in terms of one another: thus, since by the forty-seventh Proposition of the first Book of Euclid,

Again, by the similar triangles CFB, CAT,

CF : FB :: CA: AT;

In like manner we may easily deduce from the similar triangles CQt, Cg B, the values of Qt and Ct, the co-tangent and cosecant (see p. 6.) of the arc AB, (= A),

Or, we may dispense with this second set of similar triangles (CQt and Cg B) and deduce the values of the co-tangent and co-secant from the previous values of the tangent and secant, by means of their definition (see p. 6.)

Thus, the co-tangent of an arc is the tangent of the complement of that arc. If A be the arc and Q a quadrant, Q-A is the complement: now by 1. 19. of p. 8.

... co-tan. A [the same as tan. (Q-A)] = r.

Cos. A
sin. A

If it were worth the while it would be easy to express, under different terms, the preceding equalities: for instance, we may thus express the two latter.

B

The radius is a mean proportional to the secant and cosine of

an arc, and, also, to the co-secant and sine.

The radius is also a mean proportional to the tangent and co

tangent: which may be thus deduced,

and this same proportion is immediately deducible from the similar triangles CQt, CAT (fig. p. 7.).

The right line AB is (see p. 4.) the chord of the arc AB. If the right line be bisected at n and Cn be drawn perpendicularly

to AB, then the point m, where Cn produced meets the circle, bisects the arc AB (Euclid, Book III, Prop. XXX.); therefore

But An is by the definition of p. 3. 1. 24, the sine of Am. the chord (An B), therefore, of an arc A is double the sine (An) of half the arc : or, which is the same proposition, the

():

2

sine of an arc (A) is half the chord of twice that arc (2 A).

Instead of making A = the arc AB, make, for convenience, 2A to represent it, and let v represent mn the versed sine of Am or Bm (= A): then, since (Euclid, Book III. Prop. 35.),

mnxnp = A n2,

vx (2r – v) = sin.2 A,

or, 2rv = sin.2 A + v2.

But An2 + mn2 = A m2 (the line A m),

If we multiply the two last expressions (11. 13, 14.) for cos. A, we have

The supplement of an arc is the difference between it and a

semicircle: accordingly,

BQa is the supplement of AB,

ab is the supplement of AQb,

and ab is the supplement of AQad.

Now (see the definition) BF is the sine of BQa: for BF is

drawn from one extremity B of the arc BQa perpendicularly to the diameter a A passing through a the other extremity: but, as we have seen (p. 4.), or by the same definition, BF is the sine of AB. Similarly, of is the sine equally of the arc ba and the arc AQ b:ed the sine of ad and aQd: and, generally, the sine of the supplement of an arc is equal the sine of the arc itself.

**We have already (see pp. 6, 7.) in one or two instances, expressed by means of general symbols certain equalities that subsist between the sines and cosines of arcs, which, though different, may be said to be related. Such modes of expression are, in the actual

**

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The parts included within are less essential than the other parts, and may, in a first or partial perusal, be passed over by the Student.