PROP. VII..

7. The square of the sine of half any arc or angle is equal to a rectangle under half the radius and the versed sine of the whole; and the square of its cosine, equal to a rectangle under half the radius and the versed sine of the supplement of the whole arc or angle.

This will be at once obvious from the definitions, and the diagram to prop. 1, of this chapter. For 4Ан* Whence Aн2 = {AC

AB2 AL. AD = 2AC. AD.

. AD.

Also, 4ch2 = LB2 = AL . LD = 2AC . LD. Whence CM2 = AC. LD.

PROP. VIII.

8. The rectangle under the radius and the sine of the sum or of the difference of two arcs, is equal to the sum or the difference of the rectangles under their alternate sines and cosines.

Let AB and BD be two unequal arcs, of the circle whose radius is AC; and let BD' = BD.

Then is AD the sum, and AD' the difference, of the arcs AB and BD. Also, BH = sine, ci cosine, of arc AB; DF =sine, CF cosine, of arc BD; DK

=

E'

sine, CK = cosine, of AD = AB + BD; CK GHKA DK' sine, CK' = cosine, of AD' = AB — BD.

The triangles CBH, CFG, being similar, we have, CB BH :: CF: FGEK, whence CB. EK BH. CF.

Again, the triangles CBH, DFE, are equiangular; for each has a right angle, H and E, and angle CBH = DFE, the former being equal to CFG, which is the complement to EFC, and the latter being also a complement to the same angle.

Hence, CB:CH:: DF: DE, and conseq. CB.DE CH. DF. But, since as above, CB. EK = BH.CF,

and CB. DE CH. DF,

we have, by addition, CB. DK

BH.CF + CH. DF,

which is the first part of the proposition.

Again, since the triangles DFE, D'FE', are equal as well as equiangular, ED E'D: the preceding rectangles may, therefore, be expressed thus,

CB. E'K' BH. CF,

and CB. D'E′ = ch. df,

of which the difference is CB. D'K' BH.CF CH . DF, which is the second part of the proposition.

Cor. If the two arcs become equal, then we have for the sum, rad x sin 2AB sin AB × 2 cos AB, agreeing

with prop. 2 of this chapter.

PROP. IX.

9. The rectangle under the radius and the cosine of the sum or the difference of two arcs, is equal to the difference or the sum of the rectangles under their respective cosines and sines.

Recurring to the same diagram, we have from the similar triangles CBH, CFG,

CB: CH:: CF: CG; whence CB. CG

CH.CF:

and the similar triangles CBH, and DEF, give

CB: BH:: DF: EF or KG; whence CB. KG = BH.DF. The difference of these rectangles is,

CB. CK CH.CF BM. DF; which is the first part of the proposition.

The equivalent rectangles to the preceding, are

CB. CG CH.CF,

and CB. K'GBH. DF;

the sum of which gives CB. CK' CH. CF + BH, DF, which is the second part of the proposition.

Cor. When AB and BC are equal, we have from this proposition rad x cos 2AB = cos 2AB sin 2AB.

Remark. The preceding figure is adapted to the case where not only AB, and BD, but their sum AD is less. than a quadrant. But the properties enunciated in these two propositions are equally true, let the magnitudes of

2

the arcs, of their sum, and their difference, be what they may. This might be rendered evident by a suitable modification of the diagram; and will still farther appear in the fourth chapter of this work.

PROP. X.

10. As the difference or sum of the square of the radius and the rectangle under the tangents of two arcs, is to the square of the radius; so is the sum or difference of their tangents, to the tangent of the sum or difference of the arcs.

Let AB, AD, be the two arcs; AT, AR, their tangents; also let вs in the first figure be the tangent of the sum of those arcs, and BS in the second figure the tangent of their difference: and from T the point R let RH, in both figures, be drawn parallel to Bs, or perpendicular to the radius GB.

B

R A

R T

D

D

Then, because of the similar triangles TAC, THR, we shall have,

TC:CA:: TR: RH, whence TC. RH = CA. TR,

TC: TA:: TR: TH, whence TC. TH TA. TR. Each of the last equal rectangles being taken from the square of TC, there will remain, TC2 TC. TH, or TC. (TC TH), or TC. CH = TC2 TA.TR. Now, TC .CH (or тc2 TA. TR) TC. RH (or CA. TR) :: ch

: RH :: CB (or CA): BS :: CA2: CA. BS; whence, alternando, Tc2 TA. TR: CA2:: CA. TR: CA.BS :: TR: BS Of this proportion, the first term TC2 TA. TR, because TC2 = CA2 + AT2, may also be expressed by ca2 + AT2 TA. TR, or by CA2 + AT2 or lastly, by CA2AT. AC; whence the proposition is manifest.

TA (TAAR),

PROP. XI.

11. As the sum of the sines of two unequal arcs, is to their difference, so is the tangent of half the sum of those two arcs, to the tangent of half their difference. Let AE and AB be two unequal

arcs, of which EK and BG, are the sines; and let Eк be produced to cut the circle in D, and BI be drawn parallel to the diameter A AA'. Draw ID, IE, from the centre I with the radius CA of the assumed circle, describe an arc dbe, and through 6 draw Loм parallel

G K

M

E

BG

to DE. Then, it is evident that NEKE + BG, is the sum of the sines of AE and AB, and ND = KE = KD — KN, is their difference. Also, since BIE (at the circumference) BCE (at the centre); = and BID= = BCD (Euc. iii. 20); bм is equal to the tangent of (AE+AB), and b to the tangent of (AE-AB), that is, of 1⁄2(AD AB.) But, by reason of the parallels DE, LM, we have EN: DN :: bм: 6L; which is evidently the theorem enunciated above.

Cor. The sum of the cosines of two arcs, is to their difference, as the cotangent of half the sum of those two arcs, is to the catangent of half their difference.

For, the cosines being the sines of the complements, it follows from the proposition that the sum of the cosines, is to their difference, as the tangent of half the sum of the complements, is to the tangent of half their difference. But half the sum of the complements of two arcs is the complement of half the sum of those two arcs, and half the difference of the complements is the same as the complement of half the difference; whence the truth of the corollary.

PROP. XII.

12. Of any three equidifferent arcs, it will be, as

1

radius, to the cosine of their common difference, so is the sine of the mean arc, to half the sum of the sines of the extremes; and, as radius, to the sine of the common difference, so is the cosine of the mean arc to half the difference of the sines of the two extremes.

Let AD', AB, AD, (in the figure to prop. 8, of this chapter), be the three equidifferent arcs. Then Dr

DF', is the sine of their common difference, and cr its cosine. Also FG, being an arithmetical mean between the sines DK, D'K', of the two extreme arcs, is equal to half their sum, and DE equal to half their difference. By reason of the similar triangles CBH, cfe, DFE,

we have, CB CF :: BH: FG,

and CB: DF :: CH: DE;

which are the analogies in the proposition.

Cor. 1. From the preceding proportions, we have

Cor. 2. Hence, if the mean arc AB be one of 60°, its cosine CH, will (prop. 5) be equal to CB, and DK — D ́k' DF: consequently DK will in that case equal Df D'K'. From this conjointly with the preceding corollary result these two theorems:

+

(A). If the sine of the mean of three equidifferent arcs (radius being unity) be multiplied into twice the cosine of the common difference, and the sine of either extreme be deducted from the product, the remainder will be the sine of the other extreme.

(B). The sine of any arc above 60°, is equal to the sine of another arc as much below 60°, together with the sine of its excess above 60°.

Remark. From this latter corollary, the sines below 60° being known, those of arcs above 60° are determinable by addition only.