spherical triangles may always be found that have a side and the angle opposite to it the same in both, but of which the remaining sides, and the remaining angle of the one are the supplements of the remaining sides and of the remaining angle of the other, each of each. Though the affection of the arc or angle found may in all the other cases be determined by the rules in the second of the preceding tables, it is of use to remark, that all these rules, except two, may be reduced to one, viz., That when the thing found by the rules in the first table is either a tangent or a cosine; and when, of the tangents or cosines employed in the computation of it, one only belongs to an obtuse angle, the angle required is also obtuse. Thus, in the 15th case, when cos AB is found, if C be an obtuse angle, because of cos C, AB must be obtuse; and in case 16, if either B or C be obtuse, BC is greater than 90°; but if B and C are either both acute, or both obtuse, BC is less than 90°. It is evident that this rule does not apply when that which is found is the sine of an arc; and this, besides in the three ambiguous cases, happens also in other two, viz., the 1st and 11th. The ambiguity is obviated in these two cases by this rule, that the sides of a spherical right-angled triangle are of the same affection with the opposite angles. Two rules are, therefore, sufficient to remove the ambiguity in all the cases of the right-angled triangle in which it can possibly be removed. PROBLEM II. Solution of oblique-angled spherical triangles. In this Table the references (c. 4) (c. 5), &c., are to the cases in the preceding Tables (16), (17), &c., to the propositions in Spherical Trigonometry. GIVEN. SOUGHT. SOLUTION. The construction is the same as in Prop. 16. Cos A tan В tan AD, gives AD (c. 2); therefore, BD is known, and sin CD = sin A sin b (c. 1) gives CD; therefore, sin BD = tan CD cot B (c. 7); B and A are of the same or - different affection, according as AB is greater or less than BD (16). AD, BD, and CD are found as above, and cos a = cos CD cos BD (c. 13); according as the segments AD and DB are of the same or different affection, b and a will be of the same or different affection (c.13). Cot ACD = cos b tan A (c. 3), sin CD sin A sin b (c. 1); therefore, BCD and CD are known, and cos BCD = tan CD cot a (c. 5); a is less or greater than 90°, according as CD and BCD, i.e. (14) as A and BCD are of the same or different affection. BCD and CD are found as above; and cos B: = cos CD sin BCD (c. 6); B and A are of the same or different affection, according as CD falls within or without the triangle, that is, according as ACB is greater or less than BCD (16). SOLUTION. Sin a sin b : : sin A : sin B (17); the affection of B is ambiguous, unless it can be determined by this rule, that according as AC + BC is greater or less than 180°, A+ B is also greater or less than 180° (12). From C, the angle sought, draw CD perpendicular to AB; then ACD and CD may be found as in the last case; and cos BCD = tan CD cot a (c. 12). ACD ± BCD ACB, and ACB is ambiguous, because of the ambiguous sign + or —. = ACD is found as above, and Sin B sin A :: sin b: sin a CD is found as in the last case; and tan AD=tan b cos A (c. 2); and sin BD= tan CD cot B (c. 7); BD is ambiguous, and therefore c=AD+BD may have four values, some of which will be excluded by this condition that c must be less than 180°. 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. 221 NOTES ON THE ELEMENTS. DEFINITIONS. I. In the definitions a few changes have been made, of which it is necessary to give some account. One of these changes respects the first definition, that of a point, which Euclid has said to be, "That which has no parts, or which has no magnitude." Now, it has been objected to this definition, that it contains only a negative; and that it is not convertible, as every good definition ought certainly to be. It is accordingly changed here by the addition of an affirmative clause, which includes all that is essential to a point. This addition is that which is given in Scarburgh's English Euclid (fol. Oxf. 1705) as the interpretation of a sign (onusov). A point or sign is a certain position without any quantity. II. Euclid has introduced, as his third definition, the following, "the extremities of a line are points." Now, this is certainly not a definition, but an inference from the definitions of a point and a line. Accordingly, Playfair has judiciously put it down as a corollary to the second definition, and has added, that the intersections of one line with another are points, as this affords a good illustration of the nature of a point, and is an inference exactly of the same kind with the preceding. The same thing has been done with the fourth definition, where that which Euclid gave as a separate definition is made a corollary to the fourth, because it is, in fact, an inference deduced from comparing the definitions of a superfices and a line. III. Euclid has defined a straight line to be a line which (as we translate it) "lies evenly between its extreme points." Great diversity of opinion exists relative to this definition; many persons rejecting it altogether as useless, whilst others rest satisfied with condemning its obscurity. The former class manifestly regard a definition as the expression of some property characteristic of the |