are the cosines. Hence, in any formula involving sin. A, cos. A, &c.

calculated for a radius = 1, substitute instead of sin. A, cos. A; sin. A cos. A

r

r

,

and the resulting formula will belong to lines

drawn in a circle, of which the radius is r: for instance,

And if in any formula, any power of sin. A, or of cos. A, such as sin.3 A, sin." A, cos. A, or cos." A, occurs, the radius being 1, then,

ing formula will belong to a circle of which the radius is r.

This is the rule; but, since it would leave a Trigonometrical formula with fractional terms (the denominators being powers of the radius) it may, with advantage, be modified and made more convenient. Thus, suppose a form should occur such as

cos. n A = a.cos." A + b. cos." -2 A+ &c. to radius 1; then, by what had preceded, the reduced form to a radius r, is

cos. n A a. cos." A COS." -2 A

+6.

γη-2

+ &c.

r

and, cleared of fractions, is

r-1 cos. n A = a.cos." A + b.r2. cos." -2 A + &c.

Here, cos. n A is multiplied by r2-1, cos." -2 A by r2, &c.; that is, if we choose to call cos." A a quantity of n dimensions, cos." -2 A, a quantity of n - 2 dimensions, cos. A x sin. A, a quantity of two dimensions, we may announce the preceding rule under the following simple form :

Multiply each term of a Trigonometrical formula, in which the radius = 1, by such power of r, as shall make it of the same dimensions with the term of the highest dimensions; the resulting formula will be true when the radius is = r.

Thus, if

cos. 3 A=4.cos.3

3. cos. A

(rad3. = 1),

since cos.3 A, the term of the highest, is of three dimensions, and cos. 3 A, cos. A, are of one dimension, we have

r2. cos. 3 A = 4. cos. A 3 r2 cos. A.

19. The Trigonometrical symbols, such as sin. A, tan. A, &c. that have been obtained, are merely general ones, and, hitherto, no methods have been given of assigning their values in specific values of the angles. The general methods for this purpose will be given in a subsequent part of the Treatise; but, even at this stage, by peculiar artifices, we may, in certain simple cases, assign the arithmetical values of the sines and cosines of angles. For example,

If (fig. p. 14.) AB=BQ, that is, if AB=half a quadrant, or expressed in degrees, if AB or ABC=45°, since 2 BCQ=45°, also cos. 45°=sin. 45°, but cos. A + sin. A=1, (1 = radius in this case); therefore, 2(sin. 45°)=1, and sin. 45° = =.7071068, √2

1

or, (see the preceding rule)=7071.068, to a radius = 10,000.

If ACB

1

==90°=30°, since BCB'=2. ACB=60°, and since

3

∠CBB' = ∠CB' B=60°, the triangle BCB' is equilateral, and consequently BB' (the chord of 60°) = radius CA = 1, and,

BF sin. 30° =

(see Art. 10.)=

112

BB' =

1

112

=.5, and ... cos. 30° or sin. 60°

√(1-1)==.8660254, and (see p.

8660.254 to a radius = 10,000, and 8660254 to a radius = 1000000.

18.)=

Hence may be proved, what was asserted in p. 3, that the sines of arcs do not vary as the arcs themselves. For, the

1

sin. 30° = radius=,... sin. 90°; in other words, the sines

1

2

2

are as 1 to 2, whilst the arcs are as 1 to 3.

The values of the tangent may be found, in the above cases,

We may here direct the Student's attention to the superior augmentation of the tangent above that of the sine; for, we have

In the same manner as we have found, in the preceding cases, the values of the tangents, we may find those of the versed sine, secant, &c.

* We have adhered, in this Chapter, to the antient and common division of the circle. But, in most of the French scientific treatises that have, of late years, been published, the circumference is divided first, into 400 equal parts or degrees, then, each degree into 100 equal parts, or minutes, then, each minute into 100 equal parts or seconds: so that a French degree is less than an English in the proportion of 90 to 100: a French minute less than an English, in the proportion of 90 × 60 to 100 × 100: and a French second less in the proportion of 90 x 60 x 60 to 100 x 100 x 100: hence, if n be the number of French degrees, the

which last form points to an easy arithmetical operation for finding the number of degrees in the English scale from the number in the French scale: since from the proposed number we must subtract the same, after the decimal point has been moved one place to the left:

Examples: What number of degrees, minutes, &c. in the English scale correspond to 730, to 71° 15', and to 26°.0735, in the French scale.

In the next Chapter we will proceed to investigate certain expressions for the sine and cosine of the sum and difference of two arcs, in terms of the sines and cosines of the simple arcs. Such expressions are, in this science, very important, since from them, by an easy derivation, may be made to flow almost all other Trigonometrical formulæ. ***

This operation of reducing French to English degrees may be superseded, and rendered less liable to mistake, by means of the opposite Table, in which, as it is usual, the reduction is effected simply by addition.

Example to the Table.

Reduce 260.0735 to English degrees, &c.

French.

6 0

English.

By the Table, 20° Ο' Ο"....18° 0′ Ο
Ο............. 5 24 0
0.07 0 0 3 46.8
0 Ο 30......00 9.72
Ο 5.............. 0 0 1.62

0