: & so of the rest. As for example, in this figure the pole zenith of the South wall CD is B, but the pole zenith of the North wall CD, is A. Likewise of the East wall AB is D, & of the VVest wal AB, is C, for euery circle hath 2. faces, as is said in the 10. preamble. Also of the South wal declining VVest EF, the pole zenith is H, whose declination is the space HB, but of the Northwall declining East, represented by EF, the pole Zenith is G, whose declination is AG, and so of the rest. By all this you may well gather, that euery wall, flat, and horizon, ought to take his name of the place or point where his pole zenith lieth, and not of his owne scituation: for if I should say, make me a diall to the wal CD, who could tell, whether I meant a North or South diall, but if I say make me a diall to the South wall CD, it is easie to vnderstand: wherfore me thinketh as I shewed in the 5. chap. they abuse their tearmes who call that an Equinoctial dial, whose pole Zenith is the pole it selfe, mentioned in the 7. chap. which in my opinion ought to bee called a pole or polare diall. Lastly, al declining walles are deuided into 2. sorts, & eche of those againe subdiuided into other two: the diuision is into North declining, and South declining: of which the first haue their pole zenithes in the North semicircle CAD, the 2. in the South semicircle CBD, the subdiuision of the North declining is in­to North, declining East: whose pole Zenithes lie in the northeast quadrāt AGD, & North declining VVest, whose pole zenithes lie in the northwest quadrant AEC. The subdiuisiō of the South declining, is into south, declining East, whose pole zeniths are in the Southeast quadrant BFD, & into Southwest, whose poles are in the quadrant CHB.

Chapter 15. How to know the scituation of any wall, and to finde how much he declineth.

VVHat the declination of walles are with the diuersities of them, is shewed in the last chap. In this I wil shew how to take the declinations of al walles respecting the South: by which any man may do the like for walles, respecting the North, the woorking being al one.

By a wall respecting the South, I mean a South declining wall, & it is easily to be known, because the Sun being in the South, yet shineth on him though he decline neuer so farre East or VVest, by which you may alwaies knowe a South declining wall: wherfore when you come to any such wal: First de­uise to place your Iewel vpon some stoole or otherwise Leuel with the horizon: but so that the line of Leuel may directly stand perpendicu­ler or square from the wall: neither is it material whether he stande close to the wal or not: which to do one way is thus: If the wal haue any returning square neere the place, then choose your standing, so that laying the rule on the said leuel line, you may through the sights looke directly with the end of the wal, as if you would make a dy­al on the side of a house, commonly the ende of the house shutteth square vpon it, so that if you lay the Leuel line directly with the end, then he standeth square to the side. Another way is thus: get a large squire, set one side euen to the wal, and parallel to the horizon, and hāg at the other side. 2. plum lines, at ech end one, there let one hold it til you haue chosē you a stāding: so that the rule lying on the Leuel, you may see through the sightes, the one plum line hiding the other, & there fixe your Iewel on the stoole.

The third and best way as I think is, eyther to fasten your Iewel with waxe or pitch on the side of a squire so that his line of Leuel lye euē with it, or els to get a little square boord or block: or for a shift, a square trēcher may serue, and thereon draw a line through the middest, perpendiculer from one of the sides: in which place the Leuel line of your Iewel, and apply it to the wall. VVell, to come to the matter, when you haue thus placed your Iewel, then doth the Leuel line poynt directly to the pole Zenith of the wall: therefore turning about the rule till the sun pearce both the sightes, or at the least erecting a wyre in the one sight, the shade thereof beyng proiected on the other: you haue in the Limb of your Iewel, the horizōtal distāce between the Sun, & the wals pole zenith (that is to say) betweene the azimuth, wherein the Sunne is then, and the azimuth cutting the walles pole zenith: wherefore you must presently take vp your Iewel, and take the altitude of the Sun, by whi­che & the 3. booke 6. chap. you shall know in what azimuth reckoned from the South, the Sun is then, either Eastwards or VVestwardes. Nowe by the Sunnes azimuth known, and his distaunce from the pole zenith of the wall, you shal alwayes know the declination of the wall by these rules following: therefore marke thē well, together with these 4. figures, in which the circuite CLDK representeth the horizon DC, the meri­dian C being the South part thereof, N the zenith, NA the azimuth wherein the Sun is: EF the base of the declining wall, B his pole zenith, B Hithe line in which the line of Leuel, must lie, K East, L VVest. First, if the Sunne be directly with the Leuell line of the Iewell, that is to saye, hauing no di­staunce from the pole zenith of the wall, as in this figure, then is the Sunnes azimuth reckoned from the South, equall to the walles declination, that way as the Sun lyeth, East or VVest: but as this seldom happeneth, or not at all, if you choose the best times: so in all other cases you haue three thinges to bee noted, that is, whether the wall decline East, which you may bee sure of in the morning: for in al wals declining East, the Sun com­meth to them before 6. & departeth at night before 6.

In VVest declining he commeth after 6. and setteth after 6. then must you note whether the horizontall distaunce of the Sunne from the pole zenith of the wall be on the East or VVest part of the Leuel line. Lastly, whether the Sunnes azimuth found, bee East or VVest from the meridian: al which you must be sure to note downe, which done, then in all East declining walles, if the di­staunce of the Sunnes azimuth from the Leuel line, chance to be East from the Leuel line, as A is from B, and the Sunne also East from the meridian as A is from C, in this 2. fygure, then take the deg. of the distaunce, videl. AB out of the degr. of the azi­muth, videl, AC, the remainder vz. B, C is your desire, if the di­staunce be VVest, as A is from B, and the Sun VVest, as A from C, in this 3. fygure, then take the azimuth, videl. AC, out of the distance videl. AB, so resteth BC your desire: but if the distance be VVest, as A is from B, and the Sun East as A is from C, in this fourth fygure, then adde the distaunce videl. AB to the a­zimuth, videl. AC, and so euery way haue you BC the decli­nation of the wall if hee be a South wall declining East.

But if the wall decline VVestwards, then if the distaunce bee VVest, and the Sun VVest, take the distance from the azimuth, if the distance be East, and the Sun East, take the azim. out of the distance. But if the distance be East, & the Sun VVest: adde the distaunce to the azimuth: and so haue you the declination of any South wall bearing VVest. All which you may perceiue by the last 3. figures, so that the words East and VVest change places. These rules shew harde, but you shall little neede them when you are working: for the very scituation of the distaunce and the Sunne, will almost alone direct you. Onely I set down these, least you shoulde doubt in any thing. Besides that, you may ease the matter thus: if the wall decline East, come be­times in the morning, and the Sunne and distance shall be both East: if VVest, towards the euening, & they shall both be VVest: so that in both these you need but take the distaunce out of the azimuth, according to the seconde figure: whiche tymes are more conuenient then about the middest of the daye to gette the Sunnes exact azimuth by his altitude, as by proofe you shall finde.

For example, the 26. of Iulie 1581, the ☉ then beeing 12 ½. in ♌, I being requested to make a diall on the Church wall of Sooning besides Reading, to set the clocke by, did place the Leuell line of my Iewell square from the wall by helpe of a square boorde: then turning about the rule to the Sunne, it cut in the Limbe 36 ½. deg. Eastwards of the Leuell line, which I noted in a paper, and quickly tooke vp my Iewell and tooke the altitude of the Sun 45. deg. the same instant, by which & the 3. booke 6. chap. I found the Sun to bee in the 53. azimuth reckoned from the South Eastwards: wherfore according to the 2. figure, taking the distance 36. ½. out of the azimuth 53. there remained 16 ½. the true declination of the wall. For taking the declinations of North walles the [Page 103] difference is nothing but that you must reckon the Sunnes azimuth from the North, as here is done from the South.

Another example, the 14. of August 1581. the ☉ being in ♍ 1 ⅔. de. I took the declinatiō of another wal about the same church, declining VVest, to serue the after noone: wherefore comming thither towards the e­uening, I found the distance VVest 20. deg. and the azimuth VVest 73 ⅓. degrees, then taking as before the distaunce, vz. 20. out of the azimuth 73 ⅓. there rested 53 ⅓. deg. the wals true declination VVestwards. Note that if the ruler of your Iewell want necessary sights for this purpose, you may hold a threed, and plummet in your hand, so that the shade of the threed cut the centre of the Iewell, and then placing the edge of the rule e­uen with the same shade, you haue your desire.

Chapter 16. How by the declination of any wall knowne, to finde his eleuation with his angle of deflexion.

THe eleuation of a wal is alwaies the height of the cock, for as I haue often said, the cock of any oblique diall, must be eleuated in his due place, as much as the pole is eleuated aboue the wal wheron it stādeth. The angle of deflexion, I cal the angle made between the 12. a clock or plūb line of the wal, & the line where the cock must stand: which line is also the meridian of the horiz. represented by the wal. VVherefore reckon on the limbe of your Iewel the walles declination learned by the last chap. and that from the pole right­wards, if the wal declining East, or leftwards, if VVest (though it be not material which way but only for the demonstrations sake following,) and thereto lay the Zenith of your Reete, then reckon on the axetree line frō the centre the eleuation of the South wall or the cōplement of the latitude, al is one, & mark what azimuth of your Reete, doth cut the axtree line in that deg. for if on the same azimuth you reckon thence to the Finitor, you haue the poles eleuation to that wall, and the degrees of the Finitor between the same azimuth & the cen­tre is the 4. angle of deflexion.

Example, the declination of the first wall mentioned in the last chap. was there found 16 ½. degrees East­wards, which 16 ½. deg. I reckon on the limb of my Iewel frō the pole rightwards, thereto I lay the zenith of my Reete, which done, I number on the axetree line from the centre, the South walles eleuation, vz. 38 ⅓. degrees & there do I see to cut the 12 ½. azim. reckoned frō the zen. line, on which 12 ½. azi. coūnting frō the said 38 ⅓. de. to the fin. I find 36 ⅓. deg. which is the eleuation of the South pole to that declining wall, then on the finiter betweene the sayd azimuth, and the centre I find 12 ½. deg. which is the angle of deflexion of the same wall: & so much must the cock stand from the plumbe line, which in all perpendiculer walles representeth our me­ridian. This truely is a singuler conclusion which you will confesse, if you conceiue the reason of it, which is thus. The limb of the Iewel representeth our horizon the axtree line our meridiā or East wall: the Equinoctial the East azimuth or South wall: the end of the Equinoctial line rightwardes, the East point of our horizon, the other end the VVest: the centre the Zenith or rather his Nadir: the Finitor the declining wall, and the Zenith of the Reete, the pole Zenith of the same wall. All this so determined, and the distaunce of the South pole from our Nadir, vz. 38 ⅓. deg. reckoned on the axtree line frō the centre, there must needs bee the south pole. Therefore the 12 ½. azimuth cutting the same 38 ⅓. deg. must needes represent the meridian of the wall by the definition of a meridian which is to cut the poles and passe by the Zenith of his horizon, and because the height of the pole is alwaies the arch of the meridian betweene the pole eleuated and the horizon: there­fore the saide 36 ⅓. deg. must needs be it. But what may you see more, O notable singularity of an instrument that is, inasmuch as you see that the angle made betweene the said 12 ½ azimuth, and the axetree at their cros­sing in the said 38 ⅓. deg. to be VVestwards on the Finitor: therfore you might be sure if I had not tolde you, that the angle of deflexion must be made on your dial VVestwards from the plumbe line, for it is a generall rule, if the diall decline East, then must the cocke stand on the VVest side of the plumbe line, if VVest then on the East side.

Chapter 17. To perfourme the last chap. a second way by the 6. chap. of spherical triangles.

THough I tooke some more paines then needes in the laste chap. to shewe you some pleasure, I meane to take the more ease in this: for in truth the last chap. is onely performed by the 7. chap. of spherical trian­gles, inasmuch as the question demanded fel out to be the 2. sides vnknown of a right angled triangle, whose one side 38 ⅓ was knowne, and one of the angles not right, which was the complement of the decli­tion, that is to say, the angle betweene the axetree and the Finitor: and therefore the last chap. and the sayd 7. chap. of trian. wel vnderstood, it is as easie to doe it by the 6. chap thus: the declination of the wal giuen, videl. 16 ½. deg. take that from 90. so haue you the complement thereof, vz. 73 ½. whiche number from the Limbe a­mong the meridians in the same 73 ½. meridiā foūd, reckon y^e side giuē being the south poles eleuatiō, vz. 38 ⅓. deg. and thereto lay the rule, & so are the degr. of the rule betweene the saide 73 ⅓. & the Limbe the poles e­leuation, which you shall find, as in the last chap. 36 ⅓. deg. the deg. of the limbe between the rule and the pole, vz. 12 ½. as before are the deg. of the angle of deflexion: now iudge whether my worke of spher. trian. be not profitable, the study of which brought me to al this, and wil bring any man to much more that will consider wel of it.

Chapter 18. To performe the same a third way.

GOod God so infinite is the vse of this Iewell, that I should not know how to finish, but that want of time will driue me to hasten one end or other: here haue you now a third way to fynd your angles of eleua­tion, and deflexion, more apparant without wresting of the spheare then either of the other: I shewed before in the 15. chap. that the Finitor beyng sette to the latitude, the Zenith line representeth the South wall, and the azimuthes the declining walles, whose pole Zenithes are all scituate in the Finitor: wherefore now to come to the matter, set the Finitor to the latitude, and reckon among the azimuthes from the Zenith line to­wards the pole, the walles declination, and the azimuth there representeth the declining wall: then to finde his eleuation, you must seeke his meridian or some portion thereof, which is alwayes a great circle passing by the pole zenith of this wall, and one or both of the poles of the worlde. As for the poles you see where they are, and where the pole Zenith of this wall is, cannot be farre to seeke: for looke howe much the wall or azi­muthe is declined from the Zenith line Southwardes: so muche his pole zenith is departed from the South point of the Finitor, towardes the centre. VVherefore if you reckon thence inwards the degrees of the walles declination, there is, no doubt, his pole zenith: which being founde, looke what meridian circle you see to goe from thence streight to the pole, must needes be the meridian of the wall by definition of a meridian, and therefore marke where he cutteth the azimuth of the wal, for his degrees thence reckoned to the pole, is the eleuation no doubt, and the portion of the azimuth betweene this hys meridian, and the Zenith poynte of the Reete, is the angle of deflection. This chapter doth depende on the reason of the 24. chapter of spherical try­angles.

For example, in the wal declining VVest 53 ⅓. deg. mentioned in the 15. chap. the Finitor set to the latitude, reckon the 53 ⅓. azimuth from the Zenith line southwards, which azim. representeth the said wal: thē to fynd the walles pole Zenith, reckon on the Finitor 53 ⅓. deg. inwards from the South, & there shal you fynd the 60. meridian almost counted from the Limbe, thence taking his way to the pole, which must needs be the meridi­an of the horizon represented by this wall: therefore marke where he cutteth the 53 ⅓. azimuth, the distaunce on him thence to the pole, you shal fynd 21 ¾. deg. the walles eleuation, and the space betweene the said cros­sing and the zenith, counted on the 53 ⅓. azimuth, you shal fynd 32. deg. ⅓ the angle of deflexiō. But see more the forwardnesse of your Iewell, which sheweth vnasked the difference of longitude of this country from the countrey where to the wall is horizon: vz. 60. deg. almost the distaunce betweene the meridians, cutting the pole Zeniths of eyther place.

Chapter 19. How to make a diall to any declining wall, respecting the South.

THe making of declining dialles before now that this my Iewel wil ease it, hath byn very troublesome & paineful with a number of lineaments to be put out after the dial ended, of which the Theorical demō­stration was not shewed, as in Munster & Orontius de horologiis, you may see, whereby you had verye fewe but went blindly to woorke on them. I haue seene instructions in many bookes, how to make the horizon & South wall dialles, as namely in Albertus Durerus, Daniel Sambecke, Gemma Frisius, Curteise, and diuers other: but whē t hath come to the declining dials, they haue giuen it ouer in the plaine field.

But to the matter, you may note thus much of your selfe, that euery declining wal is an oblique horiz. be­cause you are driuen to seek his eleuation, wherby you wold think that the dial should differ nothing in ma­king from the oblique diall in the 9. cha. as in effect it doth not, after you haue placed the said angles, & foūd where you must begin to reckon your houres. VVherfore in Gods name, first make a plumb line on the wal, wherein appoint the centre of your diall: theron making a blind circle equal to your Iewels Limbe, as in the ob­lique diall you made a quadrant: in this blynd circle set the deg. of deflexion, (takē from your Iewels Limbe) as you are wont, and that below the centre VVestwards of the plumbe line, if the wall decline East: or East­wardes, if VVest: thereto from the centre draw a long line where the cock shal stand, & crosse the same lyne on the centre with another, sufficiēt for the diameter of your dial: vpō which diameter, make the circle or limb of your dial, in which write 12. at the plumb line, for that is alwaies the 12. a clock line: but now to proceede further, for the deg. of the other houre lines, set the Finitor so many deg. vnder one of the poles (the matter is not great which) as the wals eleuation cōmeth to: and so doth the Finitor represent the wall, & the Limbe the meridiā, to the wal or to that countrey whose horiz. it is. So that if you were in that countrey, you might pro­ceed to make this dial as in the 9. chap. But now to place the houre lines in him to serue our coūtrey, you must seek out on your Iewel or meridian: for the limbe is here the walles meridian, and from our meridian of ne­cessity we must haue al the houres reckoned: which to find nūber on the finitor placed to y^e eleuatiō as before, the angle of deflexion from the limbe inwards, marke what meridian cutteth the finitor, there the same is the meridian of our coūtrey, which when you haue, marke him wel that you may know him: for it is he that lea­deth this daunce, & is already represented on the wall by the plumb line. VVherefore if beginning from this meridian, you distinguish on your Iewel euery 15. going thence in order from 15. meridian to 15. quite roūd, returning at the limbe til your compasse be vp: these meridians or rather houre circles thus distinguished, shall cut the deg. on the finitor so standing to the walles eleuation answearable to euerye houre in all respectes, as in the oblique dial in the 9. chap. which deg. by help of the blind circle made before, you may place in your di­al, remembring that you must here begin from our new found meridian, & not frō the Limbe as there: & the spaces, deg. or houres from this meridiā towards the limbe of your new iewel, must be set in your diall towards the line where there the cock shal stand and so round: the reason is for that the limbe is now the meridiā of the wal, & on the dial is the line of the cock, wherby you may see that the cock will not stand out of the meridi­an of his horizon.

For example I will take in hand to make a Dyall to the wall, mentioned in the 15. Chapt. declining VVest 53 ⅓ deg. therefore the cocke must stande on theast side of the plumbeline by the ende of the 16. Chapt. First I draw a perpendiculer on this wall vz. AB by helpe of a plumbe rule, therein I pitch the centre of my Dy­all vz. E, whereon I make a blinde circle equall to the graduated circle on my Iewels Limbe, as in the 9. Chapt. whense I take in my compasse the quantitie or degrees of the angle of deflexion founde by the 16. 17. or 18. Chapt. vz. 32 ⅓ and set the same in the blinde circle East from the plumbeline at C, because the wall decli­neth VVest by which and the centre E. I draw the line ECD, and crosse the same square on the centre with the line FEG, which shall be the dyametre of my Dyall, to which on the centre E, I draw two semicircles or more, to serue for the Limbe of my Dyall as you see, wherein at the plumbline, I write 12. This prepared, I set the Finitor of my Iewel to the wals eleuation, founde in the last Chapter 21 ¾ deg. whereon I reckon in­wards the angle of deflexion 32 ⅓ deg. and there do I see to crosse the Finitor, the 60. meridian almost coun­ted from the Limbe, which is now become our meridian, and is he that we must trust to, for he is alreadie re­presented on the wal by the plumbeline.

Now because this 60. meridian is an houre circle alredie marked and distinguished on the Iewel, I neede not distinguish him a newe, nor any of the rest, for that if I reckon from 15. to 15. from him they all fall out in the ordinarie houre circles, which for the most part will not so fall out though it chaunced so here. But now being as it is, marke what degrees of the Finitor, the next fifteene houre circle from this 60. meridian towards the Limbe doth cutte: you shal finde 12 ⅓ degrees, which place in your blinde circle from the plumbeline towards the cocke, like as in the oblique Dyal, thereby draw an houreline from the centre of your Dyall, the seconde houre circle towards the Limbe, cutteth in the Finitor 20 ⅓ deg. counted still from the in­tersection of our newe found 60. meridian, the thirde towards the Limbe cutteth 27. deg. on the Finitor, these two notes set also in the blinde cir­cle, and by them drawe two houre lines more: well, the fourth houre circle is the Limbe it selfe cutting on the Finitor 32 ⅓ deg. being the angle of deflexion which is alreadie placed in your Dyall where the cocke must stande. But nowe for the fift houre you must returne backe a­gaine on the Finitor from the Limbe where you shall finde the 5. houre circle to cut in the Finitor 38. degrees counted from the saide 60. meridian to the Limbe, and thense backe againe: the 6. houre circle cutting 44 ⅓ deg. the 7.52 ⅓ degrees, the 8. 66. deg. all which placed in the Dyall as before, you neede go no further that way, for that the sunne at the most in our country yealdeth but 8. houres before or after 12. Now for the hourelines on the otherside of the plumb­line, begin againe from the said 60. meridian, but the contrarie way towards the centre, and you shall finde the first houre circle to cutte 24. degrees: the seconde 57 ⅔ deg. which, place in your Dyall, and you haue done: For beyonde the dyametre you neede neuer go in a perpendiculer wall. Lastly the hourelines thus drawne, their numbers a childe almost may set in by the 12. of clocke, or by this figure. And as for the cocke let him be erected verie plumbe from the wall in the line of deflexion drawne for him: his length may be equall to the Dyals semidiametre, or shorter if you will, but his height must needes be to the walles eleuation vz. 31 ¾ and thus haue you a singuler peice of skill, which being wel vnderstoode is more easie then to do it Geometricallie after Orontius or Munster, but ere I finish my Iewel I will teach to do it Geometrically more easie then theirs by farre, hauing but a modicum of my Iewels helpe.

Chapter 20. How to make a Dyall to any declining Wall respecting the North.

THere is no diuersitie of making a North declining Dyall, from the South declining, mentioned in the last Chapter, but that where as there you did drawe the circumference downewards vnder the centre towards the South pole: heere you must make him vpwards towards the Zenith, respecting the North pole, and then following on the rule in all respectes as in the last Chapt. Namely if the wall decline East, set the cocke on the VVest side of the plumbeline and so contrariwise: But now is the plumbline 12. of clocke at midnight, whereby you may number the rest of the houres, the cocke also must be set vpwards, according as I tolde you in the 11. Chapt. and the superfluous houres of the night left out. There is so little diuersitie in the making, that looke what Dyall you make to a south wal declining west the very same Dyall will serue to a North wall declining west, like much, if you turne him vpsidedowne and alter the numbers: and so like wise the South declining East, serueth the northeast as by this figure you may plainely see, being but the Dyal of the last Chapter, turned topsitoruie and the numbers of the houres altered vnto those which I haue placed [Page 106] at the endes of the prickt lines as you see, and nowe serueth to a North wall declining VVest 53 ⅓ degrees, whereto the sun commeth not before 2. of clock afternoone, and there continuing til eight of clocke at night, all the rest of the houres be superfluous.

And heere if I should conceale one thing, I should offer you iniurie and hinder my selfe of that I pretende which is breuitie and facillitie: I will therefore that you know if the patterne of your Dyal be made in paper first, as I commonly vsed, and that the paper were oyled or prickt so that you might see the figure of this Dyal on the backside, the same doth represent the cocke turned vpwards, a North Dyal declining East 53 ⅓ degrees & the cocke turned downwards a south wall Dyal declining East 53 ⅓ deg. so that you may by one paterne make 4. seuerall Dyals as by these figures you may see, vnto which the figures at the end of the prickt lines ex­presse the houres.

Chapter 21. How to order Dyals that decline very farre East and west.

I Shewed you before in the 12. Cap. that euery east & west wall is a right horizon, hauing the hourelines of the Dyall parallel, & the pole nothing eleuated to that wall, because he standeth in the meridian & cutteth the pole: wherefore it must needs appeare that the more any wal declineth to­wards the East or VVest, the lesse the pole is eleuated, euen vnto nothing at the last, whereby the hourelines neere the cock or meridian of the wall, grow so neere togither that to­wards the centre they thrust one vpon another, & farther off they shew almost parallel: for in a wal declining 80. deg. east or west, some of the houre spaces are not aboue 2. deg. or little more, which space except the dial be monstrous big, cannot be sufficiently discerned, wherefore you may cut of the dyal, cock and all, with a fashionable square like an east Dyal, as you see done in this figure, which thē sheweth almost like vnto it, you may make him first in paper, & then hauing cut off all super­fluitie, set the 12. of clock line euen with the plumbline, and so by the paper prick the other houres.

Chapter 22. Of uals reclining and inclining, and of the diuersitie of them.

I Shewed in the 14. Cap. the diuersities of declining wals and that euery perpendiculer wal declining or not declining lay in one or other of the azim. their pole zenithes being all of them in the horizon or Finitor. In like manner reclining wals are flats, whose bases lie leuell with the diametres of one or o­ther of the azim. crossing all vpon the centre of the earth, but their flats recline or beare backe from the Zenith, & their pole zenithes are eleuated aboue the horizon in the azim. standing square to their base as much as the reclination commeth to. The inclining wall or flat, for his base he is like the reclining, lying in the diametre of one of the azim. but his face or flatte is inclined forwards, lying groueling from the zenith: in so much that his pole Zenith is sunck so muchvnder the hor. as the inclination cōmeth to, which by this figure you may conceiue, wherin the circle AFBG is the horiz. the centre C the Zenith, the streight lines crossing at the [Page 107] centre represent both azimuthes and their diametres, the arch ADB is a reclining wall, whose base is the line AB, being the dyametre of the azimuth ACB, this wall ADB reclineth from the zenith as manie degrees as CD containeth, wherefore his pole zenith shalbe eleuated in the azimuth square to his base vz. FCG, as much as CD commeth to, and is the point E, so that G E is equall to CD.

Likewise the arch ADB representeth an inclining wall, whose base is AB also, bending forwards towards the point F from the zenith C, according to the quantitie of CD, whose pole zenith is the point T, which you must imagine depressed or sunke vnder the horizon AFB, in the same azimuth FCG, as much as D inclineth from C, or as the point E is eleuated aboue the horizon, and so of all the rest in this figure. And thus vpon the dia­metre of any azimuth whatsoeuer, there may be imagined both a reclining and inclining wall, nay rather a number of both kindes according to the reclination of the point D from the centre C. Then to proceed further, know that as the diuersitie of perpendiculer wals are east, west, north, south, and declining, so is there the like of these, for either the base of reclining, or inclining walles lie in the dya­metre of the East azimuth, as AB, and then are they called North or South, reclining or inclining, or in the diametre of the meridian or south azimuth, as FG, and then are they East or VVest, reclining or inclining, or else in the diametre of some other of the azimuthes, as KL, or QR, or any o­ther, and then are they called reclining walles declining and inclining wals declining: which you may deuide and subdeuide, I meane their denomination as in the perpendiculer wals in the 14. Chapt. as thus, KOL is a south reclining wall, declining East, because his pole Zenith is the point P, his reclination is the space CO, his declination RG, also KLO is a north inclining wall, declining west, whose pole zenith is the point N vnder the horizon, & thus you may spell them all, and as many other as you list to imagine, tu ipse annitere rotis.

Chapter 23. How to take the reclination, or inclination of any wall or flat, togither with his declination if any be.

VVHen you come to any reclining flatte, first by helpe of a Masons leuell or some other deuise, draw thereon a line parallel to the horizon, which by helpe of your Iewel you may also thus do, sticke a long stile or wyer vz. A plumb for the flat BCDE, and then setting your Iewel to your eye, the rule with sights lying on the line of leuell, go euen with the side of the flat through the sight, behol­ding the stile, and let some other bodie sticke vp another stile vz. F, directly betweene your eye and the first stile, plume also from the flat: then drawing a line by both these stiles vz. AF, the same shalbe leuell with the horizon: this horizontall line I will call the base line of the reclining flat, to which I drawe another crossing him square, which I will call the bases per­pendiculer, both which drawne, I then applie my Iewel by helpe of the square boorde or squire, men­tioned in the 15. Chapt. vnto this base line leuell with him horizontal wise, and so by the doctrine of the 15. Chapt. take the declination of this reclining flat in his base line, euen as though it were a perpen­diculer wall: which I note downe in a paper or ta­bles, but if this base line haue no declination, that is to say, be full East, VVest, North, or South, then is the same flat barely reclining, and not reclining de­clining, which reclination to take do thus: erect two short stiles in the bases perpendiculer, both of equal height aboue the flat and plumbe thereto: then if the flat be a slope wall, as of a buttresse or such like, hang your Iewel to your eye, remoouing your rule with sightes till you may through them see the very tops of the two stiles, and so are the deg. of your Iewels Limbe counted from the line of leuell, the mounting of this flatte aboue the horizon, the complement wherof being the deg. betweene the rule and the zenith, or handle of the Iewel is the flats reclination, but if it chaunce to be the bancke of a garden, or any other that you may go a­boue it, then is your best way to place your selfe aboue the same, and to looke downewards through your sights by the ends of the two stiles.

For example Anno Dom. 1581. A Gentleman of auncient house and great worshippe, Humphrie Fo­ster Esquire by name, being elected heigh shiriffe of the countie of Berkes, bestowed the same yeare some cost about his house at Aldermarston, where amongest other thinges, hee trimmed vp two Gardens with walkes vppon heigh slope banckes as the order is: and for because one of the banckes lay verie open to the sunne, and to the prospect of most of the Chambers and roomes of the house, it pleased him to vse my simple helpe, being his verie neere kinseman, and such a one as not without greate cause hee might commaunde, to make a greate Dyall at the whole bredth of the bancke being almost three yeardes deepe: wherefore I laide astreight boorde on the toppe of the bancke directly with the edge thereof, which [Page 108] serued me for my base line, because the banck was leueled, of which I tooke the declination by the 15. Chap. and founde it 78 ½ degrees, then did I by the first conclusion direct a perpendiculer from this base line, vnto the foote of the bancke, wherein insteede of stiles I stucke two kniues of equall height aboue the banck: and then choosing my standing on the heade of the bancke, so that I might through the sights of my Iewel see the verie toppes of both knyues, I found the end of the rule next my eye to cut on the Limbe 34. deg. which coun­ted from the line of leuel which was the mounting of this bancke aboue the horizon, the complement where­of vz. 56. degrees, was his reclination from the zenith, being the degrees betweene the rule and the handle, or toppe of the Iewel, so that I was to conclude that this bancke was a South reclining flatte 56. deg. declining West 78 ½ deg. whereto I will shew to make the Dyall in the 28. Chapt. But now for to take the declination and inclination of any flatte, the working is all one, as by experience you shall better finde then by manie words.

Chap. 24. How to make a Dyall to a south or north reclining or inclining wall, bancke or flatte.

A South or North reclining or inclining flatte is such a one by the 21. Chapt. whose base lieth in the East line of the horizon, and his pole zenith in the meridian circle, which well vnderstoode, and the reclinati­on taken and knowne by the last Chapt. the making of the Dyall thereto is most easie. First set the Fini­tor of your Iewel to the latitude: then if the flatte recline or incline Northwards from the zenith, remooue the zenith line of your Reete so many degrees towards the North pole, as the reclination commeth to, or towards the south pole, if southwards: and you shall thereon see presently which pole, and howe many degrees the same is eleuated aboue your flatte, to which eleuation make your Dyall, either by the 9. or 10. Chapt. placing the cocke and 12. a clocke in the bases perpendiculer southwards, if on your Iewel you finde the south pole e­leuated: or northwards, if the north pole.

As for example, admit that in our latitude being 51 ⅔ deg. by the 23. Chapt. I founde a south flatte recli­ning 20. deg. First I set the Finitor to our latitude by the 2. booke 19—Chapt. there I finde the zenith point of the Reete to shew 38 ⅓ deg. counted from the north pole. Thence I remoue him according to the reclination found vz. 20. deg. more from the southwards, where I finde the North pole distant, or rather depressed 18 ⅓ de. vnder this flat, being the zenith line: but looking towards the other end therof, there I finde the south pole 18 ⅓ degrees eleuated aboue this flat by the 10. preamble. For this is generall that the zenith line, I meane drawne forth vnto his Nadir, supplieth al flats, whose poles zenith & poles Nadir lye in your meridian circle. VVherefore now by the 9. or 10. Chapt. making an oblique Dyall to 18 ⅓ eleuation, his cocke and 12. of clocke standing in the bases perpendiculer turned towards the south, you haue done. And the same Dyall the 12. and cocke turned north, serueth the north flat in our latitude inclining 20. deg. But if a South wall be re­clining, or a North wall inclining according to the Equinoctials height aboue our horizon vz. 38 ⅓ degrees. then will the zenith line one your Iewel lie in the two poles, so that neither is eleuated. Wherefore by the 8. Chapt. you may know the same flat to be a right horizon, in which Chap. the Dyall is alredie made thereto. If a South wall recline or a North wall incline more then the Equinoctials height in your latitude: as admit with vs 60. deg. your Iewel first placed to the latitude, remoue the zenith line towards the north pole and be­yond, till he come to 60. deg. from his right place, and you shall there finde the north pole distant, and eleua­ted aboue the said south reclining flat 21 ⅔ deg. and the south pole eleuated or rather depressed vnder the said North reclining flat as much vz. 21 ⅔ deg. for the zenith line representeth them both: to which eleuation you may make their Dyals by the 9. & 10. Chap. as is aboue said. In the very like manner is it with north reclining and south inclining flats vice versa.

Chapter 25. How to make a Dyall to an East and West wall, bancke or flat reclining or inclining.

THe base line of euery East and West reclining and inclining flat lieth directly in the meridian line, as in the 21. Chapt. and in truth euery of themselues lyeth in the flat of one or other of the circles of positi­on: (that is to say) they all crosse vpon the intersection of the meridian with the Horizon, as the decli­ning wals do in the zenith and Nadir, their poles zenith and Nadir lying all in the circle of the East, called of some the verticall circle. What a circle of position is, looke the 3. Booke 43. Chapt. The Dyall to euerie of these is made in all respects as the declining wall dyall. And to make the matter more plaine to the learner, whom I would as willingly helpe as I onse could haue beene contented to be holpen, know that all reclining and inclining Dyals of this sort are the very declining wal Dyals to that country which hath a south latitude equall to the complement of your latitude being heere at Reading 38 ⅓ degrees. Wherefore if you did but imagine your selfe in a South latitude, of 38 ⅓ deg. & turne your Iewel vpsidedown accordingly as you do here in your owne, you may easily perceiue the zenith line to be there your verticall circle, and all the Azimuthes your declining wals, which are heere circles of position, and may make your Dyals to them by the 19. Chapter. Nowe notwithstanding I thinke it plaine ynough to some alredie, yet I will not sticke for o­ther some to prosecute this Chapter demonstratiuely, the rather because I haue set but one example of de­clining VVall Dyalles. Admitte therefore heere in our latitude being 51 ⅔ at Reading: by the 23. Chapter, I founde an East VVall reclining from our zenith 50. degrees, I call that an East reclining [Page 109] wall or flat, whose pole Zenith is in the East quadrant of the verticall circle. To make a diall to this flat, you shall in steede of the finitor, set the Zenith line to the latitude, vz. 51 ⅔. degree, vnder the North pole, so shall the azimuthes represent all the East and VVest, reclining or inclining flats, and the finitor line now supplieth the Zenith line for this turne. Then since my wall proposed, reclineth 50. degrees from the zenith, I reckon so many degrees on the finitor from the Limbe inwardes, and there doe I finde the 50. Azimuth representing my flat sought for: aboue which also I may see the North pole eleuated: but howe much, since it appeareth not, you may thus finde. Marke which of the almicantares cutteth the pole, and where he crosseth this 50. azimuth: if thereto you set the labell fixed: the degrees thereof betweene this crossing and the limbe, you shall finde 36 ¾. which is the poles eleuation aboue that flat, and the degrees of the limbe betweene the label, and the pole, 39 ⅔. the angle of deflection. Thus may you finde the eleuation and deflexion of any of the azimuths either in this constitution or any other like, as declining walles, &c. For it is but the rule of the 17. chapter, somewhat more plainely set downe for this purpose. These had, you shall draw through the middest of your reclining flat, his base line parallel to the horizon, vz. AB, which, as is said, in all these dialles, lyeth directly North and South, euen in the meridian line. In this base line AB, pitch the center of your diall, vz. D, and thereon describe a Limbe for your diall to write the numbers in, and within it a blind circle equall to your Ie­wels limbe, wherein set the note C, 39 ⅔. de. (being the angle of deflectiō) aboue the base line Northwards: be­cause the North pole is eleuated, drawing foorth the line DC, where your cocke must stande in all respects, as in the 19. chap. you did in the declining wall, from which this little had differed, if in steed of the reclina­tion, I had taken this complement being 40. degrees, vz. the mounting of this flat aboue the horizon: and re­koned this 40. from the Zenith line, where here I counted 50. from the limbe, all commeth too bee one azi­muth.

But to proceede, now you must remoue the finitor of your Iewel to the walles eleuation founde) that is to say) set him 39 ¾. deg. vnder the pole, wheron from the limbe inwards, reckon the angle of deflection, vz. 39 ⅔. deg. which angle alwayes hunteth out our meridian, for there shal you see to cut the finitor, the 54. meri­dian counted from the limb, which is he frō whence we count all our houres: being in this figure, the line AB, the limbe of the Iewell is as hath been saide in the 19. chapter, the meridian of the flatte, and representeth the line DC, where the cock stādeth. In this figure frō the crossing of this 54. merid. with the finitor, reckō to the next 15. meridian towardes the limbe, which you shall see to cut in the finitor 1.3 ⅓. deg. frō the crossing: those deg. I set in the blinde circle from the base line AB, towardes DC, the line of deflection, and thereby draw a line thorow the same to the dials limbe for my first houre on that side: then look I on my Iewell to the next 15. meridian, being 30. meridians from the said crossing he cutteth on the finitor 24 ½. deg. counted from the crossing for the second houre line. Then the next 15. meridian beyonde that, he cutteth 34. deg. of the fini­tor for the third houre line: these two houres I set likewise in my diall: then haue I but 9. meridianes more to the limbe, therfore I must reckon thence backe againe tyll 15. be vp, which are 6. deg. backe, reckoning on the finitor from the said crossing to the limbe, and thence backe againe to the same 6. meridian I finde so 43. de. for the fourth houre: which being set in my diall, falleth out beyond the line of deflection, so that heere the cocke standeth betweene two houres lines. And so proceeding for the rest, as in the declining diall: you shall finde 53. deg. for the 5. houre line, 63 ½. for the 6. 76. for the 7. 93 ½. for the 8. and so foorth if neede were. But in our countrey 8. houres from the 12. any way is enough at the most. In like maner for the houres vn­der the base line, count on the finitor from the said crossing towardes the center of the Iewel, and you shall find 17 ½. deg. for the first, 40. for the second, 65. for the third, 86 ½. for the 4. and so foorth if neede bee. All these seuerall notes made in the blind circle, and houre lines by them drawen to the limbe of your diall, and your cocke set Northwardes plumbe from the flat in the line of deflection CD, equall in height to the walles e­leuation, vz, 36 2/4. deg. hygh, your dyall is made. The base line AB, beeing that it lyeth in the meridian, must needes be 12. of clock. The rest of the numbers you may by reason set to the other houres. All which by the [Page 110] figure before you may most plainely see.

Then for East and VEest inclining flats, the dials are made in very like maner, sauing that the line of de­flection lieth Southwardes, & vnder the base line the cocke beholding the South pole, so that this diall would serue to a VVest inclining flat, 50. deg. being turned vpside downe, & the numbers altered: of all which my minde giueth me that my Iewel sheweth the reason so theorically that I neede spend no more speech thereof.

Chapter 26. How to finde the angles of the poles eleuation, dials deflection, and meridians ascension, to any reclining or inclining wall declining.

BY the title of this chap. you may see that there belongeth to the making of a reclining diall many angles which must first sphearically be gotten before the diall can bee made: namely his reclinatiō frō the Ze­nith, his declination from the North or South in his base line, his deflection from your meridian, and nowe lastly another angle lept in, which is the ascension of your meridian aboue the base line, which in the last diall were both in one selfe same line: therefore what did I say in the 19. chap. that the declining dialles were harde to make before time? It is not so perhaps, if that be true, then were inclining and reclining dials more harde, nay most hard, yea and so intricate, that I haue not seene any to haue perfourmed them, but on­ly Andreas Sconerus, yet hee without any theoricall demonstration. But it is such an intricate peece of work and the worse by misprinting that he himselfe had neede come againe to vnderstand it. There must be such a sort of circles, lineaments, such a number of quadrants and deuises drawen and diuided vpon the wall or paper in suche abundance ere the diall can beee made that a man woulde thinke it a wildernesse of lines ra­ther then any reguler peece of worke. Tell me then, if the diamond be set by, in that he is rare and shineth in the darkest places, and cutteth deepe where nothing els wyll scratche, who dare cauill that I call my in­strument a Iewell, that sheweth the most hidden & difficult matters of the speare as broad as day, and perfor­meth the hardest with plainesse & ease. But now to come to this defuse matter which hath balked a great many who haue been very busie to shew their skill in other slighter dials.

VVhen you haue the reclination of a wall, flat, or banke, together with the declination of his base line, as in the 33. chap. is shewed: First place the finitor in the axtree line of your Iewel, which here shall be your meri­dian, thence remoue the finitor frō the South pole towards the left hād, if it be of a South flat reclining, & his declination VVest or rightwards, if East according to the deg. of declination, and so shall your Iewell remayne in an excellent constitution for this purpose As in example of the banck mentioned in the 23. chap. the de­clination of his base line was 78 ½. from the South VVestwards, the reclination 56, deg. VVherefore the fini­tor, first laid in the axtree, which heere I meane shall represent our meridian: remoue the South end thereof left wards 78 ½. deg. which done, you shall finde the Zenith point of the Reete, standing 11 ½. degr. from the North pole. Now vnderstand what your Iewell doth here represent. First the limbe is our horizon, the axtree line is the East wall or meridian in whiche the North pole must needes be found, the Equinoctiall is the South wall or East azimuth, the center is here the Zenith of our horizon, the right hand at this worde meri­dies is East, and the left hand at media nox VVest, in respect the Iewell is heere imagined to lie flatte, the Zenith line representeth the side or base line of the banck, the finitor the bases perpendiculer, wherein the reclination is taken. Lastly, if on the finitor from the center you reckon the reclination of the banck, vz. 56. deg. there haue you the azimuth representing the bancke or rather the horizon, wherein the flat of the banck lyeth, whereto our dyal must be made. Now sir that you see the standing of all the circles of the spheare needfull for thys purpose in theyr due constitutiō, it is some cunning yet to get those angles, which this chap. proposeth, not­withstanding one of them is there at hande: for if you reckon from the Zenith poynt of the reete, the deg. of the said 56. azimuth vnto his crossing with the axtree line, you shal finde them 13 ⅔. deg. which is the angle of our meridians ascension, that is to say, our meridian crosseth the base line of this bancke making an an­gle about it of 13 ⅔. deg. for the other angles, first reckon in the axtree line, which now is our meridian as in the 16. chap. you did the poles eleuation, vz. 51 ⅔. deg. from the limbe beeing nowe the horizon, there make a prick with inke for the pole: for now if you can deuise with all your skil where y^e pole Zenith of the bank lyeth on your Iewel, and from him to lett downe an arch of a great circle, cutting the saide pole or pricke of inke in the axtree line, hee must needes represent the meridian of the banckes horizon, by the 1. book and 4. chap. and stande perdendiculer thereon, wherefore his portion or arch betweene the 56, azimuth, and the sayd pole or pricke must needes bee the poles eleuation & the deg. of the 56. azimuth betweene it and the axtree shalbe the angle of deflexion. Now to make this matter plaine by demonstration, reason thus. The Zenith line representeth a wall declining VVest 78 ½. deg, that is to say, a wal perpendiculer on the base line of this bancke, therefore his poles Zenith & Nadir must needes bee in the endes of the finitor: but since this bancke reclineth from the Zenith beyng the center of the Iewel 56. deg. rightwardes: therefore his pole Ze­nith doth come in as much on the other side from the limbe towardes the center in the finitor line, as in the 4. booke and 4, chapter, and 5. booke and 24. chap. hath been saide. This 56. deg. on the other side of the finitor founde, you shall nowe for necessties sake lay the label on the axtree line, and therein make the foresaid prick for the pole, and there the label fixed to the reete, you shall turne about the reete and label so fixed tyll you find some of the meridians of the mater too cut both the saide pole zenith founde in the finitor, and the saide pole pricke in the label, which you shall finde to bee done by the 65 ½. meridian counted from the limbe there stay your reer, and reckon on the same 65 ½. meridian the degrees betweene the saide pricke in the labell, and the saide 56. azimuth which you shall finde 35 ½. deg. the same is the eleuation of the pole to this bancke. Then reckoning on the 56. azimuth from his crossing with this 65 ½. meridian vnto the fiduciall line of the labell, which nowe representeth our meridian, you shall finde 28. deg. the angle of deflexion, and so much must the [Page 111] cocke stande aboue the meridian: and thus haue you gotten all those angles, beholde the figure of the nexte chapter, which if you marke well doth after a sort represent your Iewell, in the constitution of this chapter.

Chapter 27. How to perfourme the last chapter by sphearicall triangles very easily, and with great pleasure and profite.

SOme perhaps will scant giue credite to the finding of the angles in the last chapter, perhaps mistrusting my skill, since so many learned men haue been contented to passe by them: some againe may happily not well vnderstande that hath been saide, and other some, I presume loue more wayes to the wood then one who knowe well that change of pasture maketh fat calues, and therefore to all sortes I thought thys chap. very conuenient. You shall as I taught you in my booke of triangles, drawe in a figure suche circles of the spheare, as you are to deale with in this question, as here I haue done it to your hande, partly by ame, whiche for this turne serueth as well as the truest draught, where in the circle ABCD, representeth our horizon AB. our meridian, CD, our East line, EF, the base line of the banck proposed, GKH, the azimuth crossing hym square, in which the banckes pole zenith must needes be by the 22. chapter ENF, is the great circle of the spheare wherein the flat of the banke lyeth: A, the North poynt of our horizon B, the South, D East, C VVest, the center K, our zenith KN, the reclination of the banck, vz. 56. deg. I, the pole zenith of the bancke eleuated aboue G, as muche as N reclineth from K, by the 4. booke and 4. chap. G, the pole zenith of hys base line EF, or rather of his perpendiculer flat EKF, declined from the South pointe B, according too the quantitie of GB, founde in the 23. chapter to be 78 ½. degr. the pointe L is the North pole eleuated in our meridian circle AKB, aboue hys North point A, accordyng to the quantitie of AL, being 51 ⅔. degrees.

Your plate being thus laide and well vnderstood, then begin to descant on it thus. And first too get the e­leuation of this bancke since the angle GKB, is knowen to be 78 ½. deg. his complement BKF, shal then be 11 ½. deg. by the 5. booke, and first chap. to which the angle EKA, is equall by the 5. theoreme: EG is a quadrant or 90. deg. to which adding EA, being 11 ½. deg. it maketh 101 ½. the angle AKC, then AL, beeing as is saide 51 ⅔. leaueth his complement LK, 38 ⅓. deg. likewyse GI. beyng 55. deg. leaueth hys complement KI, being the distaunce of the banckes pole zenith, from our zenith 34. degrees. Heere then haue you a sphericall triangle, videlicet, IKL, whose two sides IK, 34. deg. and LK, 38 ⅓. deg. togeather wyth the angle IKL, 101 ½. deg. are knowen, wher­fore by my 13. chap. of triangles, the side IL shalbe founde 54 ½. deg. which is the distance betweene 1. the pole zenith of the banke, and L the North pole, the complement whereof beeing LM, must needes be the eleuation sought for, vz. 35 ½ degr. Then for the angle of deflexion, you shall vnderstand that the same is the arch of any flats horizon comprehended betweene the meridian of the place, and the meridiā of the flat: as in this figure AKB, is our meridian, be­cause he passeth by the pole L, and our zenith K, and ILM, is the meridian of the bancke, because it passeth by the pole L, and the banks zenith I, by the defini­tion in the first booke & fourth chapter: and therefore the arche OM shalbe the degrees of deflexion, which to finde descant thus: since by the 5. booke & 13. chap. you haue founde the side IL, of the triangle IKL, and also the angle ILK, or at least if you haue not, you may: the angle OLM, is equal to it by the 5. theo­reme: the angle OML. is a right angle by the 3. def. and 3. theoreme. VVherfore to be short, here you haue the right angled triangle LMO, whose one side LM, 35 ½ deg. & his one angle not right OLM, are kno­wen: wherfore by the 5. bo. 9. cap. the side OM, shalbe found 28. de. Now lastly to the angle of the meridiās ascen. which is the arch of any flats horiz. cōprehended between the meridiā of your place, & his base line, which in this figure is the arch EO, for which you haue the right angled triangle EOA, of which the an­gle A is a right angle by the 3. def. & 3. theor. & the angle AEO, is knowen, because by the 4. def. it is de­nominated by NH, being 34. deg. the cōplement of y^e recli. the side EA, is said before to be 11 ½ deg. where­fore by the 5. b. 9. c. you shal find the side EO, 13 ½ deg. which is the angle of ascens. sought for, & so is y^e side OA, 7 ½. de. which is the mounting of our merid. aboue our hor. in this bank or flat, if we had need therof.

Many wayes might you descant vpon this figure, to come by the premises, as by a little exercise you shall finde: as for the purpose another way, and that a shorter way, is thus: if you will begin with the meridians ascension, you may first finde all the sides and angles of the right angled triangle EAO, as I haue last before shewed. And after that haue you the right angled triangle LMO, whose one angle not right LOM, by the 5. theoreme is equall to the angle EOA, of the former triangle knowen, and the side OL, is knowē by sub­ducting the side of the former triangle AO, out of AL, the 51 ⅔. latitude. Therefore againe by the saide 9. chap. you shall know the side OM, which is the quantitie of the deflexion, and the side ML, the poles eleua­tion aboue the bank sought for: & so I cōclude this chap. leauing the rest to your diligent indeuour.

Chapter 28. The angles of reclination, delination, eleuation, deflection, and ascension had as before, how to make the dial to a reclining wall, bancke, or flat declining.

HAuing spent ouermuch speech about dialles alredie I feare mee, and yet like to spende more, the mat­ter yeelding it self in such abundance, I wil for breuities sake proceed streight with an example, which shall bee the diall I made to the bancke in my good cousins garden at Aldermarston, mentioned in he 23. chapter, where because I wold haue my work last, and be surely placed, I caused a square to be made of foure peeces of timber, eche twelue inches broade, and in length three yardes, equall too the depth of the bancke, as in this figure you see ABCD, and to be set in euen with the turfe: so that of the lower peece, the vpper edge AB, did serue for the horizontall or base line of the banke.

Then did I appoint the center of this diall in the lower angle Southwardes, vz. in A: In which point A, I placed the center of a woodden se­micircle, which I had alwayes readie for suche like purposes, his diameter being 3. foot in lēgth, which I laid euen in the base line AB, as in this figure you see: for on the grasse I could make no blinde circle: whichdon, fastening a threed in the center A, I ex­tended the same so many deg. aboue the base line AB, in this wooden semicircle, as our meridianes as­cension founde in the 25. or 26. chapter came to, vz. 13 ¾. deg. whiche threede or line crosseth the side BC, in E, this line AE, is the meridian too vs here at Reading or Aldermarston all is one, so little differeth not being but 8. miles betweene: where­fore being so, it must needs be also the 12. a clock line, for in all dialles the 12. is in the meridian of the place: that done, I extende the threede from A, to so many degrees in the semicircle aboue the me­ridian AE, as the angle of deflection founde in the 25. or 26. commeth to, vz. 28. and it crosseth the side BC, in P, this line AP, is the place for the cock, and so is this bancke prepared for this diall. Therefore nowe am I too betake mee too my Iewell, whence I must fetch all the houres spaces. VVherefore I set the Finitor vnder the pole according to the eleuatiō of the bank found in the two last chapters, wher­on then I reckon the degrees of deflection, vid. 28. degrees from the limb inwardes, there I finde to cut the 43. meridian, which as I told you in the 19, chap. becommeth our meridian, and representeth the line AE, in the bancke, the limb being in this case the banckes meridian, vz. the line AP, wherfore you may proceed euen as you did in the declining diall imagining the meridian AE. to bee the plumbe line on the wall: as for example, I reckon on my Iewel too the 15. meridian from the same 43. meridiā towards the limbe, which I find cutting in the Finitor 11 ½. degr. for the first houre, wherefore extending the threed from A to 11 ½. deg. in the said semicircle aboue the line AE, I draw there­by the first houre for one of the clocke: and so proceeding in all respectes with the rest, euen as is done in the diall of the 24. chapter, as by this present figure doth more euidently appeare, then by manifolde words: wherein I haue set to the degrees of euery houres space, for your better satisfaction, Ther is no difference but that there I began the houres from the base line, the reason was, for that the same was there the meridian. The diall being made in maner of this figure, I made the cocke of a barre of Iron issuing a slope out of the center A, according to the eleuation of the [...]cke being 35 ½. deg. standing plumbe ouer the line of deflec­tion AP. This barre was supported with two other, on ech side one: but in placing of the first barre the pretiest matter I would note vnto you is, that when as by a quadrant his center fixed in A, you lift vppe the slope barre too this 35 ½. eleuation if you let downe a plumbe line from the vpper edge thereof hee then shall be truly placed when as this plumbe line hangeth euen on the 12. a clocke line, vz. AE.

Chapter 29. Of inclining dials reclining.

THE reason, order, and making of these is all one with the reclining, but most commonly vice versa, as it was in the 19. & 20. chapter betweene North and South declining dials, the one vpwardes, the other downwardes: In which the very constitution of the Iewell vpon any cituation proposed, as hath been shewed will instruct any man of iudgement more by the bare beholding of it then I should doe if I woulde contend to write great volumes of all the diuersities which are very many. For as well of the reclining as inclining, some will fall out to bee right horizons, some oblique, some haue the South pole eleuated, some the North pole: in some our meridian lighteth aboue the base line, as all doe yet that I haue shewed, in some a­gaine our meridian wyll light vnder the base line especially in inclining dials, and then may it bee called the meridians descension. Al these thinges the Iewel sheweth for the asking, of which he that well vnderstandeth [Page 113] all this sixt booke cannot bee to seeke, and for him that vnderstandeth not the same, it were great pitie too trouble him with more. I presume that I haue so theorically, or rather vranically shewed the making of di­als in this booke, that I may turne you loose to the rest: and therefore if you giue it ouer for once reading you learne not that of me.

Chapter 30. Howe to knowe in what countrie any declining, reclining, or inclining diall, woulde or of right shoulde serue as an horizon diall, and also to make him shewe the houres of the same place together with yours.

EVery man that hath yet taken vpon him to be the authour of diall making hath delighted too adde some noueltie of deuise with them: as in Sconer, Orontius, & Munster you may see: of which the put­tyng in of the signes with the plenetarie houres, and such like hath beene most taken vp and accounted of. Therefore I am to bee borne wythall, if I augment them with a deuise or two of mine: since then there are no declining, reclining, inclining, or East or VVest dials (the North and South reclining & inclining on­ly excepted) but in very deede are dyals of other countries, wrested from their owne deliniation, to serue our turne heere, as in the sixt chapter I haue saide, and that the true meridian of the diall to his owne coun­trie, is the line where the cock stādeth: therfore to place other houre lines in any declining or reclining diall to serue his owne longitude shall bee more easie to doe then the declining houres. For when by the 19. 24. and 27. you haue placed the houres of your countrie by helpe of your newe found meridian, as there hath been shewed, you shall againe (your Iewell standing as there too the wall or banckes eleuation) count these newe houres from the limbe in very like manner as in the oblique and South wall diall you did: & place them in your dyall on each side of the line of deflexion like distaunce as in the nienth & tenth chapter, they were on eyther side of the meridian, that is to say: as you finde the houre circles distaunt from the limbe of your Iewell by degrees of the finitor, so place the houre lines of your diall like distant from the line of de­flexion by degrees of the blind circle, among the other before made, but with red or some coloured lines and figures for distinction.

This line of deflexion being as is saide the meridian of the other place must needes bee the 12. of clocke line to these new houres, from the which the other houres may easily be numbred accordingly, as by this figure you may better see then by much speech, which is the di­all made, in and by the nienteenth cha. in respect of the blacke lines and figures: and the prickte houre lines and figures are drawen and placed by the 9. or rather 10. chapter, according to the walles eleuation being 21 ¾. deg. euen as thogh he did not decline: that is to say, the finitor being set as in the nienteenth chapter 21 ¾. degr. vnder the pole all those prickte houre lines are coun­ted on the Finitor from the limbe, and not from the 60. meridian, as there the other were, and so set on each side of the line of deflexion. Then at the last to conclude our purpose, this rule is ge­nerall, that the degrees counted on your Iewel a­mong the meridians betweene the Limbe & that meridian from whence the houres are counted by the 19. 25. and 28. which in this diall are 60. is the differen. of longitude of such countries frō ours, whereto the prickt houre lines serue, or more plainely, the diall of right ought to serue, which if the diall decline East must bee added, but if VVest as this doth, then subducted from yours, vz. 60 out of 14. our longitude here at Reading: which cannot be taken: therefore adde (as in all suche caces you must) 360. vnto 14. it maketh 374. then take 60. out of that, you haue left 314 the longitude sought for, of all such places to which the shade of the cock among the prickt houre lines yeeldeth the houres. Nowe to know what countries lie vnder that longitude, you must repaire to Ptolomees, Apians, or any other mans tables (you think best) of regions, cities, prouinces, &c. and seek howe many places you can finde of the same longitude, write them in some voide roome of your diall, for to all those doe the prickt houre lines serue. I haue sought in Apians tables called Abacus regionum & can find very few of or neere this 314. long. whereby it should seem that either the world is not much discouered on that coast, or els that the Sea is large that way: only these I noted out of Carthaga Insula lying in America, whose longi. I find 315. deg. 10. min. which is very neere. Also another Iland in America called Spagnolla, where is great store of Guiacum, whose middest is in lon­gitude 315. deg. Also two other partes of America in the land of Peru, called Montana Altissima, in longi. 312. & Caput de stado 317. which are neere enough so they exceed not 3. deg. all these I write in this diall. Too bee short, you see that when the Sun giueth 12. of clock to vs it sheweth to any of those places, but 8. of clocke among the prickt houres, & this is that which in my 1. book 12. cha. I mentioned. You wil say perhaps that all these differ in lat: That is not here materiall, for the long only maketh the difference of houres. VVel, the latit. may cause the Sun to rise & set sooner or later in one of these places then an other, yet keepe they one sight of their houres. But if you will needes knowe to what countrie this diall serueth, both in respect of lon­gitude and latitude, seeke in the tables whether anye of these places before founde, bee in latitude equall [Page 114] to the walles eleuation or height of the cock, vz. 21 ¾. deg. & the same is he whō you look for: you shal find Carthaga insula in lat. 22. deg. 15. min. which is neere enough to serue your turne, & so I leaue you. Note that in this figure because the line of deflex. fell out iust in one of the houre lines, therefore all the prickt houre­lines of necessitie fall in the rest of them, by which meanes I neede but set too new figures seruing the o­ther countries, and it is done.

Chapter 31. How to place a flat representing the horizon of any countrie, citie or place in the worlde both according to the longitude & latitude of the same.

AS it is a pleasant thing by the last chap. to set the houres in your dials too the longit. of other countries whereto they serue: So no doubt, it is as pleasant, or rather more pleasant to know how to make a bo­dy, which shal haue diuers flats standing both to the longi. & lati. of what places you wil nominate. For if you make a body replenished round with sundrie flats as well reclining inclining, &c. vniformlie as the most do, it is a chaunce if any of them answeare any speciall place you desire either in long. or latit. To per­forme this matter is harder then a man would take it for: it spent me some studie at the first to compasse it: but now I hope to lay it open enough to the learner. Admit therfore that I were to make a bodie of sundry flats looking euery way some vpwards, some downwards, some this way, some that way, as in times past I haue done with great labor of seeking the angles: which now by help of my Iewel I could do with more ease, and that in the same body I would place one flat exactly lying parallell & leuell with the horiz. of Rome, that it might be seen how that horiz. lyeth vnto ours: the first thing I do, I go to Ptolomes or Apiās tables where I find the long. of Rome 37. deg. almost the lat. 41. de. 40. mi. Our longi. here at Reading is 14. well neer, our lat. 51 ⅔. deg. then taking 14 out of 37. there resteth 23. deg. the difference of long. of Rome frō Reading, & that Eastwards, bicause that lōg. was greatest. These had, you shal with waxe fixe the labell 51-⅔. deg. vnder the zenith point of the reete leftwards, which point representeth the pole, & the label the Finitor set to our lat. that done, count among the azi. ftom that part of the limb whereto the label is fixed the differ. of long. vz. 23. deg. & on this 23. azi count from the zenith point downwards, the lat. of Rome, vz. 41. 40. there make a sure prick with inke, for in that that the said 23. azi. standeth for the meridian or circle of lon­gitude of Rome, therefore that prick being that it boundeth the poles eleuation, there must needes bee the North point of the horiz. of Rome. Now for that this 23. azi. commeth in towards the center, therefore hys pole zenith or Nadir must needes bee the 23. deg. of the finit. beyond the center by the 5. bo. 24. ca. And be­cause the poles zenith & Nadir of the merid. of any horiz. lye in the East & VVest points of the same hor. therfore the same 23. deg. of the finit. so coūted beyond the center shall here be the East or VVest point of the hor. of Rome, & in truth must needes be here the VVest point if you conceiue it as wel as I do: wel then, ha­uing two pointes of the horiz. of Rome, vid. the North point being the pricke made with inke in the 23 azi. & the VVest point being the said 23. deg. of the finit. you shall turne about the reet & label fixed together as before, till you see some one merid. of the mater to cut both those points which you shall find too bee the 17. merid. counted from the axtree. This 17. merid. with his match, vz. the 17. merid. on the other side of the ax­tree, without all doubt represent the whole horiz. circle of Rome euen as in the 4. bo. 9, ca, two merid. made vp alwayes the whole eclipticke circle. Thus your Iewel beieng in an excellent constitution for perfour­mance of this chap. you shall count the degrees of the said 17. merid. from the said pricke in the 23. azimuth, which is there the merid. of Rome vnto our merid. being the limb. which deg. end in the pole point of the ma­ter which here is no pole but an intersection, you shall finde them 16. which are the quantitie of the deflexion of this dyall by the def. in the 27. cap. then from the crossing of this 17. merid. with our merid. being as I said the pole counting on his saide match merid. videlicet, the 17. merid. vnder the axtree till he crosse our horison being here the label, you shall finde 24. degr. & better, which is our meridians ascension in that flat which shalbe here set to the horison of Rome, but both these angles are more then bargaine: for the drifte of this chap. was no more but to get the declinatiō & reclin. of this proposed flat, which you shall also now haue & those into the vantange. Reckon therefore the deg. of the label between the limbe & the last intersec­tion of the said match merid. the same you shall find 23. deg. the complement whereof, vid. 67. being the de. of the label betwene the same intersection & the center of the Iewel, is the declination of the base line whiche must needes be Eastwards from the South, because Rome lyeth that way from vs. Now for the reclination of this flat you shall from the said last intersection in the label count on the said match merid. to the pole be­ing as is said 24. de. and so round thence by the deg. of the other 17. merid. tyll you come to 90. which en­deth at 66. deg. thereof counted from the pole: directly ouer that 66. degr. make a pricke with inke on the Reete, then turne about the reete & labell stil fixed till the label lye euen with the Equinoctialline, & there your shall finde the 22. ¾ merid. counted from the axtree to cut the said prick last made in the reet ouer the said 66 or rather 90. degr. whose deg. from the same prick counted to the pole, are 71 ½. the reclination of this flat sought for: the complement wherof being the deg. of the same 22 ¾. from the prick to the label, vz. 18 ½. deg. is the mounting of the same flat aboue our horiz. in his bases perpendiculer. Thus in end (since the declinat. is found 67. deg. & the reclination 71 ½. deg.) I conclude that to place a flat here at Reading, leuel or parallel with the horiz. at Rome, that is to say, to agree with it both in longi. & lat. the same must be a South flat de­clining East 67. deg. reclining 71 ½. deg. which by the 22. & 23. chap. you may easily knowe too set, and by the 28. make the dial thereto, and by the last chapter place the houres of Rome.

By the way note thus much that though in the constitution of the Iewel by this chap. the pole point of the mater chaunced aboue the labell, yet you shall in the like working manie times finde the pole fall out to bee vnder the labell. In euery such case that which here being aboue the label was your meridians ascension, shall then bee your meridians descension being vnder the label: many suche like notes you shall finde by your own practise and industrie.

Chapter 32, To perfourme the same by sphearicall triangles.

TO bring you in vse with sphearicall working by triangles, I take this needles paines, first make your platform as here you see, ABCD, is our merid. AB the axtree, A the North pole, CD the Equi. EF our horizon, K our zenith, E North, F South, the center P East and VVest, EA our eleuation, vz. 51-⅔. deg. CL the difference of longitude, vz. 23. deg. AG the eleuation at Rome, vz. 40. degr. 40. min. G the North point of that horizon PH equall too CL, and therefore H the pole Nadir of ALB by the 5. booke 25. chap. being also the VVest point of the horizon of Rome: I drawe therefore a circuler line by ame from G to H, and beyonde letting him to crosse EF, some where at aduenture in Q: This circle GHQ if hee were truly drawen repre­senteth the horizon of Rome in his right cituation to ours Now then I reason thus GL, being the complement of AG must needes be 48 ⅓. deg. which by the 4. def. denominateth the angle GHL, to the which angle DHQ is equall by the 5. Theoreme: then the angle DHQ, vz. 48 ⅓. taken out of a semicircle, vz. DBC leaueth 131. ⅔. degr. the angle PHQ, y^t angle DPF is 38 ⅓. de. the Equinoctials height aboue EF, our horizon PH is 23. deg. as before. Heere therefore haue you a sphearicall triangle PHQ, whose two angles HPQ 38 ⅓. deg. and PHQ 131 ⅔. deg. are knowen with the side lying between them, vz. PH, 23. deg. wherfore by the 5. bo. 17. ca. you shall find the side PQ to be the 67. deg. which is the declination of the base line of this flat proposed, and the angle HQP (which by the 4. def. is denominated at 90. deg. frō Q by RT to be 18 ½. deg. the flats moūting aboue our horizon EF, whose complement TK, is his reclinat. sought for. I might by this figure get the deflexion & ascension, but they are easie enough by the 26. or 27. ca. now the declina­tion and reclination are had.

Chapter 33. To describe dials to al maner of declining walles, and also to all East and West reclining and in­clining flats more at plaesure then yet hath been shewed without foreknowing the angles of eleuation or deflexion.

SO wonderfull is the copie of operation by this instrument that it is strange so much matter & so great varietie should be cōprehēded in so simple a thing as it is to behold. Truly I am so drencht in the bowels of this Iewel that I know not almost which way to turn me, or whē to determine an end. For being now re­die to send these 6. bo. to the presse: there sodainely commeth into my head a farre easier course for all decli­ning reclining & inclining dials of what sort soeuer: which for the most part are very troublesom by the 19. 20. & 28, c. in numbring the houres from the new found meridian there. Verily were it not that I challenge the first inuention of the former way, I might here euen blush to think what a compasse, I haue fetcht what extreeme paines & studie it hath cost mee too atchieue the same euen by the racking of the spheare as I may terme it. But it is written of a great clerke, facilius est addere inuentis quā de nouo componere, which shall be a sufficiēt tutele for [...] in this behalf. Neither am I to be more ashamed of this then all other writers before me euen y^e princes of Astronomie as som of them haue been termed, are to be reproued in that all they haue not taught to make these busie kind of dials by the vse of either the Spheare, Globe, Astrolabe or any other instrument as wel as they haue the other 2. easie & ordinarie dials. But now to the matter, first you shal vnderstand that all declining dials, whose making are taught in the 19. and 20. chapters, and all East and VVest reclining or inclining taught in the 25. cap. are in maner all of one sort, that is to say, their flats represent circles of po­sition to some one lati. or other: as for example, all our declining walles represent circles of position to the 38 ⅓. South latit. euen as our East & VVest reclining flats are decli. walles to the same 38 ⅓ lati. as in the 25. cha. I shewed: to al flats of both these sorts you shal thus presently make the diall without respect either of e­leuation or deflexion, which in all my other cap. must be foreknowen. And for your better instruction I wil here shew to make in this maner a diall to the South decli. wall Eastwards 16 ½. de. mentioned in the 15. c. VVhich to do I place the finit. to our lat. vz. 51 ⅔. deg. vnder the pole, then from the zenith line towardes the limbe I reckon on the finitor 16 ½. deg. rightwards, there do I finde the 16 ½. azimuth, whiche representeth this wall proposed, or more plainely, the great circle or horizon in whose flat this wall lyeth, & is the 16 ½. circle of position, to those that haue the South pole 38 ½. de. eleuated, which being found let him there rest awhile. And now I must haue you (since I can assure you that euery of the dials of this sort are oblique dials) too call to remembrance the def. of an oblique diall mentioned in the 3. chap. & the grounde & reason whēce his lineaments proceed there also shewed: & then if you returne again with this imagination to the saide 16 ½. azi. where we left, you wil be redier to tel mee then I haue been hitherto to tel you, that the verye houre lines alreadie on your Iewel distinguished by fifteenes, must needes deuide the same 16 ½. azimuth in too hys horizontall spaces seruing this diall: since the limbe is still in this constitution our merid. from whence the [Page 116] houre spaces in all dyals must be counted. VVherefore to go on: because in South declining dials the South pole is euer eleuated, and in North declining, the North pole, therefore it shall not bee amisse too turne a­bout the reete that this zenith and Nadir may chaunge places to the intent you may as well reape the rea­son as driue on your diall. For there belongeth yet more thynges too bee obserued and foreknowen: of whiche this is one generall. You must knowe although I heere teach too describe your dials without firste seeking the angles of eleuation and deflexion: yet for the height of the cocke and his place you must, the di­all finished, craue their acquaintance, and in the meane time haue a great regarde of them, of which too bee fully instructed, first you must note that on your Iewell any two azimuthes lying on eache side of the Zenith line, like distaunt from it, doe by prospectiue reason make vppe and represent one whole great circle of the spheare, as in my seconde booke and the one and twentie chapter, I haue somewhat largely touched, whe­ther resort, so that this said 16 ½. azimuth with his matche beyng the 16 ½. azimuth counted from the zenith line on the other side doe make vp the whole horizon or great circle, wherein this wall lyeth. Next to this, since you know that the limbe representeh the meridian of our horizon, & that, as I haue often before said, euery wall or flat hath a meridian proper to himself lying among the rest of the meridians, you shall know that the Iewell being set to the constitution of this chapter the same his meridian as wel in this wall as any other flat, which you wyll make any dial to, by the instruction of this chapter lyeth also betweene the zenith point of the reet, and the neerest crossing of the axtree line, with your proposed azimuth, for whiche cause all the houre spaces gathered on the same azimuth from the zenith point of the reete towardes the axtree, of necessitie must in your dial be appointed and set from the 12. a clocke line, towardes the line where the cocke shall stande, that is to say, VVestwardes of the plumbe line in this diall, because he declineth East as be­fore is saide, or Eastwardes if hee had declined VVest. These circumstances settled in your minde, then come wee neere to the matter, first drawe on this wall a plumbe line, vz. AB, in which pitch the center of your diall vz. C, wheron describe a rounde limbe bigge enough for your diall, and within it a blinde circle equall to your Iewels limb as you are wont: This done, set the finitor to our latitude, vz. 51 ⅔. his Zenithe poynte downewardes towardes the South pole, as before is saide: and from the Zenith poynt towardes the axtree looke where the next 15. houre line (counted from that part of the limbe where the Zenith point standeth) doth cut the saide 16 ½. a Zimuth, and you shal find theron 9. degrees for your first houres space towardes the cocks place, which by help of your blind circle you shal set in this diall VVestwards of the plumb line: That done, looke againe howe many degrees of the same 16 ⅓. azimuth are cut by the nexte distinguished houre line of the mater you shall finde 18 ¼. for the second houres space: likewise for the third 27 ⅔. for the 4. 39. for the 5 52. for the 6. 70. then can you go no further, except you turne againe your reete, that hys Zenith and Nadir may chaunge places: In so doing you shall get for the 7. houre 93. deg. for the 8. 118 ½ beyonde eight neuer goe in our countrie: All these spaces you shall by helpe of the blinde circle set on the VVest side

of the plumbe line of this diall. Then for the houres on the other side, you must looke to the saide matche azimuth that is the 16 ½. azimuth on the other side of the Zenith line, & you shall find the said first houre line to cut on him 10 ⅓. deg. counted frō the Zenith point: the seconde houre line cutteth 23 ⅓. degrees, the thirde 40. degrees, the fourth 60 ½. degrees, the fifth 86 ½. degrees, and so foorth if neede were. But since all walles serue but to 12. houres, therfore howe many houres of the one side of the plumbe line needeth aboue 6. the other side hath so many lesse, notwithstanding you may thus put thē in round if you thinke store is no sore. These last houres being put in by help of y^e blind circle on the other side of the plumb line y^e description of this diall is finished. For the height of the cock & his line of defle. look the 16. or 17. ca. & perform that, as [Page 117] hath beene before shewed, or rather if you will to the 18. Chapt. whereby you shall neede go no further then this present constitution of your Iewel for your whole Dyall cum pertinentiis. For if according to the instruction of the same 18. Chapt. you reckon on the Finitor from the left part of the Limbe inwards 16 ½ deg. there shalbe the pole Nadir of the said 16 ½ azimuth lying rightwards of the zenith line, I say his pole Nadir in respect of his face beholding the south pole, and at this saide 16 ½ deg. of the Finitor, you shall finde to cut the 20 ⅔ meri­dian counted from the Limbe, which in truth is the meridian of this declining wall wherefore following him towards the south pole you shall finde him to crosse the said 16 ½ azimuth in the 12 ½ deg. from the Zenith Point of the Reete, which by the definition in the 27. Chapt. is the quantitie of the angle of deflexion. And the deg. of the same 20 ⅔ meridian betweene this crossing and the pole being there 36 ¾ deg. is theleuation of the south pole aboue this wall, and is the height of the cocke. All the premises you see performed in this pre­sent figure.

In the verie like manner shall you worke, for all East and West flats reclining or inclining, mentioned in the 25. Chapt. but that insteede of the Finitor there, you must here place the zenith line to the latitude. And so doe the azimuthes serue your turne in the very selfe same sort, as by this Chapt. compared with the 25. you shall most plainely vnderstand, the base line of the bancke or flat, being 12. of clocke: note that the saide azi­muth with his match must for East and VVest reclining Dyals be counted from the Limbe.

Chapter 34. To reduce all reclining and inclining flats declining vnto East and West, reclining and inclining flattes to some one latitude or other, and thereby most easily to make the Dyall.

THe reclination and declination of any flat being proposed or else taken by the 23. Chap. you shal ther­by get the angles of eleuation and deflexion as is taught in the 26. & 27. Chapt. which had, if you finde there the North pole eleuated then by the 25. Chapt. if south pole, then by the 16. 17. or 18. Chap. you may as easily by the said angles of eleuation and deflexion get the latitude and reclination, as in the 25. chap. by the latitude and reclination, you got the deflexion and eleuation, and that in this manner. Reckon the an­gle of deflexion giuen on the Limbe of the Reete from the zenith point, thereto lay the Label whereon from the Limbe inwards count theleuation giuen, and the azimuth there crossing the Label counted from the Limbe is the circle of reclination, whose degrees to the zenith is the latitude, wherein that reclination is to be taken that is to say the latitude of that countrie whereto the flatte proposed shalbe an East or West, reclining or inclining flat, or more plainely where the same flat lyeth in one of the circles of position.

For Example the bancke at Aldermarston his reclination found by the 23. Chap. to be 56. deg. and his de­clination in his base line 78 ½ deg. whereby in the 26. and 27. Chapt. his eleuation was founde 35 ½ deg. and his deflexion 28. deg. By this eleuation and deflexion, you shal finde in what latitude this bancke were a bare East or West reclining flat, and how much that reclination there shalbe. Reckon first the deflexion vz. 28. degrees on the Reetes Limbe from the zenith point, thereto lay the Label, whereon count the eleuation vz. 35 ½ deg. inwards, and there shall you finde to crosse the Label the 56 ⅔ azimuth counted from the Limbe, which I say is this newe reclination sought for, and the degrees of this 56 ⅔ azimuth from this crossing with the La­bel to the zenith you shall finde 44 ⅙ deg. the same is this new latitude sought for wherein the said bancke is a west flatte barely reclining 56 ⅔ deg. which being had after the line of our meridians ascension and the line of deflexion are placed on the bancke as in the said 28. Chapt. is shewed: you shall insteede of the Finitor euen as was done in the 25. Chapt. set the zenith line to this new latitude, that is to say 44 ⅙ deg. vnder the north pole leftwards. And thereby the intersections of the 24. distinguished houre circles of the Iewel with the same 56 ⅔ azimuth, and his match azimuth persue the making of this Dyal in all respects as in the last Chapt. you did, placing the houres in this Dyall from the line of ascention being the 12. of clocke line, as in this last chap. you did from the plumbline, and it commeth to the very same effect with the 28. Chapt. but with more ease a great deale, as by experience you shall trie.

In the verie like manner may you performe inclining Dyals declining, taking a speciall regard by the con­stitution of the 26. Chapt. which pole it eleuated.

Chapter 35. To do the same somewhat easier.

BEcause in the working of the last Chap. I referre you to the 26. and 27. Chap. to get theleuation & de­flection of any reclining flatte declining, before you can worke your feate, and knowing the working of the 26. Chapter to be in some respect troublesome, I thought good to beate about for some better way for you, and thus much haue I founde: After you haue taken the declination and reclination of any such flat by the 23. and then set your Iewel to the first constitution of the 26. Chap. you shall easily thereby streight finde your meridians ascention, and your meridians mounting, which two note in minde, and then set the Label to your latitude fixed, and aboue it towards the pole place the zenith point of the Reete as much as the meridians mounting commeth to, and there marke where the almicantare counted from the zenith equall to the meridians ascention doth cut the Label, for the azimuth meeting him there counted from the Limbe shew­eth the new reclination sought, which with his match representeth your desired horizon. And the zenith [Page 118] point standeth there vnder the pole to your new latitude by which meanes you may proceede as in the 33. and 34. Chapt. is shewed. As for example in the same bancke.

By his reclination being 56. deg. and the declination of his base line being 78 ½ deg. found in the 23. Cap. I placed my Iewel in the first part of the 26. Chapt. to a most excellent constitution representing the same and euery necessarie circle thereto belonging whither you must repaire: and there shall you finde that the zenith point was 11 ½ deg. from the pole, where we finde the 56. azimuth representing this bancke, whose degrees betweene his crossing with the axtree and the zenith of the Reete which I called the meridians ascention, were 23 ⅔ deg. and the degrees of the axtreeline betweene the same crossing and the Limbe or pole point of the Ma­ter being in truth our meridians mounting were, 7 ½ deg. if I had made accoumpt of them there, but I then had no vse for them. These had which you may alwayes haue without any paines for the looking on, I first set the Label to our latitude vz. 51 ⅔ deg. vnder the pole, then aboue the Label I set the zenith of the Reete equal to our saide meridians mounting vz. 7 ½ deg. towards the pole, so is the pole eleuated aboue it 44 ⅙ degrees, and is our new latitude desired. Then counting from the zenith of the Reete among the Almicantares the quantitie of our meridians ascention vz. 13 ⅔ deg. I finde the same 13 ⅔ almicant. to cut the Label in the 11 ½ deg. from the Limbe, which in truth is the complement of the declination as in the 31. Chapter it was, from which this constitution now differeth not: but now in the same crossing of the saide almicant. with the Label you shall finde the 56 ⅔ azimuth to cut being the reclination desired: So that now if you keepe the zenith of the Reete fixed, you haue him readie placed at the new latitude euen as he was in the last Chapt. and the 56 ⅔ azimuth your desired horizon, wherefore let the Label now go his way, for the rest is to be done without him by the 33. Chapt.

Note that if in steede of the meridians ascention among the almicantares you had here numbred on the La­bel the base lines declination vz. 78 ½ deg. from the centre, or his complement vz. 21 ½ deg. from the Limbe, there had you found the said 56 ⅔ azimuth without respecting the ascention.

Note also when by the said constitution of the 26. Chapt. you shal at any time conceiue that your meridi­an is depressed vnder your horizon as in diuerse cases it happeneth, then that which heere is the ascention is called the descention, and that which here is the mounting is then the depression. And in euery such case the zenith of the Reete is to be placed vnder the Label as here it was aboue.

Chapter 36. To performe the 31. Chapter somwhat more easily, or at the least more pleasingly.

FIrst set the Finitor to the latitude of the place proposed, whose horizontall flat you desire to scituate, then reckon inwards among the meridians, the difference of longitude, and the meridian so found shal cut on the Finitor the angle of deflexion of the same flatte in this our countrie, euen as in the 19. Chapt. the de­flexion found the difference of longitude, which had I reckon this deflexion on the Limb of the Reete from the Zenith point, thereto I lay the Label whereon I count againe the latitude proposed inwards, and there crosseth the azimuth shewing the position reclination as I may terme it. And the degrees of the same azimuth be­tweene this crossing and the zenith point sheweth the position latitude, that is to say, the latitude where­in the horizon of the place proposed is an East or West reclining flatte equall to this reclination founde or more plainely wherein he lyeth in that circle of position, such a one as in the last two Chapters are mentio­ned. VVherefore taking this latitude out of yours, there resteth the mounting of your meridian if your latitude be the greater, or the depression if the lesse, according to which if the Label and Zenith point be set in manner of the last Chapt. the same azimuth at his crossing with the Label, if you count thense to the centre, findeth out the base lines declination of your proposed flat, and from the same crossing to the zenith point your meridians ascention. And hauing nowe these, then if by the first constitution of the 26. Chapter you place the zenith line of the Reete according to this declination as there is taught: and then reckoning from the Limbe on the axtree line, the mounting of your meridian found, there streight meeteth you the azimuth shew­ing the true reclination in his base lines perpendiculer sought for. By example this will be more plaine which shalbe the example in the 31. Chap. of the scituation or placing of the horizon of Roome, the latitude there­of being 41 ⅔ deg. and the difference of longitude from vs here at Reading 23. deg. both proposed: First I set the Finitor to the 41 ⅔ latitude vz. 41 ⅔ vnder the pole, then from the Limbe among the meridians I count the difference of longitude vz. 23. this 23. meridian cutteth in the Finitor 16. deg. which is the angle of deflexion of the meridian of Rome from ours here in our countrie. These had I then reckon this angle of deflexion vz. 16. deg. on the Limbe of the Reete from the zenith point, thereto I lay the Label, nowe on the Label inwards I reckon the latitude of Roome vz. 41 ⅔ degrees, and there do I finde to cut the 73. azimuth counted from the Limbe which I call the position reclination: And on the same 73. Azimuth from the cutting with the La­bel vnto the zenith of my Reete I finde 44 ¼ degrees, which I call the position latitude, that is to say, in that countrie where the latit. is 44 ¼ degrees. The horizon of Rome is an East or VVest reclining or inclining flatte lying in one of the circles of position there: therefore taking that latitude vz. 44 ¼ out of our latitude vz. 51 ⅔ there resteth 7 5/12 deg. the difference, which because our latitude is the greater, is the mounting of our meridian in this flatte rightly placed. These knowne you shall first set the Label to our latitude vz. 51 ⅔ deg. vnder the pole, then place the zenith of the Reete according to the other latitude vz. 44 ¼ degrees vnder the [Page 119] same pole on the same side, or according to the difference of these latitudes vz. 7 5/12 degrees aboue the Label because the meridian mounteth, all commeth to one, and then marke where the said 73. azimuth cutteth the Label and you shall finde betweene it and the centre of the Iewel 67. deg. the declination of the base line of this flatte sought for, and from the same cutting to the Limbe is our meridians ascention aboue the base line, so you count it on the same 73. Azimuth which to our purpose is little pertinent heere but that the Iewel will needs do vs the curtesie. The declination of the base line thus knowne, you shall as in the first part of the 26. Chapt. place the Zenith point of the Reete accordingly, and according to the instruction there it shall winde from the Equinoctiall 67. deg. towards the pole, where the Reete fixed I counte on the axtree line from the same pole, the foresaid mounting of our meridian here found vz. 7 5/12 deg. and there I finde to cut the 71 ½ a­zimuth counted from the zenith line which is the circle representing the horizon of Rome in his right sci­tuation to ours, and his true reclination: but if by the 33. Chapt. you woulde make a Dyall thereto, then must you (the zenith point being as before at the 44 ¼ latitude) take the saide 73. azimuth and his match to performe your purpose. Otherwise you may proceede by the 28. Chapter, or rather now by the 40.

Note that when you haue the position reclination and latitude, and thereby the meridians mounting as before: you may somwhat easier get the true reclination and declination by a Sphericall triangle imagined such a one as AEO is in the 27. Chap. For being that as I haue sinse shewed in the 37. Chap. the angle EOA is equall alwayes to the position reclination: therefore hauing of the triangle AEO the side AO knowne with the angle not right EOA you shall easily by the 5. Booke 9. Chapt. get the other sides and angle, of which the side EO shalbe the meridians ascension, and EA the declinations complement, and the angle AEO the complement of the true reclination: hereof looke more in the 37.

Chapter 37. A most easie and briefe way to reduce all reclining and inclining flats declining, vnto East and West reclining flats to a new latitude, otherwise then in the 34. & 35. Chap. by helpe of sphericall triangles.

BEhold yet againe in these 4. short Chapters fol­lowing the intricatest part of the whole Art of Dyalling most plainely performed, not thought of when I wrote the 33. 34. & 35. Chapt. but freshly came vnto my minde as I was busilie occupied in cut­ting the prints for this 6. book. Better it is that it come too late then neuer: I were best now to say that there be two courses in making of Dyals, Longation and Curtation, as the priest saide to Baluinus in Erasmus Dialogue called Alcumistica, and that it was my chance to hit first vpon Longation, as in my 19. 20. 25. & 28. Chapters full well appeareth: and yet truely they are very plesant and profitable in diuerse respectes though not so redie for Dyalling as these & others, & therfore not to be reiected, yet let no mā stand longer on them then to conceiue the reason and manner of them. Wel then, now you shall haue Curtation, which because I will make short of, behold this first figure whose linea­ments are sufficiently declared in the 27. Cap. & there­fore I neede not repeat them againe representing the first constitution of the Iewel in the 26. Chap. wherein I am in this Chapt to let you vnderstand that the arch LO is our new latitude sought for, & the angle LOM our newe reclination mentioned in the 34. and 35. Chap. which are most easily to be had because the tri­angle EAO hath all three sides knowne either by the verie constitution of the 26. Chap. or by the 27. wher­fore the angle EOA is easily had by the 5. Booke 4. Chap. vnto which the angle LOM is equall by the 5. theoreme and is our new reclination desired, called in the 34. and 35. the position reclination. Then taking the arch AO out of AL, our latitude there resteth LO this newe latitude desired called in the 34. and 35. the position latitude. Lastly you haue the right angled triangle OLM whose one angle not right LOM with the subtending side OL are knowne, wherefore by the 5. booke 9. Chapt. the other sides & angle shall be knowne, of which the side OM is the angle of de­flexion and LM theleuation as in the 27. is shewed. Now do you see how easily these arches and angles are gotten, of which diuerse Chap. haue before entreated: And therfore methinks the very lineamēts of this figure should easily shew him that hath read the rest of my book of dyalling with any iudgmēt, how to place these angles on y^e dyal euen without y^e figure of the 28. ca. which notwithstanding I haue here set for your better conceiuing, for who seing the circle EFM to represent [Page 120] the flat of the bancke proposed, and therein how the notes EOM do stande, will not easily imagine that the degrees of the arch EO being our meridians ascention are to be placed aboue the base line of the bancke, as in this second figure you see EA, and aboue that againe OM to be placed for the angle of deflexion or Gno­mons place as in this Dyall you see AP. But now for LM you may conceiue that must stand perpendiculer on the flat, being indeede the height of the cocke as in this Dyall you see.

Chapter 38. A briefe note how East and West reclining or inclining Dyals are to be made two seuerall wayes.

HAuing first placed all the angles on your Dyall, as in the last Chapter, you shall forget that ENF is a South declining flat reclining VVest, as in the 23. Chapt. he was found to be: and that AL is your la­titude, and KEN your angle of reclination as it was in the 27. and 28. Chapter. But you shall nowe take OL to be your latitude and the flat ONP to be a bare west reclining flat according to the angle KON or KPN, wherefore if nowe you take your Iewel in hande, and count on the Limbe thereof the latitude OL from the pole towards thequinoctiall, and thereto lay the zenith point of your Reete fixed, and then choosing out the azimuth counted from the Limbe equall to the angle KON your new reclination, worke your Dyall by him and his match azimuth in all respects as in the 33. Chapt. is shewed.

If this way like you not, or that you desire variety, I wil here shew you to make this dyal by the 5. Book 18. Cap. after another sort. First therfore for as much as all East or VVest reclining or inclining flats in any latit. howsoeuer seene or imagined, are verie declining wals to that countrie whose lat. is equal to the complement of the same latitude as I haue said in the 25. & 32. Chap. and by the 13. preamble is partly manifest, therfore if you conceiue not or like not of the making of the Dyals to these East and VVest reclining or inclining flats you may turne them plainly into declining walles. As for example: sinse the latitude OL is 44 ⅙ and the angle KON 56 ⅔ deg. as in the 34. Chapt. is shewed, you shall take the complements of them both, and the first vz. 45 ⅚ deg. being the complement of 44 ⅙ shall be the latitude wherein ONP is a declining wall o being the zenith there. And the complement of 56 ⅔ degrees being 33 ⅓ deg. is the declination of the wall represented by the said ONP in the said 45 ⅚ latitude, and that westwards if it were material. But sinse the angles of EO and OM be al redie placed on the flat, therefore it needeth not to seeke so carefully which way such a wall should decline in that 45 ⅚ latitude, because which of the 4. wayes so euer hee declineth the degrees of the houres will serue as I shewed in the 20. Chap. And now lastly if you do but lay before you the figure of the 5. Booke 18. Chapt. here present, and imagine the arch BD to be this foresaide latitude vz. 45 ⅚ and the circles ZAY to be this wall declining 33 ⅓ deg you may by the saide 5. Booke 18. Chapt. make the Dyall without more teaching in all respectes as there was shewed.

Chapter 39. How to make a diall to any East or West, reclining or inclining flat, by helpe of spherical triangles most easily.

FOr as much as in the last chap. to make the East or VVest reclining diall, I referred you to the 33. chapt. which doubtlesse is singuler good, but that my example there is a declining dial, or els to reduce the same to a declining wall by the complements, it may be that first you vnderstand not, and the second you vt­terly mislike. Therefore I wil here briefly teach you to performe the East and VVest reclining after the fyfte booke 18. chap. in other manner then in the last chap. I shewed, whereby you shall not choose, but fullye to vnderstand the 33. chap. Therefore behold this Figure and imagine BD to be your new position latitude mē­tioned in the 34. 35. and 37. chap. videl. 44 ⅙. deg. and let the circle BBBC represent the new position recly­nation mentioned in the same chapters, reclyning from the Zenith A, according to the arche ABB, beeyng 56 ⅔. deg. Here may you beholde on this Figure, being but a poore platforme of your pretence, in what man­ner the houre circles of the spheare doe cut your proposed flat BBBC. The distances between euery inter­section are to be had in all respectes, as was done in the saide 5. booke 18. chap. the case being but a very lit­tle altered. For in steed of the arch DA, which in working the same 18. chapt. lieth on the Limbe of your Iew­el,

betweene the pole point of the matter, and the Zenith point of the Reete, heere you must place the arche BD, and in steede of the angle DAZ, which in the same 18. chap. was common to the finding of euery houres space on the arche AZ, so heere the angle DBBB is common to the fynding of euerye houres space on the arche BBB and beyonde if you list vnto C. And euen as the angle BAY was there the common angle to gett the houres of the arch AY, so here the angle DBCC is the common angle to get the houres of the arch BCC, and beyond as farre as you list: which angle DBCC is easily had, seing CCBZ is equal to KBBB, the complement of the reclination proposed, vz. 33 ⅓. deg. and thereto adding the angle DBZ or ABL videl. 90. deg. there commeth 123 ⅓. deg. the angle DBCC. And when you haue all done, you shall find that the working it spherically in this manner, shall differ no iot from the working of the 33. chap. re­membring this rule in general, looke whereabouts on your platforme, or on your Iewel you see the houre cir­cles to cut your flat or horizon proposed thickest or neerest together, thereabouts bee sure the cocke or Gno­mon must stand as here you see they doe betweene B and BB, which I note because you may be sure that those houres from B towardes BB are in your diall to be placed from our meridian, being the 12. a clocke line to­wards the line where the cocke must stand.

Chapter 40. Most excellently and easily to make dialles to al maner of reclining or inclining flats declining, by the Iewell and helpe of spherical triangles.

I Knowe some will thinke it a great toyle when he hath reclining or inclining flats declining, to seeke newe position latitudes and reclinations by the 34. 35. or 37. chap. and so to goe to worke, the next way about by Lōgation lane: to satisfie them, I suppose the flat YFFZ in this figure to be a North flat reclining South­wardes [Page 122] according to the quantity of the arch AFF, vz. 30. deg. declining Eastwardes as much as the arche BEE is, vz. 40. deg. YZ being the base line of this flat, and EE the pole zenith of the base. The first thing you doe you shall seeke the angle DAAZ, and the arch AAD: both which you shall thus easily find. The arch AFF is perpendiculer on the circle YFFZ, by the 3. def. the angle FFAC is equal to EEAB, being the decli­nation, vz. 40. deg. by the 5. theor. Here now haue you the right angled triangle AFFAA, whose one side A

FF is known to be 30. deg. and one of his angles not right FFAAA 40. deg. knowne, therfore by the 5. book 9. chap. the angle AAAZ shalbe known, and also the side AAA which added to AD maketh the whole arch AAD knowne: both which had, this chapter is easily performed euen in maner of the 5. booke 18. chap. For insteed of the arch AD, which in working, the said 18. chap. you placed on the Limbe of your Iewell between the pole point of the mater, and the zenith point of the Reet, you shal here place the arch DAA. And there the Reete fixed let the angle DAAZ be here the common angle to all the houres of the arch AAZ, euen as in the said 5. book 18. chap. the angle BAZ was common to al the houres of the arch AZ. Likewise let DAAY be here the common angle to all the houres of the arch AAY, euen as there the angle BAY was common to al the houres of the arch AY in all respects as I taught in the last chap. for taking the angle ZAAD out of 180. there resteth the foresaid angle DAAY, known by the 5. theoreme. you might haue gotten the arch AAA, and the angle DAAZ. otherwise, and that in this manner, and I thinke it the surer way, you haue the spheri­cal triangle AYAA, whose angle YAAA is equall to the declinations complement, vz. 50. deg. and the an­gle AYAA 30. deg. proposed, the side lying betweene these angles, vz. AY is 90. deg. known, therefore by the 5. booke 17. chap. the angle AAAY shalbe found 124. degr. whiche taken out of the semicircle ZAAY being 180. leaueth the angle AAAZ 56. deg. and by the same 17. cha. the arch AAA shalbe found 37. deg. whereto adding the arch DA, vz. 38 ⅓. so haue you the whole arch DAA 75 ⅓. deg.

And here I thought good to note thus much for the credit of my 33. chap. that if you reckon on the Limbe of your iewel the said arch DAA, vz. 75 ⅓. deg. from the pole, and thereto set the zenith poynt of the Reet fix­ed, and there choosing ourth: azimuth counted from the Limbe equal to the angle AAAZ, viz. the 37. azim. you may by the same 37. azim. and his match worke this diall, as in the 33. chap. in all respectes, and yet in conclusion wil differ nothing from the premisses.

Chapter 41. To know in what Longitude and Latitude our horizon or any other shall represent any declining wall proposed, or els any circle of position proposed.

GOod God if I should bend my selfe to inuent but so many propositions to be performed on my Iewel as my course capacity could accomplish, I thinke I should spend al my dayes in writing. For whiles I think to make hast to an end, I fare like a dog that goeth in a wheele hauing an endlesse path to tread, let him make neuer so much haste it helpeth him not til he fynd a meane to get himselfe quite out. So I haue no other mean to come to an end, but euen here to stop and lay a straw as they say for this time, this chapter being en­ded, which cannot but he somewhat pleasurable. As if a man would know in what longitude and latitude [Page 123] our horizon here at Reading were a south wall declining East 30. deg. or els were the 30. circle of position: euery man perhaps could not performe this, notwithstanding all that which hath byn said. You shall reckon our latitude on the Label from the Limbe inwards, and lay the same deg. on the 30. azimuth counted from the Zenith line and the degrees of the same 30. azimuth counted thence to the Zenith point shall bee the latitude wherein our horizon is the 30. circle of position, and the complement thereof shalbee the latitude wherein it is a wall declining 30. deg. And the deg. of the Reetes Limbe betweene the Label and the zenith point shall bee the deflexion of the diall to that declination and circle of position. Lastly, set the Finitor vnder the pole accor­ding to our latitude, and thereon reckon inwardes the deflexion before founde, and the meridian cuttyng there the Finitor counted from the Limb sheweth the the difference of longit. which being added to ours shew­eth the longit. desired.

As for example, our horizon here at Reading is in longit. 14. in latitude 51 ⅔. deg. First I reckō our latitude videl. 51 ⅔. deg. on the Labell inwardes and apply the same degree of the Label to the 30. azim. counted from the zenith line, which cutteth off on the said 30. azim. 64 ⅔. deg. counted thence to the zenith pointe of the Reete, and in the Limbe the Label cutteth 46. ⅔. deg. coūted also from the zenith point, which are the deg. of de­flexion of which I will speake anon. But before I proceede further, I conclude that the said 64 ⅔. degr. are the deg. of latitude in that countrey where our horizon shall lye in the 30. circle of position counted there tow­ardes the zenith as the circles of position are alwayes reckoned: then if you take the complement of the said 64 ⅔. deg. vz. 25 ⅓. deg. the same shalbe the latitude wherein our horizon shall represent a wall declining 30. deg. yet by your leaue you may be in doubt whether this be a South latitude or a North, but you shall haue this rule general, if you once knowe whiche pole is eleuated aboue anye South or South declining wall, you may be sure the contrary pole is eleuated in the latitude whereon that wal standeth, as here seeing the North pole is eleuated 51 ⅔. deg. you may be sure where this our horizon shall be a south or south declining wal, that countrey hath a South latitude: and therefore this 25 ⅔. is a South eleuation. But in what longitude withal he shal be a South wal, declining VVestwards 30. deg. that shall the deflexion helpe vs too, being 46 ⅔. as I be­fore saide. Therefore you shall set the Finitor according to the sayde South latitude videl. 25 ⅓. deg. vnder ey­ther pole, and thereon rrckon inwardes the saide angle of deflexion vz. 46 ⅔. deg. where you shall finde the said 68. meridian counted from the Limbe to crosse the Finitor being the sayde difference of longit. from ours. Now because I propose our horizon to decline VVestwards from the meridian of that 25 ⅓. latitude, therfore I adde those 68. deg. to our longi. being 14. it maketh 82. deg. the long. desired. So that hereby I conclude my purpose, affirming that our horizon here at Reading shal represent a South wall declining VVest 30. degr. in that countrey where as the longit. is 82. deg. and the South pole eleuated 25 ⅓. deg. & you do not beleue me me, goe thither and trye it. It is a prety walke for a recreation: wel, yet if you subduct 68. out of 14. (whiche without adding first 360. thereto you cannot doe) so is it 374. there resteth then 306. the longi. of that coū­trey where our said horizon is a South wall declining East 30. deg. the same latit. vz. 25 ⅓. remayning stil. In the very same maner by hauing regard of the longitude, you may knowe where it is a North wall declining East or VVest 30. deg. since those two are but opposite to these.

Note that the said 30. circle of position is the 60. reclining or inclining flat in the said 64 ⅔. latit. but whe­ther you will haue him East or VVest reclining or inclining, that shal you euen as easily doe by taking regard of the differences of longitude in maner as before.

VVel then though in the 37. I likened my 19. 20. 25. and 28. chapters vnto longation, hauing since in some respectes lighted on Curtation, yet commeth this fruite (if it be ought worth) out of them: and they are in­troductions to diuers such like, for which cause I thought good not to omit this chapter although the 30. 36. and diuers other chapters came wholly of them, and therefore I repent not much my paynes.