Ens fictum Shakerlaei: OR THE ANNIHILATION of Mr. Jeremy Shakerley, his inartificiall Anatomy of Urania Practica. Wherein his fallacies or ignorance, are demonstratively detected, his malice in its groundless colours displayed, and the Authors of the said Urania Practica justly vindicated from his unjust Aspersions. By Vin. Wing, and Will. Leybourn, Philomathematicis. Scientia non habet inimicum, nisi ignorantem. LONDON, Printed by Robert Leybourn, 1649. Philomathematicis omnibus verè ingenuis, praesertim Astronomicae facultatis studiosis, necnon ejusdem laborisque pristini nostri Fautoribus semper honorandis. VRaniam istam (quae sub nostro nomine (heu) indigno sed vestro patrocinio libero jamjudum liberè prodiit) coelum summo verticis cacumine prius pulsantem, ad communem fere omnium mensuram horis subsecivis (dum per occupationes liceret) faeliciter cohibuimus: quam quidem postquam ita depressam, & infimo captui manum porrigentem (quod intentionis nostrae fuit ipsum culmen) diuturnis Vigiliis explicatè; enucleatimque exposuimus Zoilorum quorundam pennae quàm acriter acuminatae, mordaces Sciolorum linguae, Mathematicastrorum manus pedesque malevoli (porcorum instar) Margaritas praetiosissimas calcârunt, lacerârunt, violârunt. Inter quos Shakerlaeus homuncio inprimis pertinax coelestem illam ill otis ut aiunt manibus publicè, sed tamen pessimè (ut lectoribus statim innotescet) attractavit: qui etiamsi praejudiciò obcoecatus supra omnes mortalés 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 Gloriosus, coram videretur, dum conatus nostros alioqui satis gratos, (absit verbo 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉) chartis ejus indignis, accrbissimo felle impressis inanes reddere conatur, futuramque prolem Progymnasm. Astron. (de quibus in Uraniâ Practicâ mentionem fecimus) sublimioribus ingeniis magis adaequatam, & jam ferè limatam (nemine obstetricante) obstrueret; nos tamen ipsi (cum nemo alius manum forsitan admovere, & serram contentionis reciprocare velit) alata ejus sophismata, Ventosasque fallacias duabus aut tribus paginis refutare facilè possumus, & justè debemus: ita (ut I carius alter) alis Platonicis non bellè instructus in irrequietum aliquem Oceani Chimaeroplastici Euripum praeceps decidet. itaque in manus nostras Uraniae Anatomia incidit, ad hoc opus merito accingimus, calamum in manum assumpsimus, quo Uraniae veritatem & nosmet ipsos tueremur: quod quidem tam abundè fecimus, ut totam ejus Anatomiam resecavimus, atque ipsum Shakerlaeum Anatomicum minimè artificiosum aequo Lectori & praejudicio haud praeoccupato patefecimus, qui etsi errores praeli, nimis frequentes, facili manu corrigendos, non publicè refutandos, passim forsan inveniet; tamen nihil ponderis (etsi humanum est) cujus conscii sumus, vir melancholicus reprehendere possit. Sed ne nimis improbâ praesatione vestra studia moremur, humilimè orandi est is (Lectores candidi) ut utriusque partis (quia opposita oppositis magis elucescunt) argumentationes aequâ judicii vestri trutinâ perpenderetis, & non dubitamus, quin sacram Uraniam eo magis, magisque indies amplectemini: & contra prosternatura invidorum tormenta, satis validum praesidium vosmet profitemini: quibus suffragiis nos suffulti, & promissa nostra maturius praestare, & vestrae expectationi liberius respondere (Deo opt. max. volente) decrevimus Vale. Boni publici studiosissimi V-W-L. To the judicious Readers. THe World, that 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 which from our Master Aristotle hath received the Title of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, cannot challenge a freedom from contrariety and contradiction; the Ancients observed the Heavens had their Anomalies and Obliquities, the glorious Planets their passions of Retrogradation and Obscuration, the Elements their Mutations, yea this Earth whereon we live, move, and have our being (one would think the most stable creature) his motions, and herein fair Kingdoms after some revolutions, their periods, & non plus nltra? Semblably the clearest and most evident truths have suffered martyrdom in the furnace of dispute, and though naked, have not a Mandamus to be embraced. Those Communia effata in Metaphysics are guilty of opposition. We know not then how the foundation, Arithmetic and Geometry, in that stately Pyramid in the Mathematics can be privileged from a creeping hole, which may admit of Aliens, false ingredients, at least opposers, from which our Antagonist would seem (but it must be by a more potent and skilful hand) to deliver. The young Gentleman is not so simply elemented, but he hath heard of Parallogismes in Euclid, and a naeve in his dignities, read of irrationals (a doubt whether numbers or no) in Arithmetic, and for all M. Shakerley, room enough for descent: well, therefore, may Astronomy situated on the vertex, and Optics too, (for here we must needs shake hands with him (which consists in medio) be gradually diminished in their stabilities, and decurtated in their certainties, yet by the way note, that Madam Urania would hardly entertain those for her servants which are so towering in their airy imaginations, and want a foundation to the sublime Aedifices of their own futile conceits whereof the devasted servant stands doubly guilty, by taking his knowledge too much upon the public faith, and hath need of an Army for his protection, being guilty of so much treachery to so royal a Mistress, as we shall presently attest. But our Antagonist now being too much conscious of bloodshed, would willingly turn over a new leaf, and to lessen his cruelty like an Impostor turn the other end of the glass, and would make a universal diminution (to magnify himself) of that which a man with one eye would easily confess to be consentaneous to truth and demonstration; and then like another Oedipus or Dedalus, would lead us in with both hands into his enigmatical labyrinths, and thinks none can have the assistance of the third but himself. But, lest we should lose ourselves too, we will tract him in his progress, and great gate to his small City, and find how (like a cunning Sophister) he seems to insinuate by interweaving Sophistry with more specious argumentations: to this purpose, those who have the advantage of precedent knowledge and their own, from them greater performances are expected, the Authors of Urania Practica enjoy that, etc. ergo, greater: ye, to tell the vulgar of the Rotation of the Sun about his own Axis, and the Mountains and Seas in the Moon, etc. Sir, Plutarch is no new writer, though you are pleased to take it at the second hand. Oh! these are useful inventions and tickle his fancy, and what can we despair of? marry of any profitable or pertinent invention from you since you are out of your wits, and herein, O, wonderful Heavens and Stars! how hath the subtle Anatomist triumphed over the Authors of Urania Practica? How hath the miracle of men, to his own perpetual disgrace, by the perverting of his teachers, especially the learned Kepler, scarce triumphed over his palpable folly? The Gentleman, it seems, is popular, and a great enemy to antiquity, and disrelisheth the sovereignty of his Ancestors, against whom he fortifies himself with Scripture, which as light to an ill favoured Picture makes it become more odious to its beholders: For our own parts, we are not so much affected with novelty that we can despise the honest labours of so noble Worthies. Now, we have done with him, we must repair to our judicious and friendly Readers, whom we desire that they would be pleased candidly to interpret our honest meaning and endeavours for the propagating of these commodious and pleasant Sciences for the glory of the Creator, and that nothing might obstruct our real intentions and further labours, we have set pen to paper, and have sufficiently (although briefly) illustrated his mistakes and misconstructions of us and his Authors, and by consequence vindicated ourselves from the venom of his pen. Rumpatur, quisquis rumpitur invidiâ. V-W-L. Amicissimo suo Vincentio Wing. QVis tamen ille vapor foedavit nubibus atris Solem pallentem: fulmina nescit iners? Non superos timuit? non maxima lumina mundi? Infaelix decidet, cum redit orbe jubar. Te non con Vincent hostes (praenobilis Ala,) Et si Theoninis morsibus arma gerunt. Ingeniosa manus rumpet sucosque dolosque Haeresews Cynicae dexteritate tuâ. Ergo superstes eris (car pit te Zoile pulvis), Non malè defixus tu super astra nites. Sydera te servent, irrupto foedere, Mundi Cardine: libantes litibus astra parent. Amoris & Gratulationis ergô F. R. B. Mathemat. Joan. Cantabrig. To his Friend Master Vincent Wing. Do Vipers gnaw their passage through the Press To come in print? that youth, would do no less. Some Fury-piper got him o'er a Hearse: I vow his name will hardly * Shaker stand in Verse. Call him Anatomist? a Cutter sure. There's none but madmen, these times, can endure. But stay spectator think he did indent, * lie Not show his skill, but an experiment. As you may further view. His Teacher here Will plainly make it to the World appear. Detect his Errors then, whilst we thy fame Do stamp in verse, that none outfly thy name. By your Friend and Countryman. W. Billingsley. THE ANNIHILATION of M. Jeremy Shakerley, his inartificiall Anatomy of Urania Practica. §. 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 arrogancy, and large promises: That whatsoever he hath writ is undoubtedly true, and of learned Readers will be so approved, etc. But we, doubt not, to make the contrary appear, and to manifest to the world, that his aim 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 Vraniae. §. II. WE shall not meddle with that Peccadillo he mentions in the Foreign Account, in regard it's not worth the labour, neither shall we, till an on, dispute the inequality of the Equinoctial 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 perpetual, but to continue them for some years for the making the Ephemeris more useful, which may fitly be done notwithstanding the inequality of the Equinoctial points, which are there considered: and yet M. Shakerley to make an Error 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 been to set down any particular Tables to attein it (to make the thing more difficult) sigh it is far better for speed in calculating, and altogether as exact to unite it with the Sun's mean motion: But such is his rashness, and misguided zeal to Urania, that he would even disrobe her of her comely furniture. His next objection is against the Table of the Sun's Aequation, affirming we have followed the Theory of Longomontanus, or some equivalent thereunto (but it seems he is not certain of it) only a little (though almost insensibly) increasing the Sun's Excentricity, but for what reasons themselves do not show, nor can he conjecture. No, we think M. Shakerley doth not know indeed, neither shall 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 civil way, we shall be ready to give him sufficient satisfaction; in the mean time, if he'll 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 Moon's 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 do: But this is not true, for there is a sensible variation, amounting sometimes to 7 or 8 minutes, nay many times more, as they who list to try shall find; but because his error may be apparent, and his insufficiency appear, we have added the following operation from Argols' Tables. ☾ being Perig. distant from the ☉ 30 degrees s d ′ ″ Centrum Lunae 2 00 00 00 Argumentum medium 6 00 00 00 Aeqnatio centri add. 9 39 00 Scrup. proport. 19 30 Argumentum verum 6 09 39 00 Excessus 34 36 Pars pro scrup. prop. 11 15 Aequatio absoluta add. 1 05 27 But according to Urania Practica, 00 56 00 differing from argol 9′ 27″, whereas according to his judgement it should be equal. Lastly, he saith, that in the Latitude of the Moon we have merely followed Lansberge, and so from the fragments of broken Authors have patched up the Tables of the Luminaries motions, attiring the divine Urania in a particoloured vesture: But this is as false as the other, and though in the latter we come near to Lansberge, (which makes him conjecture we had it from him) yet the Table is de novo of our Calculation, and the Theory itself, whereon it's grounded, which is consentaneous to the former. In the beginning of Chap. 5. we find him guilty of another error, in imagining by the quality of the Table of the Moon's Aequation, that we have followed a Theory equivalent to that of Copernicus, viz. A double Epicycle, the circumference of the one, carrying the centre of the other; But we deny this to be true, for should we grant it, it would follow that the Aequation of the Centre, or second Epicycle should be swifter in the former semicircle, 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 groundless contradiction. §. 4. SHak. Chap. 6. We now come to the Touchstone of our Author's judgement, and will (by God's help) lay open those many Absurdities which would follow, should we admit of our Author's Tables. This speculation is not ordinary, nor obvious to every young Practitioner, yea the intricacies thereof have entangled many sounder Artists, than either Mr. Wing, Mr. Leyburn, or myself, few of those many Authors, which to this day have appeared, have had a full knowledge thereof, excepting Kepler, the late Bullialdus, etc. Lo! 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 Ptolemy, Copernicus, Lansberge, Longomontanus, Tycho, Reinholde, argol, or Eichstade present, they would never suffer themselves to be thus grossly scandalised with every simple Idiot, 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 infers, yet are not we so idle as to think, 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 own (which they have made ample and excellent demonstration of) as to be guided by his fancy. We knew the same long since and have fitted those (longed for) new Tables in Progym. Astron. thereto, which (by God's blessing) is in a good step to perfection. Of the other we would here have made demonstration, but in regard of the tediousness 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, the Revolut: Pitiscus Lib. 4. Probl. 14 & Lib. 5. Prob. 12, & 13. de Probl. Astron. Lansberge, Lib. 1, 2, & 3, Vranometriae: where they may be satisfied of the verity of our Hypothesis, and may find (having respect to our suppositions) that the Tables we have inserted are exactly consentanious to demonstration, and more rational than he yet apprehends, else without doubt never would so many expert Mathematicians in all ages with joint consent have followed this and rejected that of his beloved Hyparchus. Certainly all the Learned would account that man a mere fool that should examine the Tables of Ptolemy by the Hypothesis of Copernicus, or the Tables of Copernicus by the Theory of Kepler, or that should compare equal lines by different Scales to produce an equality, & è converso, and now we leave it to the judgement of the Learned, whether this man hath not exceedingly shown his folly in the selfsame kind, 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. According to Copernicus. ☉ Apog. ☾ Perig. ′ ″ Semidiameter of the ☉ 15 50 horizontal Parallax of the ☉ 3 00 Semiangle of the Cone 12 50 horizontal Parallax of ☾ 62 21 Semidiam of the Shadow 47 52 Semiangle of the Cone 14 29 Differing from the former 1 39 According to Lansberge. ☉ Apog ☾ Perig. Semidiameter of the ☉ 16 47 horizontal Parallax of the ☉ 2 18 Semiangle of the Cone 14 29 horizontal Parallax of the ☾ 63 39 Semidiameter of the shadow 46 19 Semiangle of the Cone 17 20 Differing from the former 2 51 But according to Eichstade, Longomontanus, argol, and Urania Practica, as followeth. ☉ Apog. ☾ Apog. ☉ Ap. ☾ Per. ′ ″ ′ ″ Semidiameter of the ☉ 15 0 15 00 horizontal Parallax of the ☉ 3 0 3 00 Semiangle of the Cone 12 0 12 00 horizontal Parallax of the ☾ 59 9 62 39 Semidiameter of the Shadow 43 0 47 00 Semiangle of the cone 16 9 15 39 Differing from the former 4 9 3 3 Having thus examimed these great Masters of Astronomy, we find their Tables will not agree to the Diagram of Hyparchus, no more than Urania 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 SHak. I further obscrve from the Tables of our Authors; First, That the quantity of the Sun's semidiameter, pag. 120. cannot agree to that Hypothesis, from which the Aequations pag. 59 seem to be derived, for if the excentritie be taken 3577, to the Radius 100000, we shall have the Sun's greatest distance from the earth 103577. etc. By your favour (Sir) you are (again) 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 we have followed the proportion of bi-sected excentricity (though different from the Hypothesis of Kepler) and that his error 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, BEAUMONT 15′— 0″— 4363. so BE 1017890 to B A 4441. 2. As E D 982110. to CD (equal 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 to our Hypothesis, mathematical diagram from which our Tables never differ more, which was the reason we followed argol and Eichstade therein without further calculation, and now Master Shakerley, me thinks, you cannot have the impudence to own that Paper-kite so coursely decked in your feathers, which flies abroad, as unseemly as an Owl at noon day. Next he affirms, 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. April 9 and to the time of this Observation he forms a Caculation from our Tables, but yet he mentions not the moment of the obscurations, though Clavius there tells him it was circa meridiem about noon, which, it seems, he is loath to make known, yet he cannot deny but that (according to our Tables) the Eclipse was central 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 year 1560 at Conimbrica in Lusitania about noon he meddles not with, because he knows it speaks much to the praise of our Tables (as doth the other) though much against his will. §. VI BUt not to trifle away ink (as he hath done) to no purpose, we shall come to the substance of his 7 Chapter, concerning the Aequation of natural days, 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 fancy, yet what he delivers there, he (its 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, argol, 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 days, 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 secondary 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 exactness of demonstration. §. VII. CHap. 8. He comes to examine the distance of the Celestial bodies from the earth, wherein his malice and ignorance as much appears as before, but before we come to examine his mistakes here, let us look back to the 24 pag. where are these words, Hence would likewise follow, that the Sun's 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 exemplify, and shall here, by the verity of our calculation, sufficiently prove him a mere Butcher, and by the way advise him to turn to the 3 Book of Lansbergs Vranometria, de errantium & in errantium Stellarum dimentione, and then, if he be not too much biased 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 find what we have said, cum rei veritate ad amussim consentire, to be no less than truth. Now that the judicious may see his failings and unparallelled mistakes, we shall show him, as we promised, how to find 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 centre 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 Sun's appearance from the superficies of the Earth B, therefore A D is the distance of the Sun from (A) the centre of the earth, and the angle A D B is the horizontal 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, As Radius A B, 10,00000 to the co-tang. of ADB 3′ 13,05915 So the side AB 1 Semidi. 0,00000 to the side AD 1146 serè 3,05915 mathematical diagram 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 infiniteness, as he surmises, and therefore from our Hypothesis the distance of the Sun from the earth (in Germane miles) is 985560, and this gives the horizontal 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 ourselves; but (that noble Maecenas, 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 from the earth to be near 4 Semidiameters greater, viz. 1150 Semidiameters: Now he that shall multiply this number by 860 shall have in the product 989000 Germane miles which is the true number we have set down; so likewise in Saturn, 10571 multiplied by 860 giveth 9091060: in Jupiter 3990 multiplied by 860 gives 3431400, and in Mars (likewise 1745 multiplied 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 find some hole to creep out at, which we cannot yet espy. Concerning his three Queries (or demands) we shall here forbear to make any tedious repetition, in regard one of us intends ere long to publish something 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 himself. But leaving him herein, we next come to his Bug-bear-bundle, or brief summary of nonsense. §. VIII. SHakerley, First, I say, that by our Author's Rule; the Sun's altitude cannot be gathered universally; for though the example pag. 99 be truly wrought; yet if we turn to the sixth book for a Precept we shall find none, but only a few concise Tables, calculated for some Latitudes, which are too narrow and insussicient for him whose intentious are for generality and exactness. Although the precept to find the Súns' altitude were casually omitted in the 6 Book, yet that defect may very well be supplied by help of those Tables there inserted, which are sufficient for this Kingdom and the Regions conterminate; and besides, what is he that is but a mere 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 amplify that there promised, we have added the following Example, mathematical diagram Let the time proposed be the 2 of July 1649, at four of the clock in the afternoon, the Sun's declination being 22 deg. Northwards, at which time the Sun's 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 51d 32′, whose compliment is Z P 38d 28′, H O the Horizon, E Q the Equinoctial, D K the Sun's 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 obliqne angled Triangle Z S P, in which we have given: First, The side Z P 38d 28′ the compliment of the Latitude, Secondly, the side S P 68d, the Sun's distance from the Pole, or the compliment of his Declination. Thirdly, the angle Z P S 60d, the time from noon 4 hours. And it is required to find the side Z S, the compliment of the Sun's altitude above the Horizon H C O. As the Radius 90d 10,0000000 to the co-sine of Z P S 30d 9,6989700 So the Tangent of Z P 38d 28′, 9,9000865 to the Tangent of P R 21d 40′ 9,5990565 Which being substracted from the whole side S P, there remains the Arch S R 46d 20′ As the co-sine of P R 68d 20′ 9,9681781 to the co-sine of Z P 51d 32′ 9,8937452 So the co-sine of R S 43d 40′ 9,8391396 19,7328848 to the co-sine of Z S 35d 34′ 9,7647067 Which 35d 34′ is the altitude of the Sun above the Horizon. §. IX. SHakerley, The tedious calculation of the Moon's parallax in her Circle of altitude detracts from the praise of the Book, and might have been with far more ease, and by the only help of the Logarithmes supplied thus: As the Radius, to the Sine of the horizontal parallax; so the cousin of the Luminaries altitude, to the sine of the parallax in that altitude. This way is no less demonstrative, & far more easy than the other which our Authors have used, pag. 99 Here we find him still plunging himself into more gross absurdities then before, for he would here make the Reader believe the calculation of the Moon's parallax in altitude detracts from the praise of the Book, and might be performed with far more ease and no less 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 37d 47′55″ and the parallax of the Moon in the Horizon 1d 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 find another number, viz. 49′ 3″, differing from the truth no less than 32″, which in a business of this nature is very considerable; but that he may plainly see his error and arrogancy, and the truth of our calculation, we'll take a little pains to inform his judgement by the demonstrative example here following. mathematical diagram A represents the centre of the earth, B the place of observation, A G L the true Horizon, B H the apparent Horizon, F the place of the ☽ in her own Orb, L F her altitude, 37d 47′ 55″, A H B Her Parallax in the Horizon 62′ 4″, A F B her Parallax required. In the Rectangled Triangle B C F we are first to inquire the side B C thus. D A F 52d 12′ 5″ sine C F 79018 Sine of the compliment C A 61288 Sine of greatest parallax A H B 62′4″ subst. 1805 Rests B C 59483 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, 53d 1′ 41″, from which detracting the Angle D A F, 52d 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 45d, and her Parallax in the Horizon 62′, her Parallax in that altitude will be found to be 44′ 24″. For in the former Diagram suppose, B C is 45d 70711 Comp. C A 45d 70711 Horizon. Parallax AHB 62′ 1803 B C 68908 As BC 68908, to C F 70711, So Radius BC 100000, to C F 102617, the Tangent of the Angle C B F 45d 44′ 24″, from which taking the Angle DAF 45d, 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 Physical salve being reasonably applied, is sufficient to counterpoise these differences. §. X. TO the third and fourth Sections of his Musterroll 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 Sun's excentricity cannot cause an alteration of above 3 or 4″ in the table of the hourly motion of the Moon from the Sun, and what error 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, argol, and divers others have done before us, being loath to trouble themselves with such niceties & needless trifles. In the next, where he saith The Sun's horizontal 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 Author's Excentricity. And here, indeed, he speaks truer than he supposed, Ex falsa sequitur verum, for henever dreamt of a bisected Excentricity, but we shall 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 EBB 103577 to AB 90. 2 As ED 94423 to DC 90 (being equal 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 equal. §. XI. WE are now arrived at the fifth and last Section of his Summary, where he is doubtful Whether in our Tables we have used any reduction of the Moon from her Orb to the Ecliptic, & 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 obtain it: but we do not well conceive his meaning, where he saith, the middle of the Eclipse is not the greatest obscuration, etc. Surely this (indeed) is strange music in the ears of Urania, and is not suitable to her excellency, for the proving whereof we desire the Reader to peruse Ptol, Lib. 6. Cap, 4. etc. 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. add fol. 70. Precept. call. 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 himself, and therefore shall advise him to turn to Precept 146. Tab. Rudolph. or to pag. 864. Epit. Astron. Cop. where Kepler tells him, Quod medium Eclipse 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 centre of the Moon is either joined to the centre of the shadow, or is in the perpendicular which comes from the centre 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 shows in what respects the places of the true Conjunction and greatest obscuration differ: Differunt enim in arcu minimo (as his own Author there tells 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 Orb from her place in the Ecliptic, which always differ, unless the Moon be in the Nodes or Quarters: the other is caused by the inclination of the way of the Moon to the Zodiac, when she is in the shadow of the earth, and this is all these Authors intent, 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 error 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 skilful 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 examined this learned (or rather wrangling) Discourse wherein we find him so unadvisedly rash that we can but admire at his folly, especially that such a man as he, who professeth himself 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 himself into most infinite errors and gross absurdities, even such as may be discerned by every judicious Spectator, if he wink not on purpose; but we shall leave him as we found him even brimful of malice; his aim being (as every one may perceive) purposely to smother those tender buds which begin to appear in the fields of Urania. FINIS. Domino Jeremiae Shakerlaeo, in Mathematicis studioso. CHare tum Colende vir, 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 illud (quod pro magis seria negotia subfuratus sum) tuam (quam vocas) brevitatem compendiosius respondere liberè compulit; quid dixti? brevitatem; certo certius ultra veritatis limites multis parasangis extensam: illud etenim, quod de literis ad me missis (quibus à me responsis Anatomiam tuam coecum forsitan (proh amentia!) suspenderes) susurras, pro Commento splendidissimo (cujusmodi (defectum an redundantiam tui ingenioli culpem, compertum vix habeo) tota tua controversia nimis conscia facile à nobis refutatur) aequo Lectori invalebit. Itaque (ut omnia uno verbo expediam) dum veritatem sanctissimè solertissimèque propugnare velut luxuriasti (heu!) quantum tristi discrimine violasti. Plura charitas mea (nè tecum hyperephaneus videar) proferre vetat literas quas tibi consulta nudius septimus miseram, antequam provectior fuit dies, responsione dedignareris, orandus es, interim siat pro coronide, ut sit tibi mens sanae in corpore sano. Votum profecto perpetuum. Ab aedibus meis North-Luffenhamiae in Rutlandia 24 Junii 1649 Aetatis nostrae 30 labente. Tui Uraniaeque amicissimi Vin. Wing, seu de Alâ Mathematicâ. depiction of a star