worse of the two yet not so bad but that our Dissenting Brethren have I hope some better Arguments to justifie their Non-conformity than what I see published in a little Book without any name to it concerning two Easters in one Year by the General Table saith this learned man who owneth the Feast of Easter was to be observed Anno 1674. upon the 19 day of April so the Almanacks for that Year as well as the General Table set before the Book of Common Prayer but by the Rule in the said Book of Common Prayer given the Feast of Easter should have been upon the twelfth of April for Easter-Day must always be the first Sunday after the first Full Moon which happeneth next after the one and twentieth day of March and if the Full Moon happen upon a Sunday Easter-Day is the Sunday after Now in the Year 1674. the 19 of April being Friday was Full Moon therefore by this Rule Easter-Day should be the twelfth and by the Table and the Common Almanacks April the tenth but this learned man must know that the mistake is in himself and not in the Rule or Table set down in the Book of Common Prayer for if he please to look into the Calendar he will find that the Golden Number Three which was the Golden Number for that Year is placed against the last day of March and therefore according to the supposed motion of the Moon that Day was New Moon and then the Full Moon will fall upon the fourteenth day of April and not upon the tenth and so by consequence the Sunday following the first Full Moon after the 21 day of March was the nineteenth of April and not the twelfth And thus the Rule and the Table in the Book of Common Prayer for finding the Feast of Easter are reconciled and when Authority shall think sit the Calendar may be corrected and all the moveable Feasts be observed upon the days and times at first appointed but till that be a greater difference than one Week will be found in the Feast of Easter between the Observation thereof according to the Moons true motion and that upon which the Tables are grounded for by the Fathers of the Nicene Council it was appointed that the Feast of Easter should be observed upon the Sunday following the first Full Moon after the Vernal Equinox which then indeed was the 21 of March but now the tenth and in the Year 1674. Wednesday the 11 of March was Full Moon and therefore by this Rule Easter-Day should have been upon March the fifteenth whereas according to the Rules we go by it was not till April the nineteenth The Tables of the Sun and Moons middle motions are neither made according to the usual Sexagenary Forms nor according to the usual Degrees of a Circle and Decimal Parts but according to a Circle divided into 100 Degrees and Parts and this I thought good to do to give the World a taste of the excellency of Decimal Numbers which if a Canon of Sines and Tangents were fitted to it would be found much better as to the computing the Places of the Planets but as to the Primum Mobile by reason of the general dividing a Circle into 360 Degrees I should think such a Canon with the Decimal Parts most convenient and in some cases the common Sexagenary Canon may be very useful and indeed should wish and shall endeavour to have all printed together one Table of Logarithms will serve them all and two such Canons one for the Study and another for the Pocket would be sufficient for all Mathematical Books in that kind and then men may use them all or either of them as they shall have occasion or as every one is perswaded in his own mind What I have done in this particular as it was for mine own satisfaction so I am apt to believe that it will be pleasing to many others and although I shall leave every one to abound in his own sense yet I cannot think that Custom should be such a Tyrant as to force us always to use the Sexagenary form if so I wonder that men did not always use the natural Canon if no alteration may be admitted what reason can be given for the use of Logarithms and if that be found more ready than the natural in things of this kind where none but particular Students are concerned I should think it reasonable to reduce all things hereafter into that form which shall be found most ready and exact now the Part Proportional in the Artificial Sines and Tangents in the three first Degrees cannot be well taken by the common difference and the way of finding them otherwise will not be so easie in the Sexagenary Canon as in either of the other and this me thinks should render that Canon which divides each Degree into 100 Parts more acceptable but thus to retain the use of Sines Degrees and Decimal Parts doth not to me seem convenient and to reckon up a Planets middle motion by whole Circles will sometimes cause a Division of Degrees by 60 which hath some trouble in it also but if a Circle be divided into 100 Degrees this inconvenience is avoided and were there no other reason to be given this me thinks should make such a Canon to be desirable but till I can find an opportunity of publishing such an one I shall forbear to shew any further uses of it and for what is wanting here in this subject I therefore refer thee to Mr. Street's Astronomia Carolina and the several Books written in English by Mr. Wing The fourth Part of this Treatise is an Introduction unto Geography in which I have given general Directions for the understanding how the habitable part of the World is divided in respect of Longitude and Latitude in respect of Climes and Parallels with such other Particulars as will be found useful unto such as shall be willing to understand History in which three things are required The time when and this depends upon Astronomy the place where and this depends upon Geography and the Person by whom any memorable Act was done and this must be had from the Historical narration thereof and he that reads History without some knowledge in Astronomy and Geography will find himself at a loss and be able to give but a lame account of what he reads but after the learning of these Arts of Grammar I mean so much thereof as tends to the understanding of every ones Native Language Arithmetick Geometry and Astronomy a Child may proceed profitably to Rhetorick and Logick the reading of History and the learning of the Tongues and sure there is no studious and ingenious man but will stand in need of some Recreation and therefore if Musick in the Worship and Service of God be not Argument enough to allow that a place among the Arts let that poor end of Delight and Pleasure be her Advocate and although that all men have not Voyces yet I can
SFP 2. s PSD cs SPD Rad. cs DS PS  3. Rad. cs DS s FSD cs SFP PSF+PSD = FSD  s PSD cs SPD s FSD cs SFP PSF  1. ct PSF Rad cs SF ct SFC 3. SFP FPS 2. s SFC cs PSF Rad. cs FC SF  3. Rad. cs FC s PFC cs FPS SFC-SFP = PFC  s SFC cs PSF s PFC cs FPS CASE 11. The three Sides being given to find an Angle This Case may be resolved by the Catholick Proposition also according to the direction of the Lord Nepier as I have shewed at large in the Second Book of my Trigonometria Britannica Chap. 2. but may as I conceive be more conveniently solved by this Proposition following As the Rectangle of the Square of the Sides containing the Angle inquired Is to the Square of Radius So is the Rectangle of the Square of the difference of each containing Side and the half sum of the three Sides given To the Square of the Sine of half the Angle inquired In this Case there are three Varieties as in the Triangle FZP Fig. 3. Given Required  ZP  s ZP x s PF Rad. q. 1. PF ZPF s ½ Z-ZP x s ½ Z-PF Q FZ  s ½ ZPF ZP  s PF x s PZ Rad. q. 2. PF PFZ s ½ Z-PF x s ½ Z-FZ Q FZ  s ½ PFZ ZP  s ZP x s FZ Rad. q. 3. PF FZP s. ½ Z-ZP x ½ Z-ZF Q FZ  s ½ FZP CASE 12. The three Angles given to find a Side This is the Converse of the last and to be resolved after the same manner if so be we convert the Angles into Sides by the tenth of the third Chapter for so the Sides of the Triangle ACD will be equal to the Angles of the Triangle FZP n Fig. 3. That is AD = AEE the measure of the Angle ZPF DC = KM the measure of the Angle ZFP AC = HB the Complement of FZP to a Semicircle The Angle DAC = QR = ZP ACD = rM = Hf = Zoe = ZF ADM = sK = AEl = Ph = PF And thus the Sides of the Triangle ZPF are equal to the Angles of the Triangle ACD The Complement of the greatest Side PF to a Semicircle being taken for the greatest Angle ADC And in this Case therefore as in the preceding there are three Varieties which make up sixty Problems in every Oblique angled Spherical Triangle which actually to resolve in so many Triangles as have been mentioned would be both tedious and to little purpose I will therefore select some few that are of most general use in the Doctrine of the Sphere and leave the rest to thine own practice CHAP. V. Of such Spherical Problems as are of most General Use in the Doctrine of the Primum Mobile or Diurnal Motion of the Sun and Stars PROBLEM 1. The greatest Declination of the Sun being given to find the Declination of any Point of the Ecliptick THe Declination of the Sun or other Star is his or their distance from the Equator and as they decline from thence either Northward or Southward so is their Declination reckoned North or South 2. The Sun 's greatest Declination which in this and many other Problems is supposed to be given with the Distance of the Tropicks Elevation of the Equator and Latitude of the Place may thus be found The Sun 's greatest Meridian H ♋ 61.9916 least Altitude H ♑ 14.9416 Their difference is the distance of the Tropicks ♋ ♑ 47. 050 Half that Difference is the Sun 's greatest Declination AE ♋ 23. 525 Which deduct from the Sun 's greatest Altitude the remainer is the height of the Equator HAE 38. 467 The Complement is the height of the Pole AEZ or PR 51. 533 Now then in the Right angled Spherical Triangle ADF in Fig. 1. there being given 1. The Angle of the Sun 's greatest Declination DAF 23. 525. 2. The Sun 's supposed distance from ♈ to ♎ AF. 60 deg The Sun 's present Declination DF may be found by the 10 Case of Right angled Spherical Triangles As the Radius Is to the Sine of DAF 23. 525. 9.60113517 So is the Sine of AF 60. 9.93753063 To the Sine of DF. 20. 22. 9.53866580 PROBLEM 2. The Sun 's groatest Declination with his Distance from the next AEquinoctial Point being given to find his Right Ascension In the Right angled Spherical Triangle ADF in Fig. 1. Having the Angle of the Sun 's greatest Declination DAF 23. 525. And his supposed distance from ♈ or ♎ the Hypotenusa AF. 60. The Right Ascension of the Sun or Arch of the AEquator AD may be found by the ninth Case of Right angled Spherical Triangles As the Cotang of the Hypot AF. 60. 9.76143937 Is to the Radius 10.00000000 So is the Cosine of DAF 23. 525. 9 96231533 To the Tang. of AD. 57. 80. 10.20087596 PROBLEM 3. To find the Declination of a Planet or Fixed Star with Latitude In the Oblique angled Spherical Triangle FPS in Fig. 4. we have given 1. PS = AE ♋ the greatest Declination of the Ecliptick 2. The Side FS the Complement of the Stars Latitude from the Ecliptick at K. 3. The Angle PSF the Complement of the Stars Longitude To find PF the Complement of Declination By the eighth Case of Oblique angled Spherical Triangles the Proportions are As the Cot. of PS 23. 525. 10.3611802 Is to the Radius 10.0000000 So is the Cos. of PSF 20 deg 9.9729858 To the Tang. of SE. 22. 25. 9.6118056 FS 86 deg ES. 22. 25. = FE 63. 75. As the Cos. of ES. 22. 25. Comp. Arith. 0.0336046 To the Cosine of PS 23. 525. 9.9623154 So the Cos. FE 63. 75. 9.6457058 To the Cos. PF 64. 01. 9.6416258 Whos 's Complement is FT 25. 99. the Declination sought PROBLEM 4. To find the Right Ascension of a Planet or other Star with Latitude The Declination being found by the last Problem we have in the Oblique angled Spherical Triangle PFS in Fig. 4. All the Sides with the Angle FSP 20 deg or the Complement of the Stars Longitude Hence to find FPS by the first Case of Oblique angled Spherical Triangles I say As the Sine of PF 64. 01. Comp. Arith. 0.0463059 Is to the Sine of FSP 20. 9.5340516 So is the Sine of FS 86. 9.9984407 To the Sine of FPS 22. 28. 9.5787982 Whos 's Complement 67. 72. is the Right Asc. of a Star II. 10. North Lat. 4. PROBLEM 5. The Poles Elevation Sun's greatest Declination and Meridian Altitude being given to find his true place in the Zodiack If the Meridian Altitude of the Sun be less than the height of the AEquator deduct the Meridian Altitude from the height of the AEquator the Remainer is the Sun's Declination towards the South Pole but if the Meridian Altitude of the Sun be more than the height of the AEquator deduct the height of the AEquator from the Meridian Altitude what remaineth is the Sun's Declination
to reduce her place from her Orbit to the Ecliptick Chap. 19. To find the mean Conjunctions and Opposition of the Sun and Moon The Fourth Part or an Introduction to Geography CHap. 1. Of the Nature and Division of Geography Chap. 2. Of the Distinction or Dimension of the Earthly Globe by Zones and Climates Chap. 3. Of Europe Chap. 4. Of Asia Chap. 5. Of Africk Chap. 6. Of America Chap. 7. Of the description of the Terrestrial Globe by Maps Vniversal and Particular A Table of the view of the most notable Epochas The Iulian Calendar Page 461 The Gregorian Calendar 466 A Table to convert Sexagenary Degrees and Minutes into Decimals and the contrary 476 A Table converting hours and minutes into degrees and minutes of the AEquator 480 A Table of the Longitudes and Latitudes of some of the most eminent Cities and Towns in England and Ireland 482 A Table of the Suns mean Longitude and Anomaly in both AEgyptian and Iulian Years Months Days Hours and Minutes 484 Tables of the Moons mean motion 493 A Catalogue of some of the most notable fixed Stars according to the observation of Tycho Brahe rectified to the year 1601. 511 Books Printed for and sold by Thomas Passinger at the Three Bibles on the middle of London-Bridge THe Elements of the Mathematical Art commonly called Algebra expounded in four Books by Iohn Kersey in two Vol. fol. 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Calvert twelves Pilgrims Port or the weary mans rest in the Grave twelves Christian Devotion or a manual of Prayers twelves The Mariners divine Mate twelves At Cherry Garden Stairs on Rotherhith Wall are taught these Mathematical Sciences viz. Arithmetick Algebra Geometry Trigonometry Surveying Navigation Dyalling Astronomy Gauging Gunnery and Fortification The use of the Globes and other Mathematical Instruments the projection of the Sphere on any circle c. He maketh and selleth all sorts of Mathematical Instruments in Wood and Brass for Sea and Land with Books to shew the use of them Where you may have all sorts of Maps Plats Sea-Charts in Plain and Mercator on reasonable Terms By Iames Atkinson FINIS