CHAP. I. The occasion of this Discourse.

I Am not ignorant, how dangerous a boldnesse it is, to appear abroad in the world as an Antagonist, and how subject they are to be judged themselves who would give judgement upon others, es­pecially upon those whom worth or custome hath inthroned in the good-likings of other men; and whosoever whets his Pen against such as these, is like to be more weakned with the batteries of Envie, then strengthened with the fortifications of a good cause. For when once we have enslaved our wits to any Author, and looke upon him with the eye of subjection, we are very hardly drawne from the pressure of that yoake; [Page 2]yea, and oftentimes we murmure and repine at those, who shew us the way to cleer our selves of this burden, and free our judgements from captivity. If then I could prefer the sloathfull security of My name before the truth, I should rather have smothered these modest cri­ticisms, and confined them to the chamber of my private thoughts, then in this censorious age, expose my self to the censure of the world: for although I do beleeve that whatsoever I shall write here is undoubtedly true and certain, and I doubt not will of learned Readers be ap­proved; yet can I not promise to my self the generall consent of every Reader: for some there will be whom the specious words and large promises of my Authors have deluded, and these are most likely to cal me obstre­perous and impertinent, that after the worthy Labours of these famous men, and the sum of their endeavours herein, I should yet desire a plus ultra, and manifest my boldnesse and impudence, in that having but newly crept out of the limits of Child-hood, I dare bend my Pen against famous Artists of such a continuance.

But it is the part of a slothfull timorousnesse, to be afraid of the rash censure of the multitude, and it is doubtfull I shall not be so much commended for mode­sty, as censured for want of boldnesse, to be fearfull to restore the Truth to her proper Dignity, and cleer her noblenesse from being partaker in those counterfeits, which passe abroad under her name. It is to be imagi­ned that Urania Practica hath found her favourites, and there want not those, who have fixed their devotion on this new Deity: And I must confesse, when I heard the report of her comming, and saw the news of her so con­sidently fly through the British Isle, upon the paper­wings of severall Almanacks, I had a good hope some extraordinary performance would have confirmed those glorious reports, and upon the Theater of a Mathema­ticall Judgement, acted something worthy those Praelu­dia. [Page 3]But when in January 1648/9. I received from my worthy and honoured friend, Master William Lilly, that so much expected piece, & had with a greedy eye, survei­ed the contents, the clouds of despair over-shadowed my thoughts, and involved my hopes in a sable vesture, yea I could scarcely refrain from blaming my self, that upon such uncertain foundations as the popular voice had built my hopes to that height: for I did not onely fail of my expectation of a certain course whereby to attain a reasonable perfection in Astronomical grounds and The­ories; but on the contrary, perceived scarcely any dres­sing therein, which (if rightly considered) did not mis­become the divine Urania. I expected a compound of the best simples that could be found in the Treasuries of those Authors, which Master Vincent Wing hath put in his Muster-roll, recorded in the beginning of his Alma­nack, 1649. but finde that their errours are here renew­ed: all which I beheld with an eye, truly pittying such studious Tyro's, as understand onely English; for they are those who are most subject to swallow these insalu­brious baits, so destructive to a perfect proficiency in these Sciences. For the remedy whereof, and because I knew no other from whom any such performance might be expected, I adventured upon this task, and have here­in delivered to the eye of the world a detection of some the most notable mistakes I considered in that Treatise: but I passe by many things worthy also to be noted, if my professed brevity would grant me leave. What I have done, is done with such reverence to the sacred name of Urania, that I am confident I have not in the least any way wronged the meanest of her favourites: And in re­gard my Authors are the first that have in English ad­ventured upon such a subject, my tender Quill shall spare their infancy, whose errours otherwise merit to be vindicated with more severity.

CHAP. II. An errour in finding the Dominicall Letter and Epact in the Forraign Accompt perpetually, detected.

THough something below the levell of our intended endeavours, it will not be amisse to begin with this Peccadillo, which must necessarily proceed either from want of care, or knowledge of the true manner of the Calenders Gregorian reformation; the explication whereof, I shall chuse rather briefly to deliver from its beginning, then prefcribe to the Reader a rule he under­stands not.

Whereas the Julian yeer contained 365. dayes, six hours, and the Tropicall yeer, or time of the Sun's re­stitution to any point of the Zodiack, onely 365. dayes, 5. hours, and about 49. minutes, as we shall have occa­sion to shew hereafter; it happened that by this defect of the Tropicall yeer, dilating it self throughout every yeer, since the first establishing this Accompt in the Church, hath in processe of time, changed the places of the Equinoxes and Solstices, and altered the time of the Paschall solemnity: This defect having from the yeer of the Nicene Counsell 322. to the yeer of the Gregori­an Reformation, 1582. made an anticipation of ten dayes, which caused Pope Gregory the thirteenth to omit those ten dayes, thereby repairing what was amisse, and also to provide for the future, that no such anticipation should be; which he hath thus performed. He ordered that from the 5. of October 1582. (at what time he omit­ted his ten dayes, making that day the 15 ^th) untill the yeer 1700. there should be ten dayes added to the Juli­an Accompt; and from that yeer inclusive, every fourth Centenary of the yeers of our Lord, is onely of all the following Centenaries to be Bissextile, the rest of the [Page 5]Centenaries onely common yeers of 365. dayes; where­by it fals out, the Gregorian Accompt, every 400. yeers gains three dayes of what the Iulian loses. The differen­ces of the two Accompts in some succeeding Centenaries, we have here exhibited in this Table.

Thus by omitting the intercalation in these yeers, the Dominical Letter and Epact, which depend on the num­ber of dayes in each yeer, come to be changed; and by reason of the former, the number of direction, and con­sequently the movable Feasts, cannot by my Authors rules be truly perpetually gathered; all which, with some other inconveniencies, had been here more fully insisted on, if we had not thought this that hath been said sufficient for the ingenious, whereby to correct and amend these imperfections; and that Origanus in the first part of his Introduction to his Ephemerides, hath saved our Pen that labour, which hasts to discoveries of further concernment.

CHAP. III. The inequality of the Precession of the Equinoctiall points examined.

WHat may be happy to Urania, and gratefull to her true and legitimate Favourites, we now adventure upon the Sun and Moons Motions. A large current of considerations doth charge us, and we are likely to have more matter, then convenience to prosecute it: yet shall my unwillingnesse to trouble the Reader with more then is needfull for our present purpose, and my hope of a future fitter opportunity, to dilate my conceits upon this subject, prevail with the urgency of the matter, and confine my Discourse to its intended limits.

The first occasion that invites our Pen to consider here­of, is given by our Authors, pag. 52. where mention is made of a mean and true Equinox, in these words: ‘ So shall you have the true motion of the Sun, ab Aequinoctio vero, for in these tables, the Sun's mean motion is reck­oned from the true Equinox, and not from the mean.’ Whereby we may gather, that our Authors admit of an inequality of the Precession of the Equinoctial points: The manner whereof, with the cause of its admission in­to Astronomy; and lastly, the validity thereof, (because none of these are by our Authors so much as touched) it will not be inconvenient here in as brief a way as may be to deliver.

After that noble Dane Tycho Brahe had to the glory of Art, and joy of Artists, with incredible pains and dili­gence, perfected that elaborate table of the fixed Stars, and rectified it to his own time, a further and necessary care of perpetuating it, induced him to consider what helps might be drawn from ancient Observations to this purpose; and perceiving by those accounts that were ta­ken [Page 7]of their places, first by Hipparchus, afterwards by Ptolomey, Albategnius, Arzabel, Copernicus, and some others, that they had not onely motions, but unequall motions, and inconstancy in their latitudes, in severall ages; he was forced to devise some way whereby these motions might be regulated, to prove consentaneous to the observations of all ages. The Theory of this inequa­lity is according to the famous Astronomer, Chr. S. Lon­gomontanus, in this manner.

Let A be the Pole of tke Eclip­tick, BC that part of the Arctick circle of the Ecliptick, which the Pole of the earth in B hath run, by its equall motion, upon the center of the Ecliptick, since the Creation: This Arch measures also the Precession of the Equi­noxes, and progressive motion of the fixed Stars; BA is 23. degr. 42. min the Zodiacks mean obli­quity, DGE the small circle re­gulating the obliquity of the Zodiack, AD its radius, 10. min. 53. sec. By which it appears, both how the Zodiack changeth its obliquity, and also how the Equinoctiall points, and consequently every severall point in the Zo­diack do inequally anticipate: for when the Pole of the Aequator, which is carried in the circle EDG, is at E or G, the obliquity is in its mean deviations, and is equall to AB, 23. degr. 42. min. But the equation of the Equi­noxes is the greatest, GBA 27. min. 5. sec. and is to be substracted at G, added at E; wnen the Pole of the Ae­quator is at D, the obliquity is least, and is equall to DB, 23. degr. 31. min. 7. sec. but when the Pole of the Aequator is at F, the obliquity B F is greatest, and is 23. degr. 52. min. 53. sec. and in both these cases, there is no aequation of the Equinoxes, by reason of the coin­cidence [Page 8]of the lines BA and BG. This is the artifice Longomontanus hath used to satisfie appearances with, which if we should Phiscally consider, I doubt we should finde it more ingenuous then true; for it is scarcely tole­rable for any Astronomers to devise circles and imagi­nary motions, where with to fill the heavens, and with­draw the eye of man from a perfect consideration of the wisdome and power of his Creator, which best appears in the simplicity and uniformity of these Coelestiall es­sences: yet might this Hypothesis have been allowed, yea highly commended, if any good to Astronomy had come thereby, more then a needlesse multiplication of uncer­tainties. We will consider in a few words the validity and necessity of this Hypothesis, to perform what it pro­miseth.

Of the necessity hereof, ancient observations can give us no certainty; for from Proclus to our times, for above a thousand years, the Aequinoctiall points have made a certain and equall Precession, agreeable to that rule of motion which Timocharis and Hipparchus observed a­bove 1800. yeers agoe, if we onely except Ptolomey: Therefore if any Circulation more then annuall and di­urnall (if those be to be admitted) have befaln to the Poles of the aequator, whereby it hath been so enormi­ously removed from its scituation, it was betwixt the times of Hipparchus and Ptolomey, in the space of lesse then three hundred yeers, and was again restored in the time betwixt Ptolomey and Preclus, in other three hun­dred yeers: Wherefore without injury we may doubt of the certainty of Ptolomies observations; and the rather, for that he himself seems to imply as much, by these words.

‘ Non in tropicis tantum Observation bus, sed & in Ae­quinoctiabus error accidere po [...]est, qui ad quartam unius diei partem se extendat: Quod si [...]nim in 3600 tantum particula (as if that were little or nothing) Aequatoris [Page 9]situs, aut Instrumentis divisio, arecta raratione deficiat, illam in Latitudine sive Decliatione ac ☉ ad aequatorem accessu ad aequabit quarta circiter unius gradus pars, in Zodiaco & Longitudine, &c. Praeterquam quod & ma­jus erratum esse soleat si per instrumenta fiant Observa­tirnes, quae non illarum tempore exquisite positae sunt sed iam olim ita constituta, ut diu firmata lapsu temporum tandem commoveantur ac in situ deficiant.’ This and more, Ptolom. Almag. lib. 3 cap. 2. How sandy a foun­dation his Observations are, whereon to build Astronomy, especially seeing they disagree from others, may by his own words best be gathered.

He that desires to see a more full confutation of this inequality, may have it in Phocylides his Examen Astro­nom [...]ae Lansbergianiae, who from page 38. to page 63. he cleerly evinceth the same, from all the Observations of Equinoxes, had by the best Astronomers in every Age, and proveth a constant quantity of the Tropicall yeers in all Ages. A short Synopsis whereof, we had here pre­sented the Reader with, but that it would grow beyond our intentions; and that if the disposer of all our actions grant me ability and conveniency to prosecute the ser­vice of Urania, I may hereafter both enlarge and correct my present thoughts upon this subject.

The learned Kepler, pag. 27. Prec. tab; Rudolph. doubts not to assert, that there hath never been any other obli­quity of the Zodiack, then what is now, viz. 23. degr. 31. min. 30. sec. or by reason of his diminution of the Sun's paralax, 23. degr. 30. min. 30. sec. or consequent­ly any inequality of the Precession of the Equinoctiall points, and affirms he can demonstrate it; but methinks it is too manifest an injury to the Ancients to deny the one, so constantly evinced from their observations. But we must ever look with an indulgent eye upon that wor­thy man, whole Astronomicall performances do suffici­ently make known his worth, and memorize him to Po­sterity. [Page 10]It is not one Age, much lesse one man, that is able to restore Astronomy: His setting down five forms hereof in his Rudolphine Tables, shew the copiousnesse of his wit; his choosing of none, manifest the penury and uncertainty of former Observations. And surely these things, with many more, lye hidden in the Pandects of Posterity, not to be disclosed, untill God, the arbiter of Ages, shall open this eternall book, and disclose the se­crets hereof to mortals.

That noble French-man Ismael Bullialdus, the latest restorer of Astronomy, hath in his Astronomia Philolaica, followed Longomontanus in the obliquity of the Zodiack, but rejects the aequation of the Equinoxes, for these rea­sons.

First, there are no observations of the Ancients, which gives a sufficient exactnesse in the times of the Equinox­es, or places of the fixed Stars, whereupon to build such a fabrick of turbination, and that it were rashnesse in a­ny to attempt it.

Secondly, no circular revolution in the Heavens, ad­mits in its whole circumference, more inequalities then one, being slow in the one semicircle, swift in the other; but if we admit this inequality of the Precession of the Equinox, the simple motion is many times intended and remitted: But in other revolutions, intended but once, and remitted no oftner.

Thirdly, so small a difference is there found in distinct intervals of time, that it cannot be attributed to any true aod naturall motion, but with great boldnesse and teme­rity, whereby we impudently fasten upon the Heavens, the fictions and Chymera's of our own imperfect in­tellect.

Fourthly, that body which is furthest distant from the center of the World, would be immovable, which yet not­withsanding ought to partake of motion, as wel as other bodies which move obout the Sun, although the motion [Page 11]be very slow, by reason of its immense distance from the Center, and the amplitude of the space in which it moves; but why should other bodies move, and the whole Systeme of fixed Stars remain unmovable? every body placed about the Center of the World ought to have a motion about that, otherwise it would be a stran­ger to nature, and no partaker thereof, it keeping all things in motion, and not suffering them to be idle.

Fiftly, we ought not to think that the fixed Stars have an apparent progressive motion, according to the order of the Signs; for that cause alone, because the fixed Stars and the Equinoctiall points have a slow motion upon the terrestriall Poles, in antecedence of the Signs: For although in respect of the fixed Stars, such an Hypothesis might be true, because there is no exteriour body divers­ly proved, to which the motion of the fixed Stars may be sensibly compared; yet is it not to be admitted, because it cannot stand with the Planets motions; yet might it stand, if the Sun alone did appear: for by the annuall motion of the Poles of the Earth, the Sun, which then would not be supposed to run his annuall motion through the Zodiack, would manifest his accesse and re­cesse; but the Planets would be seen in the circle of Al­titudes, subject to irregular deviat ons, which neverthe­lesse is not: Therefore this Hypothesis were possible, were there but one Planet; but there being more, it is not possible, nor ought to be admitted.

Sixtly, this Argument is drawn a simili; we see in the Moon, a certain direction of her parts to the Earth, it is therefore likely that there is also direction of the Earths parts to the Sun, and that their axes retain alwayes the same positure, the one to the other, without any turbina­tion of either.

These are the Arguments which the learned Bullial­dus, lib. 5. cap. 2. Astron. Philol. hath brought for the dis­solution of this inequality, which I have here presented [Page 12]to the Reader in the same manner that he hath deliver­ed them: which though some of them vary from my present conceits, yet do the rest notably fortifie my opi­nion. Adde to these what Bullialdus hath demonstrated concerning the perpetuall equality of the Tropicall yeers, and I would fain see how the Authors of Urania Practica will disprove them. But such is their want of consideration, they have not sufficiently followed their own Theory herein; and though admitting of this in­equality, yet have given no rules or tables how to obtain it. The lustre of Urania hath (it may be) dazled their eyes, and the high flight of their Pen hath left their judgements behinde it; so that we may justly wonder what concert, whether the desire of being serviceable to Urania, rr enobling their names, hath drawn them to be actors upon the publick stage, where every judicious Spectator may discern their insufficiency.

CHAP. IIII. Of the Sun and Moons Tables.

THe next thing in order we should take notice of, is the Sun and Moons tables, and hereof we can say lit­tle, because our Authors have said nothing, they onely affording us Epochaes for some few yeers, without any sufficient rule whereby to perpetuate them. For those annuall motions by them set down in the end of page 65. cannot be perpetually consonant to their own Rule, un­lesse they will with us deny the inequall Precession of the Equinoctiall points, which their own words (men­tioned in the precedent Chapter) do oppose.

Yet what we can gather from the Tables themselves, and our Authors prescriptions for the use of them, we will here briefly deliver. Our Authors have in the Table of [Page 13]the Suns equations, followed the Theorie of Longomon­tanus, or some equivolent thereto (for there are divers) onely a little, though almost insensibly increasing the Eccentricity of the Sun, making the proportion of the radius of the Suns orbe, to the radius of his epicycle as is 100000. to 3577. to which Eccentricity in that hypo­thesis their equations of the Sun agree. But for what reason they have made this change themselves doe not show, nor can I conjecture. In the Tables of the Moons equations they have followed Argol, a man very labori­ous in calculations, but one who hath not to my know­ledge given any reason for what he hath done. He hath omitted the variation of the Moone, induced thereto as he saith by Observations. I will not question his doings, because I know not what Observations he used, but cer­tainly if there had not been a necessity for it, Tycho had never retained it into the Theory of the Moone, nor had it been confirmed by the after doings of I ongemontanus, Kepler, and the industrious and expert Bullialdus, espe­cially it causing so great a difference in the Moones place, extending it selfe to 40 min. 30 sec. according to Tycho, but according to Kepler a fourth part more. And although Keplers variation may be justly thought too bigge, notwithstanding he seems to deduce it from Phy­sicall and Archetypicall demonstrations, which he so much affected, yet in a Mathematicall eye, which at­tends precisenesse, the variation is not altogether con­temptible.

In the latitude of the Moon, our Authors have meerly followed Lansberge, and together with him rejected the inequality of motion in the Moones Nodes; of which I will nor dispute, the demonstration thereof depending upon such dubious Principles as Authors are not satisfi­ed thereof. Tycho making the period of this inequality menstruall (with whom herein Argol and Bullialdus al­so agre) Kepler annuall, yet all since Tycho admitting [Page 14]thereof, excepting onely Lansherg, of whose corruption and depravation of ancient Observations, so wresting them to his purpose, he that is not satisfied may finde him sufficiently characterized by Phocylides in his fore­mentioned booke. Thus from the fragments of broken Authors have our Authors patcht up their Tables of the Luminaries motions, which, however they will be suffi­cient to represent Coelestiall Observations, I much doubt and am fearfull that our Authors have done the divine Vrania wrong in attiring her simple excellence in such a particoloured vesture.

CHAP. V. Whether the second inequalitie of the Moon have de­pendance of the Sunnes mean or true and apparent place.

BY the quality of that table of the Moones equations, our Authors have set downe, occasion is given me to imagine they have therein followed a Theorie equivo­lent to that of Copernicus, viz, a double Epicycle, the circumference of the one carrying the Center of the o­ther: Yet however, the two inequalities, which are by Copernicus attributed to these Epicycles, are here by our Authors digested into one Table, which without questi­on were of great concernment to him that desires spee­dinesse in calculation, if the Artist could be assured of its exactnesse, and agreement with the Heavens, and those legitimate and Physicall Theories which may be thence deduced. But whereas it appeares by the Pre­cepts which guide us to the use of this Table, that the mean distance of the Luminaries is one of the steps wher­by we attain the Moons equation, we may (and not with­out just cause) suspect it of error.

It is true, that in those Theories of the Planets which were used before Tycho had happily confuted the solidi­ty of Caelestiall Orbs, there might be some appearance of reason why the Centers of those solid Spheres, rather then the Centers of the eccentricke Circles should re­gulate those other inequalities which depended there­on, they being supposed in that age not imaginary, but reall points, and therefore sufficient whereon to build a connexion of mosions. But after that Tycho had by the helpe of his exact Instruments, found the existence of temporary and fading Lights, within that Circuit, which was supposed to be free from generation and corrupti­on, and thereby solidly refuted the solidity of those Orbs and Spheres, so laboriously demonstrated by Pto­lemie, Proclus, Peurbachius, and others; this disclosure gaue the minde liberty to thinke of more rational wayes then the old multiplicity of Circles and Motions, where­by to salve caelestiall appearances. And hereby it came to be known, that the Causes of Motions were meerly Physicall, and dapended not upon the variety of Orbs, but followed that simple and uniforme course Nature had assigned them, and respected not those imaginary Centers which prudent Antiquity had for want of other helpes devised for them; but the very body of the Sun, the fountaine of Motion and common node of all their Orbs. Why then the Moon (which though a secondary Planet, yet hath relation to the Suns course) should re­ceive the Lawes of her extra-sysygiall inaequality from the Suns mean motion I cannot see. These reasons will evince she contrary.

1. The mean motion of the Sun, as also of any other Planet, is not in nature, but onely devised to regulate those exorbitances and deviations from equality, to which their apparent motions are subject.

2. Observations testifie that the longest line of every Primary Theory, which exactly bisects the orbe into [Page 16]two semicircles, equall in the quantity and celerity of the same parts, passeth by the center of the Sun, in which the Aphelian lines of the primary Planets concur.

3. The orbite of every primary Planet is intersected by the Eclipticke in places opposite by the center of the Sun, and not by any point without it.

4. The fountaine of motion, and the generall antece­dent to the particular relative inequalities of the Pla­nets, ought rather to be in the most excellent body then any where without it; for these reasons: first, because the moving force cannot reside in any Mathematicall point (such as this is imagined to be) but requires a bo­dy the more fully to exercise his operating power. Se­condly, it is most cnnsentaneous to reason, that the mov­ing force should be in the Center of the world (where it is evident the Sunne is) there being rest in the superfi­cies or sphere of the fixed Stars and motion in the in­termediate places.

5. The cause why Copernius and Tycho supposed the two Centers, viz. of the Orbe, and of the Eccentrick to be different things in themselves, is not sufficiently Mathematicall, they being drawn hereto by the desire of making their Hypotheses equivalent to those of Ptolo­mie. But it was not necessary to follow the steps of Pto­lomie so diligently; for Ptolomie made not every part of his Hypothesis from observations, but grounded many things upon a fore-conceived opinion, that the motions of the Planets were equall through every portion of their own circles, which Observations do sufficiently evince to be untrue, as may appear by famous Kepler in his learned Comentaries of the motions of Mars.

To these reasons I may adde also, the continued Ob­servations of Tycho in Eclipses, which cleer the Moon of any secondary inequality at the time of Defect, which is the true, not mean Syzigia of the Luminaries. But to this I cannot impute any great force, the difference be­ing [Page 17]so little, as it is hardly to be distinguished from those many irregularities, which attend on the propinquity of the Moon to the Earth, and of us in these Nothern Regi­ons, by reason of the great obliquity of our Sphear more perceived. Nor can we as yet think the manifold moti­ons of this inconstant Torrella, so throughly known as we can for the present build any certainty hereon. I doubt not but that Astronomia Brittannica (now by the blessing of God almost brought to perfection) will shortly take away many doubts herein, which have hi­thereto puzled divers Artists: And I doubt not but I shall discover some things concerning the Moons motions, which may be usefull for Astronomers in this subject, and more rationall then those impertinencies our Authors have here delivered; for they are out of all possibility of being excused, that in such a cleer Sun-shine, will impudently adventure to set out their dim Lanthorn for a guide to the young Practitioner, through these my­sterious Laborinths, especially such a worthy Lumina­ry as Kepler, having long since put all out of doubt, and taught Truth to move in her own Orb, not impedited by the adventitious remora's of humane fancy.

CHAP. VI. A demonstrative examination of our Authors Tables of Eclipses.

WE now come to the touch stone of our Authors judg­ment, and will (by God's help) lay open those many absurdities which would follow, should we admit of our Authors Tables, This speculation is not ordinary for obvious to every young Practitioner; yea, the intri­cacies hereof have entangled many profounder Artists then either Master Wing, Master L [...]y [...]ourn, or my Self, [Page 18]few of those many Authors [...] ^ch to this day have appeared the publick Champions of Urania, have had a full know­ledge hereof, excepting Kepler, the late Ballialdus, and the noble genius of our worthy Country-man, Master Jeremy Horrox, [...], from whose remains I have gathered the most of what I shall write in this Chapter.

Those who have presumed of their own sufficiency, to be able to demonstrate those dimensions which are re­quisite for the calculation of Eclipses, have used thereto a Diagram, said to be invented by Hipparchus, which they have severally commented on: Amongst the rest, the late famous Lansberge hath in his Uranometrie made a large discourse thereof; but so simply, as he may be a­shamed to spend those brasts upon so insufficient a piece. Yet hath his greatest fault been, that he hath not fitted his numbers to those Theoremes or Elements in his Ura­nometrie, as by him that will compare his numbers with the following Diagram, may be seen. But the like can­not so well be said of Copernicus, Tycho, Longomontanus, &c. by whom it is likely this demonstrative way was ra­ther omitted then not conceived: for they perceiving their Observations not to answer fully the rigid Theo­remes hereof, took more care to satisfie their Observa­tions, then Demonstrations. But the greatest cause of this difference being the inconstancy of Physical causes, which still interposed themselves, they are not altoge­ther to be excused, as not professing to deliver the acci­dentall inconstancies of the Phaenomena, but the true and demonstrative Principles of Art. But the divine Kepler both understood the excellency of this Diagram, and hath fitted the Precepts of his Rudolphine Tables hereto. He mentions oft a book of his own, entituled Hipparchus, wherein the demonstrations hereof are con­tained The Book I have not seen, perhaps it is not yet published. I shall onely at this time touch some few [Page 19]things herein, that concern my present purpose, refer­ring the Reader for the rest to Astronomia Brittannica, where it is fully demonstrated, and the manifold uses thereof declared.

In the Diagram annex­ed, let A be the center of the Sun, C of the Perigae­an shadow, H of the Apo­gaean shadow, B of the Earth; so is the apparent Semidiameter of the Sun ABE, of the Perigaean shadow CBP, of the Apo­gaean shadow HBN, the vertex of the Conical sha­dow D, the Semiangle thereof BDG, the axis BD; let the lines GF, PK NO, be paralel to DA: the Semidiameter of the Earth BG, the Center of the Moon L, the Semidia­meter of the Moon in the change LBM, the Hori­zontal Paralax of the Sun BAG, of the Moon BLG, whence we thus proceed.

I. The Semiangle of the Cone of the shadow is al­wayes lesse then the Sun's apparent Semidiameter, and the difference of these is the Sun's Horizontall Paralax.

The former part is pro­ved from an opticall prin­ciple: [Page 20] Idem objectum quo proprius cerniter, eo majus ap­paret, so that the Semidiameter of the Sun being beheld from B, appears greater then if it be beheld from D, in regard B is neerer the object then D. The Sun's Semidi­ameter apparent is ABE—AGE, the Sun's Horizontall Paralax is BAG—GAF. Now in regard the two lines AD and FG are paralel, it is necessary that ADE should be equall to FGE; therefore AGE-FGE-AGF-BAG the Sun's Horizontall Paralax.

II. The Semiangle of the Cone of the shadow is e­quall to the difference of the Moon's Horizontall Para­lax, and the Semidiameter of the shadow.

Let us take the Moon in her Apogaeum, and opposition to the Sun, her Horizontall Paralax is BHG, whereto BNG is equall; the Semidiamiter apparent of the sha­dow HBN: now BNG—HBN—ONP, which by reason of the Paralellisme of the lines NO and DA, is equall to the angle ADE, which is the Semiangle of the Cone of the shadow.

If we take also the Moon in her Perigaeum, her Hori­zontall Paralax is BCG, whereto BPG is equall; the apparent Semidiameter of the shadow is CBP: now BPG—BPK—KPG, which by reason of the Paralellism of the lines PK and DA, is equall to the angle ADE, the Semiangle of the Cone of the shadow.

Nor need any one think our demonstration invalid or insufficient, for that we have assumed the angles BCG and BPG to be equall, whereas indeed they are not ( [...]) precisely so: For I shall shew how inconsiderable a difference this is and how far it avoids our Observation. For if we take the angle ADE, or which is the same KPG to be 12 min. o sec. which follows from our Authors numbers, (for by the first Theoreme AGE 15 min. o sec. AGF 3 min. o sec. FGE—ADE 12 min. o sec.) the naturall Tan­gent thereof is 349067—KG. We shall also from our [Page 21]Authors Tables take the apparent Semidiameter of the shadow CBP, or BPK, to be 47 min. o sec. which is the greatest they set down; Its Tangent is BK—1367260. the sum of the angles is BPG—59 min. o sec. the sum of the Tangents is BG—1716327. whereto answers the angle BCG—58 min. 59. sec. 50 thirds: This angle differs from the former BPG, (to which it is assumed equall) onely 10 thirds; nor in this practice is the diffe­rence ever greater. The like might have been said of the former Theoreme, but that the difference which is here so contemptible, is there far smaller, and can in neither of them, though with the most scrupulous, leave any scruple of the certainty of our demonstration.

This Theoreme is the ground of the 148. Precept of the Rudolphine Tables, whose words run thus; Conjice in unam summam Parallaxes Horizontales Solis & Lunae; ab hac summâ abjiciatur Semidiamiter Solis apparens re­linquitur Sem diameter umbrae justa ad tempus: which is the same our Theoreme doth require, I might here shew.

III. That if the Cone of the shadow be so continued beyond the Earth, that the Diameter of the Sun be the base thereof, and the Center of the Moon be in the axes thereof betwixt the Earth and the Sun, the sum of her Horizontall Parallax, and the Semiangle of the Cone is equall to the apparent Semidiameter of the Cone in that distance.

IIII. That the difference betwixt the apparent Semi­diameter of the shadow and the Horizontall Parallax of the Moon in the shadow (that is the Semiangle of the Cone of the shadow) in the same distance of the Sun from the Earth is still the same, and receives no change from the Moons varying her distance from the Earth.

V. That it is impossible for any Planet or Star, that is scituate without the Earth, to be utterly void of Hori­zontall Parallax, if we speak Mathematically; though sensibly the fixed Stars and other remote bodies cannot [Page 22]be said to have any Parallax, in regard that the Semidi­ameter of the Earth, which is the Tangent of their Paral­lax, to the Radius of the distance, bears no evident pro­portion to that distance.

But these things to him that understands any thing of Geomitrie, need no demonstration, but are evident from the Diagram it selfe, and partly for that cause they are here omitted, partly because I hold it superfluous to use more Engines against our Authors falsities, then are needfull for the confuting thereof; and lastly, because that all those Theoremes which may be deduced from the precedent Diagram, are not to our present purpose, but have a nobler Object then any our Authors seem to have aim'd at; their rules and tables being made of such abject stusfe, they deserve not the title of Mathema­ticall or Astronomicall, as not agreeing to the pure and undoubted Principles of Art, which we will here manifest.

In the first Theoreme we demonstrated, that the diffe­rence of the Sun's apparent Semidiameter, and his Ho­rozontall Parallax was equall to the Semiangle of the Cone of the shadow: and in the second Theoreme, that the difference of the Moons Horizontall Parallax, and the apparent Semidiameter of the shadow was likewise equall to the Semiangle of the Cone of the shadow: It followes hence, that these two differences are likewise equall one to another. But how well our Authors have accorded hereunto may appear by the following Synop­si [...], The numbers whereof we have taken from our Authors Tables.

Thus are our Authors Tables unmasked, and laid o­pen to the view of every Artist, their disagreement to [Page 24]the demonstration being so great, as no Physicall Salve that can reasonably be applyed, is sufficient to counter­poise these differences. In every one of these positures, if we grant our Authors their numbers, it would follow, that the semiangle of the Cone is greater then the semi­diameter of the Sun; which is as if we should affirm, that FGE, part of the angle AGE were greater then the whole angle, which is absurd and impossible. But who can tell whether or no our Authors desire of exteri­our helps, hath furnished them of spectacles fitter for a greater age then their own, which amplifying the semi­angle of the Cone, made it appear so much too big for the room whereon it stood, and yet unlucky, did not amplifie the room withall.

Hence likewise would follow, that the Sun's distance from the Earth is not onely infinite, but (if we may so say) a degree beyond infinitenesse: And yet with much confidence can they proceed to determine (as we shall shew anon) the distance of the Sun from the earth in miles; whereas it appears by their Tables no such di­stance is ever possible to be defined, and their very di­stances there set down, are not onely disconsonant to the Truth, but also to their own erronious assumptions.

Hence would also follow, that the Sun's Horizontall Parallax were not onely nothing, but even lesse then nothing (contrary to the fift Theoreme) which how we should salve, I know not, unlesse we should imagine, that the Sun-beams passing through the Christalline Orbs of the inferiour Planets, finde in their journey a burning point wherein the severall rayes concur, and invert the species of the Object, so that hereby the pro­perty of the Sun's Parallax comes to changed: But I never heard that this was the confirmed experiment of any Optist, and am of opinion, that our Authors never dreamt they should have stood in need of such supporters as these for their new Tables; yet have they in the Title [Page 25]page spoken more truly then they thought, in these words, Nothing of this nature being extant in the English tongue: for scarcely shal we find any such absurdities drop from a Pen that professes to be able to perform so much.

I might here urge the Diagram further, and from thence shew not onely the generall, but particular de­fects of our Authors Tables, bu as it is true in all Arts, that Contra principia neg [...]ntem non est disputatio, so nei­ther Contra principia non babentem: Aliqua tamen claritas immittenda erit, ne erroris sussumig um oculos obtenebret, vapidisque n [...]gis impeditum mentis visum teneat.

I further observe from these Tables of our Authors; first, that the quantity of the Suns semidiameter, pag. 120. cannot agree to that Hypothesis from which the e­quations, pag. 59. seem to be derived. For it the eccen­tricity be taken 3577. to the Radius 100000. which fol­lows

from the greatest equation 2 d. 2 m. 59 sec. we shall in the Diagram annexed have EB the Suns greatest distance from E the earth 103577. the least distance ED 96423. the an­gle of the Apogaean [...]m diameter AEB, which we will a siume with our Author 15 m o s. and from thence proceed to finde the Per gaean semi­diameter CED, thus: Radius 1000000 Tang 15 m. o s. 4363. EB 103577. AB 4519. which is the tangent of the Apogaean semidiameter 35 m. o s. the radius being 103577. to which the line CD is equail: then ED 96423. CD 4519. Radius 1000000 Tangent of 16 m. 7 s. 4687. which is by our Authors Hypothesis the Perigaean semidiameter, and is greater by 37 sec. then our Authors have made it: Nor will the diminution of the eccentricity to make it equall with that of Longommtanus, any whit avail them, that proportion still leaving half a min. difference.

It is true, they have onely followed Argol herein, but his authority cannot shelter them, for he hath in the Equation of the Sun followed Kepler's Hypothesis of the bisected Eccenticity, whereto his Semidiameters do agree: but our Authors have confounded their Tables, by assuming severall parts of severall Hypotheses, which disagree amongst themselves, and thereby have made their work as unseemly a spectacle, as the Daw in her stolen plumes, or the Arcadian Dametas in his borrowed Armour.

II That the semediameters of the Moon, which our Authors have given us, pag. 121. are not consonant to the Observations which have been made by Artists, es­pecially in Eclipses of the Sun: for the Apogaean Semi­diameter of the Moon, 15 min. 15. sec. is almost equall to the Perigaean Semidiameter of the Sun, 15 min. 30. sec. whence it would fall out, that in most part of the Suns Eclipses, where the Moons visible latitude is small, the Sun would be totally eclipsed, which neverthelesse is contrary to all Observations of this kinde.

Chr. Clavius, in the fourth book of his Comentaries (upon my Country-man) John de Sac. Bosco, tels of an Eclipse he observed at Rome, Anno 1567. April 9. where­in he saw the obscure body of the Moon comprehen­ded all within the light body of the Sun, and so that the unobscured part of the Sun, was a bright circle about the Moons body. To this Observation, our Authors Cal­culation will not agree, the whole processe whereof we had here set down, had not our room failed us; yet the generall heads thereof we will deliver, as far as concerns our present purpose

This Eclipse falling without the limits of our Authors Epochaes, we took this course for obtaining the mean motions for the time of the Eclipse. First, we took the Radix 1667. and by the annuall motions set down pag. 65. gathered the motions of 100. common yeers, to which [Page 27]we added the motion of 25 dayes (being the number of Leap-yeers therein) the sum subducted from the said Radix, gave us the Radix anni 1567. as in the following Paradigma.

This calculation is fitted to the Horizon of Rome, whose latitude, according to our Authors, pag. 182. is 42. deg. o min.

Hereby it appears, that the Eclipse was totall, and that the body of the Moon covered the body of the Sun

on every side, the limbe of the Moon exceeding the Suns limbe on the south part, 1 m 15 sec. on the north part 1 m. 25 sec. whereas it ought to have been short of it on every side, and contain­ed it self within the luminous body of the Sun, according to the obser­vation of Clavius.

Nor can that be objected hereunto, which the learned Kepler hath discoursed in Astron. Opt. pag 297 concern­ing the possibility of this appearance: Those physicall causes having not so much power and force, as to take a­way all this distance of the limbs; and besides, never ta­king place but where the Suns light surrounds the limb of the Moon, for otherwise the splendor of the ayr which would encrease the Sun's semidiameter, quite vanisheth and leaves the Luminaries to their just bignesse.

Neither can the groundlesse limitations of Longomon­tanus or Lansberg be of any use in reconciling these dif­ficulties: For by the rule of Longomotanus, Theoric. pag. 177. there should be 33 sec. substracted from the semidiameter of the Moon; and by the rule of Lansberg, Uranometr. pag 66. there should be 45 sec. substracted from the semidiamer of the Sun, because the Moon is seen in a greater angle then the Sun. The former way could something lessen the errour, but not wholly take it away; the latter way would encrease it, and make it more notorious.

But why should I spend time in seeking such helps for our Authors, as perhaps they will not accept of; none of these limitations being so much as mentioned by [Page 29]them, their Tables freeing themselves of such un­necessary burdens, thereby to run more freely into error and falsity.

There have been had other observations of Solar Eclipses of this kinde, as that which was had by D. Jessenius, Anno 1598. February 25 at Torge in Misnia; Kepler, Astron. Optic pag 299. and that which was observed by the Fisherman, neer Bergen in Norway, upon the sea shore; where the bright circle about the Moon was 1 ½ Digit. This befell Anno 1601. Decemb. 14. Longom. Theor. pag. 165. The calculation of these, he that lists may try by our Authors Tables, and see how neer he can reconcile them with observation.

CHAP. VII. Of the aequation of Naturall Dayes.

OUr authors have, pag. 87. given us a precept for the finding of the aequations of naturall dayes, wherein they have followed Tycho, whose Table is not consenta­neous to Demonstration: For the aequation of dayes de­rives it current from two fountains, the one whereof is the motion of the Sun, in a circle inclined to the Equa­tor, which is the true measure of time, and out of this dis­agreement of the arches of the Equinoctiall, and the E­cliptick, ariseth the first inequality of time, which alone our authors have used, consisting of the difference of the right ascensions of the Zodiacall arches from the arches themselves. The other cause of this aequation is the une­quall progression of the Sun in the Zodiack, occosioned by his eccentricity; the difference of which diurnall arch­es, from the arch of his diurnall mean motion 59 m. 8 s c is the diurnall aequation: Yet must both this, and the former part of aequation be converted into time, before it be fit for use.

Yet have Tycho and the most of his followers rejected the later part of the equation, whose authority hath also drawne our Authors into their number. None of them showing any reason for their so doing herein, but this, that their restitution of the Moones motions, and the ob­servations of Eclipses, did seeme to require it; But this Empeiricall way can never be true, because it satisfies not the exactnesse of demonstration. And who can affirm that they have so restored the caelestiall motions, as that we may rather trust them then our alone senses. Expe­rience is indeed the Lady president of Vrania's repub­licke, and merits regard, so long as she prosecutes the Mandates of Demonstration, the supream Authority; but when once their results agree not, experience must yeeld and resigne her power to Demonstration.

CHAP. VIII. Our Authors determination of the distance of Cae­lestiall bodies from the earth examined.

I Shall not need to particularize the Stars and Planets severally. An example-will be sufficient to demon­strate the insufficienty of the rest.

We begin with the Earth, whose dimentions since our Authors have not here set downe, we must borrow from another place. In Master Wings Almin [...]ck 16 [...]8. the cir­cumference is given 21600. miles, and to this the con­version of degrees into miles ( Vran. Pract. par. 5 cha 3) doe agree. Whence he hath formed the Semidiamiter 3436. miles, which neverthelesse is not exactly true, for as 314159. is to 100000. (of this see Ludolphus van Cu­len and Lansberge de Cyclemetria) so is the cireumference 21600. to the Diameter 6875½. and so is the Semidiami­ter 3437¾. But we will take his number 3436. and see how the rest agree thereto.

We finde ( cap. 4. par. 4.) that the mean distance of Sa­turn from the Earth is according to Argol 10571. semi­diameters, which maketh (if we will beleeve our Au­thors) 9091960. miles. But he that shall multiply the number of semidiameters 10571. by the number of miles in one semidiameter 3436. shall finde another number, namely 36321956. So likewise in Jupiter 3990. multi­plyed by 3436. gives 13709640. not 343120 And in Mars also 1745. multiplyed by 3436. gives 5995820. not 1500700 In the Sun the distance in semidiameters of the Earth is not given, the distance is in miles 989000 which being divided by 3436. gives the distance in se­midiameters 287⅔. which gives the horizontall parallax almost 12 min. foure times as big as that which our Au­thors have given in their tables of Ecclipses. I will spend no more time in reckoning up the rest. I hope these er­rors (obvious to every School-boy) are sufficient to ma­nifest the likenesse and [...] of the rest.

But in regard our Authors are so carefull of the com­mon good, that over and besides this worke of Urania Practica, Master Wing hath publikely invited those that desire to be satisfied in any thing touching the Mathe­maticks, to repair to his judgement for satisfaction; and because he is holden a man of much dexterity in these sciences, I shall make bold to propound unto him and his [...] Master William Leybourne, a question or two, and upon their answer shall be ready to content them for their paines.

1. I shall enquire how the proportions of the fixed Stars to the Earth ( page 173.) are gathered, and a de­monstration that [...]hose of the five severall magnitudes, are so many times greater then the Earth, as is there set down; and also how those of the sixth magnitude, are found to be lesse then the Earth. I shall also demand how those of every severall magnitude, are approved to be so many in number, as they are there given, and no more.

2. I shall desire to know, how and in what manned the Ancients have demonstrated the distance of the Hea­vens by fixed Stars from the Earth to be 130715000 miles, or indeed whether any such Demonstration can be made, now that the Sphaeres are found no lesse fluid and penetrable then this ayre of ours.

3. I shall desire to know how the Planets distances from the Earth were gathered, or whether the Analo­gies of Tycho, and the so much mentioned Argol be groun­ded upon sufficient principles, or no; or whether those apparent diameters of the Planets, which our Authors have borrowed from Tycho, be consentaneous to the truth or no, it being propable that the raies of the Stars illu­minating the ayre they passe through made them ap­peare far bigger then indeed they were.

And let my Authors know, that I will not suffer my self to be over-ruled with the authority of any Writer, unlesse his Reason have a greater authority then his Name, and Judgement tread on the heels of Invention.

CHAP. IX. A brief summary of some other Defects and Imper­fections.

BUt my room grows narrow, and so tyeth my Discourse to a period: Good Reader, be pleased to take the rest of my collections in a bundle; I shall onely tell you what it contains, and would, if my professed brevity had granted me leave, have laid open the severall pieces to your view.

First, I s [...]y, that by our Authors rules, the Sun's al­titude cannot be gathered universally: for though the Example, pag 99. be truly wrought, yet if we turn to the fixt Book for a precept, we shall finde none, but [Page 33]onely a few concise Tables, calculated for some latitudes which are too narrow and insufficient for him, whose in­tentions are for narrow and insufficient for him, whose in­tentions are for generality and exactnesse.

Secondly, the tedious calculation of the Moons Pa­ralax in her circle of Altitude, detracts from the praise of the Book, and might have been with far more ease, and by the onely help of the Logarithms supplyed thus. As the Radius to the Sine of the Horizontall Parallax, so the Cosine of the Luminuries altitude, to the sine of the Parallax in that altitude. This way is no lesse de­monstrative, and far more easie then the other which our Authors have used, pag. 99.

3. That the table of hourly motion of the Moon from the Sun, pag. 118. cannot be exactly true, because it sup­poseth the Suns motion to be equal, and still of the same quantity; which neverthelesse by reason of his Eccen­tricity is not so, nor can be affirmed.

4. The herozontall Parallax of the Sun is not still 3. min. o. sec. according to the Paper adjoyned to the end of pag. 118. but if that be his Parallax in his meane di­stance, the Apogaean Parallax is 2. min. 53. sec. The Pe­rigaean Parallax 3. min. 7. sec. according to our Authors Eccentricity.

5. I affirme no Eclipse of Sun or Moon can be truly calculated, if we use no other rules then what our Au­thors have given us. For it appeares not whether they have used any reduction of the Moone from her Orbe to the Eclipticke; or of the Sun from the Ecliptick to the Moones Orbe; either of which wayes (consideratis con­siderandis) would serve. If they used the first, they finde onely the greatest Obscuration, which is not the midle of the Eclipse. If they used the second, they finde onely the midle of the Eclipse, which is not the greatest Ob­scuration, Vide Kepl. Astr. Cop. pag. 865.

Thus hath my Pen run over these imperfections which are the principall Moles in the face of Urania Practica, [Page 34]and that without all hatred or desire of Contradiction; the Causes whereof have no predominance in the least over my affections. I wish my Inke had rather been wa­ter to have washed a way these blots of Art, then markes to make them notorious. Yet I shall be glad if my Au­thors can clear themselves hereof, which for the present I see not how it may be done. My wishes are, that God the creator of the stars, and the disposer of their influen­ces, would second the weake endeavours of me and o­thers that desire the restauration of these sciences, for the common good of Man-kinde, and the glory of his Name, Amen.