minutes seconds thirds fourths and fifths into Decimalls and the contrary 10 A Table to convert the hours minutes seconds thirds fourths and fifths of a day into Decimalls and the contrary 14 A Table to convert hours parts into deg parts of the Aequator 20 A perpetual Table for the Equation of Time 21 The Suns mean motions 22 The Aequations of the Suns Excerntrick 26 The Moons mean motions 29 The Aequations of the Moons Excentrick 34 A Table for the finding of the secōd third inequalities of the Moon 37 Bullialdus his Table of Evection 40 A compounded Table of the Moons Evection and Variation 43 A Table of the Aequations of Nodes and Moons latitude 53 A Table of the Reductions to the Ecliptick 56 The difference of the true ☌ or ☠from the middle of the obscuration 57 A Table of the mean Lunations 58 The Horizontall Parallaxes Semidiameters and hourly motions of the Sun and Moon 59 The Declination and Meridian Angles 60 Tycho's Table of Refractions 61 Saturn's mean motions 62 Jupiter's mean motions 66 The mean motions of Mars 70 The mean motions of Venus 74 Mercuries mean motions 78 A Table of Declinations 82 A Table of Right Ascensions 89 A Table of Ascensional Differences 100 A Table of Oblique Ascensions 108 A Table of Positions for the latitude of 51 degrees 53 parts 138 A Table shewing the elevation of the Pole upon the severall circles of Position of the 11 12 2 and 3 houses for 60 degrees of latitude 151 A generall Table of Positions 152 A Catalogue of the more notable fixed Stars with their longitude latitude and magnitude for the yeare 1650 compleat 154 The Preface ALL Propositions Astronomical and Astrological have some dependence on the Sphere or Globe for the better understanding therefore of that which follows it is fit that the Reader be somewhat acquainted with the doctrine thereof that he know at least what a Globe is and what the lines circles and arches usually drawn thereon do represent Now a Globe or Sphere is an Analogical representation either of the Heavens or the Earth And in this Sphere or Globe there are ten imaginary circles whereof there are six great and foure small A great circle is such a one as divideth the body of the Globe into two equal Hemispheres And a small circle is that which divideth the same into two unequal Hemispheres wherof the one is more the other less then half the body of the Globe or Sphere The six great circles are these 1 The Horizon 2 The Meridian 3 The Equinoctial 4 The Zodiack The fifft and sixt are the two colures The four lesser circles are 1 The Tropick of Cancer 2 The Tropique of Capricorn 3 The circle Artick 4 The circle Antarctick And are all exprest in this annexed Diagram 1 The Horizon which is also called the Finitor is a circle which divideth the visible part of the Heavens from the not visible that is the lower Hemisphere from the higher in the figure noted with A B. 2 The Meridian is a circle which passeth by the Poles of the World and through the Zenith and Nadir and is marked with A Z B N. 3 The Equinoctial is a Circle which divideth the whole Sphere into two equal parts and is therefore equally distant from both the Poles to which when the Sun cometh which is twice in the Year the Dayes and Nights are of equal length all the World over this circle is noted with E F. 4 The Zodiack is a great circle which conteineth the 12 Signes cutting in the very middle the Equinoctial in two points which are the beginning of Aries and Libra whereof the one half viz. six Signes decline from the Aequator to the North Pole and are therefore called the Northern Signes as Aries ♈ Taurus ♉ Gemini ♊ Cancer ♋ Leo ♌ Virgo ♠The other six decline towards the South Pole and are therfore caled the Southern Signes as Libra ♎ Scorpio ♠Sagittarius ♠Capricornus ♑ Aquarius ♒ Pisces ♓ 5 The one of the Colures which dividing the Sphere into two parts passeth by the Poles of the World and the two Equinoctial points called the Equinoctial Colure and marked with C D. 6 The other Colure which dividing the Sphere also into two equal parts passeth by the beginning of Cancer and Capricorn and the Poles of the World called the Solstitial Colure and is the same with the Meridian as the Sphere is here projected 7 The Tropick of Cancer is one of the lesser circles distant from the Equinoctial towards the North Pole 23 deg 31 min. 30 seconds or in Decimal Numbers 23 deg 525 to which when the Sun cometh he causeth the longest day and shortest night to all Northern the shortest day and longest night to all Southern inhabitants and is noted with G ♋ 8 The Tropick of Capricorn is a circle distant from the Equinoctial towards the South Pole 23 deg 31 min. 30 seconds or in Decimal numbers 23 deg 525 parts to which when the Sun cometh he maketh the longest day and shortest night to all Southern the shortest day longest night to all Northern Inhabitants and is noted with H ♑ These two circles are called of the Greeks 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 à convertendo because when the Sun toucheth any of these circles he is at his greatest distance from the Aequator and returneth thither again 9 The Artick circle is distant from the North pole of the world as much as the Tropick of Cancer is distant from the Equinoctial and is noted with K L. 10 The Antartick circle is distant from the South Pole as much as the Tropick of Capricorn is distant from the Aequator and is noted with O M. Besides these circles exprest upon the Globe there are other circles not exprest that are also in familiar use but these being sufficient for our intended matter omitting the rest we will now speak of the several affections of the Sphere or Globe and so proceed to practice According to the diverse habitude of the Aequator to the Horizon which is either Paralel to it or else cutteth it and that either in right or oblique angles there is a threefold position or Situation of Spheres The first is of those that have either Pole for there Zenith or vertical point with these the Aequator and Horizon are Parallel to each other or rather indeed do make but one circle between them and this is called a Parallel Sphere and they which there inhabit if any such be see not the Sun or other Star either rising or setting or higher or lower in their diurnal revolution The third position of the Sphere agreeth to all other places else and is called an oblique Sphere in which the dayes are sometimes longer then the nights sometimes shorter and sometimes of equal length when the Sun is placed in the Equinoctial point the dayes and nights are equal but when he declineth from the Aequator the dayes are observed to
the time of the Suns continuance in every of the 12 signes in their year therefore which is Solar there are alwayes 365 dayes And the Julian yeare which is the account of all Christendome doth differ from the other onely in this that by reason of the Suns excesse in motion above 365 dayes which is about 5 hours 49 minutes it hath a day intercalated once in 4 yeares and by reason of this intercalation it is more agreeable with the motion of the Sun then the former and yet here is a considerable difference between them which hath occasioned the Church of Rome to make some further amendment of the Solar year but hath not brought it to that exactness which is desired nor will as is to be feared be over-hastily brought to that exactnes which it might taking these accounts therefore as they now stand if we will reconcile that discrepancy that is between them there must be some beginning appointed to every of these accounts and that beginning must be referred to some one as to the common measure of the rest The most natural beginning of all accounts is the time of the Worlds Creation but they who could not attain the Worlds beginning have reckoned from their own as the Romanes ab urbe condita or from some great name or notable event so the Greeks account from their Olympicks and the Assyrians from Nabonasser and all Christians from the birth of Christ the beginning of which and all other the most notable Epochaes as others formerly so we now have also ascertained to their correspondent times in the Julian Period which Scaliger contrived by the continual multiplication of three circles all in former times of good use two of them do yet remain the Circles yet in use are those of the Sun and Moon the one to wit the Sun is a Circle of 28 years in which time the Sunday Letter makes all the varieties that it can have by reason of the Bisextile or Leap-year and the Circle of the Moon is the revolution of 19 years in which time though not precisely the Lunations do recur it is called the Golden Number and was made Christian by the Fathers of the Nicene Council as being altogether necessaay to the finding out of the Neomenia Paschalis upon which the Feast of Easter and the rest of the moveable Feasts depend The third Circle which now serves for no other use then the constituting of the Julian Period is the Roman Indiction or a Circle of 15 years for if you multiply 28 the Cycle of the Sun by 19 the Cycle of the Moon the product will be 532 this by 15 the product will be 7980 the number of years in the Julian Period whose admirable condition is to distinguish every year within the whole Circle by a several certain character the year of the Sun Moon and Indiction being never the same again until the revolution of 7980 years be gon about this Period the Authour fixed in the Tohu or eternal Chaos of the World 764 Julian years before the most reputed time of Creation which being premised we will now by example shew you how to reduce the years of Forreigners to our Julian years and the contrary 1 Example I desire to know at what time in the Turkish account the 5 of June 1649 falls The work is this The years compleat are 1648 and are thus turned into Dayes by the table of Dayes and Decimals of Dayes in Julian Years 1000 Julian yeares give dayes 365250 600 years give 219150 40 years give 14610 8 years give 292â— May Compleat 151 Dayes 5 The Summe 602088 Now because the Turkish account began July the 16. Anno Christi 622 convert these yeares into dayes also thus 600 Julian years give 219150 20 years give 7305 1 year gives 365 June Compleat 181 Dayes 15 The Summe subtract 227016 From 602088 There rests 375072 900 Turkish years gives 318930 There rests 56142 150 years gives 53155 There rests 2987 8 years give 2835 There rests 152 Giumadi I. gives 148 There rests 4 Therefore the 5th of June 1649 in our English accompt falls in the Turkish accompt in the year of Mahomet or their Hegira 1058 the 4th day of Giumadi II. 2 Example I desire to know upon what day of our Julian year the 23 day of the moneth Pharmuthi in the 1912 year currant of the Aegyptian accompt from the death of Alexanders fall The beginning of this Epochâ— is from the beginning of the Julian Period in compleat dayes  1603397 1000 Egyptian years give 365000 900 yeares give 328500 10 years give 3650 1 yeare gives 365 Phamenoth compleat 2â—0 Dayes 23 The summe 2301145 6000 Julian yeares 2191500 There rests 109645 300 yeares give 109575 There rests 70 April compleat 59 There rests 11 It therefore fell out in the yeare of the Iulian period 6300 the 11 of March that is subtracting from that period 4712 in the yeare of Christ 1588. He that understands this may by like method convert the yeare of other Epochaes into our Julian yeares and the contrary Next to the tables which concern the reduction of years in general we annexed tables for the perpetual finding of the Sunday letter Golden number and Epact in both the Old Julian and New Gregorian accompt with the fixed and moveable Feasts and a Catalogue of some famous places with their latitude and distance in longitude from the meridian of London whose use is so obvious that it needs but little explanation yet to take away all difficulty we have added these directions The Cycle of Sun Sunday Letter Golden Number and the Epact in both accounts are set against the yeare of our Lord and when those years are out they may be renewed againe as oft as you please thus for the yeare 1656 the Cycle of the Sun 1513 the Sunday letters in the English account are F E in the Gregorian B A the prime or Golden number in both is 4 the Epact in the English accompt is 14 in the Gregorian 4. And now to find the movable Feast seek the English Epact in the first Columne of that Table towards the left hand and the first F that follows will shew you that the 3d. of February is L X X Sunday the 17 of February L Sunday the 20th of February Ashwednesday the first E that follows will shew that Easter day is the 6th of April Ascension day the 15th day of May Whitsunday the 25 of May Corpus Christi the fifth of June Advent Sunday November the 30th But in the Gregorian the Epact and Sunday Letters must be sought in the first Columne towards the right hand so shall the Sunday Letters B A shew the Feast of Easter to be on the 9th of their April and the rest as in that line they are set down The fixed Feasts together with the Week-day Letters are set against their proper dayes in every moneth of the Julian year knowing therefore the Sunday Letter you may easily
of Pharmuthi In which space of time There are dayes 697746 And from the death of Alexander to the 26 of Mechir 178 there are 64781 There rests 632965 From days 697746. 829â—6111 Subtract 64781. 86746111 There rests 632964. 96240000 And in this time the Earth or Sun hath gone 1733 circles 〈◊〉 623880 degrees Hence to find the mean motion for a year or 365 days I say If 632964. 9624 d Give 623880 degrees How many degrees shall 365 dayes give And the answer is 359 deg 7611456036. That is in Sexagenary numbers 359 deg 45 minutes 41 seconds 1 third 27 fourths Again to find the mean motion for a day I say If 365 dayes gives 359 degrees 7611456036 what shall one day give And the answer is 0. 9856469743. That is in Sexagenary numbers 0 deg 59 minutes 8 seconds 19 thirds 44 fourths And what is here done for the middle motion of the Earth or Sun may be done for the other planets CHAP. 2. Of the figure which the planets describe in their Motion HAving shewed in the former Chapter by what means the Annuall or Periodicall revolutions of the Planets may be knowne with their mean or equal motion for any part of those revolutions we should now shew you how by those equal motions to find their true or apparent places But we can never hope to find the true and exact Phenomenon of the planets unlesse we first know the figure in which they move And this must be collected from such affections as are by the constant observations of all ages found to be proper and naturall to them or may be rationally collected from them 1 That the planets have one onely motion in one onely line and that those motions are equal constant and perpetual hath been confirmed by the observation of all ages 2 And therefore they must needs be regular their motions must be in a circle or some other line returning into it selfe or else their motions could not be perpetuall 3 Their equall motions must have some place assigned which the planets naturally behold to be the beginning of this equall motion 4 And because the apparent place of a planet taken by observation is generally different from the place reckoned in its middle motion the inequality of this middle and apparent motion must be referred to the center of the Zodiacke â—s to the point of that inequâ—lity 5 And because the center of the Zodiacke and of the world is to out appearance the same the point of this inequality must be referred to the center of the world 6 And because of this difference between the middle and apparent motion the center of the world cannot be the true and exact center of the planets but the center of that figure which the planets describe in their motion must be some other point then the center of the Zodiacke 7 And though the planets to our appearance are observed to be sometimes swifter in motion then at other some yet the cause of this inequality of motion must not be such as shall alter the natural and equal motion of the planet it must be such as shall make the planet to be slower in its furthest distance from the center of the world and swifter at his nearest without transposing the equal motion into any other then the first place assigned whether superficies or circle 8 And further the apparent motions of the planets in their nearest and furthest distances from the center of the world being the same with their middle the way of the planets must be such that when they have gone 90 degrees from their farthest distance in their middle motions their apparent motions must be lesse then 90 by the quantity of that whole inequality between the middle and apparent motion And when the planets have gone a quadrant in their apparent motions their difference between their motions shall be that whole inequality also and therefore the center of that figure which the planets describe in their motions must be in the middle between the points of their equal and apparent motions 9 And because the mean motion from the point of a planets farthest distance from the center of the world to the first quadrant is greater then the apparent therefore the apparent motion must be greater then the mean from the first quadrant to the point of the planets nearest distance and consequently a greater portion of the line in which the planets move must be allowed to the apparent from the first quadrant to the point of nearest distance then from the point of farthest distance to the first quadrant 10 And because the equal motion must not change and that the apparent motion doth increase from the point of the planets farthest distance from the center of the world the angles of the middle motion must be reckoned in the arches of many parallel circles which shall also increase from the points of farthest to the point of their nearest distance to the center of the world and the line of the apparent motion must containe those circles in one and the same superficies and therefore that line must be excentricall from those circles of apparent motion and so placed that all the parts of apparent motion may proportionably answer to all the parts of equal yet so as that the least circles of equal motion shall agree with the point of the planets farthest distance and the greatest circles with the point of the planets nearest distance from the center of the world Seeing now that these circles of middle motion must be parallel succeeding one another in a continued series and are not one within another and that the apparent motion must in the farthest distance answer to the least circles in the nearest distance from the center of the world to the greatest there is none but a solid Superficies that can be capable of those greater and lesser circles And that an unequal sided Cone may be so cut as that the figure upon the plaine of that Section shall truly represent these affections of the planets the learned Bullialldus doth Demonstrate and for a preparation thereunto he sheweth first How two equal right lines may be so drawn in an unequal sided Triangle as that the one shall bisect the other An unequal sided Cone being cut through the Axis by a plaine perpendicular to the plain of the base shall make an unequal sided Triangle and let A B C be such a Triangle whose base B C let be bisected in I and parallel therunto draw the line P S which being within the triangle shall be also bisected in the point R and from a point taken in this line at pleasure suppose at H to the Axis of the Cone A I draw the line H M so as that the angles H M R and M R H may be equal then shall H M and H R be equal also and let the line H M being continued to the sides of the Triangle A B and A C be bisected in the point X and by