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reasons of this will be explained in a subsequent chapter.) (D). 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 + BG = 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 CDB, 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, cq: GB::cE: EM ; that is, the cotangent is a fourth proportional to the sine, cosine, and radius. And, BB : 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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These, when unity is regarded as the radius of the circle, become I. cos = y(1 — sin”). 2. tan = v(seco – 1). 3. sec = y(1 + tan”). 4, cosec = y(1 + cot”). cos sin

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 ra. dius, that correspond to arcs from 1 second of a degree, through all the gradations of magnitude, up to a quadrant, or 90°. #. of the computations are arranged in tables called Trigonometrical Tables 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 I, 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 logarithms 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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pages, are called natural sines, tangents, &c. In the small pocket tables these are usually omitted, and the logarithmic alone retained, as the most useful and expeditious in operation.

Of the various trigonometrical tables which have been published at different times, those which deserve the warmest recommendation, as most accurate and best. fitted for general use are “The Mathematical Tables” of Dr. Hution, in one vol. royal 8vo.; and the stereotyped “Tables Portatives de Logarithmes, par Francois Callet,” printed also in royal 8vo. in 1795. Dr. Hutton’s work contains a copious and valuable introduction, comprizing the history, nature, construction and use of logarithmic and trigonometrical tables. The introduction to Callet’s tables, likewise exhibits directions for their use, and some of the best formulae employed in their construction.” Of small tables for the pocket the best with which I am acquainted are those of Mr. Whiting, and the stereotyped tables of the Rev. F. A. Barker.

CHAPTER II.

General Properties, and Mutual Relations of the Lines and Angles of Circles and Plane Triangles.

PROP. I.

w 1.THE chord of any arc is a mean proportional between the versed sine of that arc and the diameter of the circle. ~

* The nature and use of logarithms being fully explained in Dr. Hutton's valuable work, and, indeed, in every collection of logarithmic tables which a student ought to possess, I think it entirely unnecessary to occupy any portion of this introduction by an elucidation of the properties of those useful numbers.

In the marginal figure, AB is the chord, and AD is the

versed sine of the arc AB. BL being Tes joined the angle LBA in a semicircle is N a right angle, and is therefore equal to angle RDA, BD being perpendicular to 11 LA. Hence, the triangles BDA, LBA, L CD A

are similar, and we have AD: AB :: AB : AL.

PROP. II.

2. As radius : cosine of any arc :: twice the sine of that arc : the sine of double the arc.

In the preceding figure ch is the cosine of the arc RN, AB is twice the sine BH of that arc, and Blo is the sine of AB the double arc. From the similar triangles AcH, ABD, we have Ac: CH :: AB : BD.

PROP. III.

3. The secant of any arc is equal to the sum of its tangent, and the tangent of half its complement.

In the annexed diagram, where AB is the proposed arc, let the tangent TA be produced downwards, till TA + AR = CT the secant. Then, since angle R is the complement of AcR, and of #ATC, it fol- T lows that Acr = }ATC, that is - A comp. /*) Act. But AR is the tangent of Ack, or C A. of the arc AG; whence the proposition is s ) manifest.

*

- ProP, IV.

4. The sum of the tangent and secant of any arc, is equal to the tangent of an arc exceeding that by half its complement.

Produce AT, the tangent of the assumed arc, till the prolongation Ts becomes equal to ct the secant, and join cs, intersecting the quadrantal arc AF in D. Then, because Ts = Tc, angle Tcs = Tsc = TCE, by reason of

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5. The chord of 60° is equal to the radius of the cir- cle; the versed sine, and cosine of 60° are each equal to half the radius; and the secant of 60° is equal to double the radius. Let AB be an arc of 60°, AB its chord, cD T its cosine, AD its versed sine, CT its secant. Then, - 1. Since ACB = 60°, and CB = CA, A = B = 4 (180° – 60°) = 60°. That is, the three * angles of the triangle ABC are equal, and therefore the triangle is equilateral. Conseq. AB = radius. C ID A. 2. CD = DA = }cA, because in an isosceles or equilateral triangle the perp. bisects the base. 3. BAT = comp. 60° = BTA ; therefore BT = BA.; and TB + BC = Tc = 2BC = 2 radius. Cor. If with the same radius an arc were described from centre B, then CD would become the sine of 30°, which is consequently half the radius.

PROP. VI.

6. The tangent of an arc of 45° is equal to the radius.

Suppose AB in the last figure but one were an arc of 45°; then would ACB be half a right angle, and consequently its complement ATc. The sides AT, and Ac, opposite to those angles, would then be equal; that is, tan 45° = radius.

Cor. From this and prop. 5, it is evident that the sine of 30°, tangent of 45°, and secant of 60°, are in the ratio of the numbers 1, 2, and 4; or that tan 45° is a mean proportional between sin 30° and sec 60°.

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