the yeers of our Lord is onely of all the following Centenaries to be Bissextile the rest of the 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  An. DoÌ„ ad da Anno Domini Add days Anno dom Ad da. From the 5. of October 1582. 10.  From the 24 of February 1700. 11. 2500. 17. 3300 23 1800. 12. 2600. 18. 3400 24 1900. 13. 2700. 19. 3500 25 2100. 14. 2900. 20. 3700 26 2200. 15. 3000. 21. 3800 27 2300. 16. 3100. 22. 3900 28 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 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 Longomontanus Theric lib. 1. 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 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 poâ—est qui ad quartam unius diei partem se extendat Quod si â—nim in 3600 tantum particula as if that were little or nothing Aequatoris 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 Astronomâ—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 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 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 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
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 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 I. ☉ Apog ☽ Apog  Min. Sec.  The Semidiameter of the Sun 15 0 AGE The Horizontal Parallax of the Sun 3 0 AGF The Semiang of the cone of the sha 12 0 FGE ADE The Horizontal Paral. of the Moon 59 9 BHG The Semidiameter of the shadow 43 0 HGN The Semiangle of the Cone 16 9 ONP ADE Differing from the former 4 min. 9 sec whereto it should be equall II. ☉ Apog ☽ Perig. The Semidiameter of the Sun 15′ 0″ AGF The Horizontall Paral. of the Sun 3 0 AGF The Semiangle of the Cone 12 0 FGE ADE The Horizontal Paral. of the Moon 62 39 BPG The Semidiameter of the shadow 47 0 BPK The Semiangle of the Cone 15 39 KPG ADE Differing from the former 3 min. 39 sec whereto it should be equall III. ☉ Perig. ☽ Apog The Semidiameter of the Sun 15′ 30″ The Horizontall Parallax of the Sun 3 0 The Semiangle of the Cone 12 30 The Horizontall Parallax of the Moon 59 9 The Semidiameter of the shadow corrected 42 32 The Semiangle of the Cone 16 37 Differing from the former 4 min. 7 sec whereto it should be equall IIII. ☉ Perig. ☽ Perig. The Semidiameter of the Sun 15 30 The Horizontall Parallax of the Sun 3 0 The Semiangle of the Cone 12 30 The Horizontall Parallax of the Moon 62 39 The Semidiameter of the shadow corrected 46 32 The Semiangle of the Cone 16 7 Differing from the former 3 min. 37 sec whereto it should be equall Thus are our Authors Tables unmasked and laid open to the view of every Artist their disagreement to 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 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
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 Alminâ—ck 16â—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 287â…” 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 â—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 sâ—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 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