THE ANATOMY OF Urania Practica. OR, A short Mathematical Discourse; Laying open the Errors and impertinencies delivered in a Treatise lately published by Mr. Vincent Wing, and Mr. William Leybourne, under the title of Urania Practica. By Jeremy Shakerley Philomath. Tota stupet natura Deum, robustaque quamvis, Viribus in nihilo deficit aegra suis. Ars stupet, & tantae spectans molimina dextrae Heret, & ipsa suas damnat egena manus. Nature's astonished at her god, though fast And far she go; her strength is tired at last. To view these works, Art doth amazed stand, Doubts, and condemns her own untutered hand. LONDON: Printed by Thomas Brudenell, 1649. To the Honourable Major General LAMBERT: The Author wisheth Felicity. Most Honoured Sir: THese few leaves more trusting to the worth of the subject they treat of, than the worthiness of their Author's performance, have made bold to become an object of your eye, and though small in bulk, yet great in substance: Its winged subject hath given it occasion to mount on high, and from the top of Art, to look down on the unartificial Fabrics of others, which would make us believe they are fixed in solid Orbs, and not subject to corruption: Yet may it be doubted, its high flight is endangered with precipices, which every where make the passage more difficult: And so much the rather, for that the vain confidence of Artists in these days, hath drawn into the number of their adherents, others who take their Art upon trust, which of how dangerous a consequence it is, will in some measure appear in the following Discourse. But I did not know any to whom I would more willingly, or could more justly presume to Dedicate these the first fruits of my endeavours, then to your worthy self; who to your many rare and singular virtues, have added the Star of Mathematical knowledge, thereby to make your excellencies a perfect Constellation. Under the fortitude of whose favourable aspect, this slender piece craves a shelter from the malevolent rays of Detraction. Thus fare your celebrated goodness hath drawn the hopes of him who is The honourer of your Virtues, And Your Servant devoted, JER. SHAKERLEY. To the Mathematical READERS. THe principal glory of the Mathematic Sciences, is their certainty whereby they are able to command the approbation of others, and impose upon their Adversaries a necessity of favouring their Demonstrations, when other Arts, built upon the experiments of former times, leave room for the refined conceits of others, who many times change and alter their dubious principles. But these with their own materials have strengthened their holds, and are of power to frustrate the designs of opposition. And of these the two corner-stones, Arithmetic and Geometry have this pre-eminence above other Arts, that they leave no hole to creep out at, no quirk for dissent, but an everlasting addition of new Inventions, to what before hath been happily demonstrated. But Astronomy deriving her current from more abstruse & hidden fountains, hath left a harder task to her Observers, to find her out, and trace her through her curious Labyrinth; and unless learned in the other two, here most subject to fall into many absurdities and untruths. So nice is Urania in the choice of her servants, and so unwilling to prostitute her Virgin excellencies to the mercenary embraces of every vulgar Professor. And hence for the most part it falls out, that in these curious speculations, the more we desire to know, the more we come to the knowledge of our own ignorance, our new experiments informing us of the insufficiency of ancient suppositions, and giving an example to our followers to handle our conceits, as we have done theirs that went before us. For although what way soever we turn the eyes of our judgement, and from the beginnings draw down our cogitations upon each particular, we shall every where find evident testimonies of the divine omnipotency; and although that our minds by the Towning raptures of sublime imaginations, exalt themselves above the Spheres, and suffer not themselves to be bounded with other limits than those that circumscribe the universe; yet when we enter into these sacred contemplations, we find as much as we can behold, more than we can understand, these mysteries being too enigmatical for our dull apprehensions, as this load of flesh tries and clogs our immortal part in its aspiring journey. Notwithstanding the most necessary care of worthy Artists, hath scarcely in any age been wanting, and the divine Urania still found those who admired and celebrated her excellencies, whose successive labours have taught us how great a thing experience is, and what danger there is in conformity to ancient rules. Little could be expected from them who were guided by so few; yet their performances were beyond expectation. More may be required from our times, whose subsidiary knowledge is greater. This last age (by the blessing of God) doth enjoy the benefit of more admirable and useful Inventions, than any, or almost all before it, and still new ones are added to the old, and the welcome tidings of Mathematical novelties daily delight us. And indeed what shall we mortals now despain of? within what bounds shall our wits be contained? Difficult, remote and envious things are now grown pervious to humane industry. We have seen the spots of the Sun, and its conversion about its own axis; we have seen the lateral Guardians of Saturn and Jupiter, the various Phases of Mars, the horns of Venus and Mercury, the mountains and Seas of the Moon; we have seen the generation of Comets, the apparition and disparition of new Stars amongst the Planetary Orbs; we have seen innumerable Cohorts of fixed Stars; yea, even the small constellation of the Pleyades, beautified with divers lights impossible to be numbered; we have seen the Rainbow like colours of divers lucid Globes, and the motley outsides of the Dog-star and Orion. O Heaven and Stars! how much hath our age triumphed over you! Neither doth our victory end here, still new miracles add to the number of the old, and no day passeth without a triumph. Why then shall we subject ourselves to the authority of the Ancients, when our own experience can inform us better? why do we not break their bands asunder, and cast their cords far from us? certainly the Ancients (were they now alive) would not condemn our choice. Aristotle that referred the studious in his time to Eudoxus and Calippus for satisfaction, could not dislike our Arguments against the corruption of the visible Heavens. Ptolomey that founded his Hypothesis upon Observations, would not be angry if our Observations persuade us to another hypothesis than he hath constituted. Tycho Longomontanus & Lansberge (unless puffed up with self glory, or sworn to contradiction) would not deny our demonstration of the insufficiency of their Theories. Lastly, M. Wing, and M. Leybourn (if they be owners of that true worth which ought to be an inseparable companion of a Mathematician) will not storm, when we show them their Writings are subject to error. Nor is it indeed possible that one age, or one man should perform all things requisite in this Science. Many things are impossible ever to be found out, and many things reserved to the discovery of after ages. How easily, and how justly may we apply the Prophetical words of sententious Seneca to our present purpose: — Venient annis Seculae seris, quibus uraniae Vincula rerum laxet, & ingens Pateat mundus, Vitrumque novos Detegat Orbs, nec fit nobis Ultima Tellus. But I have been more large and general then for such a small Treatise. At your feet then (Mathematical Readers, and my honoured Judges) I prostrate this my first. born: For it is the censure of you alone which I value; the popular voice, like other Agents, never acting beyond their proper sphere of activity. Your Servant JER. SHAKERLEY THE ANATOMY OF URANIA PRACTICA. CHAP. I. The occasion of this Discourse. I Am not ignorant, how dangerous a boldness 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 custom hath enthroned in the good-liking of other men; and whosoever whets his Pen against such as these, is like to be more weakened with the batteries of Envy, then strengthened with the fortifications of a good cause. For when once we have enslaved our wits to any Author, and look upon him with the eye of subjection, we are very hardly drawn from the pressure of that yoke; yea, and oftentimes we murmur and repine at those, who show us the way to clear ourselves of this burden, and free our judgements from captivity. If then I could prefer the slothful 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 myself to the censure of the world: for although I do believe 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 myself the general 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 call 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 boldness and impudence, in that having but newly crept out of the limits of Childhood, I dare bend my Pen against famous Artists of such a continuance. But it is the part of a slothful timorousness, to be afraid of the rash censure of the multitude, and it is doubtful I shall not be so much commended for modesty, as censured for want of boldness, to be fearful to restore the Truth to her proper Dignity, and clear her nobleness from being partaker in those counterfeits, which pass 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 confess, when I heard the report of her coming, and saw the news of her so confidently fly through the British Isle, upon the paper-wings of several Almanacs, I had a good hope some extraordinary performance would have confirmed those glorious reports, and upon the Theatre of a Mathematical Judgement, acted something worthy those Praeludia. But when in January 1648/9. I received from my worthy and honoured friend, Master William Lily, 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 myself, that upon such uncertain foundations as the popular voice had built my hopes to that height: for I did not only 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 Musterroll, recorded in the beginning of his Almanac, 1649. but find that their errors are here renewed: all which I beheld with an eye, truly pitying such studious Tyroes, as understand only 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 pass 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 errors otherwise merit to be vindicated with more severity. CHAP. II. An error in finding the Dominical Letter and Epact in the Foreign Account perpetually, detected. THough something below the level of our intended endeavours, it will not be amiss to begin with this Peccadillo, which must necessarily proceed either from want of care, or knowledge of the true manner of the Calendars Gregorian reformation; the explication whereof, I shall choose rather briefly to deliver from its beginning, then prefcribe to the Reader a rule he understands not. Whereas the Julian year contained 365. days, six hours, and the Tropical year, or time of the Sun's restitution to any point of the Zodiac, only 365. days, 5. hours, and about 49. minutes, as we shall have occasion to show hereafter; it happened that by this defect of the Tropical year, dilating itself throughout every year, since the first establishing this Account in the Church, hath in process of time, changed the places of the Equinoxes and Solstices, and altered the time of the Paschall solemnity: This defect having from the year of the Nicene Counsel 322. to the year of the Gregorian Reformation, 1582. made an anticipation of ten days, which caused Pope Gregory the thirteenth to omit those ten days, thereby repairing what was amiss, 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 days, making that day the 15th) until the year 1700. there should be ten days added to the Julian Account; and from that year inclusive, every fourth Centenary of the years of our Lord, is only of all the following Centenaries to be Bissextile, the rest of the Centenaries only common years of 365. days; whereby it falls out, the Gregorian Account, every 400. years gains three days of what the julian loses. The differences of the two Accounts in some succeeding Centenaries, we have here exhibited in this Table. An. Dom.. 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 years, the Dominical Letter and Epact, which depend on the number of days in each year, come to be changed; and by reason of the former, the number of direction, and consequently the movable Feasts, cannot by my Author's 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 Ephemerideses, hath saved our Pen that labour, which hasts to discoveries of further concernment. CHAP. III. The inequality of the Precession of the Equinoctial points examined. WHat may be happy to Urania, and grateful 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, than convenience to prosecute it: yet shall my unwillingness to trouble the Reader with more than is needful for our present purpose, and my hope of a future fit 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 only motions, but unequal motions, and inconstancy in their latitudes, in several 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. two concentric circles inscribed with several labeled angles Longomontanus Theric. lib. 1. Let A be the Pole of the Ecliptic, BC that part of the Arctic circle of the Ecliptic, which the Pole of the earth in B hath run, by its equal motion, upon the centre of the Ecliptic, since the Creation: This Arch measures also the Precession of the Equinoxes, and progressive motion of the fixed Stars; BASILIUS is 23. degr. 42. min the Zodiacs mean obliquity, DGE the small circle regulating the obliquity of the Zodiac, AD its radius, 10. min. 53. sec. By which it appears, both how the Zodiac changeth its obliquity, and also how the Equinoctial points, and consequently every several point in the Zodiac 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 equal 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 equal 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 BASILIUS and BG. This is the artifice Longomontanus hath used to satisfy appearances with, which if we should Phiscally consider, I doubt we should find it more ingenuous than 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 wisdom and power of his Creator, which best appears in the simplicity and uniformity of these Celestial essences: yet might this Hypothesis have been allowed, yea highly commended, if any good to Astronomy had come thereby, more than a needless 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 Equinoctial points have made a certain and equal Precession, agreeable to that rule of motion which Timocharis and Hipparchus observed above 1800. years ago, if we only except Ptolomey: Therefore if any Circulation more than annual and diurnal (if those be to be admitted) have befallen to the Poles of the aequator, whereby it hath been so enormiously removed from its situation, it was betwixt the times of Hipparchus and Ptolomey, in the space of less than three hundred years, and was again restored in the time betwixt Ptolomey and Preclus, in other three hundred years: Wherefore without injury we may doubt of the certainty of Ptolemy's 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 poorest, qui ad quartam unius diei partem se extendat: Quod si ●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, etc. 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 clearly evinceth the same, from all the Observations of Equinoxes, had by the best Astronomers in every Age, and proveth a constant quantity of the Tropical years 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 Zodiac, than 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 Equinoctial 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 Astronomical performances do sufficiently make known his worth, and memorise him to Posterity. It is not one Age, much less one man, that is able to restore Astronomy: His setting down five forms hereof in his Rudolphine Tables, show the copiousness of his wit; his choosing of none, manifest the penury and uncertainty of former Observations. And surely these things, with many more, lie hidden in the Pandects of Posterity, not to be disclosed, until God, the arbiter of Ages, shall open this eternal book, and disclose the secrets hereof to mortals. That noble Frenchman Ishmael Bullialdus, the latest restorer of Astronomy, hath in his Astronomia Philolaica, followed Longomontanus in the obliquity of the Zodiac, but rejects the aequation of the Equinoxes, for these reasons. First, there are no observations of the Ancients, which gives a sufficient exactness in the times of the Equinoxes, or places of the fixed Stars, whereupon to build such a fabric of turbination, and that it were rashness in any to attempt it. Secondly, no circular revolution in the Heavens, admits in its whole circumference, more inequalities than 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 oftener. Thirdly, so small a difference is there found in distinct intervals of time, that it cannot be attributed to any true aod natural motion, but with great boldness and temerity, whereby we impudently fasten upon the Heavens, the fictions and Chimaeras of our own imperfect intellect. Fourthly, that body which is furthest distant from the centre of the World, would be immovable, which yet notwithsanding ought to partake of motion, as well as other bodies which move obout the Sun, although the motion be very slow, by reason of its immense distance from the Centre, 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 Centre 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 Equinoctial points have a slow motion upon the terrestrial Poles, in antecedence of the Signs: For although in respect of the fixed Stars, such an Hypothesis might be true, because there is no exterior body diversely 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 annual motion of the Poles of the Earth, the Sun, which then would not be supposed to run his annual motion through the Zodiac, would manifest his access and recess; but the Planets would be seen in the circle of Altitudes, subject to irregular deviations, which nevertheless is not: Therefore this Hypothesis were possible, were there but one Planet; but there being more, it is not possible, nor aught 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 Earth's parts to the Sun, and that their axes retain always 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 fortify my opinion. Add to these what Bullialdus hath demonstrated concerning the perpetual equality of the Tropical years, 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) dazzled their eyes, and the high flight of their Pen hath left their judgements behind 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 public stage, where every judicious Spectator may discern their insufficiency. CHAP. FOUR 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 only affording us Epochaes for some few years, without any sufficient rule whereby to perpetuate them. For those annual motions by them set down in the end of page 65. cannot be perpetually consonant to their own Rule, unless they will with us deny the inequal Precession of the Equinoctial points, which their own words (mentioned in the precedent Chapter) do oppose. Yet what we can gather from the Tables themselves, and our Author's prescriptions for the use of them, we will here briefly deliver. Our Authors have in the Table of the Sun's equations, followed the Theory of Longomontanus, or some equivolent thereto (for there are divers) only a little, though almost insensibly increasing the Eccentricity of the Sun, making the proportion of the radius of the Sun's orb, 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 do not show, nor can I conjecture. In the Tables of the Moon's 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 Moon, induced thereto as he saith by Observations. I will not question his do, 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 Moon, nor had it been confirmed by the after do of I ongemontanus, Kepler, and the industrious and expert Bullialdus, especially it causing so great a difference in the Moon's place, extending itself 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 big, notwithstanding he seems to deduce it from Physical and Archetypicall demonstrations, which he so much affected, yet in a Mathematical eye, which attends preciseness, the variation is not altogether contemptible. In the latitude of the Moon, our Authors have merely followed Lansberge, and together with him rejected the inequality of motion in the Moons 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 menstrual (with whom herein Argol and Bullialdus also agree) Kepler annual, yet all since Tycho admitting thereof, excepting only Lansherg, of whose corruption and depravation of ancient Observations, so wresting them to his purpose, he that is not satisfied may find him sufficiently characterized by Phocylides in his forementioned book. Thus from the fragments of broken Authors have our Authors patched up their Tables of the Luminaries motions, which, however they will be sufficient to represent Celestial Observations, I much doubt and am fearful that our Authors have done the divine Urania wrong in attiring her simple excellence in such a particoloured vesture. CHAP. V Whether the second inequality of the Moon have dependence of the Sun's mean or true and apparent place. BY the quality of that table of the Moon's equations, our Authors have set down, occasion is given me to imagine they have therein followed a Theory equivolent to that of Copernicus, viz, a double Epicycle, the circumference of the one carrying the Centre 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 speediness in calculation, if the Artist could be assured of its exactness, and agreement with the Heavens, and those legitimate and Physical Theories which may be thence deduced. But whereas it appears by the Precepts which guide us to the use of this Table, that the mean distance of the Luminaries is one of the steps whereby we attain the Moon's 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 Celestial Orbs, there might be some appearance of reason why the Centres of those solid Spheres, rather than the Centres of the eccentricke Circles should regulate those other inequalities which depended thereon, they being supposed in that age not imaginary, but real points, and therefore sufficient whereon to build a connexion of mosions. But after that Tycho had by the help 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 gave the mind liberty to think of more rational ways then the old multiplicity of Circles and Motions, whereby to salve celestial appearances. And hereby it came to be known, that the Causes of Motions were merely Physical, and dapended not upon the variety of Orbs, but followed that simple and uniform course Nature had assigned them, and respected not those imaginary Centres which prudent Antiquity had for want of other helps devised for them; but the very body of the Sun, the fountain of Motion and common node of all their Orbs. Why then the Moon (which though a secondary Planet, yet hath relation to the Sun's course) should receive the Laws of her extra-sysygiall inaequality from the Sun's 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 only devised to regulate those exorbitances and deviations from equality, to which their apparent motions are subject. 2. Observations testify that the longest line of every Primary Theory, which exactly bisects the orb into two semicircles, equal in the quantity and celerity of the same parts, passeth by the centre of the Sun, in which the Aphelian lines of the primary Planets concur. 3. The orbite of every primary Planet is intersected by the Ecliptic in places opposite by the centre of the Sun, and not by any point without it. 4. The fountain of motion, and the general antecedent to the particular relative inequalities of the Planets, ought rather to be in the most excellent body than any where without it; for these reasons: first, because the moving force cannot reside in any Mathematical 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 Centre of the world (where it is evident the Sun 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 Centres, viz. of the Orb, and of the Eccentrick to be different things in themselves, is not sufficiently Mathematical, they being drawn hereto by the desire of making their Hypotheses equivalent to those of Ptolemy. But it was not necessary to follow the steps of Ptolemy so diligently; for Ptolemy 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 equal through every portion of their own circles, which Observations do sufficiently evince to be untrue, as may appear by famous Kepler in his learned Commentaries of the motions of Mars. To these reasons I may add also, the continued Observations of Tycho in Eclipses, which clear 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 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 Northern Regions, by reason of the great obliquity of our Sphere 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 puzzled divers Artists: And I doubt not but I shall discover some things concerning the Moon's motions, which may be useful for Astronomers in this subject, and more rational than those impertinencies our Authors have here delivered; for they are out of all possibility of being excused, that in such a clear Sunshine, will impudently adventure to set out their dim Lantern for a guide to the young Practitioner, through these mysterious Labyrinths, 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 remoras of humane fancy. CHAP. VI A demonstrative examination of our Author's Tables of Eclipses. WE now come to the touch stone 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 for obvious to every young Practitioner; yea, the intricacies hereof have entangled many profounder Artists than either Master Wing, Master L●y●ourn, or myself, few of those many Authors ● ch to this day have appeared the public Champions of Urania, have had a full knowledge hereof, excepting Kepler, the late Ballialdus, and the noble genius of our worthy Countryman, 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 Theorems 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, etc. by whom it is likely this demonstrative way was rather omitted than not conceived: for they perceiving their Observations not to answer fully the rigid Theorems hereof, took more care to satisfy their Observations, than 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 accidental 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, entitled Hipparchus, wherein the demonstrations hereof are contained The Book I have not seen, perhaps it is not yet published. I shall only at this time touch some few 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. a semicircle and two circles divided by several labeled angles In the Diagram annexed, let A be the centre 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 parallel to DA: the Semidiameter of the Earth BG, the Centre 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 always less than the Sun's apparent Semidiameter, and the difference of these is the Sun's horizontal Paralax. The former part is proved from an optical principle: Idem objectum quo proprius cerniter, eo majus apparet, so that the Semidiameter of the Sun being beheld from B, appears greater than if it be beheld from D, in regard B is nearer the object then D. The Sun's Semidiameter apparent is ABE— AGE, the Sun's horizontal Paralax is BAG— GAF. Now in regard the two lines AD and FG are parallel, it is necessary that ADE should be equal to FGE; therefore AGE-FGE-AGF-BAG the Sun's horizontal Paralax. II. The Semiangle of the Cone of the shadow is equal to the difference of the Moon's horizontal Paralax, and the Semidiameter of the shadow. Let us take the Moon in her Apogaeum, and opposition to the Sun, her horizontal Paralax is BHG, whereto BNG is equal; the Semidiamiter apparent of the shadow HBN: now BNG— HBN— ONP, which by reason of the Paralellisme of the lines NO and DA, is equal to the angle ADE, which is the Semiangle of the Cone of the shadow. If we take also the Moon in her Perigaeum, her horizontal Paralax is BCG, whereto BPG is equal; 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 equal 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 equal, whereas indeed they are not (〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉) precisely so: For I shall show 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 Author's numbers, (for by the first Theorem AGE 15 min. o sec. AGF 3 min. o sec. FGE— ADE 12 min. o sec.) the natural Tangent thereof is 349067— KG. We shall also from our Author's Tables take the apparent Semidiameter of the shadow CBP, or BPK, to be 47 min. o sec. which is the greatest they set down; It's 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 equal) only 10 thirds; nor in this practice is the difference ever greater. The like might have been said of the former Theorem, 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 Theorem 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 Theorem doth require, I might here show. 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 Centre of the Moon be in the axes thereof betwixt the Earth and the Sun, the sum of her horizontal Parallax, and the Semiangle of the Cone is equal to the apparent Semidiameter of the Cone in that distance. FOUR That the difference betwixt the apparent Semidiameter of the shadow and the horizontal 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 situate without the Earth, to be utterly void of horizontal 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 itself, and partly for that cause they are here omitted, partly because I hold it superfluous to use more Engines against our Author's falsities, then are needful for the confuting thereof; and lastly, because that all those Theorems which may be deduced from the precedent Diagram, are not to our present purpose, but have a nobler Object than any our Authors seem to have aimed at; their rules and tables being made of such abject stusfe, they deserve not the title of Mathematical or Astronomical, as not agreeing to the pure and undoubted Principles of Art, which we will here manifest. In the first Theorem we demonstrated, that the difference of the Sun's apparent Semidiameter, and his Horozontall Parallax was equal to the Semiangle of the Cone of the shadow: and in the second Theorem, that the difference of the Moons horizontal Parallax, and the apparent Semidiameter of the shadow was likewise equal to the Semiangle of the Cone of the shadow: It follows hence, that these two differences are likewise equal one to another. But how well our Authors have accorded hereunto may appear by the following Synopsis, The numbers whereof we have taken from our Author's 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 equal. II. ☉ Apog. ☽ Perig. The Semidiameter of the Sun 15′ 0″ AGF The horizontal 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 equal. III. ☉ Perig. ☽ Apog. The Semidiameter of the Sun 15′ 30″ The horizontal Parallax of the Sun 3 0 The Semiangle of the Cone 12 30 The horizontal 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 equal. FOUR ☉ Perig. ☽ Perig. The Semidiameter of the Sun 15 30 The horizontal Parallax of the Sun 3 0 The Semiangle of the Cone 12 30 The horizontal 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 equal. Thus are our Author's Tables unmasked, and laid open to the view of every Artist, their disagreement to the demonstration being so great, as no Physical Salve that can reasonably be applied, 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 than the semidiameter of the Sun; which is as if we should affirm, that FGE, part of the angle AGE were greater than the whole angle, which is absurd and impossible. But who can tell whether or no our Author's desire of exterior helps, hath furnished them of spectacles fit for a greater age than 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 amplify the room withal. Hence likewise would 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 can they proceed to determine (as we shall show 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 only disconsonant to the Truth, but also to their own erroneous assumptions. Hence would also follow, that the Sun's horizontal Parallax were not only nothing, but even less than nothing (contrary to the fift Theorem) which how we should salve, I know not, unless we should imagine, that the Sunbeams passing through the Crystalline Orbs of the inferior Planets, find in their journey a burning point wherein the several rays 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 than they thought, in these words, Nothing of this nature being extant in the English tongue: for scarcely shall 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 show not only the general, but particular defects of our Author's Tables, but 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 'em oculos obtenebret, vapidisque n●gis impeditum mentis visum teneat. two overlapping triangles with labeled angles It is true, they have only 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 several parts of several 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. TWO 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 equal to the Perigaean Semidiameter of the Sun, 15 min. 30. sec. whence it would fall out, that in most part of the Sun's Eclipses, where the Moons visible latitude is small, the Sun would be totally eclipsed, which nevertheless is contrary to all Observations of this kind. Chr. Clavius, in the fourth book of his Commentaries (upon my Countryman) John de Sac. Bosco, tells 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 Moon's body. To this Observation, our Author's Calculation will not agree, the whole process whereof we had here set down, had not our room failed us; yet the general heads thereof we will deliver, as far as concerns our present purpose This Eclipse falling without the limits of our Author's 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 annual motions set down pag. 65. gathered the motions of 100 common years, to which we added the motion of 25 days (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. Long. ☉. ☉ Apog. Lon. ☽ a ☉ Anom. ☽. Lat. ☽. Si. D. M. S. S. D. M. S. S. D. M. Se. Si. D. M. S. Si. D. M. Sec. 1667 9 20 0 42 3 6 51 47 6 7 44 42 2 24 2 53 10 2 38 35 100 11 6 10 50 0 1 42 20 0 2 16 40 7 21 59 19 3 21 15 0 d. 25 0 24 38 28 0 0 0 4 10 4 46 7 10 26 37 29 11 0 44 1 y 100 0 0 49 18 0 1 42 34 10 7 2 47 6 18 36 39 2 21 59 1 1567. 9 19 11 24 3 5 9 13 8 0 41 55 8 5 26 14 7 10 39 34 Hence there follows D. H. Min. Sec. The mean ☌ of ☉ & ☽ at London 1567. Apr 8 9 4 22 The intervallum to be added 13 28 30 The true Conjunction at London 8 22 32 52 The equation of days (tab. pag. 117.) add. 8 8 The apparent time of the true ☌ at London 8 22 41 0 The differ. of Merid. of Lon. & Rome ad. 1 7 0 The apparent time of the true ☌ at Rome 8 23 48 0 At which time is given S. D. Min Sec. The Anomaly of the ☉ 9 21 30 44 The Anomaly of the ☽ 3 8 4 17 The mean distance of ☽ from ☉ 0 6 50 11 The true motion of ☽ latitude 2 24 33 51 The place of the ☉ and ☽ ♈ 28 32 58 The difference betwixt the true and visible Conjunction add. 20 15 The time of the visible ☌ April 9 8 15 The Semidameter of the ☉ 0 15 9 The Semidiameter of the ☽ 16 29 The sum of the Semidiameters 31 38 The visible lati ude of ☽ North 0 5 The scruples deficient 31 33 The digits eclipsed 12 28 22 This calculation is fitted to the Horizon of Rome, whose latitude, according to our Authors, pag. 182. is 42. deg. o min. a total solar eclipse Nor can that be objected hereunto, which the learned Kepler hath discoursed in Astron. Opt. pag 297 concerning the possibility of this appearance: Those physical 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 Sun's light surrounds the limb of the Moon, for otherwise the splendour of the air which would increase the Sun's semidiameter, quite vanisheth and leaves the Luminaries to their just bigness. Neither can the groundless 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 than the Sun. The former way could something lessen the error, but not wholly take it away; the latter way would increase 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 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 kind, 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, near 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 Author's Tables, and see how near he can reconcile them with observation. CHAP. VII. Of the aequation of Natural Days. OUr authors have, pag. 87. given us a precept for the finding of the aequations of natural days, wherein they have followed Tycho, whose Table is not consentaneous to Demonstration: For the aequation of days 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 Equinoctial, and the Ecliptic, ariseth the first inequality of time, which alone our authors have used, consisting of the difference of the right ascensions of the zodiacal arches from the arches themselves. The other cause of this aequation is the unequal progression of the Sun in the Zodiac, occosioned by his eccentricity; the difference of which diurnal arches, from the arch of his diurnal mean motion 59 m. 8 s c is the diurnal 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 drawn our Authors into their number. None of them showing any reason for their so doing herein, but this, that their restitution of the Moon's motions, and the observations of Eclipses, did seem to require it; But this Empeiricall way can never be true, because it satisfies not the exactness of demonstration. And who can affirm that they have so restored the celestial motions, as that we may rather trust them then our alone senses. Experience is indeed the Lady precedent of Vrania's republic, and merits regard, so long as she prosecutes the Mandates of Demonstration, the supreme Authority; but when once their results agree not, experience must yield and resign her power to Demonstration. CHAP. VIII. Our Author's determination of the distance of Celestial bodies from the earth examined. I Shall not need to particularise the Stars and Planets severally. An example-will be sufficient to demonstrate the insufficienty of the rest. We begin with the Earth, whose dimensions since our Authors have not here set down, we must borrow from another place. In Master Wings Almanac 16●8. the circumference is given 21600. miles, and to this the conversion of degrees into miles (Vran. Pract. par. 5 cha 3) do agree. Whence he hath form the Semidiamiter 3436. miles, which nevertheless 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 find (cap. 4. par. 4.) that the mean distance of Saturn from the Earth is according to Argol 10571. semidiameters, which maketh (if we will believe our Authors) 9091960. miles. But he that shall multiply the number of semidiameters 10571. by the number of miles in one semidiameter 3436. shall find another number, namely 36321956. So likewise in Jupiter 3990. multiplied by 3436. gives 13709640. not 343120 And in Mars also 1745. multiplied 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 horizontal parallax almost 12 min. four times as big as that which our Authors have given in their tables of Eclipses. I will spend no more time in reckoning up the rest. I hope these errors (obvious to every Schoolboy) are sufficient to manifest the likeness and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 of the rest. But in regard our Authors are so careful of the common good, that over and besides this work of Urania Practica, Master Wing hath publicly invited those that desire to be satisfied in any thing touching the Mathematics, 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 pains. 1. I shall inquire how the proportions of the fixed Stars to the Earth (page 173.) are gathered, and a demonstration that ●hose of the five several magnitudes, are so many times greater than the Earth, as is there set down; and also how those of the sixth magnitude, are found to be less than the Earth. I shall also demand how those of every several 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 Spheres are found no less fluid and penetrable than this air 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 rays of the Stars illuminating the air they pass through made them appear far bigger than indeed they were. And let my Authors know, that I will not suffer myself to be overruled with the authority of any Writer, unless his Reason have a greater authority than 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 tieth my Discourse to a period: Good Reader, be pleased to take the rest of my collections in a bundle; I shall only tell you what it contains, and would, if my professed brevity had granted me leave, have laid open the several pieces to your view. First, I s●y, that by our Author's rules, the Sun's altitude cannot be gathered universally: for though the Example, pag 99 be truly wrought, yet if we turn to the fixed Book for a precept, we shall find none, but only 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 exactness. Secondly, the tedious calculation of the Moon's Paralax 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 Logarithms supplied thus. As the Radius to the Sine of the horizontal Parallax, so the Cousin of the Luminuries altitude, to the sine of the Parallax in that altitude. This way is no less demonstrative, and far more easy than 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 Sun's motion to be equal, and still of the same quantity; which nevertheless 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 adjoined to the end of pag. 118. but if that be his Parallax in his mean distance, the Apogaean Parallax is 2. min. 53. sec. The Perigaean Parallax 3. min. 7. sec. according to our Author's Eccentricity. 5. I affirm no Eclipse of Sun or Moon can be truly calculated, if we use no other rules than what our Authors have given us. For it appears not whether they have used any reduction of the Moon from her Orb to the Ecliptic; or of the Sun from the Ecliptic to the Moon's Orb; either of which ways (consideratis considerandis) would serve. If they used the first, they find only the greatest Obscuration, which is not the middle of the Eclipse. If they used the second, they find only the middle 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 principal Moles in the face of Urania Practica, and that without all hatred or desire of Contradiction; the Causes whereof have no predominance in the least over my affections. I wish my Ink had rather been water to have washed a way these blots of Art, than marks 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 weak endeavours of me and others that desire the restauration of these sciences, for the common good of Mankind, and the glory of his Name, Amen. Domino Vincentio Wing in Mathematicis studioso. Vir Clarissime, ME nec tua 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 multum scribere nequmea 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 conticescere permittit. Hot tantum utrique donabo, ut liberioris calami prolixitatem reprimam, nec sinam ulterius tendere, quam quod caeteram nostram brevitatem deceat. Privatam quidem tecum speravi literarum communionem, net forsan Anatomia haec nostra lucem vidisset, si nostras ad te literas dignat us esses agnoscere. Vnam quidem circa finem jam proxime anteacti Novembris, alteram circa medium januarii ad te miseram, sed quid de illis sentires, aut quomodo amicitiam oblat am acciperes, scire nondum potui. Verebar imprimis ne Mathematicus ille candor sublimia ingenia comitari solitus in te locum non haberet. Leviculis enim velitationibus huic Tractatulo praelusi & sperasti forsan, spretis & suppressis nostris literis, suppressa etiam fore in tuam Uraniā 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, sed aliter res cecidit, neque rei difficultas, neque Uraniae Practicae fama me ab hoc utili contradicendi munere deterruit. Veritati saltem imperium deferendum erit, ut faciliori sceptro possit tam suos tutari quam in rebelles decernere. Ipsa per se satis elucescit, nullis nubibus impedita, nullis verborum lenociniis fucata. Nec deest legitima soboles, quae ad tuos cantus aures occludit, & cognoscere nolit que probare non possit. Nobis non sufficit speciosa Tabularam Parapegmata in tuiri. Splendidulis escis vixdum capimur— Manet alta ment repostum Quam male Perpetuas jactavit Belga Tabellas. Condonabis igitur juventuti nostrae calami libertatem, quaeque hic tela libro, non solum in te, sed & in Authores tuos directa puta. Sed inexcusabilius ipse peccasti, dum post faeliciter demonstratos corum errores, hanc telam retexis, fallaciisque fallacias superstruere conaris. Egregia quidem fuisset, summisque laudibus evehenda tua cura Astronomiam Orbi Britannico asserendi, si veris insisteres fundamentis, & Cramben recoctam dedignareris introducere. Proficuum fortassis & in posterum erit, si inutiles veterum nugas veritati posthabeas, & cures ut Progymnasmata illa Astronomica quae meditari te profiteris, absurdis Authorum tuorum positionibus depurata prodeant, & ut tandem dediscas servitutem ingenii, nec ab aliorum authoritate tibi ipsi patiaris imponi. Multa quidem sunt, de quibus privatis literis tecum disceptandi occasionem datam gauderem; mulra proponere statueram de vero mundi Systemate, de causis motuum, de Harmonia Sphaerarum non sonorâ illâ quam Veterum aliquis prodidere, sed Musica motuum & Orbium contemperatione. Multa de Luminarium motibus & Theoriis; de physicis inconstantiis, quae motus reales turbare videntur: Multa denique de novis Coeli Phaenomenis, & eorum causis agere erat animus, sed silentium tuum ulteriorem progressum intercepit. Sed jamdudum definitos carceres vagus transilivit calamus. Deus Opt. Max. tuos & aliorum in Astronomiâ restituendâ conatus, pro misericordia sua adjuvare dignetur. Hoc summopere precatur Magnus quamquam minimusVrani● cultor tuique amicissimus JER. SHAKERLEY. Car juxta Colne in agro Lancastrensi 8 Martii 1648/9. Aetat. nostrae 23. labente. A Postscript to the Reader. IT is a piece of difficulty as well to avoid the just Censure of some, as the unjust Calumniation of others. That little piece of mine which my worthy and honoured friend Master William Lily was pleased to insert in his late published Book of Astrological Predictions, etc. hath been subject to the misconstruction of some pure-nosed Critics; who have (they suppose) scented out an error in the erection of the Figure, pag. 14. for that the Sun is there placed below the Ascendant; whereas my Letter mentioned only the time of Sun-rise, for the time of the first visibility; nor indeed was it likely that the Parelii should appear before the Sun. Let such therefore as have stumbled upon that Bloke, give me leave to remove it; and consider that the refraction of the Sun's rays in the Atmosphaere, or Region of vapours, cause him to appear in the Horizon, when indeed he is below it; which this Figure in some sort demonstrates. an arc and two concentric circles, connected by several labeled angles Let C be the Centre of the Earth, CR the semidiameter, R a point in the superficies intimating the place of observation. OAB the superficies of the Atmosphere, HOR the sensible Horizon, DC the true Horizon: so that when the Sun is seen from R by O in H, he is not in H but in S for the beams of the Sun emitted from S and falling upon O the superficis of the Atmosphere, proceed not on in the line OA, but are refracted in O, the point refringent and pass through the diaphanous Atmosphere, in the line OR which is a part of the sensible Horizon, and so appears the Sun in H when indeed he is but in S, and lower than the sensible Horizon by the quantity of the angle HRS, which is by Tycho aslumed 34 min. But as it is not HR but DC which is the true Horizon, so likewise is it not the angle HRS but DCS which is the Sun's depression or inferior distance form the Horizon. To find which, consider that if from the arch HDS which subtends the angle of refraction HRS you subtract HD the remainder will be DS subtending the angle DCS. Now HCD being (the Sun's horizontal Parallax) according to Tycho 3. min. subtracted from the angle of Refraction HCS which is sufficiently equal to HRS, leaves the angle DCS the Sun's depression below the Horizon at what time he appeared ther●n. And he that shall from this Depression, and the latitude I used, proceed to find the hour of the day, and the Cusps of the 12. houses, I believe will not find them much different from what I set down. Only in the cusps of the eleventh and fift houses. 17. degrees is but in stead of 7. by some mistake of the Cutter, or otherwise the minutes agreeing exactly. Thus much I thought it not amiss to certify concerning that fiigure, lest any should pass an overhasty Censure thereupon, and either condemn my unknown self of ignorance, or traduce the approbation of Master Lily, whose worthy performances have so much dignified these Sciences, and still been able to confront Envy in her most pestiferous furniture. FINIS. THE CONTENTS. CHap. 1. The occasion of this Discourse, page 1 Chap. 2. An error in finding out the Dominical Letter and Epact in the Foreign Account perpetually, detected, pag. 4 Chap. 3. The inequality of the precession of the Equinoctial points examined, pag. 6 Chap. 4. Of the Sun and Moons Tables, pag. 12 Chap. 5. Whether the second inequality of the Moon have dependence of the Sun's mean or true, and apparent place, pag. 14 Chap. 6. A demonstrative examination of our Author's Tables of Eclipses, pag. 17 Chap. 7. Of the aequation of Natural Days, pag. 29 Chap. 9 Our Author's determination of the distance of Celestial bodies from the earth examined, pag. 30 Chap. 9 A brief summary of some other Defects and Imperfections, pag. 31 FINIS.