Thus in the right angled triangle ABC, A, being the right angle, AC AB, 90°-B, 90°- BC, 90°-C, are the circular parts, hy Def. 1.; and if any one as AC be reckoned the middle part, then AB and 90° -C, which are contiguous to it on different sides, are called adjacent: parts; and 90° -B, 90°- BC are the opposite parts. In like manner B. if AB is taken for the middle part, AC and 90°-B are the adjacent parts: 90° — BC, and 90° -C are the opposite. Or if 90° — BC be the middle part, 90°-B, 90° —C are adjacent; AC and AB opposite, &c. This arrangement being made, the rule of the circular part is coutained in the following PROPOSITION. In a right angled spherical triangle, the rectangle under the radius and the sine of the middle part, is equal to the rectangle under the tangents of the adjacent parts; or to the rectangle under the cosines of the opposite parts. The truth of the two theorems included in this enunciation may be easily proved, by taking each of the five circular parts in succession, for the middle part, when the general proposition will be found to coincide with some one of the analogies in the table already given for the resolution of the cases of right angled spherical triangles. Thus, in the triangle ABC, if the complement of the hypotenuse BC be taken as the middle part, 90°-B, and 90°-C, are the adjacent parts, AB and AC the opposite. Then the general rule gives these two theorems, RXcos BC=cot B xcot C; and Rxcos BC=cos AB Xcos AC. The former of these coincides with the cor. to the 20th; and the latter with the 22d. To apply the foregoing general proposition, to resolve any case of a right angled spherical triangle, consider which of the three quantities named (the two things given and the one required) must be made the middle term, in order that the other two may be equidistant from it, that is, may be both adjacent, or both opposite ; then one or other of the two theorems contained in the above enunciation will give the value of the thing required. Suppose, for example, that AB and BC are given, to find C; it is evident that if AB be made the middle part, BC and C are the oppo ..., site parts, and therefore Rxsin AB=sin Cxsin BC, for sin C=ços (90°-C), and cos (90°-BC)=sin BC, and consequently sin AB sin Ca sin BC cos C Again, suppose that BC and C are given to find AC; it is obvious that C is in the middle between the adjacent parts AC and (90°- BC), therefore Rxcos C=tan AC xcot BC,or tan AC *cos C+ cot BC 1 tan BC; because, as has been shown above, =tan BC. 'cot BC In the same way may all the other cases be resolved. One or two trials will always lead to the knowledge of the part which in any given case is to be assumed as the middle part; and a little practice will make it easy, even without such trials, to judge at once which of them is to be so assumed. It may be useful for the learner to range the names of the five circular parts of the triangle round the circumference of a circle, at equal distances from one another, by which means the middle part will be immediately determined. Besides the rule of the circular parts, Napier derived from the last of the three theorems ascribed to him above, (schol. 29.), the solu. tions of all the cases of oblique angled triangles. These solutions are as follows: A, B, C, denoting the three angles of a spherical triangle, and a, b, c, the sides opposite to them. I, To find the angles B and C. sin i (b-c) tan á (B-C)=cot & AX c cos Š (6-0) tan (B+C)=cot $ Ax (31.) cor. 1. II. Given the two sides b, c, and the angle B opposite to one of them. To find C, and the angle opposite to the other side. sin b: sin c:, sin B: sin C. To find the contained angle A. sin (b+c) cotA=tan $ (B-C) X sin $ (6-c) (31.) cor. 1. To find the third side a. Sin B :: sin A :: sin b: sin a. III. Given two angles A and B, and the side o between them. To find the other two sides a, b. sin : (A-B) tan (b-a)=tan cx (31.) sin (A+B) cos (A-B) tan : (6+a)=tan * 6x (31.) cos (A+B) To find the third angle C. sin a : sin c :: sin A : sin C. IV. Given two angles A and B, and the side a, opposite to one of themt. To find b, the side opposite to the other. sin A: sin B :: sin a : sin b. To find e, the side between the given angles. sin } (A+B) tances (a-b) X (31.) sin (A-B) To find the third angle C. The other two cases, when the three sides are given to find the ato gles, or when the three angles are given to find the sides, are resolved by the 29th, (the first of Napier's Propositions,) in the same way as in the table already given for the cases of the oblique angled triangle There is a solution of the case of the three sides being given, which it is often very convenient to use, and which is set down here, thought the proposition on which it depends has not been demonstrated. Let a, b, c, be the three given sides, to find the angle À, contained between b and c. If Rad. = 1, and a +b+c=s, à -bs -C ✓sin 6 xsin c ✓sin bxsin c In like manner, if the three angles, A, B, C are given to find c, thie side between A and B. Let A+B+C=S, sin } = cos Sxcos (i S-A), ✓sin Bxsin C ✓cos (1 S-B)xcos (i S-C). cos i ca v sin Bxsin These theorems, on account of the facility with which Logarithms are applied to them, are the most convenient of any for resolving the two cases to which they refer. When A is a very obtuse angle, the second theorem, which gives the value of the cosine of its half, is to be used; otherwise the first theorem, giving the value of the sine of its half is preferable. The same is to be observed with respect to the side c, the reason of which was explained, Plane Trig. Schol. ; or, END OF SPHERICAL TRIGONOMETRI. NOTES ON THE FIRST BOOK OF THE ELEMENTS: simminn DEFINITIONS. I. IN cessary to give some account. One of these changes respeets 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 difinition, that it contains only a negative, and that it is not convertible, as every good definition, ought certainly to be. That it is not convertible is evident, for though every point is unextended, or without magnitude, yet every thing unextended or without magnitude, is not a point. To this it is impossible to rely, and therefore it be comes necessary to change the definition altogether, which is accord. ingly done here, a point being defined to be, that which has position but not magnitude. Here the affirmative part includes all that is essential to a point, and the negative part excludes every thing that is not essential to it. I am indebted for this definition to a friend, by whose judicious and learned remarks I have often profited: II. After the second definition Euclid has introduced the following: to the extremities of a line are points." Now, this is certainly not a definition, but an inference from the definitions of a point and of a line. That which terminates a line can have no breadth, as the line in which it is has none; and it can have no length, as it would not then be a termination, but a part of that which it is supposed to terminate. The termination of a line can therefore have no magnitude, and having necessarily position, it is a point. But as it is plain, that in all this we are drawing a consequence from two definitions already laid down, and not giving a new definition, I have taken the liberty of putting it down as a corollary to the second definition, and have added, that the intersections of one line with another Nn |