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sine (P+) 2 sine (P+Q). cos}(P+&_ sine } (P+e) 8.

sinep-sineg 2sine }(P-2).cos}(P+2) sine } (P-2) Now if two fractions be equal to each other, the numerator of the one is to its denominator, as the numerator of the other is to its denominator. The first equation is therefore the same as Prop. xiii. (or Q. 112.) The sixth is the same as Prop. xiv. (or S. 112.) (H)

If we suppose B=A, we shall have, from the third equation, (E. 116.) cos?A=žrado +(1 rad. cos 2 A); and from the fourth equation of the same article.

Sine?A= } rad—(} rad. cos 2 A) (I) But when B=A, Q=0 (F. 116.) and cos o° = rad (D. 99.)

2 cos? JP hence 1 . rad + cos P=

From the third equation

rad (F. 116.)

2 sine P 2. rad-COS P=

From the fourth equation

rad (F. 116.)

2 sine · P. cos } p. 3. sine P=

From the first equation

rad (F. 116.)

tang / P

From the fourth equation rad+cos P

rad
cot IP
(G. 116.) (Z. 103.)

rad
sine P cot P

rad
5.-

From the third rad

rad

tang IP equation (G. 116.) rad +cos P cot P

rad 6.

From the sixth rad

rad?

tango JP equation (G. 116 and Z. 103.) rad— COS P

tang? P rad2 7.

From the sixth rad + cos P

rada cot IP equation (G. 116.) &c.

rad.sine (A+B) (K) Because tang (A+B)=

(N.104.) we shall

cos (A+B) have from the first and second equations (D. 115.)tang (A+B)= (sine A. cos B)+(sine B. cos A)

rad, and since sine A = (cos A . cos B)-(sine B . sine A) cos A . tang A

and sine B

cos B . tang B rad

rad ing these values and dividing the numerator and denominator

sine P

=

COS P

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COS P

(L. 104.) by substitut

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COS A . COS B

By subtracting the third equation from the first, and dividing by 2.

COS A . sine B 2.

- i sine (A+B)- sine (A-B) rad By adding the second and third equations together, and dividing by 2. 3.

cos (A - B)+} cos (A +B) rad By subtracting the second equation from the third, and dividing by 2. sine

sine B

= cos (A-B)- cos (A+B)

rad
; (F) If in the formulæ (E. 115.) there be substituted p for
A+B, and g for A-B, then a=} (P+Q), B=}(P-e), and we
shall obtain

2
g sine ) (P+Q). cos } (P-2)

A

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2

2: Sine. P-sine grad

sine 1 (P-2). cos } (P+Q)

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tang “U. 108.)=

COS A

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2 3. Cos p+cos Q= cos } (P+2). cos } (P-2)

rad

2
4. Cos Q-COS P = - sine I (P+). sine } (P-9)

rad
sine A tang A

rad (G) And because

(103 (Z. 103.) rad

cot A we shall obtain by division and reduction.

sine p+sine e sine ) (P+Q).cos 1 (p-e)_tang ? (P+2)
sine P- sine e cos } (P+Q).sine 1 (P-2)tang 1 (P-2)

sine p+sine g sine (P+e)_tang # (P+Q)
2.
cos P + cos2 cos ž (P+Q)

rad sine P+sine

cos ž (P -2) cot ž (P-e)
3.
cos & +cos P sine } (P-2)

rad
sine P-sine e sine (p-e)_tang # (P-e)

cos P+cos e cos Ž (P-2) rad
sine P-sineg cot } (P+2) cot } (P+)
5.
COS Q-COS P
sine (P+2)

rad cOS P+cos & cos } (P+). cos } (P-9)_cot } (P+e) 6. cos Q-cos (

sine } (P+e).sine ž (P-e) tang}(P-2) sine (P + e) 2 sine i(P+2).cos}(P+0) cos } (P+2) 7. sinep+sineg 2 sine }(P+e). cos (P-9) cos } (P-9)

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=

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.

sine (P+g)_2 sine } (P+g). cos}(P+&_ sine} (P+e) 8.

sinep-sineg 2sine (P-2).cos(P+2) sine 1 (P-2) Now if two fractions be equal to each other, the numerator of the one is to its denominator, as the numerator of the other is to its denominator. The first equation is therefore the same as Prop. xiii. (or Q. 112.) The sixth is the same as Prop. xiv. (or S. 112.)

(H) If we suppose B=A, we shall have, from the third equation, (E. 116.) cosA=krad +(rad. cos 2 A); and from the fourth equation of the same article.

Sine?A= }rado – (1 rad. cos 2 A) (I) But when B=A, Q=0 (F. 116.) and cos o° = rad (D. 99.)

2 cos? ŽP hence 1 .rad+cos P=

From the third equation

rad (F. 116.)

2 sine? P 2. rad-COS P=

From the fourth equation

rad (F. 116.)

2 sine } P.cos } P.. From the first equation 3. sine P=

rad (F. 116.)

sine P

tang P 4.

From the fourth equation rad+cos P

rad
cot IP
(G. 116.) (Z. 103.)

rad
sine cot } P

rad
5.

From the third rad

rad equation (G. 116.) rad + cos P cot P

rad 6.

From the sixth rad— COS P

rad?

tang? LP equation (G. 116 and Z. 103.) rad-cos P

tang? SP

rad 2 7.

From the sixth rad +cos P

rada equation (G. 116.) &c.

rad. sine (A+B)(N.104.)we shall (K) Because tang (A+B)=

cos (A+B) have from the first and second equations(D. 115.) tang (A+B)= (sine A cos B) +(sine B

rad, and since sine A = (cos A . COS B)-(sine B. sine A) cos A . tang a

cos B . tang B
and sine B-
rad

rad ing these values and dividing the numerator and denominator

P

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COS P

tang P

cot P

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(L. 104.) by substitut

PROPOSITION XIX.

by cos A.cos B, we obtain tang (A+B)=rad – (tanga. tang B)

tang A + tang B rad', but this has been already obtained by another process (C. 115.)

tang A-tang B
tang (
AB)=-

rad (C. 115.)
rad + (tang A. tangB)

rado-(tang A. tang B) * cot (A + B)=

(from above, and

tang A + tang B L. 104.)

rada +(tang A . tang B) cot (A + B)=

(from above, and

tang A-tang B L. 104.) GENERAL PROPERTIES OF SINES, TANGENTS, &c. OF ARCS, IN ARITHMETICAL PROGRESSION.

(Plate I. Fig. 2.) Let the three arcs BO, BE, and Ba, be in arithmetical progression; viz. let Ao be bisected in E, then AESOE is half the difference between the arcs Bo and BA and therefore be is half the sum of the arcs Bo and BA (C. 35.)

But BO + BA=B0+(BO+DE+AE)=BO+(BO+200)=2BO +205=2BE, that this the sum of the extreme arcs, Bo and ВА, is equal to double the mean arc BE.

Draw the diameter ACL, and join ol; draw Bz parallel to 0A, and join xz; also from L, draw Lw perpendicular to ow.

Then the following triangles are equiangular and similar, viz. CBR, VBZ, xzv, AOP, and olw. For CBR and vBZ are right angled at R and y, and have the angle at B common. xzv is right angled at v, and the angle at z is equal to the angle at B, for XzB is a right angle, being contained in a semicircle, and vBz is the complement of vzB

to a right angle, and so is xzv.

The triangle AOP having the sides ao and op parallel to the two sides zB and By of the triangle voz, is equiangular with it; the angle aol is a right angle, being contained in a semi-circle, therefore wol, the complement of AOP, is equal to the angle OAP: hence the angle ouw is equal to the angle AOP.

The following proportions, &c. are naturally derived from these similar triangles; but first we must observe that Ag+OD =AH=wl the sum of the sines of the extreme arcs; for Ah is parallel to wl, and the angles wlc and hac being alternate angles are equal (Euclid 29 of I.), therefore the right angled triangles CGA, CYL having one side ac=lc, and the angles equal, are equal in all respects; hence LY=AG, and wy=OD, consequently lw=AG+OD. Now cg+CD is the sum of the

(L)

(N) (O)

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cosines of the extreme arcs, but on account of the equality of the triangles

. Lyc and AGC, Yc=CG, therefore Yc+CD=YD= wo, is the sum of the cosines of the extreme arcs.

CB : BR::BZ : BV radius : sine of the mean arc::double the sine of the mean arc : versed sine of double the mean arc.

CB : BR::ZX (=2RC): zv (M) radius : sine of the mean arc:: double the cosine of

the mean arc : sine of twice the mean arc.

CB : BR :: A0 (=2on): 0P (=CD-CG)

radius : sine mean arc:: double the sine of the com(

mon difference between the arcs : difference between the cosines, or versed sines of the extreme arcs.

CB : BR::OL (=2cn): LW radius : sine mean arc::double the cosine of the com, mon difference between the arcs : the sum of the sines of the extreme arcs.

CB : CR::BZ (=2BR): Zv (P) radius : cosine of the mean arc::double the sine of

the mean arc : sine of double the mean arc.

CB : CR::ZX (=2CR): XY

radius : cosine mean arc:: double the cosine of the (Q)

mean arc : versed sine of the supplement of double the mean arc.

CB : CR::A0 (=20n): AP. (=AG-OD)

radius : cosine of mean arc.: double the sine of the (R)

common difference between the arcs : difference be, tween the sines of the extreme arcs.

CB : CR::OL (=2cn) : wo (=cG+CD)

radius : cosine of the mean arc:: double the cosine of (S)

the common difference between the arcs : sum of the cosines of the extreme arcs.

BR : CR::BV : ZV

sine of the mean arc: cosine of the mean arc:: versed (T)

sine of double the mean arc: sine of double the mean arc.

BR : CR::zv : XV (U)

sine of the mean arc : cosine of the mean arc::sine of double the mean arc : versed sine of the supplement of double the mean arc.

BR : CR::PO (=CD-CG): AP (=AG-20)

sine of the mean arc : cosine of the mean arc:: dif(W) ference between the cosines (=difference between the

versed sines) of the extreme arcs : the difference between the sines of the extreme arcs.

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