¶ Of the Variation of the Compasse.
THe Variation of the Compasse, is the difference betweene the true Meridian of the world, and the Meridian of the Loadstone, which is pointed out by the Compasse or Needle; and is for the most part variable, as you sayle to different places; But fixt and permanent being the same, alwayes in one and the same place; (although there may be difference in the touch of the Stone, and in the obseruations of different men)
Now for the finding of this Difference or Variation, the vsuall and most easiest way; is by taking the Sunnes Amplitude at rising or setting, and compared with the true; But this way serueth chiefely in all places not farre distant from the Equinoctiall, whose Latitude is not great; For if you were to sayle farre to the South or North, neere or beyond the Artict or Antartict Circles, it were of no force at all;
The other way commonly vsed, is by taking the height or Almicanter of the Sunne, and at the same time the Azimuth also, which is in vse from each Pole to 30. or 40. degrees of Latitude, and at any place where the Sunne doth not vsually rise and set cleere, for in diuers places you shall not see it rise or set, yet seldome but it may be seene either forenoone or afternoone: Now the working of both these wayes are found diuersly; either by Instrument or Arethmeticks: But I will heere onely shew the worke [Page 2] by Logarithme, which is the most easiest of all Arethmaticall worke and first of the Amplitude.
To finde the Amplitude.
THe Amplitude or bredth of the Sunnes rising or setting from the true East or West point; is found by Sines thus, As the sine of the Complement 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 onely to looke the Logar: of the Complement of the Latitude, and the Logar: of the Declination; and substract one from the other, the remainer is the Logarithme of the Amplitude,
Example.
| Complement, 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 finde 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 thinges to be giuen in any Triangle, before the question can bee answered, which in this for finding the true Azimuth of the Sunne; you are to know or imagine your Latitude, the Complement thereof is [Page 3] one side of a Triangle (which is the distance betweene the Pole and the Zenith) the Complement of the Sunnes Declination is another side of the same Triangle (which is the distance between the Sunne and the Pole) then is the Complement of the Almicanter the other side (which is the distance betweene the Sunne and the Zenith.)
Heere haue you an oblique Sphaericall Triangle whose three sides are knowne, and it is desired to knowe the angle at the Zenith: whose quantitie being found is the Sunnes true distance from the North, (if the north Pole be eleuate) or the distance from the South, (if the south Pole be eleuate,) and in this question there are two cases.
The first Case.
THe first is, when you are on the same side of the Equinoctiall the Sunne is on: then are the Triangle sides all lesse then Quadrants, and may be resolued by Logarithme, 2. Booke, 6. Chapter, and 8. section: As thus,
Adde halfe the base, and halfe the difference of the containing sides together; and to the Logarithme thereof, adde the Loga: of the difference of them, out of which somme, substract the somme of the Loga: of the two sides, and the halfe of the remainer is the Loga: of an Arch, which being doubled, is the quantitie of the angle of the Zenith, or verticall angle.
Example.
| Latitude North▪ | 51.deg.30′. |
| Declination North | 20.00. |
| Almicanter | 48. 30. |
I demand the Azimuth?
Let
| P.Z. be the Complement of the Latitude | 38.d.30′. |
| P.S. being the base, by the Comp. of the Declinat. | 70.00. |
| S.Z. by the Complement of the Almicanter | 41.30. |
| And let the Angle P.Z.S. be sought for |
The forme 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 substract. | 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 Sunnes true distance from the North: which compared with the magneticall; the difference is the Variation.
The second Case.
THe other Case is, when you are on the one side of the Equinoctiall, and the Sunne on the other; then is the base P. S. more then a Quadrant, and is to be resolued by Logar: 2. Booke, 6. Chapter, 10. section: Thus,
Add the differentiall of ½ the somme of ye
legs; to the differentiall of ½ the difference of the legs: and from the Product substract the differentiall of ½ the true base, and the remainer shall be the differentiall of the alterne ½ base: which ½ alterne base added to the true ½ base is the greater case, M. S. also the same substracted from the same ½. true base is the lesser case P. M: distinguishing two right angled Triangles; which doe make knowne both their owne partes, and all the parts of the Triangle proposed.
| Data | Latitude | 51. | deg. | 30′. | North | I demaund the Azimuth? |
| Declination | 10. | 00. | South | |||
| Almicanter | 15. | 00. | — |
| P. Z. | 38. | deg. | 30′. |
| S. Z. | 75. | 00. |
| The somme | 113. | d. | 30′. | ||
| The one halfe | 56. | 45. | |||
| The difference | 36. | 30. | differentiall | 4221605− | |
| The halfe | 18. | 15. | differentiall | 11094182+ | |
| The base P.S. | 100. | d. | 00′. | 8672577+ | |
| ½ Base | 50. | 00. | differentiall | 1754259− | |
| 8626836 the differentiall of 22. d. 53′. the ½ alterne base. |
| 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. |
| 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″. |
| 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 Sunnes true distance from the North point, from which if you substract 90. the remainer leaueth 38. d. 42′. 30″ the distance from the East or West demaunded.
1. For the Sunnes Azimuth hauing no Declination.
ADde the Complement of the Latitude, to the complement of the Almicanter, which if the totall be more then a quadrant, substract 90. and set downe the sine of the remainer for the first number: Againe, adde the complement of the Latitude and the Almicanter, and adde the sine thereof to the former: from the one halfe of that totall substract your first number or sine, and set downe the remainer: Then,
As the ½ of the 2. first numbers added is in proportion to ye whole fine, so is the said remainer to the sine of the Sunnes 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, leaues 18. deg. 30′. whose sine 3173. is the first number. Againe, complement 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 therof 5849. from which substract 3173. the first number rests 2676. for the remaine: then say,
As 5849. the ½ of the 2. first numbers is to 10000. the whole sine, so is 2676. the remaine to the Azimuth desired.
Facit, 4575. whose arch 27. d. 14′. is the Azimuth from the East Southward.
2. When the Sunne hath North Declination, the 2. Complements being equall to a quadrant.
ADde the complement of the Latit. with the Almicanter only, and from ½ the sine thereof, substract the sine of the Declination, and setting downe the remainer,
As the ½ aforesaid, is to the whole sine, so is the remainer aforesaid to the sine of the Azimuth desired.
3. When the Sunne hath North Declination, the 2. Complements lesse then a quadrant.
ADde the complement of the Latit. and the complement of the Almican. setting downe the sine of the complement thereof, [Page 7] then adde the Almicanter and the complement of the Latitude, and from the sine thereof substract the former, setting downe ½. of the remaine for the first found number: againe substract the sine of the first Complement from the sine of the Declination and the remaine therof, againe substract from your first found number, and set the remaine 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 Sunne hath North Declination, and the 2. Complemements more then a quadrant.
ADde the complement of the Latitude and complement of the Almicanter, which being more then 90. substract 90. and set downe the sine of the remainer, then adde the Almicant. and complement of the Latit. and set downe the sine thereof, add both the sines together and take the ½ thereof for the first found number, then to the sine of the first 2. complements adde the sine of the Declination, and from that totall substract the first found, and set downe the remainer 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 Sunne hath South Declination, and the 2. Complements more then a quadrant.
ADde the 2. Complements, substract 90. set downe the sine of the remainer, adde also the Almicanter and complement of Latitude, adde both their sines and set downe ½ of the totall for the first found, then substract the sine of the Declination from the sine of the remaine of the first 2. Complements, and that remaine againe from the first found, which last remaine set downe and say,
As the first found is to the whole sine, so is the second found to the sine of the Azimuth desired.