less than AB (Hyp. and 10. 5), let XB be the excess of AB above AX, and let EF, EI, EK, GH, GL, GM, &c. be continued, till EK may want of AB by a magnitude less than XB (Hyp.), therefore EK is greater than AX, and the ratio of EK to CD is greater than that of AX to CD [8. 5], or [Constr.] of EF to GH, or, which is equal (Hyp.), of EK to GM; since therefore the ratio of EK to CD is greater than that of the same EK to GM, CD is less than GM (10. 5), contrary to the supposition. Therefore the ratio of AB to CD is not greater than that of EF to GH.

Let now the ratio of AB to CD be, if possible, less than of EF to GH, and let CZ be a fourth proportional to EF, GH and AB, the magnitude CZ is less than CD [Hyp. and 10. 5], let ZD be the excess of CD above CZ, and let EF, EI, GH, GL, &c. be continued, till GM may want of CD by a magnitude less than ZD (Hyp.), therefore GM is greater than CZ, and the ratio of EK to CZ greater than of EK to GM (8. 5), or (Hyp.) of EF to GH, or (Constr.), of AB to CZ; since therefore the ratio of EK to CZ is greater than of AB to the same CZ, the magnitude EK is greater than AB [10. 5], contrary to the supposition; therefore the ratio of AB to CD is not less than of EF to GH; and it has been shewn, not to be greater than it; therefore the ratio of AB to CD, is equal to that of EF to GH.

Theor. 2.-If there be two magnitudes (AB and CD), and two others (EF and GH) both severally greater than them, and the latter, by repeated diminutions, always, in such diminutions, retaining the same ratio to each other, and remaining greater than the former, approach nearer and nearer to equality with the same former, so as at length to exceed them, by magnitudes less than any given ones; the former are to each other in the same ratio.

B
A-X

D
C-Z

Let EF, EI, EK, &c. and GH, GL, GM, &c. be the continually decreasing magnitudes; and, if the ratio of AB to CD be not equal to E|-|-F G-|-|-F G|-|-H that of EF to GH, let it, if

KI

ML

possible, be greater or less than it; and first, let it be greater, and let CZ be a fourth proportional to EF, GH and AB, the magnitude CZ is greater than CD (Hyp. and 10. 5), let DZ be the excess of CZ above CD, and let EF EI, GH, GL, &c. be continued, till GM exceed CD by a magnitude less than

DZ [Hyp.], therefore GM is less than CZ, and the ratio of EK to CZ is less than of EK to GM [8. 5], or [Hyp.], of EF to GH, or [Constr.] of AB to CZ; since then the ratio of AB to CZ is greater than of EK to the same CZ, the magnitude AB is greater than EK (10. 5), contrary to the supposition, therefore the ratio of AB to CD is not greater than of EF to GH.

Let now the ratio of AB to CD be, if possible, less than of EF to GH, and let AX be to CD, as EF to GH, AX is greater than AB (Hyp. and 10. 5), let BX be the excess of AX above AB, and let EF, EI, FH, FL, &c. be continued, till EK exceed AB by a magnitude less than BX [Hyp.], therefore AX is greater than EK, and the ratio of GM to AX is less than of GM to EK (8. 5), or (Hyp.) of GH to EF, or (Constr. and Theor. 3. 15. 5) of CD to AX; since therefore the ratio of CD to AX is greater than of GM to the same AX, the magnitude CD is greater than GM (10. 5), contrary to the supposition; therefore the ratio of AB to CD is not less than of EF to GH, and it has been shewn, not to be greater than it; therefore the ratio of AB to CD, is equal to that of EF to GH.

125

ELEMENTS OF PLAIN TRIGONOMETRY.

TRIGONOMETRY, is that science, whereby, from having certain sides or angles of triangles given, the other sides and angles may be found.

As these triangles are of two kinds, namely, those which, being in a plain, are bounded by right lines, and those which, being on the surface of the globe, are bounded by arches of great circles of the globe; trigonometry is usually divided into two kinds, Plain and Spherical.

Plain Trigonometry, is that, which treats of plain triangles. Note, as there will be occasion hereafter of making other citations, besides those from the preceding elements of Euclid, citations from these will be marked Eu, from Plain Trigonometry, Pl. Tr.

DEFINITIONS.

1. A degree of a circle, is an arch thereof, equal to a three hundred and sixtieth part of its whole circumference; a minute, an arch, equal to the sixtieth part of a degree; a second, an arch, equal to the sixtieth part of a minute, and so forth.

2. An arch of a circle, described from the concourse of the legs of a rectilineal angle, as a centre, included between these legs, is said to be, the measure of the angle; which is said to be, of as many degrees, minutes, &c. as is the arch so included.

Schol. The number of degrees, minutes, &c. included between the legs of the angle, is not varied, by varying the length of the radius; since the intercepted arches whether small or great, have the same ratio to their whole circumferences, by Cor. 2. 33. 6. Eu. and therefore contain an equal number of these degrees, minutes, &c. (By converse of Theor. 2. 15. 5. Eu).

3. A quadrant of a circle, is a fourth part of its circumference.

4. The difference, of an angle from a right angle, or of an arch from a quadrant, is called the complement, of that angle or arch. Thus, if CG be perpendicular to AB, A the angle DCG is the comple-* ment of the angle BCD or ACD; and the arch GD, of the arch BD or AGD.

K

C

2

H

5. The complement, of an angle to two right angles, or of an arch to a semicircle, is called the supplement, of that angle Thus the angle ACD is the supplement of BCD, and the arch AGD of BD.

6. The sine of an arch (BD or AGD), or of the angle (BCD or ACD) of which it is the measure, is a perpendicular (DE), let fall from one extremity (D) of the arch, on the diameter (AB), passing through its other extremity (B or A).

7. The versed sine of any arch (BD), or of its corresponding angle (BCD), is the segment (EB) of the diameter (AB) drawn through one extremity (B) of the arch, between that extremity and the perpendicular (DE) let fall on the diameter from the other extremity [D].

8. The tangent of an arch (BD or AGD), or of its corresponding angle (BCD or ACD), is a right line (BF), drawn from one extremity (B) of the arch, and touching it in that extremity, to the diameter (CDF) produced, which passes through its other extremity (D).

9. And the segment (CF) of the diameter, so produced and meeting the tangent, between the centre and tangent, is called the secant of the same arch or angle.

10. The cosine of any arch or angle, is the sine of its complement to a quadrant or right angle; and the cotangent of any arch or angle, is the tangent, and the cosecant of any arch or angle, the secant of such complement.

Thus KD or CE is the cosine, GH the cotangent, and CH the cosecant, of the arch BD or angle BCD; being the sine, tangent and secant, of the arch GD or angle GCD,

Cor. 1 to these definitions.-The sine, tangent or secant, of any arch or angle, is the sine, tangent or secant of its supplement, or complement to a semicircle or two right angles.

For it is manifest from these definitions, that the same right lines are the sine, tangent and secant of the arches BD and AGD, and of the angles BCD and ACD.

Cor. 2-The sine of a quadrant or right angle, is equal to the radius.

Cor. 3. The tangent of half a quadrant or half a right angle is equal to the radius; for if the angle BCF were half a right angle, the angle CBF being right (18. 3 Eu.), the angle CFB would be half a right angle (32. 1 Eu.), whence, the angles BCF, BFC being equal, the tangent BF would be equal to CB (6. 1 Eu.), or to radius.

Cor. 4.-The radius is a mean proportional between the tangent (BF) and cotangent (GH) of any arch (BD).

The triangles CBF, HGC, having the angles at B and G right, and the angles at C and H equal, being alternate angles formed by CH meeting the parallels CB, GH (29. 1 Eu.), are equiangular (32. 1 Eu.), therefore BF is to CB, as CG or CB to GH (4. 6 Eu).

Cor. 5. The tangents of any two arches of the same circle, and, of course, of any two angles, are reciprocally as their cotangents.

For the rectangle under the tangent and cotangent of the arch BD, is equal to the rectangle under the tangent and cotangent of any other arch of the circle, each of these rectangles being equal to the square of radius [preced. cor. and 17. 6 Eu.], and therefore the tangent of D is to the tangent of that other arch, as the cotangent of the same arch is to the cotangent of BD (16. 6 Eu).

7

Cor. 6. The radius is a mean proportional, between the cosine (CE) and secant (CF), or between the sine (DE) and cosecant (CH), of any arch (BD).

Since ED and BF are parallel, CE is to CB, as CD or CB is to CF (2. 6 and Schol. 18. 5 Eu).

And since KD is parallel to GH, CK or DE is to CG, as CD or CG is to CH. [2. 6. and Schol. 18. 5. Eu].

Cor. 7. The sines of any two arches of the same circle, arc reciprocally as their cosecants; and the cosines, reciprocally as the secants.

Since the rectangle under the sine and cosecant of BD, is equal to the rectangle under the sine and cosecant of any other