§. I.

BEfore we come in point of Art to prove what we before promised, and to see whether our Antagonist have proved himself to be verus filius Ar­tis, it will not be impertinent to take notice of his specious words, high ar­rogancie, and large promises: ‘ That whatsoeuer he hath writ is undoub­tedly true, and of learned Readers will be so approved, &c.’ But we, doubt not, to make the contrary appear, and to ma­nifest to the world, that his aime hath been principally to delude them who understand not these Sciences, and we could wish his promised Astronomia Britannica, be not as desective as his Anatomia Ʋraniae.

§. II.

WE shall not meddle with that Peccadillo he men­tions in the Forreign Accompt, in regard it's not worth the labour, neither shall we, till an on, di­spute the inequality of the Aequinoctiall points (which is generally granted by the best Astronomers) but briefly come to the next thing he carps at, Chap. 4. which is touching the Tables of the Sun and Moons motions, wherein he saith ‘ he can say little, because his Authors have said nothing, they only affording Epochaes for some years, without any sufficient rules whereby to perpetuate them.’ To this we answer, it was not our intentions to make the same perpe­tuall, but to continue them for some years for the making the Ephemeris more usefull, which may fitly be done not­withstanding the inequality of the Aequinoctiall points, which are there considered: and yet M. Shakerley to make an Errour where none is, would perpetuate them contrary to the precept there set down, and besides, though we did admit of such an inequality, yet to what purpose had it bin to set down any particular Tables to attein it (to make the thing more difficult) sith it is far better for speed in calculating, and altogether as exact to unite it with the Suns mean motion: But such is his rashnesse, and mis­guided zeal to Ʋrania, that he would even disrobe her of her comely furniture.

His next objection is against the Table of the Suns Ae­quation, affirming we have followed ‘ the Theorie of Lon­gomontanus, or some equivalent thereunto (but it seems he is not certain of it) only a little (though almost insensi­bly) encreasing the Suns Excentricity, but for what reasons themselves do not shew, nor can he conjecture.’ No, we thinke M. Shakerley doth not know indeed, neither shall [Page 3]we at this time give him our reasons, and so make him understand what he is not capable of, but if he shall de­sire it in a more civill way, we shall be ready to give him sufficient satisfaction; in the mean time, if hee'l take a little pains, and have a little patience, we dare undertake he shall soon be able to make his own demonstration.

§. III.

IN the Table of the Moons Aequation he would make the Reader believe ‘we have followed Argol, a man very laborious in Calculations, but one (saith he) who hath given no reasons for his doings:’ But this is not true, for there is a sensible variation, amounting somtimes to 7 or 8 minutes, nay many times more, as they who list to try shall finde; but because his errour may be apparent, and his insufficiencie appear, we have added the following operation from Argols Tables.

But according to Ʋrania Practica, 00 56 00 diffe­ring from Argoll 9′ 27″, whereas according to his judgement it should be equall.

Lastly, he saith, ‘ that in the Latitude of the Moon we haue meerly followed Lansberge, and so from the frag­ments of broken Authors have patcht up the Tables of the [Page 4]Luminaries motions, attiring the divine Urania in a par­tie-coloured vesture:’ But this is as false as the other, and though in the latter we come neer to Lansberge, (which makes him conjecture we had it from him) yet the Table is de novo of our Calculation, and the Theo­rie it self, whereon its grounded, which is consentaneous to the former.

In the beginning of Chap. 5. we finde him guilty of ano­ther errour, in imagining ‘ by the quality of the Table of the Moons Aequation, that we have followed a Theorie aequivalent to that of Copernicus, viz. A double Epicy­cle, the circumference of the one, carrying the center of the other;’ But we denie this to be true, for should we grant it, it would follow that the Aequation of the Center, or se­cond Epicycle should be swifter in the former semi-circle, and slowest in the latter, and so the two inaequalities digested into one Table, would be sensibly disconso­nant to that which we have composed, being grounded upon a different Hypothesis, as we would here have ex­emplified, had it been pertinent to answer every ground­lesse contradiction.

§. 4.

Loe! here our Antagonist musters up Miracles, and would make the Reader believe his knowledge far sur­mounts the judgement and learning of almost all others, surely were either Ptolomie, Copernicus, Lansberge, Longo­montanus, Tycho, Reinholde, Argoll, or Eichstade present, they would never suffer themselves to be thus grosly scan­dalized with every simple Ideot, that understands not reason, nor the ground of their demonstrations; and al­though none of these Authors, nor any other that we know of, have followed the Diagram of Hyparchus; he inferrs, yet are not we so idle as to thinke, they have not had a full knowledge thereof as well as himself, could they thereby have made their Tables consentaneous to truth and observation, and if they could, had they not as much reason to credit their owne (which they have made ample and excellent demonstration of) as to be guided by his fancie. We knew the same long since and have fitted those (longed for) new Tables in Pro­gym. Astron. thereto, which (by Gods blessing) is in a good step to perfection. [Page 6]Of the other we would here have made demonstration, but in regard of the tediousnesse thereof, we cannot pos­sibly bring it within our intended limits, and therefore shall refer the Reader to Ptol. lib. 5. Cap. 14, 15, & 16. Almagesti, Copernic. Lib. 4. Cap. 18, 19, 20, 21, 22, & 23, de Revolut: Pi­tiscus Lib. 4. Probl. 14 & Lib. 5. Prob. 12, & 13. de Probl. Astron. Lansberge, Lib. 1, 2, & 3, Ʋranometriae: where they may be satisfied of the verity of our Hypothesis, and may finde (having respect to our suppositions) that the Tables we have inserted are exactly consentanious to demonstra­tion, and more rationall then he yet apprehends, else without doubt never would so many expert Mathematici­ans in all ages with joynt consent have followed this and rejected that of his beloved Hyparchus. Certainly all the Learned would accompt that man a meer fool that should examine the Tables of Ptolomie by the Hypothesis of Co­pernicus, or the Tables of Copernicus by the Theorie of Kepler, or that should compare equall lines by different Scales to produce an equality, & è converso, and now we leave it to the judgment of the Learned, whether this man hath not exceedingly shewed his folly in the self-same kinde, by comparing our numbers with the Diagram of Hyparchus. But we speak not against the Demonstration thereof, though it be of above 1780 years antiquity, but a­gainst his silly and improper application of it, as we have here mentioned, and more fully may appear by the fol­lowing Synopsis.

Having thus examimed these great Masters of Astro­nomie, we finde their Tables will not agree to the Dia­gram of Hyparchus, no more then Ʋrania Practica doth, and therefore let us not from thence conclude rashly (with Master Shakerley) their Tables are false, errone­ous, and uncertain, before we be well ascertained of the ground and verity thereof.

§. V.

By your favour (Sir) you are (againe) mistaken in sup­posing we have taken the whole excentricity 3577. which had we done, we should willingly have acknowledged an offence, and undergone your censure, but wee have fol­lowed the proportion of bi-sected excentricitie (though different from the Hypothesis of Kepler) and that his errour herein may be apparent, we will here compare our numbers with his own Diagram. Pag. 25. and use his own manner of operation thus.

1. As Radius B. E. 1000000 to the Tangent of the Angle, BEA 15′—0″—4363. so BE 1017890 to B A 4441.

2. As E D 982110. to CD (equall to B A (4441. So the Radius E D 1000000 to the Tangent of the Angle, DEC 4522. whose Arch is 15′. 31″. (and not 16′. 7″. as he saith) which if the true Semidiameter of the Sun ac­cording [Page 9]to our Hypothesis,

from which our Tables never differ more, which was the reason we followed Ar­goll and Eichstade therein without further calculation, and now Master Shakerley, me thinks, you cannot have the impudence to own that Paper-kite so coursely deckt in your fea­thers, which flyes abroad, as unseem­ly as an Owl at noon day.

Next he affirmes, ‘ That the Se­midiameters of the Moon are not consonant to the observations which have been made by Artists, especially in Eclipses of the Sun,’ and for an ex­ample instances the Observation of Clavius at Rome, Anno 1567. Aprill 9. and to the time of this Observation he formes a Caculation from our Ta­bles, but yet he mentions not the mo­ment of the obscurations, though Clavius there tels him it was circa meridiem about noone, which, it seems, he is loath to make known, yet he cannot deny but that (according to our Tables) the Eclipse was centrall there at the very moment of Observation, to which few Tables that have yet appeared, do better agree; but for the other Eclipse of the Sun, which Clavius a little (before in the same page) speaks of, which was in the yeer 1560 at Conimbrica in Lusitania about noon hee meddles not with, because he knowes it speaks much to the praise of our Tables (as doth the other) though much against his will.

§. VI.

BUt not to trifle away inke (as he hath done) to no pur­pose, we shall come to the substance of his 7 Chapter, concerning the Aequation of naturall dayes, wherein he saith, ‘ we have followed Tycho, (and here he speaks true by chance) which (saith he) is not consentanious to demonstration,’ though we may boldly conjecture he can­not tell, but we have no reason to be guided by his fancie, yet what he delivers there, he (it's true) borrows it from Bullialdus, fol. 8. Tab. Philo. where he admits of a second aequation, Ab inaequalitate diurnarum Terrae revolutionum circa axem, from the inequality of the daily revolotions of the Earth about her Axis, which peradventure others may admit of, but what of this? are we bound to follow him in every respect? hath not Eichstade, Argoll, and others since Tycho, allowed and approved of the former? And under favour, Sir, if you be a legitimate Son of Art, you cannot be ignorant of what the Ancients have delivered to posterity, how they have observed the aequation of dayes, even to this present age; compounding therewith an aequation for the motion of the Sun, without any reason or demonstration, which the Mathematicians of our time (not without good reason) have rejected as we have done, and if we should admit of a secondarie aequation, yet Eam ex passionibus obliquitatis arcuum Eclipticae cum Aequatore desumendam, Sicut (si res optimè trutinetur) parvam quan­dam posse consurgere ex Aequinoctiorum inaequalitate prae­cessionis: And this is all we can admit of, it being suffici­ent for the exactnesse of demonstration.

§. VII.

CHap. 8. He comes to examine the distance of the Coe­lestiall bodies from the earth, wherein his malice and ignorance as much appears as before, but before we [Page 11]come to examine his mistakes here, let us look back to the 24 pag. where are these words, ‘ Hence would likewise follow, that the Suns distance from the earth is not only infinite, but, if we may so say, a degree beyond infiniteness, and yet with much confidence they can proceed to determine the distance of the Sun from the earth in miles, whereas it appears by their Tables, no such distance is ever possibly to be defined, and their very distances there set down, are not only disconsonant to the truth, but also to their own er­roneous assumptions.’

What we have there said concerning the intervals and distances of the Sun, Earth and other Planets, we are a­ble to make the truth thereof demonstratively appear, as we shall exemplifie, and shall here, by the verity of our calculation, sufficiently prove him a meer Botcher, and by the way advise him to turn to the 3 Book of Lansbergs Ʋranometria, de errantium & in errantium Stellarum di­mentione, and then, if he be not too much byassed to his own opinion, we dare undertake he shall soon be able, by those simple Elements to make his own demonstration, if he will have but reason to hang his dimensions upon their proper and true Hypothesis, and then he shall finde what we have said, cum rei veritate ad amussim consentire, to be no lesse then truth. Now that the judicious may see his failings and unparalleld mistakes, we shall shew him, as we promised, how to finde the true distance of the Sun from the earth according to our Tables, which for brevity sake take thus.

In the following Scheme, A denotes the center of the earth, B C B the circumference thereof, A B its Semi­diameter, D the place of the Sun in the Horizon, B D the line of the Suns appearance from the superficies of the [Page 12]Earth B, therefore A D is the distance of the Sun from (A) the center of the earth, and the angle A D B is the Horizontall parallax of the Sun, therefore in the rectangle Triangle A B D is given (1) the side A B, the Semidia­meter of the earth 1 part, (2) the angle opposite A D B 3′0″, hence is found the side A D: For,

And this is the true distance of the Sun from the earth in Semidi­ameters according to our Tables, which is not infinite, nor a degree beyond infinitenesse, as he surmi­ses, and therefore from our Hypo­thesis the distance of the Sun from the earth (in German miles) is 985560, and this gives the horizon­tall parallax of the Sun 3′ as before, and not 12′ as he imagines: and herein we desire not to be our own judges, but shall refer it to the cen­sure of profounder Artists then ei­ther Mr. Shakerley or our selves; but (that noble Mecaenas, and restaura­tor of Astronomy) Tycho Brahe, whom we followed therein (as ap­pears pag. 174 and following,) he observed by his large and curious Instruments, his distance [Page 13]from the earth to be neer 4 Semidiameters greater, viz. 1150 Semidiameters: Now he that shall multiply this number by 860 shall have in the product 989000 Ger­man miles which is the true number we have set down; so likewise in Saturn, 10571 multiplyed by 860 giveth 9091060: in Jupiter 3990 multiplyed by 860 gives 3431400, and in Mars (likewise 1745 multiplyed by 860, gives 1500700.

Hence it appears how unjustly he hath charged us with that he can no way make good, but we could wish (because he pretends to these Sciences) he could finde some hole to creepe out at, which we cannot yet espie.

Concerning his three Queries (or demands) we shall here forbeare to make any tedious repetition, in regard one of us intends ere long to publish somthing of that nature wherein we shall fully discuss that matter; in the interim, we can but laugh at his folly, in demanding the Observations of others from us, which he understands not himselfe. But leaving him herein, we next come to his Bug-bear-bundle, or briefe summary of non-sense.

§. VIII.

Although the precept to finde the Súns altitude were casually omitted in the 6 Book, yet that defect may very [Page 14]well be supplyed by helpe of those Tables there inserted, which are sufficient for this Kingdom and the Regions con­terminate; and besides, what is he that is but a meer Tyro in these Arts that cannot perceive how to work it, having so plain and perspicuous an example as that is, Pag. 99. but to supply that defect, and to amplifie that there promised, we have added the following Example,

Let the time proposed be the 2 of July 1649, at four of the clock in the afternoon, the Suns declination being [Page 15]22 deg. Northwards, at which time the Suns Altitude a­bove the Horizon is to be enquired: Therefore in the Di­agram annexed, let the outward Circle thereof represent the Meridian of London, O P the Latitude thereof 51 ^d 32′, whose complement is Z P 38 ^d 28′, H O the Horizon, E Q the Aequinoctiall, D K the Suns parallel of Declination Northward, and Z S C N the Azimuth that the Sun is in at the time of the question. In which Diagram (by the intersection of three great Circles) we have limited the oblique angled Triangle Z S P, in which we have given: First, The side Z P 38 ^d 28′ the complement of the La­titude, Secondly, the side S P 68 ^d, the Suns distance from the Pole, or the complement of his Declination. Third­ly, the angle Z P S 60 ^d, the time from noon 4 houres. And it is required to finde the side Z S, the complement of the Suns altitude above the Horizon H C O.

Which being substracted from the whole side S P, there remains the Arch S R 46 ^d 20′

Which 35 ^d 34′ is the altitude of the Sun above the Horizon.

§. IX.

Here we finde him still plunging himselfe into more grosse absurdities then before, for he would here make the Reader believe the calculation of the Moons parallax in altitude detracts from the praise of the Book, and might be performed with far more ease and no lesse demonstrati­on, which is altogether false, and contrary to the pure rules of Art, as we shall here demonstrate. In the example of ours, pag. 100, the altitude of the Luminaries is 37 ^d 47′55″ and the parallax of the Moon in the Horizon 1 ^d 2′ 4″ from whence we there gather, her parallax in the Circle of altitude 49′ 35″, and this exactly agrees with all Authors of any account whose works are extant: but if we work according to Mr. Shakerleys prescriptions we shall finde another number, viz. 49′ 3″, differing from the truth no lesse then 32″, which in a businesse of this nature is very considerable; but that he may plainly see his errour and arrogancie, and the truth of our calculation, wee'l take a little pains to informe his judgement by the demonstra­tive example here following.

In the Rectangled Triangle B C F we are first to en­quire the side B C thus.

Then in the rectangled Triangle B CF, say, As B C 59483, to C F 79018, so B C Radius 100000, to C F 132840, which is the Tangent of the Angle C B F, 53 ^d 1′ 41″, from which detracting the Angle D A F, 52 ^d 12′ 5″, it leaveth the Angle required, A F B 49′ 35″, which is the true Parallax of the Moon in her circle of altitude, differing from Mr. Shakerley's computation 32″, as before.

Again, suppose her altitude be 45 ^d, and her Parallax in the Horizon 62′, her Parallax in that altitude wil be found to be 44′ 24″. For in the former Diagram suppose,

As BC 68908, to C F 70711, So Radius BC 100000, to C F 102617, the Tangent of the Angle C B F 45 ^d 44′ 24″, from which taking the Angle DAF 45 ^d, there remains the Angle AFB 44′ 24″, whercas according to Mr. Shakerley's rule it is but 43′ 50″, differing from the truth 34″, and now if the young Gentleman can tell us how this can be performed with more brevity and exactly, we shall wil­ligly give him the better of it: But, alas, it cannot be, for Mr. Shakerley steers by a false Chart, yea, his proposition being so disconsonant both to Demonstration and true Calculation, that (to use his own words) no Physicall salve being reasonably applyed, is sufficient to counterpoise these differences.

§. X.

TO the third and fourth Sections of his Muster-roll we shall answer with brevity, in regard they are not worth the view of an Artist. To the first whereof (being the third in order) we say, and dare affirm by the pure Rules of undoubted Art, that the Suns excentricity [Page 19]cannot cause an alteration of above 3 or 4″ in the table of the hourely motion of the Moon from the Sun, and what errour this can produce in the use thereof let himself judge: and so the difference being insensible gave us good cause to omit it, as Copernicus, Maginus, Purbachius, Lansberge, Argoll, and diverse others have done before us, being loth to trouble themselves with such nicities & needlesse trifles.

In the next, where he saith ‘ The Suns horizontall pa­rallax is not always 3′, but if this be his parallax in his mean distance, the Apogaean parallax is 2′ 53″, the Peri­gaeon parallax 3′ 7″, according to our Authors Excentri­city.’ And here, indeed, he speaks truer then he supposed, Ex falsa sequitur verum, for henever dreamt of a bisected Excentricity, but we shal examine whether it be so accord­ing to the Excentricity which he sets down, therefore in the Diagram of the 5 § reason thus.

1 As the Radius 100000, to the Tangent of the angle E 3′, ( viz. 87,) So EB 103577 to AB 90.

2 As ED 94423 to DC 90 (being equall to AB,) So ED the Radius to the Tangent of the Angle at E 95, whose arch 3′ 13″ should (according to the proportion he sets down) be his Perigaeon Parallax, differing from his own judgement 6″, whereas it should be equall.

§. XI.

WE are now arrived at the fifth and last Section of his Summary, where he is doubtfull ‘ Whether in our Tables we have used any reduction of the Moon from her Orbe to the Ecliptique,’ & contra, which he might have observed pag. 62, and pag. 87. and therefore might have saved this labour as well as all the rest, for what he saith here, we knew long since, and have taught how to obtein it: but we doe not well conceive his mean­ing, where he saith, the middle of the Eclipse is not the greatest obscuration, &c. [Page 20]Surely this (indeed) is strange musick in the ears of Ʋra­nia, and is not sutable to her excellencie, for the proving whereof we desire the Reader to peruse Ptol, Lib. 6. Cap, 4. &c. Copernicus Lib. 4. Cap. 21, & 30. Purbachus Prop. 15. Tab. Eclip. Reinhold in Theor. Geo. Purbach. Stofl. Prop. 9, & 10. Eichstad. Cap. 2, 3, 4, & 5, Paed. Astron. contin. Lans­berg à fol. 56. ad fol. 70. Precept. cal. motuum. But if he be not satisfied with these, we doubt not, but the learned Kep­ler and the expert Bullialdus will do it, for, we hope, he will have so much modesty as to credit them though it be against himselfe, and therefore shall advise him to turn to Precept 146. Tab. Rudolph. or to pag. 864. Epit. Astron. Cop. where Kepler tels him, Quod medium Eclipsis est maxima obscuration that the middle of the Eclipse is the greatest obscuration, and that is, Quando centrum Lunae est vel jun­ctum centro umbrae, vel in perpendiculari illâ, ex centro umbrae in viam Lunae: when the center of the Moon is ei­ther joyned to the center of the shadow, or is in the per­pendicular which comes from the center of the shadow, and falls upon the way of the Moon; the same saith Bulli­aldus, Lib. 5. fol. 214. Astron. Philol. yet are we not ignorant of that he seems to stumble at, pag. 865 Epit. Astron. where Kepler most excellently shews in what respects the places of the true Conjunction and greatest obscuration differ: Differunt enim in arcu minimo (as his own Author there tels him) duploreductionis Lunae loci ad Eclipticam, cujus area Luna in obscuratione maximâ semper est vicinior nodo, quam centrum umbrae: and hereunto assents Bullialdus, Lib. 4. Cap. 7. De Reduct. Temp. where he also admits of a redu­ction of time from the true Opposition or Conjunction with the Sun to the greatest obscuration: one cause where­of is the difference of the place of the Moon in her Orbe [Page 21]from her place in the Ecliptique, which always differ, unlesse the Moon be in the Nodes or Quarters: the other is caused by the inclination of the way of the Moon to the Zodiack, when she is in the shadow of the earth, and this is all these Authors intend, and this we approve of, but (by Mr. Shakerleys favour) not of that he speaks of. And in case we should not observe this nice reduction he cavils at, what errour could it breed? Nay, did the learned Co­pernicus; Reinholdus, Noble Tycho, Eichstade, Lansberge, or Longomontanus, ever so much as observe it? although as able and skilfull as our Antagonist, seeing reasons may be given pro & con, as we could instance, but we doubt not but the judicious are already satisfied of the verity of our calculations, and also observe the fallacies of his erro­neous affertions.

Thus have we diligently examin'd this learned (or ra­ther wrangling) Discourse wherein we finde him so un­advisedly rash that we can but admire at his folly, especi­ally that such a man as he, who professeth himselfe to be an Artist, should so contumeliously and inconsiderately strive to confute others, before he hath any ground for his so doing, and so plunge himselfe into most infinite errours and grosse absurdities, even such as may be discerned by every judicious Spectator, if he winke not on purpose; but we shall leave him as we found him even brim-full of ma­lice; his aime being (as every one may perceive) purpose­ly to smother those tender buds which begin to appear in the fields of Ʋrania.