Published by the same Author, and to be had of BLACK, KINGSBURY, & Co. Leadenhall - Street, London, and DEIGHTON & SONS, Cambridge. 1. The Principles of ANALYTICAL CALCULATION, 1 Vol. 4to. 1803. Price 8s. 2. A Treatise on ISOPERIMETRICAL PROBLEMS, and the CALCULUS of VARIATIONS, 8vo. 1810. Price 68. 3. The Elements of PLANE and PHYSICAL ASTRONOMY, 2 Vols. 8vo. Price £1. 13s. PLANE TRIGONOMETRY. CHAP. I. On the Division of the Circle into Degrees, Minutes, Seconds, &c. Definitions of Sines, Cosines, &c. ARCS THE MEASURES OF ANGLES. ART. 1. It is proved in the 33d Proposition of the sixth Book of Euclid, that, in equal circles, angles have the same ratio to each other, as the arcs on which they stand. Hence also, in the same circle, the angles vary as the arcs on which they stand; and consequently we may assume arcs as the measures of angles. In the circle ABDE, the arcs AB, AD, are measures of the angles ACB, ACD; and of the same angles, in the smaller circle abde, ab, ad, are the measures; which latter arcs have the same ratio to the arcs AB, AD, that the radius Ca has to CA, For, since, in the circle abde, the measure of four right angles is the whole circumference abde, LaCb:4 right ∠s :: ab: abde, therefore LaCb = 4 right ∠ s x ab abde AB Similarly, 4 ACB, or Lab = 4 right 4S × ABDE Hence, ab AB = abde ABDE = Ca • since, (Playfair's Geometry, edit. 2. p. 219.) the circumferences of circles are to one another as their radii. 2. If from the points B, b, two lines BF, bf be drawn making with CA equal angles CFB, Cfb, then, by the similar triangles CFB, Cfb, we have and hence, if, in a circle ABDE, we have determined the value of a line such as BF, we can always assign the value of a similar line bf, in another circle abde, provided the ratio of the radii CB, Cb, be known: for instance, if CB be called 1, and Cb, r, 3. * It is usual to divide the circumference of a circle into 360 equal parts, which parts are called degrees, and of which the symbol is no or 5o, if nor 5 be their number: each degree is also divided into 60 equal parts, which parts are called minutes; and of which the symbol is m' or '7', if mor 7 be their number : and, finally, each minute is divided into 60 equal parts, which parts are called seconds, and of which the symbol is t" or 35", if tor 35 be their number: thus, if AB (Fig. p. 1.) equals onefourth of the circle ABDE, AB contains 90 degrees, or, symbolically, AB=90°. If AB=th of the circumference ABDE, See the Note at the end of the Chapter. |