A MOST PLAIN and easy way for the finding of the Sun's Amplitude and Azimuth, and thereby the Variation of the Compass, by Logarithme. Written by W.B. Also another plain way for the Azimuth, by the Table of Sines in five several Cases. By I.T. ¶ Of the Variation of the Compass. THe Variation of the Compass, is the difference between the true Meridian of the world, and the Meridian of the Loadstone, which is pointed out by the Compass or Needle; and is for the most part variable, as you sail to different places; But fixed and permanent being the same, always in one and the same place; (although there may be difference in the touch of the Stone, and in the observations of different men) Now for the finding of this Difference or Variation, the usual and most easiest way; is by taking the Sun's Amplitude at rising or setting, and compared with the true; But this way serveth chiefly in all places not far distant from the Equinoctial, whose Latitude is not great; For if you were to sail far to the South or North, near or beyond the Artict or Antartict Circles, it were of no force at all; The other way commonly used, is by taking the height or Almicanter of the Sun, and at the same time the Azimuth also, which is in use from each Pole to 30. or 40. degrees of Latitude, and at any place where the Sun doth not usually rise and set clear, for in diverse places you shall not see it rise or set, yet seldom but it may be seen either forenoon or afternoon: Now the working of both these ways are found diversely; either by Instrument or Arethmeticks: But I will here only show the work by Logarithme, which is the most easiest of all Arethmaticall work and first of the Amplitude. To find the Amplitude. THe Amplitude or breadth of the Sun's rising or setting from the true East or West point; is found by Sines thus, As the sine of the Compliment of the Latitude, is to the sine of the Declination, so is the Radius to the sine of the Amplitude; But in Logarithme, you are only to look the Logar: of the Compliment of the Latitude, and the Logar: of the Declination; and subtract one from the other, the remainder is the Logarithme of the Amplitude, Example. Data Compliment, Latitude— 40. deg.30′. North Declination— 20.— 40. North I demand the Amplitude? Comp: Lat: 40. d.30′. Loga: — 4316323 Declinat: 20. 40′. Loga: 10414836 6098513 the Logar: of the Amplitude 32. deg. 55′. and somewhat more. To find the Azimuth. IT is to be considered that in the Doctrine of Triangles, it is required in the solution of any question there are three things to be given in any Triangle, before the question can be answered, which in this for finding the true Azimuth of the Sun; you are to know or imagine your Latitude, the Compliment thereof is one side of a Triangle (which is the distance between the Pole and the Zenith) the Compliment of the Sun's Declination is another side of the same Triangle (which is the distance between the Sun and the Pole) then is the Compliment of the Almicanter the other side (which is the distance between the Sun and the Zenith.) here have you an oblique Spherical Triangle whose three sides are known, and it is desired to know the angle at the Zenith: whose quantity being found is the Sun's true distance from the North, (if the north Pole be elevate) or the distance from the South, (if the south Pole be elevate,) and in this question there are two cases. The first Case. THe first is, when you are on the same side of the Equinoctial the Sun is on: then are the Triangle sides all less than Quadrants, and may be resolved by Logarithme, 2. Book, 6. Chapter, and 8. section: As thus, Add half the base, and half the difference of the containing sides together; and to the Logarithme thereof, add the Loga: of the difference of them, out of which some, subtract the some of the Loga: of the two sides, and the half of the remainder is the Loga: of an Arch, which being doubled, is the quantity of the angle of the Zenith, or vertical angle. Example. Data Latitude North▪ 51. deg.30′. Declination North 20.00. Almicanter 48. 30. I demand the Azimuth? Let P.Z. be the Compliment of the Latitude 38. d.30′. P.S. being the base, by the Comp. of the Declinat. 70.00. S.Z. by the Compliment of the Almicanter 41.30. And let the Angle P.Z.S. be sought for The form of the Worke. P. Z. 38. d.30′. Loga. 4739880 S. Z. 41.30. Loga. 4115535 8855415 added. The difference 3. d. 0′.—— ½ Difference 1. d. 30′. ½ the base, P.Z. 35. 00. The 2. former added 36. d.30′. Log. 5194916 The same subtract. 33.30. Loga. 5943212 11138128 8855415 2282713 ½ The former 1141357 the Logarit: of the arch 63. deg. 8′. 30″. which doubled is 126. deg. 17′. the Sun's true distance from the North: which compared with the magnetical; the difference is the Variation. The second Case. THe other Case is, when you are on the one side of the Equinoctial, and the Sun on the other; then is the base P. S. more than a Quadrant, and is to be resolved by Logar: 2. Book, 6. Chapter, 10. section: Thus, Data Latitude 51. deg. 30′. North I demand the Azimuth? Declination 10. 00. South Almicanter 15. 00. — P. Z. 38. deg. 30′. S. Z. 75. 00. The some 113. d. 30′. The one half 56. 45. The difference 36. 30. differential 4221605 − The half 18. 15. differential 11094182+ The base P.S. 100 d. 00′. 8672577+ ½ Base 50. 00. differential 1754259 − 8626836 the differential of 22. d. 53′. the ½ alterne base. Again, The ½ true base 50. deg. 00′. The ½ alterne base 22. 53. Added 72. deg. 53′. the greater case M.S. Substracted 27. deg. 7′. the lesser case M.P. For the Triangle P, M, Z. P.M. 27. d. 7′. Loga: 7862605 P.Z. 38. 30. Loga: 4739880 3122725 the Logar: of the angle P, Z, M. 47. deg. 2′. 30″. For the Triangle M, Z, S. M, S. 72. d. 53′. Loga: 453035 Z, S. 75. 00. Loga: 346683 106352. the Logar: of the angle, M, Z, S, 81. deg. 40′. Which two Angles so found and added together, maketh 128. deg. 42′. ½. the Sun's true distance from the North point, from which if you subtract 90. the remainder leaveth 38. d. 42′. 30″ the distance from the East or West demanded. 1. For the Sun's Azimuth having no Declination. Add the Compliment of the Latitude, to the compliment of the Almicanter, which if the total be more than a quadrant, subtract 90. and set down the sine of the remainder for the first number: Again, add the compliment of the Latitude and the Almicanter, and add the sine thereof to the former: from the one half of that total subtract your first number or sine, and set down the remainder: Then, As the ½ of the 2. first numbers added is in proportion to the whole fine, so is the said remainder to the sine of the Sun's true Azimuth. Example. Latit. 51. d. 30′. the Comple. 38. d.30′. Almicant. 20. d. Complem. 70.0. Added makes 108 d. 30′. 90. substracted, leaves 18. deg. 30′. whose sine 3173. is the first number. Again, compliment of the Latitude 38. d. 30′. Almic. 20. d. added makes 58. 30′. whose sine 8526. is the second number, those 2. numbers added makes 11699. the ½ thereof 5849. from which subtract 3173. the first number rests 2676. for the remain: then say, As 5849. the ½ of the 2. first numbers is to 10000 the whole sine, so is 2676. the remain to the Azimuth desired. Facit, 4575. whose arch 27. d. 14′. is the Azimuth from the East Southward. 2. When the Sun hath North Declination, the 2. Compliments being equal to a quadrant. Add the compliment of the Latit. with the Almicanter only, and from ½ the sine thereof, subtract the sine of the Declination, and setting down the remainder, As the ½ aforesaid, is to the whole sine, so is the remainder aforesaid to the sine of the Azimuth desired. 3. When the Sun hath North Declination, the 2. Compliments less than a quadrant. Add the compliment of the Latit. and the compliment of the Almican. setting down the sine of the compliment thereof, then add the Almicanter and the compliment of the Latitude, and from the sine thereof subtract the former, setting down ½. of the remain for the first found number: again subtract the sine of the first Compliment from the sine of the Declination and the remain thereof, again subtract from your first found number, and set the remain thereof down for your second number: and then, As the first found is to the whole sine, so is the second to the Azimuth desired. 4. When the Sun hath North Declination, and the 2. Complemements more than a quadrant. Add the compliment of the Latitude and compliment of the Almicanter, which being more than 90. subtract 90. and set down the sine of the remainder, then add the Almicant. and compliment of the Latit. and set down the sine thereof, add both the sins together and take the ½ thereof for the first found number, then to the sine of the first 2. compliments add the sine of the Declination, and from that total subtract the first found, and set down the remainder for the second found: and then, As the first found is to the whole sine, so is the second found to the sine of the Azimuth desired. 5. When the Sun hath South Declination, and the 2. Compliments more than a quadrant. Add the 2. Compliments, subtract 90. set down the sine of the remainder, add also the Almicanter and compliment of Latitude, add both their sins and set down ½ of the total for the first found, then subtract the sine of the Declination from the sine of the remain of the first 2. Compliments, and that remain again from the first found, which last remain set down and say, As the first found is to the whole sine, so is the second found to the sine of the Azimuth desired.