The three sides, 11 AB, AC, and BC. SOUGHT. The side to the given an- The side The third angle ACB. One of the angle's A. SOLUTION. Sin B: sin A:: sin AC: sin BC, (24); the affection of BC is uncertain, except when it can be determined by this rule, that according as A+B is greater or less than 180°, AC+BC is also greater or less than 180°, (10.). From the unknown angle C, draw CD perpendicular to AB; then R cos A: tan AC: tan AD, (c. 2.); tan B: tan A:: sin AD sin BD. BD is ambiguous; and therefore AB = AD ± BD may have four values, some of which will be excluded by this condition, that AB must be less than 180°. From the angle required, C, draw CD perpendicular to AB. R: cos AC:: tan A: cot ACD, (c. 3.), cos A: cos B :: sin ACD: sin BCD, (25.). The affection of BCD is uncertain, and therefore ACB ACD + BCD, has four values, some of which may be excluded by the condition, that ACB is less than 180°. From C one of the angles not required, draw CD perpendicular to AB. Find an arc E such that tan AB : tan (AC+BC):: tan (AC— BC) tan E; then, if AB be greater than E, AB is the sum, and E the difference of AD and DB; but if AB be less than E, E is the sum and AB the difference of AD, DB, (29.). In either case, AD and BD are known, and tan AC: tan AD: R cos A. In the foregoing table, the rules are given for ascertaining the affection of the arc or angle found, whenever it can be done: Most of these rules are contained in this one rule, which is of general application, viz. that when the thing found is either a tangent or a cosine, and of the tangents or cosines employed in the computation of it, either one or three belong to obtuse angles, the angle found is also obtuse. This rule is particularly to be attended to in cases 5 and 7, where it removes part of the ambiguity. It may be necessary to remark with respect to the 11th case, that the segments of the base computed there are those cut off by the nearest perpendicular; and also, that when the sum of the sides is less than 180°, the least segment is adjacent to the least side of the triangle; otherwise to the greatest, (17.), The last table may also be conveniently expressed in the following manner, denoting the side opposite to the angle A, by a, to B by b, and to C by c; and also the segments of the base, or of opposite angle, by a and y. APPENDIX ΤΟ SPHERICAL TRIGONOMETRY, CONTAINING NAPIER'S RULES OF THE CIRCULAR PARTS. THE rule of the Circular Parts, invented by NAPIER, is of great use in Spherical Trigonometry, by reducing all the theorems employed in the solution of right angled triangles to two. These two are not new propositions, but are merely enunciations, which, by help of a particular arrangement and classification of the parts of a triangle, include all the six propositions, with their corollaries, which have been demonstrated above from the 18th to the 23d inclusive. They are perhaps the happiest example of artificial memory that is known. DEFINITIONS. 1. If in a spherical triangle, we set aside the right angle, and consider only the five remaining parts of the triangle, viz. the three sides and the two oblique angles, then the two sides which contain the right angle, and the complements of the other three, namely, of the two angles and the hypotenuse, are called the Circular Parts. Thus, in the triangle ABC right angled at A, the circular parts are AC, AB with the complements of B, BC, and C. These parts are called circular; because, when they are named in the natural order of their succession, they go round the triangle. 2. When of the five circular parts any one is taken, for the middle part, then of the remaining four, the two which are immediately adjacent to it, on the right and left, are called the adjacent parts; and the other two, each of which is separated from the middle by an adjacent part, are called opposite parts. Thus in the right angled triangle ABC, A, being the right angle, AC, AB, 90°-B, 90°-BC, 90°-C, are the circular parts, by Def. 1.; and if |