2. If the circumference of a circle be divided into 360 equal parts called degrees, each degree into 60 equal parts called minutes, and each minute into 60 equal parts called seconds; and so on; then, as many degrees, minutes, and seconds, &c. as are in any arch; so many degrees, minutes, seconds, &c. are said to be in the angle measured by that arch.

Corollary 1. Any arch is to the whole circumference of which it is a part, as the number of degrees and parts of a degree, in it, is to 360. And any angle is to four right angles, as the number of degrees and parts of a degree, in the arch which is the measure of the angle, is to 360.

Corollary 2. Hence also, it is evident, that the arches which measure the same angle, whatever be the radii with which they are described, contain the same number of degrees, and parts of a degree.

3. The supplement of an arch, or an angle, is, what that arch or angle wants of 1809; that is of a semicircle; thus the supplement of the arch DB, is DCI, the remaining part of the semicircle BEI.

4. The complement of an arch or angle, is, what that arch, or angle, wants of 90°, that is, of a quadrant; thus the complement of the arch BD, is BCA, the remaining part of the quadrant, DBA.

5. A sine or right sine of an arch, is a right line drawn from one extremity of that arch, perpendicular to the diameter touching the other

extremity. Thus BG is the sign of the arch AB, and of the angle ACB.

6. The versed sine of an arch, is that part of the diameter which is intercepted between its extremity and the sine. Thus, AG is the versed sine of the arch AB, or of the angle ACB.

7. The tangent of an arch, is a right line touching the circle, at one extremity of the arch, and meeting a diamcter of that circle passing through the other extremity of the arch. Thus AH, touching the circle at A, one extremity of the arch AB, and meeting the diameter CB, produced to H, after having passed, through B the other extremity of the arch AB, is the tangent of that arch, or of the angle ACB.

Corollary. The tangent of half a right angle is equal to the radius.

The secant of an arch is the diameter of the circle of which that arch is a part, which is produced to meet the tangent. Thus CH is the secant of the arch AB, or of the angle ACB.

The sine, tangent, and secant of any angle, as ACB, are also the sine, tangent and secant of its supplement, BCE. For, by the definition, BG is the sine of the angle BCE; and if BC be produced to meet the circle in I, then AH is the tangent, and CH the secant of the angle ACI, or BCE.

The sine, versed sine, tangent and secant of an arch, which is the measure of the angle, ACB, are to the sine, versed sine, tangent, and secant of any other arch which is also the measure of

the same angle, as the radius of the first arch is to the radius of the second arch.

Hence it appears that if tables be constructed, exhibiting, in numbers, the sines, tangents, and versed sines of certain angles to a given radius, they will exhibit the ratios, or relations, of the sines, tangents, and versed sines, of the same angles, to any radius whatever.

Such tables are constructed, calculated to a given radius for every degree, minute, and second of the quadrant.

The natural sines, tangents, cosines, &c. are calculated to a radius of 1, but the logarithmic sines, tangents, &c. are calculated to a radius of 10,000,000,000 or 1, with ten ciphers; so that the latter are the logarithms of the former, with ten, added to the index.

9. The cosine, cotangent, or cosecant of any angle, is the sine, tangent, or secant of the complement of that angle. Thus, supposing the angle ACD to be a right angle, then BF=CG, the sine of the angle BCD, is the cosine of the angle BCA; DK, the tangent of the angle, BCD, is the cotangent of the angle BCA; and CK, the secant of the angle BCD, is the cosecant of the angle BCA.

QUESTIONS.

What is trigonometry, and whence is the name derived? What are its two principal divisions? What is plane trigonometry? What is spherical trigonometry? What are the uses of trigonometry? Who, according to what is known of the matter, first cultivated trigonometry? By whom has trigonometry been successively improved? How does plane trigonometry suppose the circle to be divided?

Have any mathematicians divided the circle in a different way, than into 360°?, What inference is drawn from the geometrical demonstration, that any angles at the centre of a circle, have to one another, the same proportion as the arches intercepted between the lines that contain the angle? What is the first definition in trigonometry? What is the second definition, and what corollaries flow from it? What is the supplement of any arch, or angle? What is the complement of any arch or angle? What is a sine, or right sine of an arch? What is the versed sine of an arch? What is the tangent of an arch? What is the secant of an arch? What proportion is there between the sine, versed sine, tangent, and secant of one arch which is the measure of a certain angle, and the same parts of any other arch which is also the measure of the same angle? What do trigonometrical tables exhibit, and to what are they calculated? What are the cosine, and cotangent, and cosecant of any angle?

CHAP. VI.

TRIGONOMETRY- continued.

EXAMPLES OF THEOREMS IN PLANE TRIGONO

METRY.

Theorem.

In a right-angled plane triangle, as the hypothenuse is to either of the sides, so is the radius to the sine of the angle opposite to that side; and as either of the sides is to the other side, so is the radius to the tangent of the angle opposite to that side; or,

If, in a right-angled triangle, the hypothenuse

VOL. I.

be made the radius, the sides become the sines of the opposite angles; and if one of the sides be made the radius, the other side becomes the tangent of the opposite angle, and the hypothenuse becomes its secant.

Let ABC, in the preceding figure, be a rightangled plane triangle, of which AC is the hypothenuse. On A, as a centre, with any radius, describe the arch DE; draw EG at right angles to AB, and draw DF touching the arch at D, and meeting AC in F. Then EG is the sine of the angle A, to the radius AD or AE; and DF is its tangent.

The triangles AGE, ADF, are manifestly similar to the triangle ABC. Therefore, AC is to CB, as AE is to EG; that is, AC is to CB, as the radius to the sine of the angle A.

Again; AB is to BC, as AD is to DF. That is, AB is to BC as the radius is to the tangent of the angle A.

Corollary. In a right angled plane triangle, as the hypothenuse is to either of the sides, so is the secant of the acute angle adjacent to that side, to the radius.