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. Hutton, 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 o 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 joined the angle LBA in a semicircle is §s a right angle, and is therefore equal to <!) i angle BDA, BD being perpendicular to o LA. Hence, the triangles BDA, LBA, LTC D 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 BN, AB is twice the sine BH of that arc, and BD 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 S the proposed arc, let the tangent TA be produced downwards till TA + AR = CT the secant. Then, since angle R is the 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 5. The chord of 60° is equal to the radius of the circle; 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 = . (180° - 60°) = 60°. That is, the three * angles of the triangle ABC are equal, and therefore the triangle is equilateral. Conseq. AB = radius. C D 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 = to - 2 BC = 2 radius. . Cor. If with the same radius an arc were described from centre B, then cI would become the sine of 30°, which is consequently half the radius. PROP, WI. 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°. PROP, VII. . . 7. The square of the sine of half any arc or angle is equal to a rectangle under half the radius and the versed sine of the whole; and the square of its cosine, equal to a rectangle under half the radius and the versed sine of the supplement of the whole arc or angle. This will be at once obvious from the definitions, and the diagram to prop. 1, of this chapter. For 4An” = A B* = AL. AB = 2Ac. AD. Whence AH* = }Ac • AD. Also, 4ch” = LB = AL . LD = 2Ac. Ld. Whence c11° = }AC. L.D. we have, by addition, cb . DK = BH. CF + CH. DF, which is the first part of the proposition. Again, since the triangles DFE, D'FE', are equal as well as equiangular, Ed = E'D: the preceding rectangles may, therefore, be expressed thus, CB. E'K' = BH. CF, and cB. D'E' = ch. DF, of which the difference is cB. D'K = BH. CF – CH. DF, which is the second part of the proposition. Cor. If the two arcs become equal, then we have for the sum, rad x sin 2AB = sin Ah x 2 cos As, agreeing with prop. 2 of this chapter. PROP. IX. 9. The rectangle under the radius and the cosine of the sum or the difference of two arcs, is equal to the difference or the sum of the rectangles under their respective cosines and sines. Recurring to the same diagram, we have from the similar triangles CBH, CFG, - o cB : CH :: CF : cq; whence ce. c6 = CH. CF: and the similar triangles cBH, and DEF, give CB : BH :: DF: EF or kg; whence CB'. KG = BH. DF. The difference of these rectangles is, cB. cK = ch. CF – BH. DF; which is the first part of the proposition. he equivalent rectangles to the preceding, are CB. CG = CH. CF, and cB. K’g = BH. DF; the sum of which gives CB. CK = CH. CF + BH. DF, which is the second part of the proposition. Cor. When AB and Bc are equal, we have from this proposition rad x cos 2AB = cos’AB – sin *AB. Remark. The preceding figure is adapted to the case where not only AB, and Bro, but their sum AD is less than a quadrant. But the properties enunciated in these two propositions are equally true, let o magnitudes of |