450 will represent half a quadrant AD, or half a right angle. 60° will represent one-third of the semi-circumference AG, or two-thirds of a right angle. [NOTE A.]

Obs. 3. Instead of determining the magnitude of any given angle, such as ACD, by the ratio of its arch AD to the whole circumference, it is preferable to determine that magnitude by the ratio which some right E line dependent upon the angle bears to the radius of the goniometrical circle, inasmuch as arches and circumferences do not en

ter into computation so easily as right lines. For instance, if GD, AH, be perpendicular to AC, meeting CD in D and H respectively, then the ratio of GD, or AH, to AC, determines the magnitude of the angle ACD. The reason is plain; because, by ART. 114, GEOM., all right-angled triangles which have their sides about the right angles in the same ratio, are equiangular to each other: thus, if ach be any right-angled triangle, whose sides ah, ac, about the right angle have the same ratio to each other as AH to AC, the angle ach must be equal to the angle at ACH. Consequently, if in any angle ach, the ratio of the perpendicular ah to the line ac (between this perpendicular and the vertex of the opposite angle) be given, the angle ach is determinate, or fixed, and we may assign its magnitude by means of the goniometrical circle. Ex. gr. Suppose the ratio of ah to ac be given that of Mn to PQ, we have only to raise Az perpendicular to AC, and take AH in the same ratio to AC as MN to PQ; the angle ACH will be equal to the angle ach (ART. 114, GEOм.), and

therefore the magnitude of the latter is assigned by specifying the ratio of the former to the angular unit, or, in other words, the number of degrees, minutes, and seconds it contains. Upon this principle rest the fundamental theorems of the present Science. We proceed to develope them in order.

LESSON II.

FUNDAMENTAL THEOREMS, ETC.

IN the goniometrical circle, it is convenient to make a distinction between a tangent of the circle and a tangent of an arch; likewise between a secant of the circle and a secant of an arch. Thus, the indefinite right line, az, is a tangent of the circle, while only a portion of it, AH, is the tangent of the arch AD; the line, HL, is a secant of the same circle, while only a portion of it, HC, is the secant of the same arch AD. We speak also indifferently of the tangent, secant, &c. of the arch, and of the angle to which the arch corresponds. Hence,

DEFINITION I. The tangent of an arch, or angle, is that portion of the geometrical tangent at one extremity of the arch, which is intercepted between the sides of the corresponding angle.

Thus, AH is the tangent of the arch AD, or of the angle ACD; and is generally written tan AD, or tan ACD.*

DEF. II. The secant of an arch, or angle, is that portion of a geometrical secant through one extremity of the arch, which is intercepted between a tangent at the other and the centre.

Thus, CH is the secant of the arch AD, or of the angle ACD; and is generally written sec AD, or sec ACD.

ARTICLE I. The square of the secant of an angle is equal to the sum of the squares of the tangent and of the radius.

* The square of tan ACD is thus written: tan2 ACD; and in general the index is put, for brevity, over the right shoulder of the goniometrical word, instead of that of the whole quantity.

Let ACD be any angle represented by A. represent AC by R, sec2 A = tan2 A + R2.

Then, if we

DEMONSTRATION. By ART. 47, Geoм. ch2 = AH2 + AC2. This was the assertion of the present Article.

DEF. III. The complement of an arch, or angle, is the difference between it and a quadrant, or right angle. Thus, in the last figure, the arch DB is the complement of the arch DA; and the angle DCB is the complement of the angle DCA.

Hence, if a represent any arch or angle, its complement will be represented by 90°

A.

Obs. 1. If BK be a tangent to the circle at в, then вK is the tangent of the arch BD, or of the angle BCD, i. e. bk is the tangent of the complement of ad, or of acd. This tangent is written tan. co. ACD, meaning the tangent of the complement of ACD; or more usually it is written thus: co-tan. ACD, meaning the same thing. In the same manner ck is the secant of the complement of ACD, and is written co-sec ACD.

Obs. 2. If the complementary angle BCD be called B, it follows from this Article, that

sec2 B = tan2 B + R2;

i. e. as the secant of B is the cosecant of A, and the tangent of в is the cotangent of A,

cosec2 A cotan2 A + R2.

ART. 2. The rectangle under the tangent and cotangent of an arch, or angle, is equal to the square of the radius. Let ACD (fig. prec.) be any angle, represented by a. Then tan A X cotan A = R2.

DEM. The angles HAC, ACB, CBK, being all right angles, AH is parallel to CB, and AC to вK. Consequently, by ART. 12, GEOM. the angle ACH = the angle CKB, and the angle AHC = the angle HCB. Therefore, by ART. 112, Geom., ah: BC:: AC: BK, .. (ART. 100, Geom.) AH X BK BC X AC AC2. This, &c.

DEF. IV. The sine of an arch, or angle, is the perpendicular drawn from one extremity of the arch to the opposite side of the angle.

Thus DG (fig. prec.) is the sine of the arch AD, or of the angle ACD, and is usually written sin AD, or sin ACD.

Obs. If DI be perpendicular to CB, then DI is the sine of the arch BD, or of the angle вCD; i. e. DI is the sine of the complement of AD, or of Acd. This sine is usually written co-sin ACD, meaning, as before, the sine of the complement of ACD; it is yet more briefly written cos ACD.

As the triangles DGC, DIC, are equiangular to each other, and have the radius CD a common side, they have all their corresponding parts equal, by ART. 7, GEOM. Hence, GC is equal to DI, and therefore Gc may be taken as the sine of BCD; i. e. as the cosine of ACD.

ART. 3. The square of the radius is equal to the sum of the squares of the sine and cosine of an arch or angle. Let ACD (fig. prec.) be the angle represented by a. Then, R2 = sin2 A + cos2 A.

DEM. BY ART. 47, GEOM., CD2 = DG2 + GC2. This, &c.

ART. 4. The ratio of the sine to the cosine of an angle is the same as that of the tangent to the radius.

Let ACD (fig. prec.) be the angle, represented by a. sin A tan A

Then,

DEM. BY ART. 112, Geom., DG AH: CG AC; alternando, LG ; CG:: AH: AC, (Application of Algebra

Obs. Hence, if the complementary angle BCD be called B, it follows by this ART. that

i. e. as the sine, cosine, and tangent of в are respectively the cosine, sine, and cotangent of A,

In other words, the ratio of the cosine to the sine of an angle is the same as that of the cotangent to the radius.

ART. 5. The rectangle under the secant and cosine of an angle is equal to the square of the radius.

Let ACD (fig. prec.) be the angle, represented by A. Then sec A X cos A = R2.

DEM. BY ART. 112, GEOM., CH: CD :: CA: CG; .. by ART. 100, Geom., ch × cg = CD × CA = CA2. This,

&c.

Obs. Hence it follows that, representing the angle BCD by B,

sec B X COS B =

R2;

i. e. as the secant and cosine of в are respectively the cosecant and sine of A,

cosec A X sin A = R2.

DEF. V. The versed sine of an arch or angle is the intercept of the diameter between the sine and tangent. Thus GA (fig. prec.) is the versed sine of the arch AD, or of the angle ACD, and is written ver. sin acd.

Obs. 1. It will easily be perceived that all the above definitions of goniometrical lines are applicable, whatever be the magnitude of the given angle, although we appear to have considered those only, such as ACD, which are less than a right angle.

Thus, if ACD' be the given angle, AH' will be its tangent, according to DEF. I.; because AH' is "that portion of the geometrical tangent intercepted between the two sides" CA, CD', (the latter being produced backwards through c, in order to meet the geometrical tangent at H'). Also BCD' being the difference between the angle

ACD' and a right angle, is the complement of the former, by DEF. III., and therefore BK' is the cotangent of ACD'. Again: CH' is the secant of ACD', according to DEF. II.; and CK', its cosecant.

Finally: D'G' is the sine of ACD', CG' its cosine, agʻ its versed sine, according to DEF. IV. and Obs., and DEF. V.