Otherwise,

Let D be the difference of the fides AB, AC, then, (prop. 7.)

:: R: (fin.

4AB.AC : (BC+D)(BC—D) : : R2 : (fin.

2.

BAC)2, or

2√AB.AC : ✔ (BC+D)(BC—D) :: R : fia. - BAC.

Otherwise,

Let S be the fum of the fides BA and AC. Then (prop. 8.) 4AB.AC: (S+BC)(S—BC) :: R2 : (fin. 1⁄2 BAC)2 or 2√√AB.AC : √(S+BC)(S—BC) :: R. cof.

BAC.

Other folutions of this cafe might also be given from the corollaries to the 7th and 8th propofitions, but the above are fufficient.

When this cafe is refolved by either of the two laft rules, or without letting fall a perpendicular, the firft of the two is to be used when the angle fought is acute; the fecond when it is obtufe; and it will always be known whether it is acute or obtufe, by the fquare of the fide oppofite to it being lefs or greater than the fquares of the other two fides.

SCHOLIU M.

In all the calculations performed by the preceding rules, tables of fines and tangents are neceffarily employed, the construction of which it is therefore proper to explain. These tables ufually contain the fines, &c. to every minute of the quadrant from 1' to 90°, and the first thing required to be done is to compute the fine of 1'.

1. In the circle ABC the radius AD being unity, let BC be any arch, and CE its half; then, if DG the co-fine of BC be known, the co-fine of CE may be found. From E draw EH perpendicular to DC, so that EH may be the fine, and DH the co-fine of the arch CE. Join AB and BC, and let DQ be perpendicular to AB, then AB is bifected in Q. Now, the

triangles DHE, AQD are equal, the angle DAQ being equal to the angle EDH, (because each of them is the half of the angle CDB), and the angles DHE, AQD being right angles, and also the fide AD equal to the fide DE; therefore DH is equal to AQ. And because ABC is a right angle, the rectangle CA, AG is equal to the fquare of AB, (8. 6.)

that

that is, to four times the fquare of AQ or of DH; and therefore, half the rectangle DA, AH is equal to the fquare of DH, that is, half the rectangle contained by the radius, and the fum of the radius and the co-fine of an arch, is equal to the fquare of the co-fine of half that arch. And hence, this arithmetical rule, to the co-fine of any arch add the radius or 1, the fquare root of half the fum is the co-fine of half the arch.

2. If therefore the straight line CB be equal to the radius, the arch CB will be 60°, and DG will be equal to half the radius as was formerly fhewn (8. 8.), and therefore DH the co-fine of CE or of 30°, will be found by this propofition. And if CE be bifected in F, if to DH 1 be added, and the fum divided by 2, its fquare root, will be equal to DK, the co-fine of 15°. In the fame manner is found the co-fine of 7° 30′, of 3o, 45', and fo on till, after twelve bifections, the co-fine of 52'. 44". 03""'. 45'''' is found. Now, from the co-fine, the fine of the fame arch is found; for, from the fquare of the radius, that is, from 1, take away the fquare of the co-fine, the remainder is the fquare of the fine, and therefore its fquare root is the fine itself. Thus, the fine of 52′′. 44"". 03′′. 45""""' is found.

3. But it is manifeft, that the fines of very fmall arches are to one another nearly as the arches themselves. For it has been shewn, that the number of the fides of an equilateral polygon infcribed in a circle may be fo great, that the perimeter of the polygon and the circumference of the circle may differ by a line lefs than any given line, or which is the fame, may be nearly to one another in the ratio of equality. Therefore their like parts will alfo be nearly in the ratio of equality, fo that the fide of the polygon will be to the arch which it fubtends nearly in the ratio of equality; and therefore, half the fide of the polygon to half the arch fubtended by it, that is to fay, the fine of any very small arch will be to the arch itself, nearly in the ratio of equality. Therefore, if two arches are both very small, the firft will be to the second as the fine of the first to the fine of the fecond. Hence, from the fine of 52′′. 44"". 03''''. 45""" being found, the fine of 1' becomes known; for, as 52". 44". 03"". 45"""" to 1', fo the fine of the former arch to the fine of the latter. Thus, the fine of 1' is found 0.0002908882.

4

4. The fine of 1' being thus found, the fines of 2', of 3' or of any number of minutes are found by the following propofition.

THE ORE M.

Let AB, AC, AD be three fuch arches, that BC the dif ference of the firft and fecond is equal to CD the difference of the second and third; the radius is to the co-fine of the common difference BC as the fine of AC the middle arch, to half the fum of the fines of AB and AD the extreme arches.

D

M

Draw CE to the centre; let BF, CG, and DH perpendicular to AE, be the fines of the arches AB, AC, AD. Join BD, and let it meet CE in I; draw IK perpendicular to AE, alfo BL and IM perpendicular to DH. Then, because the arch BD is bifected in C, EC will be at right angles to BD, and will bifect it in I, and BI will be the fine, and EI the co-fine of BC or CD. And, fince BD is bifected in I and IM is parallel to BL, (2.6.) LD is alfo bifected in M. Now BF is equal to HL, therefore, BF+DH-DH+HL= DL+2LH=2LM+2LH2MH, or 2KI; and therefore, IK is half the fum of BF and DH. But because the triangles CGE, IKE are equiangular, CE: EI:: CG: IK, and it has been fhewn, that EI cof. BC, and IK (BF+DH); therefore R: cof. BC:: fin. AC : 1 (fin. AB+ fin. AD). Q. E. D.

=

B

AF GK

II

COR.

COR. Hence, if the point B coincides with A,

R: cof. BC:: fin. BC: fin. BD, or the radius is to the co-fine

2

of the half of any arch, as the fine of half the arch is to halfthe fine of the whole arch.

Therefore, the fine of 1' being given, and also its co-fine, because by this Cor. R: cof. 1':: fin.1' :— fin. 2′; and fo the fine of 2' is given. To find the fine of 3'; by the preceding Theor. R: cof. i' : : fin. 2' : 1⁄2 (fin. 1'+fin. 3′); therefore,^

fin. 1'+ fin. 3' is given, and fin. 1' being already known, fin. 3' is found.

In like manner,

for the fine of 4',

R: cof. 1: fin. 3':: (fin. 2′+fin. 4'); and so on for any number of minutes; and as the ratio of R to the co-fine of 1^ remains always the fame, the calculation by means of this Theorem is very easy.

It may be convenient to have the fame rule otherwife expreffed, let a, b, c, be three arches that differ by 1', then,

R: cof. 1':: fin. b : 1⁄2 (fin. a+fin. c), and if R = 1,

(fin. a + fin. c) = cof. 1' x fin. b, and confequently,

fin. c = 2 cof. 1' × fin. b

fin. a.

In this manner, the table of fines is computed. Then, because the co-fine of any arch is to the fine as the radius to the tangent of the arch, the table of tangents may be calculated from this proportion. But it must be observed, that when the tangents of the arches under 45° are known, the tangents of thofe above 45o may be more easily found by another rule.

For the tangents of the arches above 45° being the co-tangents of the arches under 45°, and the radius being a mean proportional between the the tangent and co-tangent of any arch, (1. Cor. def. 9.) if the difference between any arch and 45° be called d, tan. (45°-d) : 1 :: I: tan. (45°+d).

Laftly, The fecants are calculated by help of Cor. 2. Def. 9* where it is fhewn, that the radius is a mean proportional beween the co-fine and the fecant of any arch. If therefore, a be any arch, cof. a : 1 :: 1: fec. a.

The