6675373 To finde the second inequality of Mercury We must have given 1. The Angle NAS which is to be found by subductiug the Suns place from the eccentrick place of Mercury reduced or this from it so that less then 6 signes may remain this remainer is the Anomaly of the orbe and the complement thereof is the Angle NAS or the halfe is the halfe sum of the opposite angles Example The eccentrick of Mercury reduced 220. 26940 The Suns true place 154. 07347 Anomaly of the orbe 96. 19593 Complement is NAS 83. 80407 Halfe Anomaly 48. 09796 These given with the sides NA and SA the Analogies are As the greater side SA 100895 co ar 4. 9961293 Is to Radius  10. 0000000 So is the lesser side NA 46509 4. 6675373 To the tangent of 24. 74799 9. 6636666 Adde 45.  As Radius   To the cotang of 69. 74799 9. 5669785 So tang halfe summe 48. 09796 10. 0470559 To tang halfe difference 22. 35160 9. 6140344 Summe 70. 44956 Angle ANS  Difference 25. 74636 Angle ASN  Because the Suns place was subtracted from the eccentrick of Mercury reduced therefore the angle of Elongation ASN must be added to the Suns place 154. 07347 Elongation ASN adde 25. 74636 True place of Mercury 179. 81983 To finde the distance of Mercury from the Earth As the sine of ANS 70. 44956 co ar 0. 0257891 To the side AS 100895 5. 0038707 So the sine of NAS 83. 80407 9. 9974556 To the side SN 106442 5. 0271154 To finde the Latitude of Mercury from the Earth As the side SX 106442 co ar 4. 9728846 Is to Radius  10. 0000000 So is XL 1749 3. 2429402 To the tang of XSL 0. 94169 8. 2158248 Which is the south Latitude of Mercury   CHAP. 18. Of the Semidiameters of the Sun Moon and shadow of the Earth THe angle of the Suns apparent Semidiameter in his nearest distance to the Earth Bullialdus hath by observation found to be 16′ 45″ or in Decimall numbers 27917. And by an Eclipse of the Moon December 1638 he found her Semidiameter to be 16′ 54″ or 28167 and the Semidiameter of the Earth shadow 44′ 9″ or 7 583 at which time being the time of incidence her distance from the Earth by his computation was 97908 parts of the Semiaxis of the Elipsis 100. 000. By this and another observation in the same Eclipse he sheweth how to finde her apparent semidiameter in all the other intervalls The inferiour limbe of the Moon and the first Starre in the foot of the former Twin whose place then according to Tycho was Gemini 28. 25′ 17″ or Gemini 28. 42138 with South Latitude 0 d. 58′ or 0. 96667. being in the same Azimuth was 8′ or 13333 higher then the Star and the Altitude of the heart of Hydra then taken by him at Paris was 30 deg 37′ or 30 d. 61667. From whence the hour was found 30 h. 40′ or 13 h. 66667 and the houre being given the altitude of the Starre is also given deg 56. 42′ 15″ or deg 56. 70416. The apparent altitude of the center of the Moone was deg 57 7′ 9″ or deg 57 11916 but by her latitude and place it should have beene deg 57 40′ 4″ or deg 57 66778 and therefore her parallax of altitude 32′ 55″ or 54861. The situation of the Moone and Azimuth in which her interiour limbe and the Stars were being given her aparent Longitude was almost in Gemini deg 28 38′ 30″ or Gemini deg 28 64167 her parallax of longitude 18 min. or 30000 and therefore the center of the Moon in her true motion in Gemini 28 d. 57 min. fere or in Gemini 28 d. 95000 her parallax of Latitude is 19 min. or 31667. to which 21′ or 35007 the difference of the observed latitude of the Moon and Stars being added the true difference is 50 min. or 83333 min. and thence the Moons Latitude 8 min. or 13333 S. Now then to finde the distance of the Moon from the Earth in this Eclipse the Earths semidiameter being one degree Let FEC represent the true Horizon BDE the vertical at Paris E the center of the earth D the City of Paris the Moons true altitude AEF deg 5766778 the observed altitude ADG deg 5711916. The parallacticall angle DAE deg 0. 34861. Therefore in the Triangle ADE we have given all the angles and the finde DE one Semidiameter of the Earth to finde AE for which the anolagy is As the sine of DAE 0 d. 54861 co ar 2. 0188745 To the side DE 1  0. 0000000 So is the sine of ADB 32 d. 88084 9. 7347147 To the side AE 56. 70 1. 7535892 This foundation being laid he proceedeth to the rest and to shew how we may possibly fall into some absurditie he supposeth the Moons distance from the Earth in this Ecclipse to be but 55 semidiameters or the side BC in the following figure the apparent angle of the semidiameter of the Earths shadow CHI 0. 73583 AEF represents the Sun his semidiameter AE the angle of his apparent Semidiameter when he is Perig. AGE 16. 45 or in decimalls 27916 BHG represents the Earth BG the Semidiameter thereof hence to finde HI in the triangle HIC the proportion is As the sine of HIC 89. 26417 co ar 0. 0000358 To the side HC 54 1. 7323937 So is the Radius HCI 90 10. 0000000 To the Hypothenusal HI 54. 004 1. 7324295 2. In the triangle HBI we have given the sides BH 1. and HI 54. 004 with the angle BHI 179. 26417 hence to finde the Angle BHI the Analogie is As the greater side H I 54. 004 co ar 8. 2675705 Is to the less H B 1. 0. 0000000 So is Radius  10. 0000000 To the tangent of 1. 06082 8. 2675705 Adde 45.  As Radius  10. 0000000 To the Cotang of 46. 06082 9. 9839145 So Tang. halfe summe 36791 7. 8075980 To tang halfe differ 35454 7. 7915125 Angle B I H 01337. And the angle C B I 72246 3. In the triangle C B I we have given the angles and the side B C 55 to find C I. Therefore say As Radius To the tangent of C B I 72246 8. 1007064 So is B C 55 1. 7403627 To C I 0. 6935 = to B K 1. 8410691 and therefore K G 3065 and the angle K I G 0. 31857. For â—â— I K 55 co ar 8. 2596373 To Radius  10. 0000000 So is K G 0. 3065 1. 4864305 To the tangent of K I G 0. 31857 7. 7450678 and the angle B D G is equal thereunto but so the angle of the Suns apparâ—nâ— Semidiameter A G E 27916 by observation is lesse then the angle A D E which is absurd and therefore some part assumed is false The Semidiameters of the Sun and Moon must not be changed constant experience agreeing with these observations In this Eclipse therefore Bullialdus doth take
G K 100. 000 5. 0000000 Is to G K deg 2. 50 8. 6396795 So is G D 9. 7854 4. 9905785 To G D or the latitude 2. 44632 8. 6302580 whereas it should have been 2. 46667. For the latitude agreeing to the second observation As G K 100. 000 5. 0000000 Is to K G deg 2. 50 8. 6396795 So is G E 59624 4. 7754210 To G E or the latitude 1. 49029 8. 4151005 whereas it should have been 1. 56667. For the latitude agreeing to the third observation As G K 100. 000 5. 0000000 Is to G K or the angle of inclin 2. 50 8. 6396795 So is G F 47954 4. 6808248 To G F or the sine of latit 1. 19856 8. 3205043 whereas it should have been but 1. 19 CHAP. 21. To find the mean Conjunction and Opposition of the Sun and Moon FOr this purpose the Table which Shakerley transcribed from Bullialdus we have here exhibited in Decimall numbers the use whereof as he hath explaind the same is this Set down first the Epochâ— next preceding the yeare given then the yeares and moneths compleate having a care of the yeare Bisseâ—tile and to every one set downe the time answering in the Table then adde them altogether and the summe subtract from the next greater in the Canonian under the titleâ— if you â—eeâ— a Conjunction or ☌ if an opposition the remainer sheweth the time required compleat from the beginning of the moneth current Example I would know the time of the meane opposition of the Sun and Moon in March 1652. The worke is this  Hou Parts The Epocha 1640 701. 96639 Yeares compleat 11 20. 14722 February compleat Bissextile 022. 53539 The summe subtract 744. 64500 From the Opposition next greater 1063. 10139 Rests the meane opposition 318. 45639 that is 13 dayes 6 houres and 45639 parts CHAP. 22. To find the true Opposition or Conjunction of the Sun and Moon FOr the time of the meane Conjunction or Opposition given find the true place of the Sun and the eccentrick place of the Moon and compare them if they either be precisely the same or precisely opposite the time of the true Conjunction or Opposition agrees with the meane but if they differ take the difference by subtracting the lesse from the greater and that call the distance of the Sun and Moon 2 Out of the Table of Semidiameters and hourly motions with the meane Anomalies of the Sun and Moon take out their hourly motions and subtract the hourly motion of the Sun from the hourly motion of the Moon by the remainer which is the hourly motion of the Moon from the Sun divide the distance of the Sun and Moone before kept the quotient gives the time which must be added to the mean time of Conjunction or Opposition if the excesse be in the Suns place or subtracted if in the Moones place 3 At this time thus corrected find againe the true place of the Sun and â—â—centrick place of the Moon together with their distance and repeat your former work till you find them absolutely to concurre and the time thus found shall be the true time of Conjunction or Opposition As in the Example  D. Hou parts At the time of the meane ☠March 13 6. 45639 The true place of the Sun is  4. 85039 The Eccentrick place of the Moone  180. 38631 The Distance of the Sun from ☠Moon  4. 46408 Mean Anomaly of the Sun  266. 40860 His hourly motion  04112 Meane Anomaly of the Moone  30. 26681 Her hourly motion  51827 Hourly motion of the Moone from the Sun  47715 By which dividing the distance  4. 46408 The quotient gives Hours 9. 3558 to be added   So the time first corrected March 13 15. 81279 The true place of the Sun is  5. 23520 The eccentrick place of the Moone  1â—5 15798 The distance of the Sun from ☠of the Moon  07722 Which divided by the hourly motion of the Moon from the Sun  48114 Gives in time to be added  16049 So the time secondly corrected 13 15. 97328 The true place of the Sun is  5. 24180 The eccentrick place of the Moon  185. â—4010 The distance of the Sun from ☠the Moon  00170 Which divided by the hourly motion of the Moon from the Sun  48121 Gives in time to be added  00353 So the true time 13 15. 97681 The true place of the Sun is  5. 24194 The eccentrick place of the Moon  185. 24190 4 For this time find out the true motion of the Moons Latitude and thereby the Reduction which divide by the hourly motion of the Moon from the Sun and the quotient contrary to the title of Reduction apply to the last corrected so have you the true time In our Example The true motion of Latitude  174. 04881 The Reduction Adde  02398 The quotient Subt.  04983 So the true opposition 13 15. 92698 5 Lastly apply the equation of time to this equal time to make it apparent The true time of the ☠13 15. 92698 The equation of time Subt.  02884 The apparent time of the ☠13 15. 89814 CHAP. 23. To find whether there will be an Eclipse or not THere are two wayes to know this of which the one is more easie the other more certain The first is this At the true conjunction if the true motion of latitude be within 17 degrees backward or forward of 6 or 12 signes or at the opposition within 12 degrees there is a possibility of an Ecclipse otherwise not In our Example the Moons true motion of latitude is 174 deg 04881 which being not fully 6 degrees distant from 6 signes shewes the necessity of an Eclipse The other way is this If at the visible conjunction the visible latitude of the Moon be lesse then the aggregate of the Semidiameters of the Sun and Moon there must be an eclipse or else not 2. If at the true opposition the true latitude of the Moon be lesse then the summe of the Semidiameters of the Moon and the earths shadow there must be an Eclipse otherwise not This latter way is most certain onely subject to this inconvenience that a great part of the calculation is performed before we come to the 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 or power to judge of the possibility CHAP. 24. To find the Quantity of a Lunar Ecclipse BY the true motion of the Moons Latitude find her true latitude according to the former directions this in our example is 0. 51496 North Descendant 2 Find out the Semidiameter of the Moon by her meane anomaly out of the Table as also hâ—â— Horizontall parallax and with the meane anomaly of the Sun take out the Semiangle of the Cone of the shadow and this subtract from the Moons Horizontall parallax there rests the Semid of shaddow 3 Adde together the Semidiameter of the shadow and Semidiameter of the Moon and
there be a concurrance Example At the visible Conjunction March 27 21. 90999 The true distance of the Moone from the Sun 0â—847 The Parallax of Longitude 02775 Their difference 00072 which being so small sheweth that the visible Conjunction is precisely enough found CHAP. 32. To finde the visible Latitude of the Moon at the time of the visible Conjunction IN these Northerne regions which we inhabit the parallax of latitude allwayes makes the Moon to appeare more South then indeed she is to find the visible latitude therefore observe these rules 1 At the time of the visible conjunction find out the true latitude of the Moon thus If the Eclipse happen in the orientall quadrant adde the parallax of longitude to the motion of the Sun agreeing to the difference between the true and visible Conjunction and the summe subtract from the true motion of latitude at the time of the true Conjunction or if the Eclipse happen in the occidentall quadrant adde the said summâ— thereto and you have the true motion of latitude at the visible Conjunction by which as formerly taught finde out the true Latitude of the Moone Example Motion of the Sun agreeing to 04943 00202 Parallax of Longitude at visible ☌ 02775 The Summ Subâ— 02977 Motion of Latitude at true ☌ 8. 80745 Motion of Latitude at visible ☌ 8. 77768 True Latitude at visible ☌ North 75808 At the same time find the parallax of latitude and compare it with the true latitude If the latitude be South adde them together the summe is the South visible latitude of the Moone but if North subtract the lesse from the greater there remaines the visible latitude of the Moon which shall be North when the latitude is greater then the parallax otherwise South Example The true latitude of the Moon North 0. 75808 Parallax of Latitude 0. 73633 The visible latitude North 02175 CHAP. 33. To find the quantity of a Solar Eclipse THis differs very little from that in the 24 Chapter for finding the quantity of a lunar Eclipse for it with their meane Anomalieâ— you enter the Table and thence take out the Semidiameter of the Sun and Moone and adde them together and from the summe subduct the visible latitude of the Moone at the visible Conjunction there rests the Scruples of the Suns body deficient which as in the Moon so here in the Sun convert into digits Example Semidiameter of the Sun 27386 Semidiameter of the Moon 27815 Summe of the â—emidiameters 55201 Visible latitude Subtracted 02175 Scruples deficient 53026 So the digits eclipsed 11. 61500 CHAP. 34. To find the beginning and ending of the Suns Eclipse BY the visible latitude of the Moon and the summe of the Semidiameters of the Sun and Moon find the Scruples of incidence as in the Moones Eclipse Chap. 25. 2 For one hour before the visible Conjunction find by the 30 Chapter the visible hourly motion of the Moon from the Sun by which divide the Scruples of incidence the quotient is the time of incidence which subtracted from the time of the visible Conjunction leaves the beginning of the Eclipse 3 For one hour after the visible Conjunction finde the visible hourly motion of the Moon from the Sun by which divide the Scruples of incidence the quotient is the time of Repletion which added to the time of the visible Conjunction gives the end of the Eclipse Example Summe of the Semidiameters 55201 Visible latitude 02175 Scruples of Incidence 55158 At 1 ho. before the visible ☌ March 27 20. 90999 Parallax of longitude Orient 13209 True hourly motion of the Moon from the Sun 56140 Visible hourly motion 45174 Time of incidence 1. 22150 Beginning of the Eclipse March 27 20. 68849 At 1 ho. after the visible ☌ 27 22. 90999 Parallax of longitude Occid 10119 Visible hourly motion of the Moone from the Sun 48796 Time of repletion 1 h. 1303â— End of the Eclipse 27 23. 04029 The whole duration 2. 35180 CHAP. 35. To find the Visible latitude of the Moon at the beginning and end of the Suns Eclipse FOr the beginning adde to the minutes of Incidence the motion of the Sun agreeing to the time of Incidence and the summe subtract from the true motion of latitude at the time of the visible Synod so have you the true motion of latitude at the beginning by which find the true latitude and by these according to the second rule of the 32 Chapter may be had the visible latitude Example The Scruples of incidence 5515â— Motion of the Sun answering to the time 05016 The summe subt 60174 Motion of latitude at visible ☌ 8. 77768 Motion of latitude at beginning 8. 17594 True latitude North 70648 Parallax of latitude 82004 Visible latitude South 11353 2 For the end adde to the minutes of incidence the motion of the Sun agreeing to the time of repletion and the sum adde to the true motion of latitude at the time of the visible Conjunction so have you the true motion of latitude at the end by which proceed as before to find the visible Latitude Example Scruples of Incidence 55158 Motion of the Sun agreeing to the time of repl 04642 The Summe Adde 59800 Motion of Latitude at the visible ☌ 8. 77768 Motion of Latitude at the ending 9. 37568 True Latitude North 80925 Parallax of Latitude 65218 Visible Latitude North 15707 CHAP. 36. To Delineate the Eclipses of the Sun and Moon FOr the Moon draw the lines AC and BD to intersect one another at right angles in E which point of intersection is the place of the Ecliptique where the Eclipse happens upon which as a Center draw the Peripherie ABCD of the quantity of the summe of the Semidiameters of the Moon and the earths shadow which may be done by helpe of a Scale or Sector of equal divisions also to the quantity of the Semidiameter of the earths shadow draw upon the same center another Peripherie Then because the Moones Eclipse begins on the east part of her body you must upon the west side of your plane note downe the latitude of the Moon in the arch BCD which here represents the west part and may be thus done From E upon the line BD prick out the latitude at the beginning towards B if the Latitude be North towards D if South and it terminatâ—s at G from which draw a parallel to AC and in the arch BC it marks out F. Also for the end of the Eclipse proceed in like manner on the other side and you have the latitude terminated at I and the parallel falling at H. Then draw a line between F and H and where it intersects BD marke it with K. Lastly upon the centers F K and H draw three equal circles having for Radius the Semidiameter of the Moone and the worke is done Typus Eclipseos Lunae praedictae Example of the forementioned Eclipse of the Moon March 15. 1652 Summe of the Semidiameters EB 97385 Semidiameter of shadow EM 70954 Initial
latitude of the Moon EG North 59223 Final latitude of the Moon EI North 43746 Semidiameter of the Moon MB 26431 2 For the Eclipse of the Sun it differs nothing at all from this of the Moon but onely that instead of the Semidiameter of the shadow of the earth you use the Semidiameter of the Sun and the visible latitude for the true Example of the forementioned Solâ—â— Eclipse March 28. â—652 Summe of the Semidiameters EB 5520â— Semidiameter of the Sun EM 27386 Initiall visible latitude EG South 11353 Finall visible latitude EI North 15707 Semidiameter of the Moon MB 27815 Typus Eclipseos Solis prâ—dictâ— CHAP. 37. The use of the Table of Refractions ALthough the Table of Refractions belongs not to the calaulation of these Tables yet will it not be amisse to shew its use in comparing of observation with calculation Know then that Refraction causeth the stars to appeare higher then really they are Therefore with an observed altitude enter this Table and take out the Refraction which subtract from the observed altitude and you have the true altitude or having the true altitude the apparent is found by adding the Refraction thereto FINIS This Scheame hath particular relation to page 103 and is there printed in most Copies but in some Copies there is another Scheame placed instead thereof the Reader is therefore desired where it is wanting to insert it Errata Page 7. line the last for the North read C the North. Page 8. line 10 read or the arch H ♋ Page 21. line 20 for else of that read else that Page 29. line 39 for ED 38 read ED 30. Page 37. line 23 for paraallctical read parallactical Page 76. line 14 for plane parallel read plane is parallel In the Tables Page 26 against degree 11 for 0 38836 read 0 38336. Page 27 against degree 90 for 5 0201387 read 5 0001387. Page 27 against degree 117 for 4 9961795 read 4 9965795. Page 34 against degree 56 for 3 04 c. read 4 04 c. Page 35 against degree 87 for 5 0010809 read 5 0018089. Page 35 against degree 119 for 4 163 c. read 4 463 c. Page 36 against degree 132 for 3 874 c. read 824 c. Page 36 against degree 139 for 3 38920 read 3 38902. Page 36 against degree 126 for 4 9895915 read 4 9895925. Page 36 against degree 179 for 0 69126 read 0 09126. Page 86 In the Title for A Table of Declinations read A Table of Right Ascensions Let this leafe be folded in at page 168 which is between the second and third Books This Diagram having particular relation to the 18 Chapter of the second Book will be in use for diverse leaves together I thought it therefore convenient to place it so that when the Book is opened in any part the Dirgram might be in sight and have therefore ordered it to be folded in Astronomia BRITANNICA The third Part Exhibiting Tables for the converting of Sexagenary Numbers into Decimal and the contrary for Astrononomical Chronologie with the Ecclesiastical Computation and the Calculation of the places of the Planets Eclipses of the Luminaries and Doctrine of the Sphere LONDON Printed by R. and W. Leybourne Anno Domini 1656. A view of the more notable Epochae Epochae Yeares of the Julian Period Moneths The Iulian Period 1 Ianuar. 1 Creation of the World 765 Ianuar. 1 Aera of the Olympiades 3938 Iuly 8 The building of Rome 3961 April 21 Epochae of Nabonassor 3667 Febru 26 The beginning of Meton's Circle 4281 Iune 26 The beginning of the Periods of Calippus 4384 Iune 28 The Death of Alexander the great 4390 Nove. 12 Aera of the Chaldees 4403 Octob. 15 The Aera of Dyonisius 4429 Marc. 25 The beginning of the Christian Aera falls in the 4713 Yeare of the Julian period compleat Yeares of Christ Moneth The Diocletian Aera 284 Augu. 29 The Turkish Aera or Negyra 622 Iuly 16 The Persian Aera from Jesdagird 632 Iune 16 The Aera of the Persian Sultan 1079 Marc. 14 Dayes in yeares of the Julian accompt Egypt and Persian accompt 1 0 0 0 365 2 5 0 1 0 0 0 365 0 0 0 2 0 0 0 730 5 0 0 2 0 0 0 730 0 0  3 0 0 0 1095 7 5 0 3 0 0 0 1095 0 0  4 0 0 0 1461 0 0 0 4 0 0 0 1460 0 0  5 0 0 0 1826 2 5 0 5 0 0 0 1825 0 0 0 6 0 0 0 2191 5 0 0 6 0 0 0 2190 0 0 0 7 0 0 0 2556 7 5 0 7 0 0 0 2555 0 0  8 0 0 0 2922 0 0 0 8 0 0 0 2920 0 0 0 9 0 0 0 3287 2 5 0 9 0 0 0 3285 0 0 0 10 0 0 0 3652 5 0 0 10 0 0 0 3650 0 0 0 Dayes in Moneths of the Julian Com. B ss Egyptian Persian accompt Januar. 31 1 Thâ—th 30 Pharvadin 30 Februar 5â— 60 Paophi 60 Aripehest 60 March 90 91 Athyr 90 Chortat 90 April 120 121 Chaeac 120 Tyrma 120 May 151 152 Tybi 150 Mertat 150 June 181 182 Mechir 180 Sachriur 180 July 212 213 Phamenoth 210 Mecherma 210 August 243 244 Pharmuthi 240 Apanma 245 Septem 273 274 Pachon 270 Wahak 245 Octob. 304 305 Payni 300 Adâ—rma 275 Novem 334 335 Ephephi 330 Dima 305 Decem. 365 366 Mesori 360 Pechmam 335   Epagomena 365 Asphander 365 Dayes in Turkish or Arabical Years 1  354  2  709  3  1063  4  1417  5  1772  6  2126  7  2480  8  2835  9  3189  10  3543  11  3898  12  4252  13  4607  14  4961  15  5315  16  5670  17  6024  18  6378  19  6733  20  7087  21  7442  22  7796  23  8150  24  8505  25  88â—9  26  9213  27  9568  28  9922  29  10276  30 0 10631 0 60 0 21262 0 90 0 31893 0 120 0 42524 0 150 0 53155 0 180 0 63786 0 210 0 74417  240 0 85048 0 270 0 95679 0 300 0 106310 0 Dayes in Turkish Moneths Muharram 30 Sahahen 236 Sephar 59 Ramadhan 266 Rabie I 89 Schevall 295 Rabie II 118 Dulkadati 325 Giumadi I 148 Dulhajati 354 Giumadi II 177 Dsilhitâ—sche Turâ— 354 Regeb 207 In anno abundanti 355 A Table shewing the Dominical Letter in both accompts Years of our Lord Cycle of the Sun Julian accom Greg. accom 1644 1 G F C B 1645 2 E A 1646 3 D G 1647 4 C F 1648 5 B A E D 1649 6 G C 1650 7 F B 1651 8 E A 1652 9 D C G F 1653 10 B E 1654 11 A D 1655 12 G C 1656 13 F E B A 1657 14 D G 1658 15
for the distance of the Moone from the earth B C 57. 85 Semidiameters of the earth and the Semidiameter of the earths shadow C B I 75111. Hence to find C I the analogie is As Radius To the side B C 57. 85 1. 7623034 So the tang of B C 0. 75111 8. 1176019 To C. I. 0. 75841 1. 8799053 Let B K be equal to C. I. So is K G 24159. Theâ— As I K 57. 85 co ar 8. 2376066 To Radius  10. 0000000 So is K G 24159 1. 3830789 To the tang of K I G. 23928 or B D G 7. 6207755 equal to E D A and the Suns apparent Semidiameter being given A G E 27916 the angle G A B or the difference between the angles A G E and E D A shall be given also viz. 03988. the Suns Horizontal parallax when he is Perigaeon And the Moones Perigaeon distance from the earth in Syzigiis 56. 50 Semidiameters of the earth For as 97908 co ar 5. 0091819 To 57. 85 1. 7623034 So is 95638 4. 9806304 To B C 56. 50 1. 7521157 Hence to find the Moones Horizontal parallax when she is perigaeon the analogie is in the preceding Diagram As E G or B C 56. 50 co ar 8. 2478843 Is to Radius So is D E 1. 0. 0000000 To the sine of E G D 1. 01399 8. 2478843 The Horizontal parallax of the Sun when he is perigaeon or the angle B A G was found to be 03988 The Moones Horizontal parallax is 1. 01399 Their aggregate 2. 05387 Semidiameter of the Sun subtract 27916 There rests the angle C B I 77471 or the apparent semidiameter of the earths shadow in loco transitus Lunâ— Perig. In the triangle therefore B C I we having the angles and the side B C given C I shall be also given For As the sine of 90 deg  Is the side B C 56. 50 1. 7521157 So tang of C B I 77472 8. 1310339 To the side C I 7640 = B K 1. 8831496 And therefore K G â—360  And in the triangle A G B having the angles and B G given the side A B is also given for As the tang of B A G 03988 co ar 3. 1585620 Is to B G 1. So is Radius  10. 0000000 To A B 1440. 66 3. 1585520 which is the distance of the Earth from the Sun when he is Perihelion And because the Suns eccentricity is 1784 his Apogaean distance is 101784 hence to find his distance in Semidiameters of the earth say As his Perigaean distance 98216 co ar 5. 0078178 Is to his distance 1440. 66 3. 1585620 So is his Apog distance 101784 5. 0076794 To his Apog dist 1493. 03 3. 1740692 Then as B S 1493. 03. or E G 6. 8259308 To Radius so is E D. 1 10. 0000000 To the sine of E G D 0. 3855 6. 8259308 The Suns Horizontal parallax when he is Apogaeon As Radius to A B 1440. 66 3. 1585620 So is tang of A B E 27916 7. 6877120 To A E 70189 0. 8462740 Then as B S 1493. 03 co ar 6. 8259308 To Radius so is S T 70189 ☉ Semid 0. 8462740 To the tang of S B T 26936 7. 6722048 The apparent Semidiameter of the Sun when he is Apogaeon The Sun being Perigaean we have given B G 1. K B. 75841. KG 24159 and B C. 56. 50 the distance of the Moon from the Earth when she is Perigaean from whence the longitude of the earths shadow may thus be found As K G 2360 co ar 10. 6270880 To K I 56. 50 1. 7521157 So is C I 7640 1. 8831496 To C D 182. 93 2. 2623533 Add B C 56. 50 then is B D 239. 43. the longitude of the earths shadow Let B S be the Apogaean distance of the Sun 1493. 03 The angle of the Suns apparent Semidiameter S G T 26936 The Perigaean Semidiameter or the angle A G E 27916 Their difference is the angle Z G E 00980 Let TG be produced to N then shall the angle I G N be equal to the angle Z G E but the Sun being Perigaean the angle B D G was found to be o. degrees 239â—8 whose complement is the angle B G D 89. 76072 therefore when the Sun is Apogaean it shall be 89 77052 therefore B X G 0. 22948 equal to K N G. Hence to find K G the analogie is As Radius 90 10. 0000000 To K I 56. 50 1. 7521157 So tang of K I G 0. 22948 7. 6006035 To K G 22632 1. 3547192 And K B 77368. Then to find C B N. Say  As B C 56. 50 co ar 8. 247â—843 To Radius 90 10. 0000000 So C I or rather C N. 77368 1. 8885613 To the tang of C B N 7844â— 8. 1364456 The Sun being Apogaean and the angle C B I the Sun being Paerigaean was before found to be 77471 and therefore the difference of the earths shadow between the Suns Apogaean and Perigaean is 00971. Then As K G 22632 co ar 10. 6452808 To K I 56. 50 1. 7521157 So is C N 77368 1. 8885613 To C X 19â— 18 2. 2859578 Add B C 56. 50. Then is B X 249. 68. The semidiameter of the Moon when she is Perigaean is greater then the semidiameter of the Sun being Apogaean and therefore Bullialdus doth make it 17. or 28333 and because the eccentricity of the Moon is given 4362 her Apogaean distance in Syzygiis 104362 the Moon being Perigaean her distance from the earth is 95638 and in semidiameters of the earth 56. 50 and therefore her Apogaean distance in semidiameters of the earth by the analogy following As 95638 co ar 5. 0193696 To 56. 50 1. 7521157 So is 104362 5. 0185423 To 61. 66 1. 7900276 As her Apogaean dist 61. 66 Co. ar 8. 2099724 To the Moons Perig. semid 28333 1. 4522925 So is the Moon Perig. dist 56. 50 1. 7521157 To the Apog semid 25964 1. 4143806 We have the semidiameter of the Cone C I 76400 and her Perigaean distance 56. 50 and D C 182. 93 but when the Moon is Apogaean D C will be no more then 177. 77. found by abating K I or K N 61. 66. from B D 239. 43. Hence to find C I or C N in the same parts say As D C 182. 93 co ar 7. 7376467 To C I 7640 1. 8831496 So is C D 177. 77 2. 2498584 To C I 7424 1. 8706547 Then as B C 61. 66 co ar 8. 2099724 Is to Radius 90 10. 0000000 So is C I 7424 1. 8706547 To the tang of C B I 68981 8. 0806271 Here then we have determined   Apogaeon 26936 The Suns Semidiameter    Perigaeon 27916  Apogaeon 1493. 03 His distance from the earth    Perigaeon 1440. 66  Apogaeon ☉ 249. 68 The Axis of the earths shadow    Perigaeon ☉ 239. 43 The Semidiameter of the shadow when the Sun is Apogaeon In loco â—ransitus Lunae Apog 78442 Perig. 77471.  Apogaeon 25964 The Semidiater of the Moone in
Syzygiis    Perigaeon 28333  Apog 61. 66 The distance of the Moon from the earth in Syzygiis    Perig. 56. 50  Perig. 23928. Semiangle of the Cone    Apog 22948. CHAP. 19. Of the Proportion and Magnitude of the three great bâ—dies the Sun Moon and the Earth THat it is a hard matter exactly to determine the true Magnitude of the coelestial bodies is not I beleeve denied by any it will be therefore sufficient if we shall determine them so as that there be no sensible errour in them and to such exactnesse we may traine by the rules and proportions following As the Semidiameter of the Earths shadow C B I. Is to the Semidiameter of the shadow in parts of the Earths Semidiameter C I = B K So is the apparent Semidiameter of the Moon To the Semidiameter of the Moone in parts of the Earâ—hs Semidiameter that is As C B I 77471 co ar 10. 1108609 To C I 7640 1. 8831496 So is the Moones semid 28333 1. 4522925 To the Moones semid 27945 1. 4463030 And Sphears being in triplicated proportion of their diameters the proportion of the earth to the Moone will be as 1. 00000. 00000. 00000. the Cube of the earths Semidiameter to 02182. 28939. 33625. the Cube of the Moones semid 27945. and therefore dividing the earths Semidiameter by the Moons the quotient will be 45. 823 and so many times is the body of the Moon contained in the Earth The proportion between the Semidiameter of the Earth and the Semidiameter of the Sun may be found by this analogy As Radius 90  To A B 1440. 66 3. 1585614 So is tang A B E 27916 7. 687706â— To A E 7. 0189 0. 8462683 But if to A B 1440. 66  You adde B D 239. 43  Their summe is A D 1680. 09  And then As Radius 90  To the side A D 1680. 09 3. 2255415 So is tang of A D E or K I G 23928 7. 6207755 To A E 7. 0197 0. 8463170 And now if you take the lesser semidiameter of the Sun the Cube thereof will be â—45 781 but taking the Semidiameter of the Sun to be but 7 semidiameters of the earth the Sun will be 343 times bigger then the Earth The proportion of the Semidiameter of the earth and the Moone is as 1 to 27945 of the Sun and the Earth as 7 to 1 and therefore of the Sun and the Moon as 7 to 27945. The Cube of 7 is 343 the Cube of 27945 is 02182 c by which dividing the Cube of the Suns semidiameter the quotient will be 15717. 47 and so many times is the Moone contained in the Sun CHAP. 20. Of the proportion between the Orbs of the superiour and inferiour Planets and the Orb of the Earth WHat proportion the Orbs of these Planets have to the earths Orb we have set down in those Chapters in which we have shewed the manner of computing their places and by what meanes the truth of those proportions may appeare we shall set downe in this and because we have used those proportions which Bullialdus hath with great diligence computed we shall exhibit here an Example in Saturne according to which method the proportions between the orbe of the earth and the orbes of the other planets are also to be found And Saturns proportion to the earths orbe as Bullialdus hath determined it and which we have used Chap. 13. is as 100. 000 to 10480. The observation from whence this proportion is gathered was made Anno Christi 1587. January the 9th Hour 9th 75 parts at which time Saturn was observed to be in Aries 26 13333. with South latitude deg 2. 46667. The Suns true place then was in ♉ deg 29 41778. and his distance from the earth 98374. Saturns true place from the Sun by calculation was in ♉ deg 2 31416. whose difference from his observed place deg 6 18083. is the parallax of the orbe or the angle A N S and the angle N A S 87. â—0362 is found by deducting Saturns place from the place of the Sun which with his distance from the Sun or side A N 95596 being given the side A S will be found to be 10310. Now as the Suns distance from the earth 98374 Is to the distance 10310 So is the Semidiameter of ♄ orbe 100. 000 To the Semidiameter of the earths orbe 10480 By a second Observation made Anno Christi 1590 February 8 about 8 of the clock in the evening Saturn was in Gemini deg 7. 53333. with South latitude deg 1. 50. The true place of the Sun at the same time was in Pisces deg 0. 02805. and his distance from the earth 98953. And Saturns place from the Sun by calculation was in Gemini deg 13. 82167. from which deducting his place taken by observation their difference 6. 28834 is the parallax of his orbe represented by the angle A I S. And subtracting Saturns place 73. 82167 from the Suns place 330. 02805 their differences 256. 20638 reject a semicircle is the angle I A S 76. 20638 and Saturns distance from the Sun represented by A I 94338 and hence the side A S 10423. And now as the Suns distance from the earth 9893 Is to the distance A G 10423. So is the Semidiameter of Saturns orbe 100. 000 To the Semidiameter of the earths orbe 10533 By a third observation made in the same year of Christ 1590 Septemb. 7 at midnight Saturns place was in Gemini deg 28. 1 with South latitude deg 1. 18333. The Suns true place at the same time was in Virgo deg 24. 49833. And his distance from the earth 100300. Saturnes place from the Suns by calculation was in Gemini deg 21. 76722 which being deducted from his place taken by observation their difference is the parallax of his oâ—â—e or the angle A K L 6. 33278 and deducting Saturns place from the place of the Sun the angle A L K is 9â— 73111 and therefore the side A L 10415. Now as the Suns distance from the earth 100300 Is to the distance A L 10415 So is the Semidiameter of Saturns Orbe 100000 To the Semidiameter of the earths orbe 10383 But Bullialdus whom we follow doth retaine the first of these 10480 as being the meane and most agreeable to Tycho's observation And from these three observations the inclination of Saturns orbe may thus be found The Triangles L D. N. I D O and D G M of the following Diagram have their sides and angles equal with the triangles N A S. I A S and L A K in the Diagram preceding being drawne from the same observations in every of which we are to compute ♄ distance from the earth for which in the triangle L D N by the first of these observations we have given the angle L N D 86. 71556. The angle L D N 87. 10362 and ♄
from the summe subtract the Latitude of the Moon the remainder is the scruples of the Moons diameter ecclipsed Example Horizontall parallax of the Moon 94409 Semiangle of the Cone Subt. 23455 Semidiameter of the shaddow 70954 Semidia meter of the Moon 26431 Summe of the Semidiameters 97385 Latitudâ— of the Moon Subt. 51496 Scruples deficient 45889 4 Convert these Scruples into digits or parts whereof the Moones body containes 12 thus As the Moones diameter 52862 co ar 5. 276857 Is to the Scruples deficient 45889 4. 661708 So is 12 digits 1. 079181 To the digits ecclipsed 10. 417 1. 017746 Yet note that Lunar Eclipses are of three sorts 1 Partiall when the Scruples deficient are lesse then the diameter 2 Totall without continuance when they are equall 3 Totall with continuance when the scruples deficient are greater then the diameter and in these the digits eclipsed are more then 12 which are so to be understood as that they shew how far the ecclipse is over the body of the Moon CHAP. 25. To find the duration of a Lunar Ecclipse or the continuance of the totall darknesse where the Ecclipse is totall FInd the scruples of Incidence thus Take the Logarithmes of the summe and difference of the Moones latitude and the summ of the Semidiameters of the Moon and the shadow halfe the summe of the two Logarithmes shall be the Logarithme of the Scruples of incidence required Example Summe of the Semidiameters 97385  Latitude of the Moone 51496  Their summe 1. 48881 5. 172836 Their difference 45889 4. 661308 Summe of the Logarithmes  9. 834544 Scruples of incidence 82656  4. 917272 2 Divide the Scruples of incidence by the hourly motion of the Moon from the Sun the quotient gives the time of incidence or halfe duration of the Eclipse This subtracted from the true time of the opposition gives the beginning of the Eclipse or added to it gives the ending Example The Scruples of incidence 82656 4. 917272 Divided by the hourly motion of the Moon from the Sun 48121 4. 682326 Time of incidence hours 1. 7177 0. 234946 The true time of the opposition 13 d. 15. 89814 Time of incidence subt 1. 71770 The beginning of the Eclipse 14. 18044 Time of incidence added gives the end 17. 61584 The whole duration 3. 43540 3 If the Eclipse be total and you desire to know the continuance of total darkenesse take the difference of the Semidiameters instead of the sum and thereby worke as you are directed in the first example of this Chapter and you have the halfe tarrience in the shadow whose double is the thing sought CHAP. 26. To find the Moons Latitude at the beginning and end of the Eclipse MUltiply the Suns hourly motion by the time of incidence the product being added to the scruples of incidence gives you the motion of the Moone agreeing to the time of incidence 2 From the true motion of latitude at the true opposition subtract this motion of the Moon there rests the true motion of latitude at the beginning of the Eclipse or if you adde it you have the motion of latitude at the ending with which out of the Table of latitude you may find out the latitude answering to the beginning and end as in our Example The time of incidence 1. 7177 The Suns hourly motion 0411 Their product is 07059 The scruples of incidence 82656 The summe 89715 Motion of Latitude at true ☠1â—4 04882 Motion of Latitude at beginning 173. 15166 Latitude at beginning North Desc. 59223 Motion of Latitude at ending 174. 94596 Latitude at ending North Desc. 43746 CHAP 27. To find the middle of the Eclipse or greatest darkenesse THe time of the true Conjunction or Opposition the most received is when the Sun and Moon are in one line perpendicular to the Ecliptique to find this with the Moons true Latitude at the true opposition enter the little Table of the difference of the true Conjunction or opposition from the greatest obscuration and you shall find the difference with the title which divide by the hourly motion of the Moon from the Sun and the quotient according to the title apply to the time of the true opposition so have you the time of the greatest darkenesse or middle obscuration Example The Moons Latitude North Descend  51496 The difference adde  04489 Which divide by the hourly motion of the Moon  48121 Gives the difference in time to be added  09328 To the true opposition March 13. 15  89814 So the middle of the Eclipse 13. 15h. 99142 The Calculation of the forementioned Ecclipse according to the preceding directions  d. ho. Meane opposition March 13 6. 45639 Interval Adde  9. 35580 True opposition 13. 15. 97681 True place of Sun  5. 24194 Eccentrick place of the Moone  185. 24190 Meane Anomaly of the Sun  266. 79954 Meane Anomaly of the Moone  35. 44378 True motion of Latitude  174. 04881 True Latitude North descend  51496 Reduction Adde  02398 Hourly motion of the Sun  04111 Hourly motion of the Moon  0. 52232 Hourly motion of the Moon from the Sun  48121 Reduction in time Subt.  04983 True opposition corrected 13 15. 92698 Equation of time Subt.  02884 Tâ—uest opposition 13. 15. 89814 Horizontall Parallax of the Moone  94409 Semiangle of the Cone  23455 Semidiameter of the shadow  70954 Semidiameter of the Moone  26431 Sum of the Semidiameters  97385 Scruples deficient  45880 Digits eclipsed  10. 4197 Scruples of incidence  82656 Time of incidence  1. 71770 Beginning of the Eclipse 13 14. 18044 End of the Eclipse 13 17. 61584 The whole duration  3. 43540 Latitude of the Moone at the beginning North D.  59223 Latitude of the Moone at the ending North D.  43746 Difference from the middle Added  09328 The middle of the Eclipse 13. 15 99142 CHAP. 28. Of the Calculation of the Suns Eclipse THis Eclipse is not properly the Eclipse of the Sun but of the earth in regard it is not the Sun but the earth which looseth light the Sun being only apparently darke the earth in truth we will how ever use the name that others have given it and shew you the manner of the Calculation Find the meane conjnction and from thence the true which correct by the Reduction and Equation of time in all things as in the Moon Example of a Solar Eclipse which happened March 28. 1652.  D  Mean Conjunction March 28 0. 82333 Suns place  19. 36150 Eccentrick place of the Moon  20. 89832 Distance of the Moon from the Sun  1. 53682 Meane Anomaly of the Sun  280. 96132 His hourly motion  04090 Meane Anomaly of the Moon  223. 17513 Her hourly motion  60109 Hourly motion of the Moon from the Sun  56019 which dividing the distance  1. 53682 Gives in the quotient subt  2.