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8. Lemma II. Let AcB be a rectilinear angle: if, about c as a centre with any two distances ca", CA, two circles be described, meeting CA, cB, in A', Bo, A, B ; the arc AB will be to the whole circumference of which it is a part, as the arc A'B' to the whole circumference of which it is a part. By, lem. 1, arc AB : whole circum. :: angle AcB : 4 right angles, and, arc A'B' : its whole circum. :: angle A'CB’:4 right angles. Therefore, arc AB : whole circum. :: arc A'B' : whole circum. of its respective circle.


9. Let AcB be a rectilinear angle, if about c as a centre, with any radius CA, a circle be described, intersecting CA, CB, in A, B, the arc AB is called the measure of the angle AcB.

10. The circumference of a circle is supposed to be divided, or to be divisible, into 360 equal parts, called degrees; each degree into 60 equal parts, called minutes; each of these into 60 equal parts, called seconds ; and so on, to the minutest possible subdivisions. Of these, the first is indicated by a small circle, the second by a

single accent, the third by a double accent, &c. Thus,

47° 18' 34” 45", denotes 47 degrees, 18 minutes, 34 seconds, and 45 thirds. So many degrees, minutes, seconds, &c. as are contained in any arc, of so many degrees, minutes, seconds, &c. is the angle of which that arc is the measure said to be. Thus, since a quadrant, or quarter of a circle, contains 90 degrees, and a quadrantal arc is the measure of a right angle, a right angle is said to be one of 90 degrees.

11. The complement of an arc is its difference from a quadrant; and the complement of an angle is its difference from a right angle.


12. The supplement of an arc is its difference from a semicircle; and the supplement of an angle is its difference from two right angles.

13. The sine of an arc is a perpendicular let fall from . extremity upon a diameter passing through the other.

14. The versed sine of an arc is that part of the diameter which is intercepted between the foot of the sine and the arc.

15. The tangent of an arc is a right line which touches it in one extremity, and is limited by a right line drawn from the centre of the circle through the other extremitv.

#. The secant of an arc is the sloping line which thus limits the tangent.

17. These are also, by way of accommodation, said to be the sine, tangent, &c. of the angle measured by the aforesaid arc, to its determinate radius.

18. The cosine of an arc or angle, is the sine of the complement of that arc or angle: the cotangent of an arc or angle is the tangent of the complement of that arc or angle. The co-versed sine, and co-secant are defined similarly.

To ofty these definitions by the annexed diagram: let AB be an assumed arc of a circle described with the E M. radius Ac, and let AE be a qua- BNLs? drantal arc; let BD be demitted r G T perpendicularly from the extre- A'HP 3-Dī;A mity B upon the diameter AA'; \ Bs y


parallel to it let AT be drawn T.
and limited by cr: let GB and E’
EM be drawn parallel to AA’,

the latter . limited by CT or ct produced. Then Be is the complement of BA, and angle BoE the complement of angle BCA : BEA' is the supplement of BA, and angle BCA’ the supplement of BcA; BD is the sine, DA the versed sine, At the tangent, CT, the secant, GB the cosine, GE the coversed sine, EM the cotangent, and cM

the cosecant, of the arc AB, or, by convention, of the

angle AcB.

: Note. These terms are indicated by obvious con


Thus, for sine of the arc AB we use sin AB,

tangent.... ditto . . . . . . tan AB,
secant .... ditto...... sec AB,
versed sine ditto...... versin AB,
cosine...... ditto...... cos AB,
cotangent ... ditto...... cot AB,
cosecant.... ditto...... cosec AB,
coversed sine ditto...... coversin AB,

Corollaries from the Definitions.

19. (A). Of any arc, less than a quadrant, the arc is less than its corresponding tangent; and of any arc whatever, the chord is less than the arc, and the sine less than the chord. For, in the preceding diagram, the circular sector eAB is less than the triangle cAT, the former being contained within the latter. That is, by the rules for mensuration of surfaces, #CA x arc AB is less than cA x tan AT; whence, dividing by #cA, there results arc AB less than tangent At. In a similar way it may be seen that chord AB, is less than arc AB, and sine BD, less than chord AB. The same is also evident from the consideration that a right line AB is less than any curve line terminated by the same points, and the perpendicular BD less than the hypothenuse AB, of a right angled triangle ADB. (B). The sine BD of an arc AB, is half the chord Br of the double arc BAF. (c). An arc and its supplement have the same sine, tangent, and secant. (The two latter, however, are affected by different signs, + or -, according as they appertain to arcs less or greater than a quadrant: the reasons of this will be explained in a subsequent chapter.) (b). When the arc is evanescent, the sine, tangent, and versed sine, are evanescent also, and the secant becomes equal to the radius, being its minimum limit. As

the arc increases from this state, the sines, tangents,

secants, and versed sines, increase; thus they continue, till the arc becomes equal to a quadrant AE, and then the sine is in its maximum state, being equal to radius, thence called the sine total; the versed sine is also then equal to the radius; and the secant and tangent becoming incapable of mutually limiting each other, are regarded as infinite. (e). The versed sine of an arc, together with its cosine are equal to the radius. Thus, AD + Bo = AD + Dc = Ac. (This is not restricted to arcs less than a quadrant, as will be seen in the chapter on analytical plane trigonometry.) (F). The radius, tangent, and secant, constitute a right angled triangle cAT. The cosine, sine, and radius, constitute another right angled triangle cbb, similar to the former. So again, the cotangent, radius, and cosecant, constitute a third right angled triangle Mec, similar to both the preceding. Hence, when the sine and radius are known, the cosine is determined by Euc. i. 47. The same may be said of the determination of the secant, from the tangent and radius, &c. &c. &c. (G). Further, since cd : DB :: cA: AT, we see that the tangent is a fourth proportional to the cosine, sine, and radius. Also, CD : CB :: CA: CT; that is, the secant is a third proportional to the cosine and radius. Again, ca: GB::ce: EM ; that is, the cotangent is a fourth proportional to the sine, cosine, and radius. And, BD : Bc::ce : CM; that is, the cosecant is a third proportional to the sine and radius. (H). Thus, employing the usual abbreviations, we should have

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20. From these and other properties and theorems, some of which will be demonstrated as we proceed, mathematicians have computed the lengths of the sines, tangents, secants, and versed sines, to an assumed radius, that correspond to arcs from 1 second of a degree, through all the gradations of magnitude, up to a quadrant, or 90°. The results of the computations are oarranged in tables called Trigonometrical Talles for use. The arrangement is generally appropriated to two distinct kinds of these artificial numbers, classed in their regular order upon pages that face each other. On the left hand pages are placed the sines, tangents, secants, &c. adapted at least to every degree, and minute, in the quadrant, computed to the radius 1, and expressed decimally. On the right hand pages are placed in succession the corresponding logarithms of the numbers that denote the several sines, tangents, &c. on the respective opposite pages. Only, that the necessity of using negative indices in the o may be precluded, they are supposed to be the logarithms of sines, tangents, secants, &c. computed to the radius 10000000000. The numbers thus computed and placed on the successive right hand pages are called logarithmic sines, tangents, &c. The numbers of which these are the logarithms, and which are arranged on the left hand

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