Almicantar or Altitude of the Sun at 6. a clock PROB. VI. By the Hour of the Night and a known Star Observed Rising or Setting to find the Heigth of the Pole REctifie the Hour Index by Prob. 2. of the former Book and turn the Globe Westwards till the Hour Index points at the Hour of the Night fasten the Globe there and turn the Meridian through the notches of the Horizon till the knowâ— Star come to the East side the Horizon if the Star be Rising 〈◊〉 the West if it be Setting so shall the degrees of the Poles Elâ—vation be cut by the Horizon under the Elevated Pole and 〈◊〉 North or South according as the Elevated Pole of the Globe 〈◊〉 PROB. VII Two Places given in the same Latitude to find 〈◊〉 Difference of Longitude BRing the first Place to the Meridian and note the number of degrees of the Equinoctial that comes to the Meridan with it then Bring the other place to the Meridian and note the number of degrees of the Equator that comes to the Meridian with it and by substracting the lesser number from the greater you have the difference of Longitude This needs no Example PROB. VIII Two Places given in the same Longitude to find the Difference of Latitude BRing the Places to the Meridian and the degrees of the Meridian over the two Places is the Latitudes of them both and by substracting the lesser number of degrees from the greater you will have the difference of Latitude PROB. IX Course and Distance between two Places given to find their Difference in Longitude and Latitude SEek the Rhumb you have failed upon as in Prob 34 of the last Book and upon that Rhumb make a mark for the Place you departed from then with your Compasses take off from the Equinoctial the number of Leagues you have failed upon that Rhumb by allowing a degree for every 20. Leagues and place one foot of your Compasses upon that mark and where the other foot falls on that Rhumb make a second mark then by bringing the first mark to the Meridian you will see on the Meridian the Latitude of that mark and in the Equator the Longitude as in Prob. 2. of the last Book and by bringing the second mark also to the Meridian you will as before find the Longitude and Latitude of the second mark also Then by substracting the lesser Latitude from the greater Latitude and the lesser Longitude from the greater Longitude you will have the difference remaining both of Longitude and Latitude you are arived into PROB. X. To find how many Miles are contained in a Degree of any Parallel EVery Degree of the Equinoctial contains 20. English Leagues and every League 3. English Miles But in every Parallel to the Equinoctial the Degrees diminish more and more even to the Pole where they end in a point Therefore a Degree in any Parallel cannot contain so many Miles as a Degree in the Equinoctial Now that you may know how many Miles are contained in a Degree of any Parallel to the Equinoctial Do thus Measure with your Compasses the width of any number of Degrees in any given Parallel suppose for Examples sake 10. Degrees in the Parallel of 51½ Examine in the Equator how many Degrees of the Equator they will make and you will find 6â…• Therefore 1. Degree in the Equator making 60 Miles 6. Degrees makes 360 to which add for the 〈◊〉 part 12 Miles makes â—72 Miles to be the Measure of 10 Degrees in the Parallel of 51½ So that by dividing 372. by 10. you have â—7 Miles for the length of a Degree from East to West in the Parallel of 51½ Degrees PROB. XI The Rhumb you have sailed upon and the Latitudes you deâ—arted from and are arived to given to find the Difference of Longitude and the number of Leagues you have Sailed FIrst seek the Rhumb you have sailed on and pass it through the Meridian till it cuts in the Meridian the Latitude you departed from and keeping the Globe there stedâ—y make a mark close by the Meridian under that Latitude and in that Rhumb on the Globe and note in the Equinoctial the degree of Longitude at the Meridian then pass that Rhumb through the Meridian again till it cuts in the Meridian the Latitude you are arived to and in that Rhumb and Latitude make on the Globe another mark and examine in the Equinoctial the Longitude of the second mark for the difference between the first and second mark is the difference of Longitude Then open your Compasses to one Degree of the Equinoctial and by measuring along in the Rhumb count how many times that Distance is contained between the two points in that Rhumb for so many times 20. Leagues is the Distance you have sailed Example I sail upon the North West Rhumb from the Latitude of 1â— degrees into the Latitude of 30. degrees 40. minutes Therefore I find the North West Rhumb and turn the Globe through the Meridian till this Rhumb cut the Meridian in the first Latitude viz. in 10. degrees and directly under 10. degrees upon the Rhumb I make a prick and also find 10 degrees 3 minutes of the Equator at the Meridian for the Longitude of the First Place Then I turn the Globe again through the Meridian till the same Rhumb cut the Meridian in the second Latitude viz in 30 degrees 40 minutes and directly under those 30 degrees 40 minutes upon the same Rhumb I make another prick which represents the Place I am arrived to I examine the Longitude of this prick as before and find it 32 degrees 10 minutes Therefore I substract the first Longitude viz. 10 degrees 3 minutes from the second Longitude viz. 32 degrees 10 minutes and there remains 22 degrees 7 minutes for the Difference of Longitude Then for examining the Distance I open my Compasses to 1. degree on the Equinoctial and measure upon the Rhumb how oft that Distance is contained between the two pricks and find 29¼ that is 29 degrees 15 minutes which multiplyed by 2â— gives 585 for the number of Leagues failed upon that Râ—umb The reason why I open the Compasses no wider then to 1 degree is because the Rhumbs being Circular or 〈◊〉 lines the distance on them may be measured more exactly by often counting that 1 degree in them theâ— if the 〈◊〉 had bin opâ—red to many degrees Thus if the Compasses had been opened wide enough to reach between the two pricks aforesaid I should not have had above 583 Leagues for the distance between the two Places neither is there indeed more great Circle distance between them But I sailed upon a Rhumb that is I followâ—d the Course of a Circular winding line and so fetcht a Compass about to come to these two pricks and therefore I have in truth sailed 585. Leagues For the segment of a Rhumb between two Places is alwaies greater then a straight line drawn betwixt them yea sometimes by half
or more in Places neer either Pole Note If you be not very curious in opening your Compasses to this smal distance you may in oft turning them about upon the Rhumb commit error in your measuring therefore when you have taken the Distance of one degree try if you neither gain or loose any thing in measuring 10 or 20. degrees of the Equinoctial by them for then your Compasses are opened to a width exact enough for your purpose PROB. XII The Longitudes and Latitudes of two Places given to find Course and Great Circle distance between them FInd on the Globe the Longitudes and Latitudes given and make pricks to either Longitude and Latitude If any Rhumb pass from one place to the other that is without more a doe the Rhumb sought But if no Rhumb pass through Take the Rhumb that runs most Parallel to the two pricks for that shall be the Rhumb or the neerest Rhumb that these two pricks Bear on An Example of this see in Prob. 34. of the Last Book And the Great Circle Distance between these two pricks you may find as by Prob 33. of the same Book PROB. XIII The Latitude you departed from and the Latitude you are arrived to and the number of Leagues you have sailed given to find the Rhumb you have sailed on and difference of Longitude MAke a prick on the Globe in the Latitude you departed from then open your Compasses to the number of Leagues you have sailed by taking for every 20. Leagues 1. degree of the Equator half a degree for 10 Leagues a quarter of a Degree for 5 Leagues and so proportionably for any other number of Leagues Place one foot of your Compasses in the prick made for the Latitude you departed from and extend the other towards the Latitude you are arived to and discribe an occult Arch Turn the Globe till this occult Arch come to the Latitude on the Meridian and where the Latitude cuts this occult arch make another prick to represent the Latitude you are arived to so shall the Rhumb passing through those two pricks or that is most Parallel to those two pricks be as in the last Prob. the Course or the Rhumb those two pricks Bears on The difference of Longitude you may find as by Prob. 11. PROB. XIV To find by the Globe the Variation of the Needle commonly called the Variation of the Compass OBserve by a Compass whose wyer is placed just under the Flower deluce what point of the Compass the Sun Rises or Sets on Morning or Evening Then examine by Prob 10. of the second Book what degree of the Horizon the Sun Rises or Sets on by the Globe also and if the Rising or Setting be the same both on the Globe and Compass there is no Variation in your Place But if there be difference between the Rising or Setting by the Compass and the Globe then is there Variation in your Place If the point the Sun Rises upon in the Compass be neerer the North point then the point the Sun Rises upon by the Globe the Variation is Westwards If the point the Sun Sets upon in the Compass be neerer the North then the point it Sets upon by the Globe the Variation is Eastwards If the point the Sun Sets upon in the Compass be further from the North point then the point the Sun Sets upon by the Globe the Variation is Westwards If the point the Sun Rises upon in the Compass be further from the North point then the point the Sun Rises upon by the Globe the Variation is Eastwards And so many degrees as there is between the point of Rising or Setting found by the Compass and the point of true Rising or Setting found by the Globe so many degrees is the Variation from the North towards the East or West point Otherwise when the Sun hath Altitude Having the Altitude of the Sun find by Prob. 22. of the second Book its Azimuth Then examine by a Compass whether the true Azimuth found by the Globe agree with the Azimuth found by a Nautical Compass If they agree there is no Variation But if the Azimuth of the Compass before Noon be neerer the North then the true Azimuth found by the Globe the Variation is Westwards If the Azimuth by the Compass Afternoon be neerer the North the Variation is Eastwards If the Azimuth by the Compass Afternoon be further from the North the Variation is Westwards If the Azimuth by the Compass before Noon be further from the North the Variation is Eastwards And this Variation shall be as aforesaid so many degrees as there is between the Azimuth Observed by the Compass and the true Azimuth Observed by the Globe PROB. XV. To keep a Journal by the Globe BY some of these foregoing Problemes you may Dayly when Observations can be made find both the Longitude and Latitude on the Globe of the Places you are arived to and also the Way the Ship hath made and make pricks on the Globe in their proper Places for every Daâ—â—s Journey so truly and 〈◊〉 naturally that if you kept your reckoning aright you may be sure you cannot miss any thing of the truth it self and that with less trouble and greater advantage then keeping a Book of every Daies Reckoning PROB. XVI To Steer in the Night by the Stars REctifie the Globe and Hour Index as by Prob. 2. of the last Book and turn about the Globe till the Index of the Hour Circle points to the Hour of the Day or Night Then turn the Globe till the Difference of Longitude between the Place you depart from and the Place you sail to pass through the Meridian and if any Star in the Latitude of the Place you sail To come to the Meridian or neer the Meridian with the degree of the difference of Longitude that Star is at that time in or neer the Zenith of that Place you sail to and by finding the same Star in Heaven as by Prob 44. of the last Book you may direct your ship towards that Star and fail as confidently saies M r Blagrave as if Mercurie were your Guide But because this Star moves from the Zenith of this Place you must often examine what Star is come to the Zenith and so often charge the Star you Steer by as the length of your Voyage may require PROB. XVII How to platt on the Globe a New Land never before Discovered THese two following Problemes are 2. Chapters of M r Wrights delivered by him as follows It may sometimes fall out in new Discoveries or when your Ship by means of a Tempest is driven out of her right Course that you shall come to the sight of some Isle Shoald or new Land whereof the Mariner is utterly ignorant And to make some relation of the same or to go unto it some other time if you desire to set it down on your Globe in the true place you may do it after this manner So soon as you have sight
motion of the Sun by only a Beetle rowling his pill of dung backward as we may read in Clemens Alexandrinus and then came Eudoxus who having learned that variety of motions among them was the first who invented Hypotheses of various Orbs for the Solution of the Phenomena Again they were very far from attaining the determinate places of the Fixt Stars according to Longitude and Latitude or according to their Right Ascension and Declination so that neither could they define the true places of the Planets by Comparation to the Fixt Stars nor consequently designe any Observations with due exactness And truely this was the Cause why Hipparchus met with no Observations either of the Egyptians or Babylonians by which he could receive the least help or advantage toward his composing either Hypotheses or Tables to represent the motions of the Five errant Stars and Ptolomy was the first who partly by the benefit of Observations left him by Hipparchus and partly by those he made himself became able to attempt such a Work as stands recorded in his Almagest There were only the Eclypses which both these Nations had set down as observed in their Commentaries and those only so as that from Past they might be able to conjecture somthing of what were to Come Not from the motions of Sun and Mon exactly calculated by the help of Tables but having learnd from common experience that every ninetneenth Year Eclypses did return again upon the same Day for the most part thereupon they endeavoured to praedict what Eclipses would happen and the time when and this after they had perceived not any Anomaly in the Sun but some certain Inequality in the Moon which reducing to a medium they concluded that the Moon did every Day run througâ— thirteen Degrees and a little more than one sixth part of a degree as Geminus delivers of the Chaldaeans But in their predictions of Lunar Eclipses they were somwhat more confident aswell because these Eclipses usually uturn for the three Ages next succeding within the compass of the same Dayes as because it is very rare in respect of the greatness of the Earths shadow but the Moon either in the whole or some part of her more or less falls into it but because as to Solary Eclypses the Moon is both so small and hath so large a Parrallax as that she doth not for the most part intercept the light of the Sun from the Earth therefore was it as Diodorus witnesseth specially of the Babylonians that they durst not determine Eclypses of the Sun to come to any certaine time but if they predicted any with limitation of time they alwayes to save their credit in case of failing annexed this Condition If the Gods be not prevailed upon by Sacrifices and Praiers to avert them Truth is these Astronomers were also Priests and it was their interest to cast in this Proviso For being ambitious to be reputed interpreters of the Will of the Gods to the People and so both knowing in things to come and skillfull in such Ceremonies wherewith their respective Deities were most attoned and delighted unwilling to be thought able to predict nothing and as unwilling again to be found erring in their chief predictions they wrapt up all in Misteries and amused the vulgar with superstitious opinions and rites The Egyptians in a great part of their sacred Worship had recourse to the Astrological Books of their Mercurius one of the Order of the Fixt Stars a second of the Conjunction of Sun and Moon a third and fourth of their rising which with what ceremonious Pomp they used to carry about with them in a kind of solemne Procession you may find amply described by Clem. Alexandrinus Nor is it strange that those Priests accounted so sacred and knowing should also be estemed for Prophets Further you meet with no mention of the Five Errant Stars all this while and the reason seems to be because they attributed an energie of them only as they were referrable to the Inerrant or Fixt and particularly as they possest this or that part of some Signe in the Zodiack and together with it had their rising or setting For so much did they ascribe to the Zodiack as that the Babylonians and in imitation of them the Persians and Indians thought that each decimal of degrees or thirds of the Signes and the Egyptians came as low as to each single degree could not be varied in the rising but some eminent variation most happen especially in him who should be borne at that time And hereupon was it that the Egptians made that great Circle of Gold described in Diodorus of a cubit in thickness and three hundred sixty five cubits in circumference plundred at last by Cambyses that upon each cubits space might be inscribed each Day of the Year 365. Dayes in the whole round and also what Stars did rise what set upon each Day nay the very hour of their respective rising and setting and what they did signifie and whereas others used to assigne the form of some Animal or other to each ten degrees they assigned one to each single degree and so made their harsolations or conjectural predictions accordingly For Example to the first degree of Aries they assigned the figure of a Man holding a Sicle or hook in his right hand and a Sling in his left to the second a Man with a Dogs-head his right hand stretcht forth and a staff in his left and so of the rest then annexing the signification to each they determined that he who should have the first degree of Aries for his Horoscope should be some part of his life a Husbandman and the rest of it a Soldier that he who should be born under the second should be contentious quarrelsom and envious and so of the rest all which Scaliger hath fully deduced from Aben Ezra In a Word what ever knowledge either the Egyptians or Chaldeans had of the Stars certain it is they referred it wholly to Astronomantie or Divination by Stars and therefore among them there flourisht no true and genuine Astronomy but a spurious and false one i. e. Astrology Divinatory or the fraudulent Art of Fortune-telling by the Heavens Berosus whom we formerly mentioned coming into Grece a little after the death of Alexander is discovered to have brought with him nothing sollid touching Astronomy but only Judicial Astrologyâ— for which as a thing new and strange to the people he was highly esteemed as Vitruvius and Pliny remark And Eudoxus who had returned out of Egypt before that well knew what sort of Astrology this was the principal Contrivers and Founders of which are said to have been Petosires Necepsus Esculapius but he highly contenmed it as Cicero remembers and brought home no other fruit of his tedious Travells beside a list of some Eclipses and the varieties of the motions of the wandering Stars by which he first essaied to compose accommodate Hypotheses as we have formerly hinted Nay
DUCTOR ad ASTRONOMIAM GEOGRAPHIAM vel usus GLOBI Celestis quam Terrestris In Libris sex viz. Astron. Geogr. Rudimē Astrō Geogr. Problē Nautica Problē Astrologica Problē Gnomonica Problē Sphaeric Triang Problē Per Josephum Moxon LONDINI Sumptibus Josephi Moxon A TUTOR to ASTRONOMIE and GEOGRAPHIE Or an Easie and speedy way to know the Use of both the GLOBES Coelestial and Terrestrial In six BOOKS The First teaching the Rudiments of Astronomy and Geography The 2. Shewing by the Globes the solution of Astronomical Geographical Probl. The 3. Shewing by the Globes the solution of Problemes in Navigation The 4. Shewing by the Globes the solution of Astrological Problemes The 5. Shewing by the Globes the solution of Gnomonical Problemes The 6. Of Spherical Triangles More fully and amply then hath ever been set forth either by Gemma Frisius Metius Hues Wright Blaew or any others that have taught the Use of the Globes And that so plainly and methodically that the meanest Capacity may at first reading apprehend it and with a little Practise grow expert in these Divine Sciences By Joseph Moxon Whereunto is added the Antient Poetical Stories of the Stars shewing Reasons why the several shapes and forms are pictured on the Coelestial Globe Collected from Dr Hood As also a Discourse of the Antiquity Progress and Augmentation of Astnonomie Psal. 111. 2. The Works of the Lord are great sought out of them that have pleasure therein Job 26. 13. By his Spirit he hath garnished the Heavens His hand hath framed the crooked Serpent LONDON Printed by Joseph Moxon and fold at his Shop on Corn-hill at the signe of Atlas 1659. A Catalogue of Books and Instruments Made and sold by Joseph Moxon at his shop on Corn-hil at the Signe of Atlas GLobes of all sizes Coelestial and Terrestrial Sphears according to the Ptolemean Tychonean Copernican Systeme The Catholick Planisphere call'd Blagrave's Mathematical Jewel made very exactly on Past-boards about 17. inches Diameter And a Book of the Use of it newly set forth by I. Palmer M. A. The Spiral Line Gunters Quadrant and Nocturnal Printed and pasted c. Stirrups Universal Quadrat Printed and Pasted c Sea-Plats Printed on Paper or Parchment and Pasted on Boards Wrights Corrections of Errors in the Art of Navigation The third Edition with Additions Vignola or the Compleat Architect useful for all Carpenters Masons Painters Carvers or any Gentlemen or others that delight in rare Building A new Invention to raise Water higher then the Spring With certain Engines to produce either Motion or Sound by the Water very useful profitable and delightful for such as are addicted to rare curiosities by Isaac de Caus. A Help to Calculation By J. Newton A Mathematical Manuel shewing the use of Napiers bones by J. Dansie A Tutor to Astrology with an Ephemeris for the Year 1659. intended to be Annually continued by W. E. Also all manner of Mathematical Books or Instruments and Maps whatsoever are sold by the foresaid Joseph Moxon To the Reader Courteous Reader IFormerly Printed a Book of the Use of the Globes Intituled A Tutor to Astronomy and Geography The Book was Composed by William Blaew but the Title was mine own and therefore I hope I may be the bolder to use it when and where I list The sale of that Impression had almost perswaded me to have Printed it again But when I considered it wanted many necessary Problemes both in Astronomy Navigation Astrology Dyalling and the whole Doctrine of Triangles by the Globe And also that the Examples throughout that Book were made for the Citty of Amsterdam which by the general sale of the Book I found rendred it less acceptable then it would have been if they had been made for London And when I considered that to add so many Problemes and alter all the Examples would both Metamorphose that Book and be as Laborious a work to me as if I should write a new one Then I resolved to take this Task upon me which at length with Gods Assistance I have finished And now expose it to thy acceptance The Globes is the first Studie a Learner ought to undertake for without a competent knowledge therein he will never be able to understand any Author either in Astronomy Astrology Navigation or Trigonometry Therefore my aim hath been to make the Use of then very plain and easie to the meanest Capacities In prosecution of which Designe I doubt the Learneder sort may be apt to Censure me guilty of Prolixity if not Tautology Because the Precepts being plain they may account some of the Examples Useless But I desire them to consider that I write not to expert Practitioners but to Learners to whom Examples may prove more Instructive then Precepts Besides I hope to encourage those by an ample liberal plainness to fall in love with these Studies that formerly have been disheartned by the Crabbed brevity of those Authors that have in Characters as it were rather writ Notes for their own Memories then sufficient Documents for their Readers Instructions The Globes for which this Book is written are the Globes I set forth about four years ago which as I told you in my Epistle to the Reader of Blaew's Book differs somewhat from other Globes and that both the Coelestial and the Terrestrial mine being the latest done of any and to the accomplishing of which I have not only had the help of all or most of the best of other Globes Maps Plats and Sea-drafts of New Discoveries that were then extant for the Terrestrial Globe but also the Advice and directions of divers learned and able Mathematicians both in England and Holland for Tables and Calculations both of Lines and Stars for the Coelestial upon which Globe I have placed every Star that was observed by Tycho Brahe one degree of Longitude farther in the Ecliptick then they are on any other Globes So that whereas on other Globes the places of the Stars were correspondent with their places in Heaven 58. Years ago when Tycho observed them and therefore according to his Rule want about 47. minutes of their true places in Heaven at this Time I have set every Star one degree farther in the Ecliptick and Rectified them on the Globe according to the true place they will have in Heaven in the Year 1671. On the Terrestrial Globe I have inserted all the New Discoveries that have been made either by our own or Forraigne Navigators and that both in the East West North and South parts of the Earth In the East Indies we have by these later Times many spacious Places discovered many Ilands inserted and generally the whole Draft of the Country rectified and amended even to the Coast of China Japan Giloli c. In the South Sea between the East and West Indies are scattered many Ilands which for the uncertain knowledge former Times had of them are either wholely left out of other Globes or else laid down so
the Night and Day 42 Prob. 8. To find the Hour of Sun Set. 42 Prob. 9. To find how long it is Twilight in the Morning and Evening 43 Prob. 10. The Suns Place given to find its Amplitude And also to know upon what point of the Compass it Riseth 44 Prob. 11 The Hour of the Day given to find the Height of the Sun 45 Prob. 12. The Altitude of the Sun and Day of the Moneth given to find the Hour of the Day fol. 46 Prob. 13. How to know whether it be Before or After Noon 47 How to take Altitudes by the Quadrant Astrolabe and Cross-staff 47 To take Altitudes by the Astrolabe 50 To take Altitudes by the Cross-staff 51 Prob. 14. To observe with the Globe the Altitude of the Sun 52 Prob. 15. To find the Elevation of the Poleâ— by the Meridian Altitude of the Sun and Day of the Moneth given 53 Prob. 16. To take the Altitude of any Star above the Horizon by the Globe 54 Prob. 17. By the Meridian Altitude of any Star given to find the Height of the Pole 54 Prob. 18. Another way to find the Height of the Pole by the Globe if the Place of the Sun be given and also to find the Hour of the Day and Azimuth and Almicanter of the Sun 56 Prob. 19. To observe by the Globe the Distance of two Stars 57 Prob. 20. How you may learn to give a guess at the number of degrees that any two Stars are distant from one another or the number of degrees of Altitude the Sun or any Star is Elevated above the Horizon only by looking up to Heaven without any Instrument 58 Prob. 21. The Day of the Moneth and Altitude of any Star given to find the Hour of the Night 59 Prob. 22. The Place of the Sun and Hour of the Day given to find its Azimuth in any Latitude assigned 60 Prob. 23. The Place of the Sun and Hour of the Day given to find the Almicantar of the Sun 61 Prob. 24. The Place of the Sun given to find what Hour it comes to the East or West and what Almicantar it then shall have 61 Prob. 25. To know at any time what a clock it is in any other Part of the Earth 62 Prob. 26. To find the Right Ascension of the Sun or Stars 63 Prob. 27. To find the Declination of the Sun or Stars 64 A Table of the Right Ascensions and Declinations of 100. Select fixed Stars Calculated by Tycho Brahe for the Years 1600 and 1670. As also their Difference of Right Ascensions and Declinations in 70. Years 65 Prob. 28. The Place of the Sun or any Star given to find the Right Descension and the Oblique Ascension and the Oblque Descension fol. 71 Prob. 29. Any Place on the Terrestrial Globe being given to find its Antipodes 72 Prob. 30. To find the Perecij of any given Place by the Terrestrial Globe 73 Prob. 31. To find the Antecij of any given Place upon the Terrestrial Globe 73 Prob. 32. To find the Longitude and Latitude of the Stars by the Coelestial Globe 73 Prob. 33. To find the Distance between any two Places on the Terrestrial Globe 74 Prob. 34. To find by the Terrestrial Globe upon what point of the Compass any 〈◊〉 Places are scituate one from another 75 Prob. 35. To find by the Coelestial Globe the Cosmical Rising and Setting of the Stars 76 Prob. 36. To find by the Coelestial Globe the Acronical Rising and Setting of the Stars 77 Prob. 37. To find by the Coelestial Globe the Heliacal Rising and Setting of the Stars 78 Prob. 38. To find the Diurnal and Nocturnal Arch of the Sun or Stars in any given Latitude 79 Prob. 39. To find the Azimuth and Almicantar of any Star 81 Prob. 40. To find the Hour of the Night by observing two known Stars in one Azimuth or Almicantar 81 Prob. 41. The Hour given that any Star in Heaven comes to the Meridian to know thereby the Place of the Sun and by consequence the Day of the Moneth though it were lost 82 Prob. 42. The Day of the Moneth given to find in the Circle of Letters on the Plain of the Horizon the Day of the Week 83 Prob. 43. The Azimuth of any Star given to find its Hour in any given Latitude 84 Prob. 44. How you may learn to know all the Stars in Heaven by the Coelestial Globe 84 Prob. 45. How to hang the Terrestrial Globe in such a position that by the Suns shining upon it you may with great delight at once behold the demonstration of many Principles in Astronomy and Geography 89 Prob. 46. To know by the Terrestrial Globe in the Zenith of what Place of the Earth the Sâ—â— is 91 Prob. 47. To find in what different Places of the Earth the Sun hath the same Altitude at the same time 92 Prob. 48. To find the length of the Longest and shortest Artificial Day or Night 95 Prob. 49. To find how much the Pole is Raised or Depressed where the longest Day is an Hour longer or shorter then it is in your Habitation 96 Prob. 50. The Suns Place given to find what alteration of Declination he must have to make the Day an Hour longer or shorter And in what number of Daies it will be 97 Prob. 51. Of the difference of Civil and Natural Daies commonly called the Equation of Civil Daies And how it may be found by the Globe 99 Prob. 52. How to find the Hour of the Night when the Moon shines on a Sun Dyal by help of the Globe 101 Prob. 53. To find the Dominical Letter the Prime Epact Easter Day and the rest of the Moveable Feasts for ever 102 Prob. 54. The Age of the Moon given to find her place in the Ecliptick according to her mean motion 104 Prob. 55. Having the Longitude and Latitude or Right Ascension and Declination of any Planet or Comet to place it on the Globe to correspond with its place in Heaven 105 The Contents Of the Third Book Prob. 1. THe Suns Amplitude and difference of Ascension given to find the Height of the Pole and Declination of the Sun 108 Prob. 2. The Suns Declination and Amplitude given to find the Poles Elevation 108 Prob. 3. The Suns Declination and Hour at East given to find the Heigth of the Pole 109 Prob. 4. The Declination of the Sun and his Altitude at East given to find the Heigth of the Pole 110 Prob. 5. By the Suns Declination and Azimuth at 6 of the Clock given to find the Heigth of the Pole and Almicantar at 6. 11â— Prob. 6. By the Hour of the Night and a known Star Observed Rising or Setting to find the Heigth of the Pole fol. 112 Prob. 7. Two Places given in the same Latitude to find the Difference of Longitude 112 Prob. 8. Two Places given in the same Longitude to find the difference of Latitude 113 Prob. 9. Course and Distance between two Places given to find their
difference in Longitude and Latitude 113 Prob. 10. To find how many Miles are contained in a Degree of any Parallel 114 Prob. 11. The Rhumb you have sailed upon and the Latitudes you departed from and are arived to given to find the Difference of Longitude and the number of Leagues you have Sailed 114 Prob. 12. The Longitudes and Latitudes of two Places given to find Course and Great Circle distance between them 116 Prob. 13. The Latitude you departed from and the Latitude you are arrived to and the number of Lagues you have sailed given to find the Rhumb you have sailed on and difference of Longitude 116 Prob. 14. To find by the Globe the Variation of the Needle commonly called the Variation of the Compass 117 Prob. 15. To keep a Journal by the Globe 118 Prob. 16. To Steer in the Night by the Stars 119 Prob. 17. How to platt on the Globe a New Land never before Discovered 119 Prob. 18. Seeing two known points or Capes of Land as you sail along how to know the distance of your Ship from them 120 Prob. 19. Of Tides and how by help of the Globe you may in general judge of them 121 The Contents Of the Fourth Book Prob 1. To Erect a Figure of the 12 Houses of Heaven 123 Prob. 2. â—o Erect a Figure of Heaven according to Campanus 128 Prob. 3. To find the length of a Planetary Hour 129 Prob. 4. The length of a Planetary Hour known to find what Planet Reigneth any given Hour of the Day or Night fol. 131 Prob. 5. To find Part of Fortune by the Globe 132 Prob. 6. To find in what Circle of Position any Star or any degree of the Ecliptick is 133 Prob. 7. To find the Right Ascensions the Oblique Ascensions and the Declinations of the Planets 133 Prob. 8. How to Direct a Figure by the Globe 134 Prob. 9. Of Revolutions and how they are found by the Globe 135 Prob. 10. How a Figure of Heaven may be erected by the Revolution thus found 135 The Contents Of the Fifth Book Prob. 1. HOw by one position of the Globe to find the distance of the Hour lines on all manner of Planes fol. 143 Prob. 2. To make an Equinoctial Dyal 147 Prob. 3. To make an Horizontal Dyal 149 Prob. 4. To make an Erect Direct South Dyal 153 Prob. 5. To make an Erect Direct North Dyal 156 Prob. 6. To make an Erect Direct East Dyal 156 Prob. 7. To make an Erect Direct West Dyal 159 Prob. 8. To make a Polar Dyal 159 Prob. 9. To make Erect South Dyals Declining Eastwards or Westwards 160 Prob. 10. To make a North Erect Dyal declining Eastwards or Westwards 163 Prob. 11. To make Direct Reelining or Inclining Dyals 164 Prob. 12. To make Declining Reclining or Declining Inclining Dyals 164 Prob. 13. To make a Dyal upon a Declining Inclining Plane 168 Prob. 14. To find in what Place of the Earth any manner of Plane that in your Habitation is not Horizontal shall be Horizontal 171 Prob. 15. To make a Dyal on the â—eâ—ling of a Room where the Direct Beams of the Sun never come 175 Prob. 16. To make a Dyal upon a solid Ball or Globe that shall shew the Hour of the Day without a Gnomon 178 Prob. 17. To make a Dyal upon a Glass Globe whose Axis shall cast a shadow upon the Hour of the Day 180 The Contents Of the Sixth Book Of the Parts and Kindes of Spherical Triangles fol. 183 Prob. 1. The Legs of a Right Angled Spherical Triangle given to find the Hypothenusa and the two other Angles 184 Prob. 2. A Leg and the Hypothenusa given to find the Rest. 187 Prob. 3. The Hypothenusa and an Angle given to find the Rest. 187 Prob. 4. A Leg and Angle adjoyning given to find the Rest. 188 Prob. 5. A Leg and the Angle opposite given to find the Rest. 188 Prob. 6. The Angle given to find the Sides 189 Prob. 7. Oblique Triangles The three Sides given to find the Angles 190 Prob. 8. Two Sides and the Angle contained between them given to find the Rest. 193 Prob. 9. Two Sides and an Angle opposite to one of them given to find the Rest. 194 Prob. 10. Two Angles and the Side comprehended between them given to find the Rest. 194 Prob. 11. Two Angles and a Side opposite to one of them given to find the Rest. 195 Prob. 12. Three Angles given to find the Sides 196 Prob. 13. How to let fall a Perpendicular that shall divide any Oblique Spherical Triangle into two Right Angled Spherical Triangles 198 The Poetical Stories of the Constellations in Heaven 200 The First BOOK Being the first RUDIMENTS of Astronomy Geography Or A Description of the Lines Circles and other Parts of the GLOBE PRAEFACE THe Students of all Arts and Sciences have ever proposed a Maxime whereon as on an allowed Truth the whole Science hath dependance and by so much the more demonstrable that Maxime is so much the more of Excellency the Science may claim This of Astronomy and Geography comes not behind any for herein we shall only admit with the Ancients that the Form of the visible World is Spherical Neither shall we beg our Ascertion any farther then Occular Appearance will demonstrate every Mans Ey being his Judge if he be either on a Plain field or at Sea where nothing can hinder a free inspection of the Horizon Vpon good grounds therefore they ascerted the Spherical form of the Whole and also concluded the Parts to be Round I meane very intire Subsistence as the Stars Planets and the Earth In the Celestial Bodies as the Stars and Planets this is also visible and therefore un-controullable But that the Earth is Round proves with the unskilfull matter of dispute they frequently objecting with S. Austine the words of the Scripture which say He hath stretched forth the corners of the Earth not considering whether those words were spoken as alluding to the amplitude of Gods Omnipotence or that the Corners were meant Capes of Land which indeed are stretched forth into the Sea But that the Earth is Round is proved by divers certain and infallible Reasons As first By the Navigations of our Age Divers able and honest Mariners having Sailed and continued an Easterly Course have at length arrived without turning back to the same place from whence they set forth witness Magellanicus Sr. Francis Drake Tho. Cavendish Oliver vander Noort W. Schouten c. Secondly By the length of degrees in every Parallel for it is found by Dayly observation that the degrees of every Parallel upon the Earth hold the same proportion to the degrees of the Equinoctial as the degrees of the same Parallel upon an Artificial Globe or Sphear do to the degrees of the greatest Circle of the same This Argument alone is sufficient yet take one more from Visible Appearance And that is this The shadow which the Earth and Water together make in the Eclipse of
these figures and not into other being moved therto by these three reasons first these Figures express some properties of the stars that are in them as those of the Ram to bee hot and dry Andromeda chained betokeneth imprisonment the head of Medusa cut off signifieth the loss of that part Orion with his terrible and threatning gesture importeth tempest and terrible effects The Serpent the Scorpion and the Dragon signifie poyson The Bull insinuateth a melancholy passion The Bear inferreth cruelty c. Secondly the stars if not precisely yet after a sort do represent such a Figure and therefore that Figure was assigned them as for example the Crown both North and South the Scorpion and the Triangle represent the Figure which they have The third cause was the continuance of the memorie of some notable men who either in regard of their singular paines taken in Astronomy or in regard of some other notable deed had well deserved of man kind The first Author of every particular Constellation is uncertain yet are they of great antiquity we receive them from Ptolomte and he followed the Platonicks so that their antiquity is great Moreover we may perceive them to be ancient by the Sciptures and by the Poets In the 38 Chapter of Job there is mention made of the Pleiades Orion and Arcturus and Mazzaroth which some interpret the 12 Signes Job lived in the time of Abraham as Syderocrates maketh mention in his Book de Commensurandis locorum distantiis Now besides all this touching the reason of the invention of these Constellations the Poets had this purpose viz. to make men fall in love with Astronomy And to that intent have to every Costellation invented strange conceited stories as you may read at the latter end of this Book therein imitating Demosthenes who when he could not get the people of Athens to hear him in a matter of great moment and profitable for the Commom-wealth he began to tell them a tale of a fellow that sold an Ass by the which tale he so brought on the Athenians that they were both willing to hear his whole Oration and to put in practice that whereto he exhorted them The like intent had the Poets in of those Stories They saw that Astronomy being for commodity singular in the life of man was almost of all men utterly neglected Hereupon they began to set forth that Art under Fictions that thereby such as could not be perswaded by commodity might by the pleasure be induced to take a view of these matters and thereby at length fall in love with them For commonly you shall note this that he that is ready to read the Stories cannot content himself therewith but desireth also to know the Constellation or at leastwise some principal Star therein There are in Heaven yet twelve Constellations more posited about the South Pole which were added by Fredericâ— Hoâ—tmanno inhabiting on the Island Sumatra who being accommodated with the Instruments of that immortal Tycho hath observed the Longitude and Latitude of those Stars reduced them into Constellations and named them as follows 1 The Crane 2 The Phenix 3 The Indian 4 The Peacock 5 The Bird of Paradice 6 The Fly 7 The Camelion 8 The South Triangle 9 The Flying Fish 10 Dorado 11 The Indian Fowl 12 The Southern Serpent XI Of the Number of the Stars Although in Heaven there be a very great number of visible Stars which for their multitude seem innumerable yet no wise man will from thence infer that they are impossible to be counted for there is no Star in Heaven that may be seen but its Longitude and Latitude may with meet Instruments for that purpose be exactly found and being once found it may have a name allotted it which with its Longitude and Latitude may be Catalogized either for the memory of the Observer or the knowledge of Posterity Now therefore if any one Star may be observed they may all be observed and then may they all have Names given them which tho to the ignorant it seem uncredible yet to the sons of God as Josephus call Astronomers who herein participate of their fathers knowledge it is easie to number the Stars and call them all by their Names Psal. 97 4. But tho all the Stars in Heaven may be numbred and named yet have not the Ancient Astronomers thought fit to take notice of more then 1025 of the chiefest that are visible in our Horizon they being sufficient for any purpose that we shall have occasion to apply them unto Yet of late the industry of Frederick Houtman aforesaid hath added to the Catalogue 136 Stars with their Longitude Latitude and Magnitude and given Names unto them which upon my New Globes I have also ascerted as may be seen about the South Pole thereof So that with these 1025 observed by the Ancients and these 136 the whole number of the Catalogue is 1161. Some other Stars of late have been also observed by Baiâ—rus among the several Constellations aforesaid but none of any Considerable Magnitude and therefore I think fit to pass them by and come to their scituation in Heaven according to Longitude and Latitude XII Of the Scituation of the Stars The Stars are Scituate in Heaven according to their Longitude and Latitude As the Longitude of any Place upon the Terrestrial Globe is an Arch of the Equator Comprehended between the first Meridian and the Place So the Longitude of any Star upon the Celestial Globe is an Arch of the Ecliptick contained between the first point of ♈ and the Star inquired after But yet because the Ecliptick is divided into twelve Signes the Longitude of a Star is therefore in the most Customary account an Arch of the Ecliptick comprehended between the Semi-circle of Longitude passing through the beginning of the Signe the Star is in and the Semi-circle of Longitude passing through the Center of the Star The Latitude of a Star is either North or South North if on the North side of the Ecliptick South if on the South side of the Ecliptick As the Latitude of any Place upon the Terrestrial Globe is an Arch of the Meridian contained between the Equator and the Parallel of the Place So is the Latitude of any Star upon the Celestial Globe an Arch of a Semi-circle of Longitude comprehended between the Equinoctial and the Star inquired after XIII Of the Magnitudes of the Stars For the better distinction of the several sizes of Stars they are divided into six several Magnitudes The biggest and brightest Stars are called Stars of the first Magnitude Those one degree inferiour in light and bigness are called Stars of the Second Magnitude Those again one degree inferiour to the Stars of the second Magnitude are called Stars of the Third Magnitude and so the Stars gradually decrease unto the sixth Magnitude which is the smalest some few obscure Stars only excepted which for their Minority and dimness are called Nebula These several Magnitudes of
incline in motion to and other times to decline in motion from the Equinoctial But how long time it will be â—re the Star inclines to or declines from the Equinoctial you may know by finding the distance of Longitude in degrees it hath from either the Solsticial or Hyâ—mnal Colure and with respecting the foregoing Rules in its Position you may by the Table in Book 1 Chap. 3. Sect. 3. satifie your self Example The most Northerly Star in the Girdle of Orion doth yet decrease in Declination But I would know how long it shall decrease Therefore by the 32. Probleme I find the Longitude of that Star to be for the Year 1670. 77. deg 51. min. which subducted out of 90 the distance of the Solsticial Colure from the Equinoctial leaves 12. 9 for the distance of that Star from the Solsticial Colure Therefore by the Table aforesaid I find what number of Years answers to the motion of 12. deg 9. min. And because I cannot find exactly the same number of degrees and minutes in the Table I take the number neerest to it which is 14. degrees 10. minutes and that is the motion of the Ecliptick in 1000. Years But because this 14. degrees 10. minutes is 2. degrees 1. minute too much I seek 2. degrees 1. min. in the Table and the number of Years against it I would subduct from the number of Years against 14 deg 10. min. and the remainder would be the number of Years required But 2. deg 1. min. I cannot find neither therefore I must take the number of degrees and minutes neerest to it which is 2. deg 50. min. and that yeelds 200. Years which subducted out of 1000. leaves 800. Years But because this is also too much by the motion of 49. min. Therefore I seek for 49. min. in the Table and subduct the number of Years against it from 800 and the remainder would be the number of Years required But 49. min. is not in the Table neither Therefore I take the neerest to it which is 51. min. and that yeelds 60. Years which subducted out of 800. leaves 740. But this is likewise too much by the motion of two min. Therefore I seek 2. min. in the Table but cannot find it neerer then 2½ and against it I find 3. Years which 3. Years I subduct out of 740 and the Remainder is 737. the number in Years required You may if you please for exactness subduct for the ½ min. 8. Moneths so have you 736 Years 4. Moneths for the Time that the most Northerly Star in the Girdle of Orion will decrease in Declination after the Year 1670. which will be till An. Dom. â—406 after which time it will increase in Declination for 12706. Years together till it come to have 47. degrees 8. min. of Declination at which time it will be in or very neer the place of the most Southerly Star of the Southern Crown and that Star in its place And thus the Pole Star is now found to increase in Declination and will yet this 421 Years after which time it will decrease in Declination for 12706 Years together till it come to be within 42. degrees 42. minutes of the Equinoctial in the void space now between Draco and Lyra at which time Lyra will be almost as neer the Pole as the Pole Star now is and then the most proper to be the Northern Pole Star And the last Star in the Stalk of the Doves mouth will be then very neer the Southern Pole and therefore most fit to be the Southern Pole-Star PROB. XXVIII The Place of the Sun or any Star given to find the Right Descension and the Oblique Ascension and the Oblique Descension BRing the Place of the Sun or the Star to the Meridian under the Horizon and the degree of the Equator that comes to the Meridian with it is the Degree of Right Descension For the Oblique Ascension Bring the Place of the Sun or the Star to the East side the Horizon and the degree of the Equator cut by the Horizon is the Degree of Oblique Ascension of the Sun or Star For the Oblique Descension Bring the Place of the Sun or Star to the West side the Horizon and the degree of the Equinoctial cut by the Horizon is the Degree of Oblique Descension They need no Examples PROB. XXIX Any Place on the Terrestrial Globe being given to find its Antipodes BRing the given Place to the Meridian so may you as by the first Probleme see its Longitude and Latitude then turn about the Globe till 180. degrees of the Equator pass through the Meridian and keeping the Globe to this Position number on the Meridian 180. degrees from the Latitude of the given Place and the point just under that degree is the Antipodes Example I would find the Antipodes of Cuida Real an Inland Town of the West Indies which lies upon the River Parana an Arm of Rio de la Plata Therefore I bring Cuida Real to the Meridian and find as by the first Probleme its Latitude 23. 50 South and its Longitude 333. degrees Then I turn about the Globe till 180. degrees of the Equator pass through the Meridian and keeping the Globe to that position I number so many degrees North Latitude as Parana hath South viz. 23 50 and just under that degree I find Lamoo a Town lying upon the Coast of China in the Province of Quancij Therefore I say Lamoo is just the Antipodes of Cuida Real Another way Bring the given Place to the North or South point of the Horizon and the point of the Globe denoted by the opposite point of the Horizon is the Antipodes of the given Place PROB. XXX To find the Perecij of any given Place by the Terrestrial Globe BRing your Place to that side the Meridian which is in the South notch of the Horizon and follow the Parallel of that Place on the Globe till you come to that side the Meridian which is in the Northern notch of the Horizon and that is the Perecij of your Place PROB. XXXI To find the Antecij of any given Place upon the Terrestrial Globe BRing your Place to the Meridian and find its Latitude by the first Probleme If it have North Latitude count the same number of degrees on the Meridian from the Equator Southwards But if it have South Latitude count the same number of degrees from the Equator Northwards and the point of the Globe directly under that number of degrees is the Antecij of your Place PROB. XXXII To find the Longitude and Latitude of the Stars by the Coelestial Globe THe Quadrant of Altitude will reach but 90. degrees as was said Prob. 9. Therefore if the Star you enquire after be on the North side the Ecliptick you must elevate the North Pole 66½ degrees above the North side the Horizon If on the South side the Ecliptick you must elevate the South Nole 66½ degrees above the South side the Horizon Then bring the
what Elevation of the Pole the Daies shall be an Hour shorter By this Probleme may be found the Alteration of Climates for as was said in the Definition of Climates Book 1. fol. 28. Climates alter according to the half-hourly increasing of the Longest Day therefore the Latitude of 56½ degrees having its Daies increased an whole Hour is distant from the Latitude of London by the space of two Climates PROB. L. The Suns Place given to find what alteration of Declination be must have to make the Day an Hour longer or shorter And in what number of Daâ—es it will be REctifie the Globe to the Latitude of the Place and bâ—ing the Suns place to the East side the Horizon and note against what degree of the Horizon it is then bring one of the Colures to intersect the Horizon in that degree of the Horizon and at the point of Intersection make a prick in the Colure and observe what degree of the Equator is then at the Meridian Then turn the Globe Westward if the Daies shorten but Eastwards if they lengthen till 7½ degrees of the Equator pass through the Meridian and where the Horizon intersects the same Colure make another prick in the Colure Afterwards bring the Colure to the Meridian and count the number of degrees between the two pricks for so many degrees must the Suns Declination alter to lengthen or shorten the Day an Hour Example The Suns Place is ♉ 10. I would know how much he must alter his Declination before the Day is an Hour longer here at London Therefore I rectifie the Globe to the Latitude of London and bring ♉ 10. to the East side the Horizon and find it against 24½ degrees from the East point therefore I bring one of the Colures to this 24½ degrees and close by the edge of the Horizon I make a prick with black lead in the Colure then keeping the Globe in this position I look what degree of the Equator is then at the Meridian and find 250¼ and because the Daies lengthen I turn the Globe Eastwards till 7½ degrees from the foresaid 250¼ pass through the Meridian then keeping the Globe in this position I make another prick in the Colure and bringing this Colure to the Meridian I find a little more then 5 degrees of the Meridian contained between the two pricks therefore I say when the Sun is in ♉ 10. degrees he must alter his Declination a little more then 5 degrees to make the Day an Hour longer Now to know in what number of Daies he shall alter this Declination you must find the Declination of the two pricks on the Colure as you found the Suns Declination by Prob. 5. and the Arch of the Ecliptick that passes through the Meridian while the Globe is turned from the first pricks Declination to the second pricks Declination is the number of Ecliptical degrees that the Sun is to pass while he alters this Declination and the degree of the Ecliptick then at the Meridian is with respect had to the Quarter of the Year the place the Sun shall have when its Declination shall be altered so much as to make the Day an Hour longer Thushaving the Suns first place given and its second place found you may by finding those two places on the Plain of the Horizon also find the number of Daies comprehended between them as you are taught by the fourth Probleme This Probleme thus wrought for different Times of the Year will shew the falacy of that Vulgar Rule which makes the Day to be lengthned or shortned an Hour in every Fifteen Daies when as the lengthning or shortning of Daies keeps no such equality of proportion for when the Sun is neer the Equinoctial points the Daies lengthen or shorten very fast but when he is neer the Tropical points very slowly PROB. LI. Of the Difference of Civil and Natural Daies commonly called the Equation of Civil Daies And how it may be found by the Globe THe Civil Day is that space of Time containing just 24. Hours reckoned from 12 a clock on one Day to 12 a clock the next Day in which space of Time the Equinoctial makes upon the Poles of the World a Diurnal Revolution The Natural Day is that space of Time wherein the Sun moveth from the Meridian of any Place to the same Meridian again These Daies are at one time of the Year longer then at another and at all Times longer then the Civil Daies There is but smal discrepancy between them yet some there is made by a two-fold Cause For first The Suns Apparent motion is different from his true motion He being much slower in his Apogeum then he is in his Perigeum For when the Sun is in his Apogeum he scarce moves 58 minutes from West to East in a Civil Day but when he is in his Perigeum he moves above 61 minutes in a Civil Day and therefore increases his Right Ascension more in equal Time The second Cause is the difference of Right Ascensions answerable to equal parts of the Ecliptick for about ♋ and ♑ the differences of Right Ascensions are far greater then about ♈ and ♎ for about ♈ and ♎ the Right Ascension of 10. degrees is but 9. degrees 11. minutes but about ♋ and ♑ the Right Ascension of 10 degrees will be found to be 10. degrees 53. minutes as by the Globe will appear But because of the smalness of the Globes graduation you cannot actually distinguish to parts neer enough for the solution of this Probleme if you should enquire the difference in length of two single Daies it will be requisite to take some number of Daies together Suppose 20. Therefore find by Prob. 3. the Places of the Sun for the beginning and ending of those Daies you would compare and find the Right Ascensions answerable to each place in the Ecliptick and also the differences of Right Ascensions answerable to the Suns motion in each number of Daies Then compare the differences of Right Ascensions together and by substracting the lesser from the greater you will have the number of degrees and minutes of the Equator that have passed through the Meridian more in one number of Daies then in the other number of Daies which degrees of the Equator converted into Time is the number of minutes that the one number of Daies is longer then the other number of Daies Example I would know what difference of Time there is in the length of the first 20. Daies of December and the first 20 Daies of March I find by Prob. 3. the Suns place December 1 is 〈◊〉 19. 45. at the end of 20 Daies viz. on the 21 Day his place is 〈◊〉 10. 11. The Suns place March 1. is ♓ 21. 16. at the 20. Daies end viz. March 21 his place is ♈ 11. 3. I find by Prob. 26. the Right Ascension answerable to â™ 19. 45 is 258. 10. ♑ 10. 11 280. 25. ♓ 21. 16 352. 00. ♈ 11. 3 9. 40. and
the difference of Right Ascensions contained between the first Day in each Moneth and the 21 of the same Moneth by substracting the lesser from the greater is for 258. 10. And for 352. 00. 280. 25. 9. 40. 22. 15 17. 40. But note because the Vernal Colure where the degrees of Right Ascension begin and end their account is intercepted is the Arch of the Suns motion from the first to the 21. of March therefore instead of substracting the lesser number of degrees of Right Ascension from the greater viz. 9. 40 from 35. 2. I do for finding the difference of the Right Ascensional arch of the Suns motion in those 20 Daies sustract the foresaid 352 degrees from 360 and the remains is 8. which is the difference of Right Ascension from ♓ 21 16. to the Equinoctial Colure to which 8 adding 9 degrees 40 minutes the Right Ascension from the Equinoctial Colure to ♈ 11. 3. it makes 17 degrees 40. minutes for the difference of Right Ascensions between ♓ 21 16. and ♈ 11. 3 Then I find the difference of this Difference of Right Ascension by substracting the less from the greater viz. 17. 40. from 22. 15. and the remains is 4. degrees 35. minutes for the number of degrees and minutes of the Equator that pass through the Meridian in the first 20 Daies in the Moneth of December more then in the first 20 Daies of the Moneth of March which 4. degrees 35. minutes converted into Time gives 19. minutes that is a quarter of an Hour and 4 minutes that the first 20 Daies of December aforesaid are longer then the first 20 Daies of March. PROB. LII How to find the Hour of the Night when the Moon shines on a Sun Dyal by help of the Globe REctifie the Globe and find by Prob. 54. or an Ephemeris the Moons place at Noon Bring it to the Meridian and the Index of the Hour Circle to 12. and turn about the Globe till the Index of the Hour Circle points to the same Hour the shade of the Moon falls on on the Sun Dyal Then by Prob. 3. find the Suns place at Noon and see how many degrees of Right Ascension are contained between the Suns place and the degree of the Equator at the Meridian when the Index of the Hour Circle is brought to the Hour the Moon shines on in the Sun Dyal for those number of degrees converted into Time shall be the Time from Noon or the Hour of the Night Only note Respect must be had to the motion of the Moon from West to East for so swift is her mean motion that it is accounted to be above 12. degrees in 24. Hours that is 6 degrees in 12 Hours 3 degrees in 6 Hours c. and this also converted into Time as aforesaid you must add proportionably to the Time found from Noon and the sum shall give you the true Hour of the Night Example Here at London I desired to know the Hour of the Night January 6. this present Year 1658. The Moons place found by an Ephemeris or for want of an Ephemeris by Prob. 54. is in ♊ 21. degree 22 minutes Therefore I rectified the Globe to Londons Latitude and brought ♊ 21. 22. minutes to the Meridian and the Index of the Hour Circle to 12. then by Prob. 3. I found the Suns place in ♑ 26. degrees 46. minutes and by Prob. 26. I found his Right Ascension to be 300 degrees Then I turned about the Globe till the Index of the Hour Circle pointed at 10 Hours and at the degree of the Equator at the Meridian I made a prick then I counted the number of degrees of the Equater contained between the foresaid 300 deg and this prick and found them 111¼ degrees which converted into Time by allowing 15 degrees for an Hour gives 7 hours 25 minutes Time from Noon which if the Moons motion were not to be considered should be the immediate Hour of the Night But by the Rule aforesaid the Moons motion from West to East in 7 hours 25 minutes is 3 degrees 42 minutes and this 3 degrees 42 minutes being converted into Time is 14 minutes more which being added to 7 hours 25 minutes make 7 hours 39 minutes for the true Hour of the Night PROB. LIII To find the Dominical Letter the Prime Epact Easter Day and the rest of the Moveable Feasts for ever THough these Problemes cannot be performed by the Globe because of the several changes and irregular accounts that their Rules are framed upon yet because they are of frequent and Vulgar use and for that the solution of many other Questions will have dependency on the knowledge these Therefore I have thought fit here to inserte this Table of M r Palmers by which you may find them All. I shall not insist upon the Reasons of the several changes of Letters and Numbers Himself having already very learnedly handled that subject in his Book of the Catholick Planisphear Book 1. Chapter 11. to which I refer you Neither shall I need to give you any other Instructions for finding what is here proposed then what himself hath given in his fourth Book Chapter 66 and part of 67. Therefore take it as he there delivers it An Example shall serve here instead of a Rule For the Year 1657. I would know all these wherefore I seek the Year 1657. in the Table of the Suns Cycle and over against it I find 14. for the Year of the Cycle of the Sun and D for the Dominical Letter And note here that every Leap-year hath 2 Dominical Letters as 1660 hath A G and the first viz. A serveth that Year till February 25 and the second G for the rest of the Year And note that these Letters go alwayes backwards when you count forwards as B A then G F c. not F G and then A B as you may see by the Table To find the Age of the Moon Remember first that the Epact begins with March which must be here accounted the first Moneth Then if you add to the Epact the number of the Moneth current and the number of the day of the Moneth current the sum or the excess above 30 is the Moons age Example January 20. 1656. According to the accompt of the Church of England who begin the Year with March 25. which was the Equinoctial day about Christ time the Epact is 14. January is the 11 th Moneth and the 20 th day is proposed now add 14. 11. and 20. together they make 45. out of which I take 30. and there remains 15 the Moons age PROB. LIV. The Age of the Moon given to find her place in the Ecliptick according to her mean motion THis Probleme may be performed exact enough for Common uses by the Globe but in regard it only shews the Moons place in the Ecliptick according to her meat motion it will often fail you some few degrees of her true Place The work is thus First set figures
to every twelth degree of the Equinoctial accounted from the Equinoctial Colure marking them with 1 2 3 4. c. to 30 which will end where you began viz. at the Equinoctial Colure again so shall the Equinoctial be divided into 30 equal parts representing the 30 Dayes of the Moons Age These figures to distinguish them from the degrees of the Equator were best be writ with Red Ink. When you would enquire the Moons Place Elevate the North Pole 90 degrees that is in the Zenith so shall the Equator ly in the Horizon Then bring the Equinoctial Colure against the Day of the Moneth in the Horizon so shall the Moons Age written in Red figures stand against the Signe and degree in the Horizon that the Moon is in at that Time Example September 28. 1658. I would know the Moons place in the Ecliptick she being then 12 Daies old Therefore I Elevate the North Pole 90 degrees above the Horizon and turn the Globe about till the Equinoctial Colure come to September 28. in the Circle of Daies on the Horizon then looking against what Signe and degree of the Ecliptick Circle in the Horizon the 12 th division in Red figures stands I find ♓ 9. which is the Signe and degree the Moon is in according to her mean Motion This Probleme may be applyed to many Uses for having the Moons Place you may find the Time of her Rising Southing Setting and Shining c. by working with her as you were taught to work with the Sun in several fore-going Problemes proper to each purpose PROB. LV. Having the Longitude and Latitude or Right Ascension and Declination of any Planet or Comet to place it on the Globe to correspond with its place in Heaven PLanets and Comets cannot be placed on the Globe so as their places will long retain correspondence with their places in Heaven Because as was said Chap. 44. they have a continual motion from West to East upon the Poles of the Ecliptick yet never-the-less you may by having their Longitude and Latitude or Right Ascension and Declination for any set Time place a Mark for them on the Globe either with Ink if your Globe be Varnisht for then you may with a wet finger wipe it off again or with Black-lead if it be not Varnisht and then you may rub it out again with a little White Bread which Mark for that Time will as effectually serve you to work by as any of the Fixed Stars placed on the Globe will do Therefore if the Longitude and Latitude of any Planet or Comet be given Do thus Elevate the North Pole if the Latitude given be North but if the Latitude given be South Elevate the South Pole 66 ½ degrees and place the Pole of the Ecliptick in the Zenith and over it screw the Quadrant of Altitude so shall the Ecliptickly in the Horizon and the Quadrant of Altitude being turned about the Horizon shall pass through all the Degrees of Longitude Then find the point of given Longitude in the Ecliptick and bring it to the Quadrant of Altitude and hold it there Then count upwards on the Quadrant of Altitude the number of degrees and minutes of given Latitude and at the point where the number ends close to the Quadrant of Altitude make a smal Prick and that Prick shall represent the Planet or Comet you were to place on the Globe If it be the Right Ascension and Declination of a Planet or Comet that is given you must find the degree and minute of Right Ascension on the Equinoctial and bring it to the Meridian and keep the Globe there steddy then find the degree and minute of Declination on the Meridian and under that degree and minute on the Globe make a Prick and that Prick shall represent the Planet or Comet as aforesaid If it be ♄ or ♃ that this Prick is to represent it may stand on the Globe sometimes a Week or a Fortnight without much difference from the Planets place in Heaven But if the Prick were to represent the other Planets you must in regard of their swift motion alter it very often especially for the Moon for so swift is her motion that in every two Hours she alters about a degree in Longitude Having thus placed this Mark on the Globe you may find out the Time of its several Positions and Aspects if you work by it as you are directed to work by the Sun in the several respective Problemes throughout this Book The End of the Second Book The Third BOOK Being the Practical Use of the GLOBES Applyed to the Solution of Problemes In the Art of NAVIGATION PRAEFACE BEcause the Art of Navigation consists aswell in the knowledge of Astronomical and Geographical Problemes as in Problemes meerly Nautical Therefore I must desire the Artist to seek in the last Book such Problemes as are only Astronomical or Geographical For my Designe is here to collect such Problemes as are only used in the Art of Navigation some few particulars excepted as for finding Latitude Longitude Course Distance c. Which though they are handled in than Book yet for their exceeding Vtility in the Art of Navigation and for that what there is given cannot alwayes be had to work by therefore in this Book I have mentioned divers other Observations which being made or had you may by the Rules proper for each Observation find what shall be proposed PROB. I. The Suns Amplitude and Difference of Ascension given to find the Heigth of the Pole and Declination of the Sun ELevate the Pole so many degrees as the Difference of the Suns Ascension is and screw the Quadrant of Altitude to the Zenith and bring the first point of ♈ to the Meridian then number on the Quadrant of Altitude upwards the complement to 90. of the Suns Amplitude and move the Quadrant of Altitude till that number of degrees cuts the Equator So shall the Quadrant cut in the Horizon the degree of the Pole Elevation and in the Equator the degree of the Suns Declination Example The difference of Ascension is 27. degrees 7. minutes Therefore I Elevate the Pole 27. degrees 7. minutes above the Horizon and screw the Quadrant of Altitude to 27. degrees 7. minutes which is in the Zenith then I bring the first point of ♈ to the Meridian and number on the Quadrant of Altitude upwards 56. degrees 40. minutes the Complement of the Suns Amplitude and bring that degree to the Equator then I see in what degree of the Horizon the Quadrant cuts the Horizon and find 51 ½ which is the Elevation of the Pole then looking in what degree of the Equator the Quadrant of Altitude cuts the Equator I find 20 degrees 5 min. which is the Declination of the Sun at the same Time PROB. II. The Suns Declination and Amplitude given to find the Poles Elevation ELevate the Pole so many degrees as the Complement of the Suns Amplitude is and screw the Quadrant of Altitude
the Meridian cut the degree of the Ecliptick on the Cusp of the tenth House The Western Semi-Circle of the Horizon shall cut the degree of the Ecliptick on the Cusp of the Seventh House and the Semi-Circle of the Meridian under the Horizon shall cut the degree of the Ecliptick on the Cusp of the fourth House If you have the day of the Moneth you may by Prob. 3. of the second Book find the Suns Place and if you have the Hour of the Day you may by first rectifying the Globe as by Prob. 2. of the same Book turn about the Globe till the Index of the Hour-Circle point to the same Hour in the Hour-Circle and you will then at the Eastern Semi-Circle of the Horizon have the degree of the Ecliptick that is Rising and by Consequence as aforesaid all the Cardinal points in their respective places Now to find what degree of the Ecliptick occupies the Cusps of the other eight Houses of Heaven Do thus The Globe rectified as aforesaid Move the Semi-Circle of Position upwards till 30 degrees of the Equator shall be contained between it and the Eastern Semi-Circle of the Horizon so shall the Semi-Circle of Position cut in the Ecliptick the degree and minute of the Ecliptick on the Cusp of the twelfth House and its opposite degree and minute in the Ecliptick shall be the Cusp of the sixth House for you must note that if you have but the degree and minute of the Ecliptick upon the Cusps of six of the Houses the opposite degrees and minutes of the Ecliptick shall immediately possess the Cusp of every opposite House Then move the Circle of Position over 30. degrees more of the Equinoctial so shall the degree of the Ecliptick cut by the Circle of Position be the degree of the Ecliptick upon the Cusp of the eleventh House and its opposite degree in the Ecliptick shall be upon the Cusp of the fifth House The degree of the Ecliptick upon the Cusp of the tenth and fourth Houses was found as before Then remove the Circle of Position to the Western side of the Meridian and let it fall towards the Horizon till 30. degrees of the Equator are contained between the Meridian and it so shall the degree of the Ecliptick cut by the Semi-Circle of Position be the degree of the Ecliptick on the Cuâ—p of the Ninth House and the opposite degree of the Ecliptick shall be upon the Cusp of the third House Let the Semi-Circle of Position fall yet lower till it pass over 30. degrees more of the Equator so shall the degree of the Ecliptick cut by the Semi-Circle of Position be the degree of the Ecliptick on the Cusp of the eighth House and the opposite degree of the Ecliptick shall be upon the Cusp of the second House The degrees of the Ecliptick on the Cusp of the seventh House and Ascendent were found as before Example I would erect a Figure of Heaven for July 27. 5. hours oâ— minutes Afternoon 1658. in the Latitude of London viz. 51½ degrees North Latitude I first place the Planets ☊ and ☋ on the Globe as by Prob. 55. of the Second Book was directed yet not exactly as I find them in the Ephemeris for that shews only their place in the Ecliptick at Noon Therefore I consider how many degrees or minutes each Planets motion is in a whole Day or 24. Hours by substracting the Ecliptical degrees and minutes of the Planets place that Day at Noon from the Ecliptical degrees and minutes of the Planets place the next Day at Noon or contrarily if the Planet be Retrograde for the remains of those degrees and minutes is the motion of the Planet that Day Therefore proportionably to that motion I place the Planet forward in the Ecliptick or backward if it be Retrograde As if the Sun should move forward 1 degree that is 60 minutes in a whole Day or 24 Hours then in 12 hours he should move 30 minutes in 6 hours 15 minutes in 4 hours 10 minutes in 1 hour 2½ minutes and so proportionably for any other space of Time which I consider before I place the Planets on the Globe PROB. II. To Erect a Figure of Heaven according to Campanus REgiomontanus as aforesaid makes the beginning of every House to be the Semi Circle drawn by the side of the Semi Circle of Position according to the succession of every 30 th degree of the Equator from the Horizon But Camp 〈◊〉 make it to be the Semi-Circle drawn by the side of the Semi-Circle 〈◊〉 Position according to the succession of every 30 th degree of 〈◊〉 Prime Verticle or East Azimuth which is represented by the Quadrant of Altitude placed at the East point The four Cardinals are the same both according to Regiomontanus and Campanus but the other eight Houses differ Therefore when you would find them according to Campanus Rectifie the Globe and Quadrant of Altitude and bring the lower end 〈◊〉 the Quadrant of Altitude to the East point in the Horizon Then count from the Horizon upwards 30 degrees oâ— the Quadrant 〈◊〉 Altitude and bringing the Circle of Position to those 30 degreeâ— examine where the Circle of Position cuts the Ecliptick which 〈◊〉 the aforesaid time is in 〈◊〉 29. 40 for that degree and minute upon the Cusp of the twelfth House and its opposite degree 〈◊〉 minute in the Ecliptick viz. ♉ 29. 40. is upon the Cusp of 〈◊〉 sixth House Lift up the Circle of Position 30 degrees highâ— upon the Quadrant of Altitude viz. to 60 degrees and 〈◊〉 Circle of Position will cut the Ecliptick in 〈◊〉 15. degrees for the Cusp of the eleventh House and its opposite degree and minute in the Ecliptick viz. ♉ 15. is upon the Cusp of the first House The degree and minute of the Ecliptick on the Cusp 〈◊〉 the Tenth and Fourth Houses is at the Meridian Then transfering the Circle of Position to the West side of the Meridian and the Quadrant of Altitude to the West point in the Horizon Let the Semi-Circle of Position fall 30 degrees from the Meridian on the Quadrant of Altitude and it will cut in the Ecliptick ♎ 16 degrees for the Cusp of the ninth House and its opposite degree and minute in the Ecliptick viz. ♈ 16. is upon the Cusp of the third House Let fall the Circle of Position 30 degrees lower on the Quadrant of Altitude and it will cut the Ecliptick in 〈◊〉 2 degrees for the Cusp of the eight House and its opposite degree viz. ♓ 2. degrees is on the Cusp of the second House The Cusps of the Seventh and Ascendent is the same with Regiomontanus viz. 〈◊〉 27. 47 and â™ 27. 47. The Figure follows PROB. III. To find the length of a Planetary Hour AStrologers divide the Artificial day be it long or short into 12 equal parts and the Night into 12 equal parts These parts they call Planetary Hours The first of these Planetary Hours takes its
placed at the East or West point of the Horizon Therefore when you would find what Circle of Position any Star or degree of the Ecliptick is in Rectifie the Globe and Quadrant of Altitude and bring the lower end of the Quadrant of Altitude to the East or West point of the Horizon and lift up the Circle of Position till it come to the Star or degree of the Ecliptick proposed and the number of degrees the Circle of Position then cuts in the Quadrant of Altitude is the number of the Circle of Position that the Star or degree of the Ecliptick is in If the Star or degree of the Ecliptick be under the Horizon turn the Globe about till 180 degrees of the Equator pass through the Meridian then will the Star or degree of the Ecliptick be above the Horizon Lift up then the Circle of Position as before to the Star or degree of the Ecliptick and the number of degrees of the Quadrant of Altitude the Circle of Position cuts on the East side is the number of Circles of Position the Star was under the Horizon on the West side Or so many degrees as the Circle of Position cuts on the Quadrant of Altitude in the West side the Horizon is the number of the Circles of Position the Star or degree of the Ecliptick was under the Horizon on the East side PROB. VII To find the Right Ascensions the Oblique Ascensions and the Declinations of the Planets EXamine the Right Ascensions and Declinations of those pricks made to represent each Planet in Prob. 1. of this Book and work by them as you were directed to work by the Sun in Prob. 26 27 28. of the second Book PROB. VIII How to Direct a Figure by the Globe TO Direct a Figure is to examine how many degrees of the Equinoctial are moved Eastwards or Westwards while any Planet or Star in one House comes to the Cusp or any other point of any other House When you would Direct any Promittor to any Hylegiacal point examine the degree of the Equator at the Meridian then turn about the Globe till the Promittor come to the Hylegiacal point and examine again the degree of the Equator at the Meridian and by substracting the lesser from the greater you will have the number of Degrees that passed through the Meridian whiles the place of the Promittor was brought to the Hyâ—â—gâ—â—cal point and that number of degrees shall be the Arch of Dâ—rection Example I would Direct the Body of the Moon in our Figure aforesaid to Medium Câ—â—â— or the tenth House I find by the Globe 20â— degrâ—es 30. minutes of the Equator at the Meridian with the â—eath House and turning the Globe till the prick made to represent the Moon come to the Meridian I find 227 degrees 20 minutes of the Equator come to the Meridian with it Therefore I 〈◊〉 the lesser from the greater viz. 2â—3 degrees 3 minutes from 227. degrees 2â— minutes and have remaining 2â— degrees 50 minutes This 〈◊〉 degrees 50. minutes shews that 23. Years 1â— Moneths must expire ere the Effects promised by the Moons present position shall opperate upon the signification of the 〈◊〉 House If the Body of the Moon had been Directed to any other point the◠〈◊〉 Meridian or Horizon you must have Elevated the Circle of 〈◊〉 〈◊〉 the point proposed and have under-propped it to that 〈◊〉 and 〈◊〉 have turned about the Globe till the prick 〈◊〉 the Moon had come to the Circle of Position and then 〈◊〉 degrees of the Equator that should have passed through the Meridian whiles this motion was making should be the number of degrees of Direction and signifie in Time as foresaid PROB. IX Of Revolutions and how they are found by the Globe BY Revolution is meant the Annual Conversion of the Sun to the same place he was in at the Radix of any Business When you would find a Revolution by the Globe first find the Right Ascension of Mâ—d Heaven at the â—â—adix of the Business as by Prob 26. of the second Book you were directed to find the Right Ascerâ—â—on of the 〈◊〉 and 〈◊〉 add 87 degrees for every Yâ—aâ— since the Radix Then substract 360 so oâ—â— as you can from the whole and the Râ—mâ—â—â—s shall be the Right Ascension oâ— Mid Hâ—aven for the Aâ—â—â—al Revoluâ—â—on Iâ— yâ—u 〈◊〉 the number of degrees of the Equator contained between the Râ—ght Aâ—cension of the Mid Hâ—aven and the Right Ascension of the Sun and convert that number of degrees 〈◊〉 Time by allowing for every 15. degrees 1 Hour of Time it will shew if the Suns place be on the Western side of the Meridian the number of Hours and minutes Afternoon the Revolution shall hâ—ppen on but if on the East side the Meridian the number of Hours and minutes Before-noon the Revolution shall happen on PROB. X. How a Figure of Heaven may be erected by the Revolution thus found SEek the degree of Right Ascension of Mid Heaven and bring it to the Meridian so shall the four Cardinal points of the Globe be the same with the four Cardinal points in Heaven at the time of the Revolution The other Hâ—â—â—ses are 〈◊〉 by the Circle of Position as in the first Probleme of this Book The End of the Fourth Book The Fifth BOOK Shewing the Practical Use of the GLOBES Applying them tâ— the Solution of Gnomonical Problems PRAEFACE DYals are of two sorts Pendent and Fixed Pendeâ— are such as are hung by the hand and turned towards the Sun that by its Beams darting througâ— smal Pin-holes made for that purpose the hour of the Daâ— may be found These are of two sorts Vniversal and Pâ—â—ticular Vniversal Dyals are those commonly called Equiâ—ocâ—â—â— or Ring-Dyals They are used by Sea-men and Trâ—vellers that often shift Latitudes Particular are such as are made and only serve for Particular Latitudes Of these sorts are the several Dyaâ—â— discribed on Quadrants Cilinders c. Fixed Dyaâ—s shall be the matter of this discourse and they are such as are made upon fixed Planes and shew the Hour of the Day by a Stile or Gnomon made Parallel to the Axiâ— of the World Of the several Kinds of Dyal Plains and how you may know them A Plain in Dyalling is that flat whereon a Dyal is discribed There is some disagreement among Older and Later Authors in the naming of Plains for some name them according to the Great Circle in Heaven they ly in and others according to the scituation of the Poles of the Plains Thus they which name them according to the Great Circle in Heaven their Plains ly in call that an Horizontal Plain which others call a Vertical Plain those Vertical which others will call Horizontal and those Polar which others call Equinoctial However they be called it matters not so you can but distinguish their kinds which with a little consideration you may easily learn to do For remembring but upon what grounds either the
that shadow shall be a Meridian liâ—e Secondly on the backside the Clinatory discribe a Circle and draw a line through the Center to both sides the Circumference cross this line with an other line at Râ—ght Angles in the Center so shall the Circle be divided into four equal parts These four parts you must maâ—k with East West North South and divide each of them into 90. degrees In the Center of this Plain erect a straight wyer prependicularly when you would find a Meridian line examine by the tenth Prob. of the second Book the Amplitude of the Suns Rising or Setting from the East or West points and waiting the just Rising or Setting that Day turn the Instrument about till the shadow of the wyer falls upon the same degree from the East or West the Amplitude is of for then the North and South line in the Instrument will be the same with the North and South line in Heaven Thirdly by the Suns Azimuth Find the Azimuth of the Sun by Prob. 22. of the second Book and at the same instant turn the Instrument till the shadow of the wyer fall upon the degree on the Instrument opposite to the degree of the Suns Azimuth so shall the Meridional line of the Instrument agree with the Meridional line in Heaven You may the same way work by the Azimuth of any Star Only whereas the shadow of the wyer should fall upon the opposite degree aforesaid Now you must place a Sight or Perpendicular upon that opposite degree and turn the Instrument about till the wyer at the Center the Sight in the opposite degree of the Stars Azimuth and the Star in Heaven come into one straight line so shall the Meridian line of the Instrument agree with the Meridional line in Heaven Fourthly It may be found by any Star observed in the Meridian if two Perpendiculars be erected in the Meridian line of your Instrument for then by turning the Instrument till the two Perpendiculars and the Star come into a straight line the Meridian line of your Instrument will be the same with the Meridian line in Heaven See more waies in Mr. Palmer on the Planisphear Book 4. Chap. 9 If your Plain either Recline or Incline apply one of the sides of your Clinatory Parallel to one of the Semi-diameters of the Quadrant to the Plain in such sort that the Plumb-line hanging at liberty may fall upon the Circumference of the Quadrant for then the number of degrees of the Quadrant comprehended between the side of the Quadrant Parallel to the Plain and the Plumb-line shall be the number of degrees of Reclination if thâ— Center of the Quadrant points upwards or Inclination if thâ— Center points downwards If your Reclining or Inclining Plain Decline draw upon it a line Parallel to the Horizon which you may do by applying the back-side of the Clinatory and raising or depressing the Center of the Quadrant till the Plumb-line hang just upon one of the Semi-diameters for then you may by the upper side of the Clinatory draw an Horizontal line if the Plain Incline or by the under side if it Recline If it neither Incline or Recline you may drawâ— an Horizontal line both by the upper and under sides of the Clinatory Having drawn the Horizontal line apply the North 〈◊〉 â— of the Clinatory to it and if the North end of the Needle 〈◊〉 directly towards the Plain it is then a South Plain If the 〈◊〉 point of the Needle points directly from the Plain it is a Norâ—â— plain but if it points towards the East it is an East Plain if towards the West a West Plain If it do not point directly 〈◊〉 East West North or South then so many degrees as the 〈◊〉 declines from any of these four points to any of the other of 〈◊〉 four points so many degrees is the Declination of the Plain 〈◊〉 respect as aforesaid had to the Variation of the Compass Or if you find the Azimuth of the Sun by its Altitude observed just when its beams are coming on or going off youâ— Plain that Azimuth shall be the Azimuth of your Plain Or you may erect a wyer Perpendicularly on your Plain and wait till the shadow of that wyer comes to be Perpendicular with the Horizon which you may examine by applying a Plumb-line to it for then the shadow of the Plumb-line and the shadow of the Perpendicular will be in one then taking the Altitude of the Sun you may by Prob. 22. of the second Book find its Azimuth and thereby know in what Azimuth the Plain of your Dyal lies for the Azimuth your Plain lies in is distant from the Azimuth of the Sun just 90. degrees PROB. I. How by one position of the Globe to find the distances of the Hour-lines on all manner of Plains YOu may have Meridian lines drawn from Pole to Pole through every 15. degrees of the Equinoctial to represent the Horary motion of the Sun both Day and Night and when the Pole of the Globe is Elevated to the height of the Pole in any Place and one of these Meridian lines be brought to the Brazen Meridian all the rest of the Meridian lines shall cut any Circle which you intend shall represent the Plain of a Dyal in the number of degrees on the same Circle that each respective Hour-line is distant from the Noon-line point in the same Circle Thus if you should enquire the distance of the Hour-lines upon an Horizontal Plain in Londons Latitude The Pole of the Globe as aforesaid must be Elevated 51½ degrees and one of the Meridian lines you may chuse the Vernal Colure be brought to the Brazen Meridian which being done you are only to examine in the Horizon Because it is an Horizontal Plain at what distance from the Meridian which in Horizontals is the Noon-line the several Meridians drawn on the Globe intersect the Horizon for that distance in degrees shall be the distance on a Circle divided into 360. degrees that each respective Hour-line must have from the Meridian or a Noon line chosen in the same Circle and lines drawn from the Center of that Circle through those degrees shall be the Hour lines of an Horizontal Plain If your Plain be not Direct but declines East or West 〈◊〉 must number the Declination Eastwards or Westwards reâ—pectively in the degrees of the Horizon and the Quadrant 〈◊〉 Altitude screwed to the Zenith as aforesaid bring the lower end of the Quadrant of Altitude to the said degrees of Declination and the number of degrees cut by the Meridians in the Quadrant of Altitude numbred downwards is the number of degrees that the Hour-lines are distant from the Noon line in a Circle of 360 degrees And lines drawn from the Center of that Circle through those degrees be the Hour lines of half the Day And if you turn about the Quadrant of Altitude upon the Zenith point till the lower end of it come to the degree of the Horizon
be numbred so that what so ever decimal degree of the Equator you light on at the Meridian or else where you will find its number from that Colure already set down to your hand without either adding to or substracting from it Bring this Colure therefore to the Meridian and the Index of the Hour Circle to 12. in the Hour Circle Then turn the Globe Westwards and so oft as 15 degrees of the Equator passes through the Meridian so oft you must examine what degrees of the Horizon the Vernal Colure cuts and those degrees and minutes so cut by the Vernal Colure must be found in the Circle C B D E beginning your account or reckoning at B towards D and markt with Pricks through which Pricks you must draw lines from the Center A and those lines shall be the Hour lines after noon Then bring the Colure to the Meridian again to find the Fore-noon Hour-lines and turn the Globe Eastwards and so oft as 15 degrees of the Equator passes through the Meridian so oft you must examine what degrees of the Horizon the Vernal Colure cuts and those degrees and minutes so cut by the Vernal Colure must be found in the Circle C B D E begining your reckoning from B towards C and markt with Pricks through which Pricks you must draw lines from the Center A and those lines shall be the Fore-noon Hour-lines These Hour-lines must be markt from the Meridian line viz. the line A B which is the 12 a clock line towards D with I II III c. till you have numbred to the Hour of Sun set found by Prob. 7. of the second Book the longest Day and from the Meridian line towards C with XI X. IX c. till you have numbred to Sun Rising the longest Day The Stile must be placed in the Center and Elevated so many degrees above the Plane as the Pole is elevated above the Horizon of the Place Example of the whole I would make an Horizontal Dyal for Londons Latitude Therefore I E evate the North Pole 51½ degrees above the Horizon and bring the Vernal Colure to the Meridian and the Index of the Hour Circle to 12 on the Hour Circle And turning the Globe Westwards till the Index points to 1 a clock or till 15 deg of the Equator pass through the Meridian I find the Colure cut the Hori in 11. 4 from the Meridian 2 24. 15 3 38. 4 4 53. â—6 5 71. 6 6 90. These are the distances of the Hour lines from Noon till 6 at Night and to these distances on the Plane counting from B towards D I make pricks and from the Center I draw lines through these Pricks and these lines are the Hour lines from 12 to 6 Afternoon But the Sun in the longest Day shines till past 8 at Night as you may find by Prob. 48. of the second Book therefore here wants the two Evening Hour lines which though they may be found after the same way I found the former viz. by continuing the turning of the Globe Westwards yet that I may the sooner reduce my work to the Plane I Count the number of degrees between the 6 a clock line and the 5 a clock line in the Circle on the Plane for the same number of degrees counted from D towards E is the distance of the 7 a clock Hour line from the 6 a clock Hour line and the number of degrees contained between the 6 a clock Hour line and the 4 a clock Hour line is the distance of the 8 a clock Hour line from the 6 a clock Hour-line Or I need not draw the 7 and 8 a clock Hour lines till I have drawn the forenoon Hour lines for then by laying the edge of a Ruler that will reach through the opposite side of the Plane to the Morning 7 and 8 a clock Hour lines I may by the side of that Ruler draw lines from the Center through the opposite side of the Plane and those lines shall be the 7 and 8 a clock Hour lines Afternoon Having thus all the Afternoon Hour lines I bring the Vernal Colure to the Meridian again so shall the Index again point to 12. Therefore as before I turned the Globe Westwards so now turning it Eastwards till the Index points to 11 a clock or till 15 deg of the Equator pass through the Meridian I find the Colure cut the Hori in 11. 40 from the Meridian 10 24. 15 9 38. 4 8 53. 36 7 71. 6 6 90. These are the distances of the Hour lines from Noon to 6. a clock in the Morning and these distances I seek in the Circle of the Plain counting from the Noon line B towards C and mark them with Pricks through which pricks as before I draw lines from the Center to the outside the Plane and those lines shall be the Hour lines Or having the distance of all the Afternoon Hour-lines I have also the distance of all the forenoon Hour lines from the Meridian as you may see by comparing the two former Tables For the 1 a clock Hour line Afternoon is equidistant from the Meridian or Noon line with the 11 a clock Hour line before Noon viz. they are both 11 degrees 40 minutes distant from the Noon line and the 2 a clock Hour line Afternoon is from the Noon line equidistant with the 10 a clock Hour line Beforenoon for they are both 24. degrees 15. minutes distant from the Meridian or Noon line and so all the other Morning Hour lines are distant from the Noon line by the same space that the same number of Afternoon Hour lines told from the Meridian on the contrary side the Noon line are distant from the Meridian Whence it follows that since as aforesaid the same number of Hour lines after 6 at Night and before 6 in the Morning have the same distance from the 6 a clock line that the same number of Hour lines before 6 at Night and after 6 in the Morning have from the 6 a clock line and since the same number of Hour lines before Noon are equidistant from the Meridian or Noon line by the same space of degrees that the same number of Hour lines Afternoon are It follows I say that having found the distance of the six Hour lines either before or after Noon you have also given the distance of all the other Hour lines If you will have the half Hour lines placed on your Dyal you must turn the Globe till the Index points to every half Hour in the Hour Circle as well as to the whole and examine the degrees of the Horizon cut by the Vernal Colure as you did for the whole Hours and in like manner transfer them to your Plane Having thus drawn all the Hour lines I count from the Noon line 51½ degrees the Elevation of the Pole here at London and from the Center A I draw a straight line as A F through these 51½ degrees for the Gnomon or Style
Westwards Having thus found out where this Plane becomes Horizontal make your Dyal to this Plane as by the second Rule in this Probleme Find also the Style as is there directed 5. If your Plane be a Declining Incliner The Globe and Quadrant of Altitude Rectified Bring the Colure to the Meridian and the Quadrant of Altitude to the degree of the Horizon opposite to the degree of the Planes Declination and count upwards on the Quadrant of Altitude the degrees of Inclination and make a 〈◊〉 there For in the 〈◊〉 of that prick found as by 〈◊〉 〈◊〉 of the Second Book that Declining In 〈◊〉 shall become an Horizontal Plane Then find the Latitude and difference of Longitude of this 〈◊〉 by the 〈◊〉 〈◊〉 and make a â—yal to that 〈◊〉 by the second 〈◊〉 in this Probleme Find also the Style as therein is directed PROB. XV. To make a Dyal on the Ceeling of a Room where the Direct Beams of the Sun never come FInd some convenient place in the Transum of a Window to place a smal round peece of Looking-Glass about the bigness of a Groat or less so as it may ly exactly Horizontal The point in the middle of this Glass we will marke A and for distinctions sake with Mr Palmer call it Nodus Through this Nodus you must draw a Meridian line on the Floor Thus Hang a Plumb line in the Window exactly over Nodus and the shadow that that Plumb line casts on the Floor just at Noon will be a Meridian line Or you may find a Meridian line otherwise as by the Preface Having drawn the Meridian line on the Floor find a Meridian line on the Ceeling thus Hoâ—d a Plumb line to the Ceeling over that end of the Meridian line next the Window If the Plumbet hang not exactly on the Meridian line on the Floor remove your hand on the Ceeling one way or other as you see cause till it do hang quietly just over it and at the point where the Plumb line touches the Ceeling make a mark as at B that mark B shall be directly over the Meridian line on the Floor then remove your Plumb line to the other end of the Meridian line on the Floor and find a point on the Ceeling directly over it as you did the former point as at C and through those two points B and C on the Ceeling strain and strike a line blackt with Smal Cole or any other Culler as Carpenters do and that line B C on the Ceeling shall be a Meridian line as well as that on the Floor Then examine the Altitude of the Equinoctial as by Prob. 6. of the Second Book you did the Meridian Altitude of the Sun and fasten a string just on the Nodus and remove that string in the Meridian line on the Ceeling till it have the same Elevation in a Quadrant that the Equinoctial hath in your Habitation and through the point where the string touches the Meridian line in the Ceeling shall a line be drawn at right Angles with the Meridian to represent the Equinoctial line Thus in our Latitude the Elevation of the Equator being 38½ degrees I remove the string fastned to the Nodus forwards or backwards in the Meridian line of the Ceeling till the Plumb line of a Quadrant when one of the sides are applyed to the string falls upon 38½ degrees and then I find it touch the Meridian line at D in the Ceeling therefore at D I make a mark and through this mark strike the line D E as before I did the Meridian line to cut the Meridian line at Right Angles This line shall be the Equinoctial line Then I place the Center of the Semi-Circle of Position upon Nodus and under-prop it so that the flat side of it may ly Parallel to the string when it is strained between the Nodus and the Equinoctial and also so as the string may ly on the division of the Semi-Circle marked o when it is help up to the Meridian line in the Ceeling Then removing the string the space of 15. degrees in the Circle of Position to the Eastwards and extending it to the Equator on the Ceeling where the string touches the Equator there shall be a point through which the 1 a clock Hour-line shall be drawn and Removing the string yet 15. degrees further to the Eastwards in the Semi-Circle of Position and extending it also to the Equator where it touches the Equator there shall be a point through which the 2 a clock Hour-line shall be drawn Removing the string yet 15. degrees further to the Eastwards in the Semi-Circle of Position and extending it to the Equator there shall be a point through which the 3 a clock Hour-line shall be drawn The like for all the other After-Noon Hour lines so oft as the string is removed through 15. degrees on the Semi-Circle of Position so oft shall it point out the After-Noon distances in the Meridian line on the Ceeling The scituation of the Semi-Circle of Position cannot conveniently be shewn in this Figure unless it be drawn by the Rules of Perspective Neither if it were would it suit with the other demonstrations expect they were drawn by the same Rules also which to do would be hard for young Learners to understand Therefore I have left out the Semi-Circle of Position in this Figure and refer you for a demonstration thereof to the sixth Probleme For even as the lines drawn through every 15 degrees of the Semi-Circle there denote in a Contingent line the distance of any Hour line from the Meridian line even so a line drawn through every 15. degrees of the Semi-Circle of Position posited as aforesaid point out in the Equinoctial line on the Ceeling the distance of each respective Hour line from the Meridian line Having thus found out the points in the Equator through which the After-Noon Hour-lines are to be drawn I may find the Fore-Noon Hour distances also the same way viz. by bringing the string to the several 15. degrees on the West side the Semi-Circle of Position or else I need only measure the distances of each Hour distance found in the Equator from the Meridian line on the Ceeling for the same number of Hours from 12 have the same distance in the Equinoctial line on the other side the Meridian both Before and Afternoon The 11 a clock Hour distance is the same from the Meridian line with the 1 a clock distance on the other side the Meridian the 10 a clock distance the same with the 2 a clock distance the 9 with the 3 c. And thus the distances of all the Hour lines are found out on the Equator Thus upon the point markt for each Hour distance in the Equinoctial line on the Ceeling I discribe the Arches I II III IIII as in the Figure and finding the distance from the Meridian of the Hour-lines of an Horizontal Dyal to be according to the third Probleme
Thus The 1 a clock Hour-line 11.40 whose Complement 90. is 78.20 2 24.15 65.45 3 38.4 51.56 4 53.36 36.24 I measure in a Quadrant of the same Radius with those arches already drawn from the Equinoctial line for the 1 a clock Hour 78.20 2 65.45 3 51.56 4 36.24 and transfer these distances to the Arches drawn on the Ceeling For then straight lines drawn through the mark in the Arch and through the mark in the Equator and prolonged both waies to a convenient length shall be the several Hour-lines a foresaid And when the Sun shines upon the Glass at Nodus its Beams shall reflect upon the Hour of the Day PROB. XVI To make a Dyal upon a solid Ball or Globe that shall shew the Hour of the Day without a Gnomon THe Equinoctial of this Globe or which is all one the middle line must be divided into 24 equal parts and marked with 1 2 3 4 c to 12. and then beginning again with 1 2 3 c. to 12. Then if you Elevate one of the Poles so many degrees above an Horizontal line as the Pole of the World is Elevated above the Horizon in your Habitation and place one of the 12 s directly to behold the North and the other to behold the South when the Sun shines on it the Globe will be divided into two halfs the one enlightened with the Sun-shine and the other shadowed and where the enlightned half is parted from the shadowed half there you shall find in the Equinoctial the Hour of the Day and that on two places on the Ball because the Equinoctial is cut in two opposite points by the light of the Sun A Dyal of this fort was made by M r John Lâ—â—k and set up on a Composite Columne at Leaden Hall Corner in London in the Majoralty of S r John Dethick Knight The Figure whereof I have inserted because it is a pretty peece of Ingenuity and may perhaps stand some Lover of Arâ— in stead either for Imitation or help of Invention PROB. XVII To make a Dyal upon a Glass Globe whose Axis shall cast a shadow upon the Hour of the Day FIrst divide the Equinoctial of your Globe into 24 equal parts and having a Semi-Circle cut out of some Brass plate or thin Wood to the same Diameter your Globe is of or a very little wider Apply this Semi-Circle to the Globe so as the upper edge of each end of the Semi Circle may touch the Poles of the Globe and the middle of the Semi Circle may at the same edge cut through some division made in the Equinoctial for then a line drawn by the edge of the Semi Circle thus posited shall be a Meridian line The same way you must draw Meridian lines through every division of the Equinoctial and set figures to them beginning with 1 2 3 4 c. to 12 and then beginning again with 1 2 3 4 c. to 12 again This Globe being made of Glass and having an Axis of Wyer passing through â—t from Pole to Pole will be an Horizontal Dyal all the World over if its Axis be set Parallel to the Axis of the World in the same Place and one of the Meridians marked 12 be set so as it may directly behold the North point in Heaven and the other the South point in Heaven for then the Axis of the Globe shall cast a shadow upon the Hour of the Day And if you divide the upper half of the Glass Globe from the under half when the Axis stands Parallel to the Axis of the World by a Circle drawn round about the Globe that Circle shall represent the Horizon and the Meridian lines drawn on the Globe shall be the Hour lines and have in the Horizontal Circle the same distance from the 12 a clock line that the same respective Hour line was found to have as by Prob. 3. of this Book But because the shadow of this Axis will not be discerned through the Glass Body therefore you may with Water and white Lead ground together lay a Ground on the Inside of the under half of the Glass to the Horizontal Circle as Looking-glass makers do their Looking Glasses with Tinfoil for then the shadow will appear Such a Glass Globe Dyal hath the Lord Robert Titchborn standing in his Garden supported by Atlas The End of the Fifth Book The Sixth BOOK Shewing the Practical Use of the GLOBES Applying them to the Solution of Spherical Triangles PRAEFACE THe Solution of Spherical Triangles is to know the length of its Sides and the width of its Angles These have already by many learned Men been taught to be performed by a Canon of Sines and Tangents and also by many Instruments some serving as Tables of Sines and Tangents such as are the Sectors Scales the Spiral line c. and others serving to represent the Globe such as be the Mathematical Jewel Astrolabium Catholicum and several other Projections of the Sphear But none hath as yet taught the Solution of Spherical Triangles by the Globe it self though it be the most natural and most demonstrative way of all and indeed ought first to be learnt before the Learner enters upon any other way To this Authors of Trigonometry agree for the most of them in their Books give Caution that the Learner be already sufficiently grounded in the Principles of the Globe For those Lines or Circles which either in Tables or other Instuments your force your Imagination to conceive represents your Line or Circle in question those Lines and Circles I say you have Actually and Naturally discribed on the Globe and therefore may at a single Operation or perhaps only by a sudden inspection have an Answer annexed according as the nature of your Question shall require and that more Copiously then by Tables of Sines and Tangents For therein you find but one Question at once resolved but by the Globe you have alwaies two resolved together Of the Parts and Kindes of Spherical Triangles THEOREMS 1. ALL Spherical Triangles are made of six parts Three Sides and three Angles The Sides are joyned together at the Angles and measured by degrees of a Great Circle from one end to the other The Angles are the distance of the two joyned sides and they are also measured by an Arch of a Circle discribed on the Angular point If any three of these parts be known the rest may be found 2. All Spherical Triangles are either Right Angled or Oblique Angled A Right Angle contains 90. degrees An Oblique Angle either more or less 3. If a Spherical Triangle have one or more Right Angles it is called a Right Angled Spherical Triangle But if it have no Right Angle it is called an Oblique Angled Spherical Triangle 4. If an Oblique Spherical Triangle have one Angle greater then a Right Angle it is called an Obtuse Angled Spherical Triangle But if it have no Angle greater it is called an Accute Angled Spherical Triangle 5. In Right
degrees 30. minutes I screw the Quadrant of Altitude and count in the Horizon from the Intersection of the Meridian with the Horizon on that side the Pole is Elevated the measure of the given Angle B viz. 142. degrees 42. minutes and to this number of degrees and minutes of the Horizon I bring the edge of the Quadrant of Altitude then I turn about the Globe till the first Meridian is distant from the Brasen Meridian 17. degrees 15. minutes of the Equinoctial which is the measure of the other given Angle So shall the Triangle be made on the Globe and the Arch of the Quadrant of Altitude comprehended between the first Meridian and the Zenith shall be the measure of the side B C 25. degrees and the Arch of the first Meridian comprehended between the Pole and its Intersection with the Quadrant of Altitude shall be the measure of the side A C 60. degrees The measure of the Angle C is found by turning the Triangle as in Prob. 7. PROB. XI Two Angles and a Side opposite to one of them given to find the Rest. EXample In the Triangle of Prob. 7. the Angles given are A 17. degrees 15 minutes and B 142. degrees 42. minutes the side given is B C 25. degrees being the side opposite to the Angle A the Angle A is made at the Pole of the Globe and measured in the Equator Therefore I separate the first Meridian from the Brasen Meridian 17. degrees 15. minutes so doth the Pole represent the Angle A the Angle B is made at the Zenith and measured in the Horizon therefore I count in the Horizon 142. degrees 42. minutes and there I make a prick to this prick I bring the edge of the lower end of the Quadrant of Altitude not minding to what degrees of the Meridian the upper end of it is placed Then I count from the upper end of the Quadrant of Altitude 25. degrees downwards the measure of th e side B C and there I make a prick and keeping the lower end of the Quadrant of Altitude to the prick made in the Horizon I slide the upper end of it forwards or backwards till the prick on the Quadrant of Altitude come to the first Meridian so shall the Triangle be made on the Globe Then the Arch of the Brasen Meridian comprehended between the Pole and the upper end of the Quadrant of Altitude shall be the measure of the side A B 38. degrees 30. minutes and the Arch of the first Meridian comprehended between the prick on the Quadrant of Altitude and the Pole shall be the measure of the side A C 60. degrees But the Angle C you must find by turning the Triangle as in Prob. 7. In the working this Probleme I would have placed the given fide B C 25. degrees upon the Brasen Meridian between the Pole and Zenith but then the Angle B being so Obtuse would have had that side which would be intersected by the Quadrant of Altitude viz. the first Meridian under the Horizon which the Quadrant of Altitude cannot reach PROB. XII Three Angles given to find the Sides THis Triangle is taught to be resolved by M r Palmer on the Planisphear Book 3. Chap. 19. It is to be known saith he That if you go to the Poles of the three great Circles whereof your Triangle is made these Poles shall be the Angular points of a second Triangle and the two lesser sides of this second Triangle shall be equal to the two lesser Angles of your first Triangle the greatest side of the second Triangle shall be the supplement of the greatest Angle of the first Triangle that is shall have as many degrees and minutes as the greatest Angle of the first Triangle wanted of 180. degrees see Pitiscus Trigonometry Lib. 1. Prop 61. This second Triangle therefore all whose sides are known from the Angles of the first you shall resolve by Prob. 7. And having by that Probleme found the Angles of this second Triangle know that the two lesser Angles of the second Triangle shall be several and respectively equal to the two lesser sides of the first Triangle and the least Angle to the least side the middle Angle to the middle side and the greatest Angle of this second Triangle being subtracted out of 180. degrees shall leave you the greatest side of your first Triangle Example If the Angles be given 142. degrees 42. minutes 17. degrees 15. minutes and 25. degrees 24. minutes and the sides be enquired Draw by aim a rude Scheam of this first Triangle writing in the Angle A 17 degrees 15. minutes in B 142. degrees 42. minutes in C 25. degrees 24. minutes supposing the sides yet unknown then draw under this by aim also a Scheam of the second Triangle setting his Base Parallel with the Base of the first and making the Base of the second shorter then the Base of the first Set also B at the Vertical Angle and A C at the Base as in the first Triangle Then say Because A in the first Triangle is 17 degrees 15. minutes therefore in the second Triangle B C which subtendeth A shall be 17. degrees 15. minutes and because C in the first Triangle is 25. degrees 24. minutes therefore in the second Triangle the side A B which subtendeth C shall be 25. degrees 24. minutes and because B the greatest Angle in the first Triangle is 142. degrees 42. minutes therefore in the second Triangle the side A C which subtendeth B shall be the complement thereof to 180. degrees viz. 37. degrees 18. minutes Write now upon the sides of this second Triangle the quantities of the sides so is your second Triangle ready to be resolved as by Prob. 7. Whereby you shall find the Angles of the second Triangle as I have expressed them in the Scheam A 25. degrees C 38. degrees 30. minutes B 120. degrees Nâ—w lastly I say these Angles of the second Triangle thus found give me the sides of the first Triangle which I seek in this manner In the second Triangle  In the first Triangle A is 25. degrees Therefore B C is 25. degrees C is 38. 30. A B 38. 30. B is 120. 00. A C 60. 00. Complement of 120. degrees to 180. And thus by all the Angles given we have found out all the sides which was required Having then the Angles of your first Triangle given and his sides now found you shall find his scituation on the Globe thus Place him as in Prob. 7. A B 38. degrees 30. minutes between the Pole and Zenith A C 60. degrees in the first Meridian separated from the Brasen Meridian 17. degrees 15. minutes of the Equinoctial B C 25. degrees on the Quadrant of Altitude counted from the Zenith when it s lower end is applyed to the 142. degrees 42. minutes of the Horizon you shall say Because the East point of the Horizon is the Pole of the Arch A B therefore at the East point of the Horizon shall
stand the Angle C which A B subtendeth Next follow the 142. degrees 42. minutes of Azimuth which maketh B C of your Triangle to the Horizon and from thence number in the Horizon towards the East point 37 degrees 18. minutes the Complement of the Angle A to 180. degrees and number yet further 52. degrees 42. minutes beyond the East point to make up 90 and there is the Pole of the Arch B C Therefore there shall stand the Angle A which B C subtendeth Then count in the Equator from the first Meridian 90. degrees which will end under the Horizon and there make a prick for there is the Pole of the Arch or side A C. Therefore at that prick shall stand the Angle B which A C subtendeth Here you see your second Triangle made by the Poles of the first adjoyning to the East point of the Globe only the side A B is wanting To get that make a prick upon the Globe against the 52. degres 42. minutes from the East point of the Horizon found before to represent the Angle A Then turn about the Globe and Qudrant of Altitude till that prick and the prick made before for the Angle B are both at once cut by the side of the Quadrant of Altitude and you will find 25. degrees 24. minutes of the Quadrant of Altitude comprehended between the two pricks for the measure of the side A B PROB. XIII How to let fall a Perpendicular that shall divide any Oblique Spherical Triangle into two Right Angled Spherical Triangles THis Probleme is much used when an Oblique Triangle having two sides and an Angle given is to be solved by the Cannon of Sines and Tangents but by the Globe it may be solved without it as was shewed Prob 8 9. Yet because letting fall a Perperdicular is so frequent in all Authors that treat of Trigonometry I have inserted this Probleme also In the Oblique Triangle of the fromer Problemes there is given the sides A B 38½ degrees and B C 25. degrees and the Angle C 25. degrees 24. minutes It is required to let fall a Perpendicular as B a from the Angle B. upon the Base A C and to know both the measure of this Perpendicular and the parts it divides the Base into Therefore Elevate the Pole of the Globe above the Horizon so much as is the measure of the Angle C which in this Example is 25. degrees 24. minutes and bring the intersection of the first Meridian with the Equinoctial to the East point of the Horizon so shall the Angle at the East point of the Horizon comprehended between the Horizon and the first Meridian be equal to the Angle C then count in the first Meridian from the East point of the Horizon the measure of the side B C 25. degrees and having the Quadrant of Altitude screwed to the Zenith bring the graduated edge of it to these 25. degrees so shall the Arch of the Horizon comprehended between the East point and the lower end of the Quadrant of Altitude be the number of degrees that the Perpendicular falls upon the Base counted from the Angle C to a which in this Example is 21¾ degrees and the Arch of the Quadrant of Altitude comprehended between the Horizon and the first Meridian is the measure of the Perpendicular B a 11. degrees And thus by letting fall this Perpendicular you have two Right angled Spherical Triangles made the one B a C wherein is found C a 21¾ degrees B C 25. degrees B a 11. degrees C 25. degrees 24. minutes and a the Right Angle There remains only the angle B unknown which you must find by turning the Triangle as was taught Prob. 1. The other Right angled Spherical Triangle made is B a A wherein is found A a Complement of 21 2 4 degrees to 60 degrees the whole Base before given 38¼ degrees A B 38. degrees 30. minutes B a 11. degrees and a the Right Angle which is more then enough to find the Angl◠A and B as was shewed in the Preface Theorem 1. The End of the Sixth Book Here follows the Ancient STORIES of the several STARS and CONSTELLATIONS Shewing the Poetical Reasons why such Various Figures are placed in HEAVEN Collected from Dr HOOD And First Of the Northern Constellations 1. URSA MINOR This Constellation hath the preheminence because it is neerest of all the rest unto the North Pole And is called of the Greeks 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 whereupon the Pole is called the Pole Arctick for that it is neer unto that Constellation It is also called Helice minor because of the smal revolution which it maketh round about the Pole or rather of Elice a Town in Arcadia wherein Calysto the great Bear and mother to the less was bred It is called Cynosura because this Constellation though it carry the name of a Bear yet it hath the taile of a Dog Last of all it is termed Phoenice because that Thales who first gave the name to this Constellation was a Phoenician And therefore the Phoenicians being taught how to use it in their Navigations did call it by the name of the Country wherein Thales was born It consisteth of 7. stars which the Latines call Septemtriones because by their continual motion those seven stars do as it were wear the Heavens The Spaniards call them all Bosina that is an Horn because they may be very well brought into that form whereof that which is in the end of the tail is called the Pole-star by reason of the neareness thereof unto the Pole of the world for it is distant according to the opinion of most from the true Pole but 23. deg 30. min. The Arabians call it Alrukaba And of the Scythians it is said to be an Iron nail and is worshipped by them as a God The two stars that are in the sholders of the Bear are called Guards of the Spanish word Guardare which is to behold because they are diligently to be looked unto in regard of the singular use which they have in Navigation The reason why this Constellation was brought into the Heavens is diversly set down and first in this manner Saturn having received of the Oracle that one of his Sons should banish him out of his kingdom determined with himselfe to kill all the men children that he should beget whereupon he gave command to Ops his wife being then great that she should shew him the child so soon as ever it was born But she bringing forth Jupiter and being greatly delighted with his hair gave the child unto two Nymphs of Crete dwelling in the mount Dicte whereof this was one and was called Cynosura the other was Helice Jupter after that according to the Oracle he had bereft his Father of the kingdom in recompence of their paines and courtesie translated them both into the Heavens and made of them two Constellations the Lesser Bear and the Greater Bear Othersome say that it was Arcas the son of
he that is ready to read the Stories cannot content himselfe therewith but desireth also to know the Constellation or at leastwise some principal Star therein FINIS A Discourse OF THE Antiquity Progress AND Augmentation OF ASTRONOMIE FIRST it seems not to be doubted but that there was some kind of observation of Bodies Coelestial as soon as there were Men considering that the spectacle which the Heavens constantly present is both so glorious and so usefull that men could not have eyes to see and not fix them attentively and considerately thereupon For among other Apparences when they saw the Sun dayly to change the places of its rising and setting at certain times of the Year to approach neerer to the Earth in its Diurnal arch and at others again to mount up to a height much more sublime and remote from it and that his coming neerer to the Earth made Winter and his remove higher made Summer we say when they beheld these things doubtless they could not but seriously remark and consider this vicissitude according to which they might expect the Season would be more hard or mild to them in this lower Region of the World Again so admirably various did the Moon appear in her several shapes and dresses of light that she could not but invite mens eyes and engage them to frequent Speculations specially when she assumed those various faces or apparences at set and certain Times in respect whereof it came to pass that every Nation measured their times and Seasons by those her constant and periodical circuits and this because those periods succeded much more frequently than the Erections and Depressions of the Sun To these we may add that beautifull shew of the Nightly Stars undergoing likewise their Variations according the variety of Seasons and more particularly that bright star of Lucifer rising sometimes later sometimes earlier and sometimes not at all before the Sun and the like But what we shall principally note is only this that though Mankind was long before they came to make inquiries into the Causes of these Coelestial changes and variations restrained to set periods yet they observed them from the very first Age and not only admired but also accommodated what they observed to the uses of their Lives and their Successors Here it might not be fruitless to remember that PROMETHEUS who was imagined to have framed the first Man was also imagined to have given him an erected Figure and sublime Countenance to the end he might the more advantageously advance his eyes to the Heavens and contemplate the glory and motions of the Coelestial Lights But because this is too General and rude a way of observation and it is our business to look back into those times wherein men first made such Observations of Sydereral bodies as gave hem the hint and occasion of reducing them to Method and founding the principles of the Art or science of Astronomy thereupon we must have recourse to the monuments in Sacred Writ for the understanding of that obscure matter And indeed the light we expect from Sacred Leaves would soon be clear enough to discuss all the darkness wherein the Original of Astronomy seems involved could we but from them deduce the least evidence for that which the learned Antiquarie among the Jewes Josephus affirms of the Sons of Seth viz. that they invented the science of the Heavens before the Flood and engraved the same on two Pillars the one of Brick the other of Stone that so it might be preserved in the one in case the fury of the Deiuge to come should demolish and deface the other or if there remained to us any the most slender testimony of the Reason he there gives of the so great Longevity of men in those dayes namely that the duration of their lives was sufficiently long to perfect the knowledge of Astronomy which requires full 600 Years at the least to the observation of all the Varieties of Coelestial motions Whereupon He notes that the Great Year as they call it doth consist of six hundred common Years the vulgar opinion being that the Celestial Motions do continually vary Donec consumpto Magnus qui dicitur Anno Rursus in 〈◊〉 antiquum redeant vaga sidera cursum Qualia praeteriti steterant ab origine mundi Again the business might be deduced from not long after the Flood if in Scripture we could find but the least word from whence might be argued the truth of what the same Author writes namely that the Egyptians were taught Astronomy by Abraham Probable enough it is we confess that Berosus and others quoted aswell by Josephus as Eusebius had read some such thing in some Books of the old Rabbins but that the same should be fetched from Holy Writ is most improbable therein being no mention at all of any such thing Besides there are pious and learned Doctors and among them Salianus who will not allow it to be so much as probable that Abraham should instruct the Egyptians in Astronomy because of the very smal time of his stay among them in Egypt It is written indeed that Abraham came from Ur of the Chaldeans but not that he received Astronomy from the Chaldeans or that he delivered it from them to the Egyptians And therefore they conclude that what Josephus said of Abrahams reading Astronomy in Egypt may with more probability be imputed to his Great-grand child Joseph Concerning Him therefore we read in truth that he was singularly favourable to the Priests in Egypt at such time as all the rest of the people mortgaged their lands to the King for bread during that wofull and long Famine For He excepted the Lands belonging to the Priesthood and as the Text saith assigned them certain portions out of the publick Granaries so that from hence may be proved what Aristotel tells us from other Authority that amongst the Egyptians the most ancient Nation the Priests were exempted from labour and left to the easy imploiment of their minds and that this gave them occasion to invent and constitute the Mathematiques and yet for all this it is not written either that Joseph taught those Egyptian Priests the Mathematiques or that they taught them to him And perhaps that Favour He shewed the Priests was an argument not only of the Respect and Veneration born them by the King and all his people but also of his particular Gratitude toward them in that He who had been bred up only to Sheppardry and Country imployments and was wholy ignorant of all Arts and Sciences at his first comming among them being afterwards advanced to the height of a Courtier and lustre of a Favorite had bin instructed by them in something more noble and sublime And truely the Divine Moses not long after admitted into the same Court is not delivered so much to have erudited any others as to have been himself learned in all the Wisdom of the Egyptians Nevertheless considering
Plato himself who was Companion to Eudoxus for thirteen Years together in Egypt profest that he could attain nothing sollid and satisfactory touching those Stars and therefore placed all his hope only in the sagacity and industry of the Grecians such as he knew Eudoxus to be For having first recounted what ever he knew concerning them he saith It is to beleeved that the Grecians make more perfect whatsoever they receive from Barbarians and therefore is it fit we allow the same touching the argument of which we have discoursed Truth is it is difficult to find out the way how all these Apparences so involved in obscurity may be explicated nevertheless there is great hope that things of that sort will be better and more advantageously handled than they were delivered to us by Barbarians From the Egyptians and Chaldeans therefore as Astronomy her self while young and rude we come to the Graecians and the most antique record of Syderal Observations to be found among them seems to be that of Hesiod who in his Book of Weeks and Dayes teacheth Husbandmen the most opportune times of reaping sowing and other labours of Agriculture from the rising and setting of the Pleiades and Hyades and Arcturus the Dog-star and Orion 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 Donec Pleiades quae Atlantiades exoviuntur c. And I cannot tell whether it were that book or some other that Pliny meant when speaking of Hesiod he sayes Hujus quoque nomine extat Astrologia there is extant an Astrology of his However we are here to remark two things in order to our more exact disquisition the First is that the Ancient Greeks principally attended to these risings and settings aswell that they might distinguish the several Seasons of the Year as that they might fore-know Rain Winds and other dispositions of the Air usually attending those Seasons And hereupon Thales Anaximander Democritus Euctemon Meton Eudoxus and many others composed certain Parapegmata Tables as Ephemerides or Diaries in which they inscribed each Day of the Year with the particular Stars rising or setting on each Day and what mutations of the Air each one did portend Such a Darapegme as these was composed likewise by Julius Caesar himself for the Horizon of Kome in allusion where to he might justly own what Lucan said for him Nec meus Eudoxi fastis superabitur Annus And him doubtless did Ovid translate into his Fasti promising in the beginning that he would sing of the Stars and Signes that rose and again descended under the Earth But to keep close to the Grecians among them he was held a great Astrologer who had discovered and observed only these risings and settings here spoken of and so of whom that might be spoken which Catullus said of Conon Omnia qui magni dispexit lumina Mundi Stellorumque ortus comperit atque obitus For before the Advent of Berosus this was the only 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 Praesignification or Divination by the Stars the Grecians had among them unless what Hesiod hints in his 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 Primùm prima dies quarta septima sacra c. where he points out what Dayes of the Moon were accounted Lucky and what Unlucky The Second observable is that among the Grecians and indeed among divers other Nations beyond all Memorials of either Traditions or books the Stars were reduced to certain Images or Constellations and denominated accordingly as their names yet shews as it pleased the fancies of Husbandmen Shepheards Matiners and the like who used to be vigillant and gizing upon the Heavens in clear Nights Though there have been some Constellations added of latter times as that of the lesser Wain by Thales which Lacrtius and Tatius recite out of Callimachus who also took the same elswhere and that of Berenices Hair removed into Heaven by Conon as Catullus relates Cleostratus likewise as we have it from Hyginus found out the Kidds though which Pliny moreover attributes to him his invention of the Signes in the Zodiack is so to be understood as that he taught men through what Signes the Sun and other Planets passed But that we may couch also upon this at first the Grecians had only Eleven Signes in their Zodiack and it was long after ere they came to add the twelfth in imitation of the Egyptians who as may be collected from Servins Marcianus and others instead of the Clawes of the Scorpion placed Libra the place destined to Augustus by Virgil Ipse tibi jam brachia contrahit ardens Scorpius They added the Twelfth we say to the end that as the whole Compass of the Zodiack was divided into Dodecatemoria as they call them twelve equal parts so it might consist also of twelve Signes Albeit being as it were necessitated to make use of such Signes as had been brought up rather by chance than Art those 12. Signes were not exactly proportionate to the 12. Divisions of the Zodiack but took up more space some than others as in particular Leo possest more room than Cancer Taurus than Gemini I say than Gemini which though composed of Castor and Pollux in so little space as is allowed them it is impossible the one should rise when the other Sets and both in the East but this Empiricus interprets of the two Hemisphears I omit to insist upon this that all Nations had not the same Constellations as among the Egyptians was no Bear no Cepheus no Dragon but other formes or representations as Tatius reports and shall add only that Eudoxus seems to have been the first who partly out of the Egyptian Figures partly out of the Grecian furnished the whole Zodiack with Images resembling the Asterismes as men had fancied at least and caused them to be drawn on a Globe or solid Sphear For Aratus upon whose Poem intitul'd 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 Apparences there have been so many Commentaries set forth as that no fewer than forty have been extant in Greek besides those of Cicero Germanicus Avienus and other Latin Interpreters did no more but only express in verse what Eudoxus had said before in prose of this argument as Hipparchus Bythinus demonstrates I know not whether it would be seasonable for me here to advertise that it is no wonder Aratus erred so grosly in many particulars considering that as is written in his life he Living with Antigonus Gonata in the quality of his Physician and Nicander in the quality of his Astrologer and both were good at Poetry Antigonus commanded the Physician to give him a tryall of his Poesie upon an Argument in Astrology and the Astrologer to give another of his upon somthing in Physick delivering to the one the Book of Eudoxus and to the other all that was extant of Treacles Antidotes or Counterpoisons So each wrote of what he did not well understand One thing I shall not forget and that is that the Phenomena of Euclid who lived neer
time of the Nicene Councel which was one of the two main causes of the Reformation of the Kalender in the eighty second Year of the last Age. And now we have an opportunity to speak more expresly of Eudoxus so frequently mentioned This man well understanding after his return out of Egypt that not only the Sun and Moon but also the five Errant Stars did keep their courses round in the Zodiack and so as that aswell the Sun and Moon as those wandering Stars did sometimes vary their latitude or deviate from the Ecliptick Line in the midle of the Zodiack for he thought that the Sun was also extravagant as well as the rest and again that the other Planets did not only go forward but were also some times upon their retreat backward and somtimes made a hault or stood still we say pondering all these various motions in his mind and casting about what might be the reasons thereof in nature he at last imagined to himself that besides the Aplanes or Sphear of Fixt Stars which being supreme carried all the rest toward the West there ought also to be allowed three other Sphears as well to the Sun as to the Moon and four to each one of the other Errant Stars of which one and that the highest should follow the Impression of the Fixt Stars or rather of the Primum Mobile the next to that should move counter to the First or toward the East the third make the deviation from the Ecliptick or midle of the Zodiack and the fourth or lowest cause in the Stars their Direction Station and Retrogradation and that by a certain Vibration or Waving to and agen So that he supposed in all twenty seven Sphears and all those Concentricall that the Superior might carry on the Inferior and these might be turned round within those Afterwards Callippus adjoyned two Sphears to the Sun two to the Moon and one a peece to Mars Venus and Mercury and so made thirty three And Aristotle to all the Sphears which did not follow the motion of the Aplanes or Primum Mobile excepting only the Lunar Sphears added as many more which he called the Revolvent ones to the end he might conform them to the motion of the Inerrant Sphear or Primum Mobile and so in the whole he constituted Fifty six Sphears for as much at least as we can collect from his own context Now all these and even Plato himself likewise thought that the Moon was the lowest of all the Planets next to her the Sun and above the Sun the five wandering Stars Nor indeed doth it appear that Archimedes himself Living a whole Age or two after them represented the Planets in any other than this very order in that so famous Sphear of his In which though Claudian tells us that no more was represented but only the motions of the Sun and Moon Percurrit proprium mentitus Signifer annum Et simulata novo Cynthia mense redit Yet Cicero adds other motions when speaking of Archimedes he saith when he collected together the motions of the Moon Sun and five wandering Stars he did the same as that God who in Platoes Timaeus framed the World that one and the same Conversion might regulate sundry motions most different each from other in slowness and swiftness But Hipparchus afterwards finding that aswell the Sun as the Moon and the other five Stars did come somtimes neerer to the Earth and sometimes again mounted up farther from it and plainly perceiving that that particular apparence could not possibly be explicated by those Sphears that were all Concentrical to the Earth therefore wholly rejecting them he resolved that the motions of the Planets were to be accounted Eccentrick and though he could not himself determine each particular he yet demonstrated the way in which Ptolomy afterwards insisting accomplisht the Invention But before wee advance further we are to commemorate two or three Persons of note by whose Observations both Hipparchus and Ptolomy profited very much One was Timocharis who about three hundred Years before Christ among other things relating to the Fixt Stars observed that that Star which is called Spica Virginis doth antecede the point of the Atumnal Equinox by eight degrees And with him are we to joyn Aristâ—llâ—s whose Observations of something about the Fixt Star Ptolomy made great use of in order to his demonstrating that the Fixt Stars never change their latitude Afterwards scarce in Age succeded Eratosthenes who being Library-keeper to Ptolomy Evergeta the fomer perswaded him to set up the Armillae in the Porticus of Alexandria which Hipparchus and Ptolomy afterwards made use of and himself among other things observed that the Obliquity of the Zodiack was of twenty three degrees and fifty one minutes which account Hipparchus and Ptolomy constantly adhered to Now that we may at length remember the great Hipparchus who florish't neer upon an hundred and forty Years before Christ truely we find it no easy task to recount how highly Astronomy was beholding to him For in the first place examining that foresaid Observation of Timocharis with some others albeit he could not conceive them to be in all points exact yet because himself had found that Spica Virginis did not antecede the Equinoctial point by more than six degrees and the other Stars in the like Proportion he thence understood that the Fixt Stars also were moved Eastward according to the Zodiack and thereupon wrote a Book of the Transgression of the Solstices Equinoxes And being that in his time as not long agoe in Tycho Brahes there appeared a certain New Star he therefore came to doubt to speak the language of Pliny concerning him whether the like happened often or not and whether those Stars that were thought to be fixt had also some certain motion peculiar to themselves Wherefore as the same Pliny goes on he attempted a task of difficulty sufficient even for the Gods themselves namely to number the Stars for Posterity and reduce the heavenly Lights to a rule so that by the help of Instruments invented the particular place of each one together with its magnitude might be exactly designed and whereby men might discern not only whether they disappeared or newly appeared but also whether they removed their Stations as likewise whether their magnitude encreased or diminished Leaving Heaven for and Inheritance for the Witts of succeding Ages if any were found acute and industrious enough to comprehended the mysterious orders thereof And this was the first time when the places of the Fixt Stars were observed and markt out according to Longitude and Latitude and that Catalogue of the Fixt Stars which he composed is the very same which Ptolomy afterward inserted into his Almagest In the next place he denoted was positions sundry Stars had in respect each of other whether they were posited in a right Line or in a triangular form or in quadrate or square c. as is manifest even