and the expressions, which have been indicated, are all of them adapted to the best mode of computation, namely, that of logarithms. It would not, then, be very unreasonable to exact, from the student, so much labour as is required, to fix in the memory the substance of seven different theorems; of which the first, in the order here assigned to them, can hardly ever be forgotten; the second and the third have not only a strong resemblance to one another, but a close connexion with the fourth, and the fifth; so that the effort of the memory is chiefly confined to the perfect retention of the first analogy: and the sixth of the requisite theorems may, likewise, in some measure, serve to recall the seventh. The three fundamental equations, cos S = cos S' cos S" + sin S' sin S" cos A, cos Scos A' = cot S" sin S sin A' cot A", deserve, also, to be committed to memory, although the solution of spherical triangles may be effected without them. The forms, which have been recommended to be used, in general, for the solution of oblique-angled sphe rical triangles, are not, it must be acknowledged, the best that can be employed in solving right-angled triangles. It happens, however, that the second mode of relieving the memory comes specially in aid of this defect. (277.) The six theorems, which are best adapted to the solution of right-angled spherical triangles, are all comprehended in two rules, invented by Naper, and afterwards explained and improved by Gellibrand and Manduit. Calling the hypotenuse, the two angles adjacent to it, and the complements of the two remaining sides, the circular parts, taking any one of these as the middle part, and calling those two of the circular parts, which are adjacent to it, the adjacent extremes, and the two remaining circular parts, the separate extremes, then, the tabular radius being unity, "The cosine of the middle part is equal, first, to the rectangle contained by the sines of the separate extremes ; and secondly, to the rectangle contained by the cotangents of the adjacent extremes." Or, if a part which is unknown, but which is not required to be found, be taken as the middle part, the two rules may be reduced to this one: "The rectangle contained by the co-tangents of the adjacent extremes is equal to the rectangle contained by the sines of the separate extremes. Thus, if A be put for the right angle, and S for the hypotenuse, then, beginning with the oblique angle A' at the foot of the hypotenuse, the circular parts are A', S, A", (S' ~ 90), (S" ~90). And it is manifest that, in any combination of three, out of the five circular parts, two of the parts will either be adjacent to, or separate from the remaining part, A' and (S" ~90) being supposed to adjoin to one another*. If, therefore, the rules be true, and any two of the parts be given, the rest may be found: because the rules will always furnish three independent equations, in each of which one of the three parts, required, is the only unknown quantity. It is unnecessary to enter upon any direct investigation of these rules: because when all the results, which they produce, are expressed at length, they are found to be merely the enunciations of six theorems, which have already been demonstrated. If, however, Forms V. and VII. (Art. 232.) be successively applied to any assumed right-angled spherical triangle, and to its two complemental triangles, it will be made evident, that the rules are true †. * If the space between two concentric circles be divided into five compartments, and a Circular Part, as it has been called, be placed in each, in the order in which they are set down in the text, then A' and (S”~90) will adjoin to another, and all the definitions, relative to the circular parts, will be clearly understood. † Naper's rules are, in reality, nothing more than a contrivance to express in few words, of extensive signification, what is otherwise enunciated in a greater number of words of more limited signification: and the merit of the contrivance is founded on this principle, that it is easier to retain the extended meaning of the new terms employed, and the short form of words, in which they are used, than the substance of the six theorems, which the rules comprehend. An eminent French Astronomer has however avowed, that it has always been less irksome to him to retain the six theorems themselves, than to call to mind, and to apply, Naper's rules. There is, he thinks, an R (278.) The method of relieving the memory, which yet remains to be explained, is founded on the analogy, which exists between the two branches of Trigonometry, Plane and Spherical. The principal theorems of Plane Trigonometry have never been accounted hard to be retained, or even to be investigated anew, if they have been forgotten: and, when they have once been attentively compared with the corresponding theorems of Spherical Trigonometry, to which they bear a very striking resemblance, they may serve, even afterward, to revive these latter theorems in the memory, Thus, if A, A', A", be put for the angles S, S', S", for the opposite sides, and P for the semi-perimeter, of a triangle, either plane or spherical, then, an inconvenience, in having to substitute, for the base and for the perpendicular, their complements; in being obliged to consider which parts are adjacent to, and which are separate from, the middle part; and lastly, in having the quantity which is sought, sometimes a middle part, and sometimes one of the extremes: and he asserts, that an attention to these details consumes more time than the, calculation itself. Certainly, neither in his opinion, nor in his experience, is he altogether singular. It may, nevertheless, be doubted, whether a person who, from constant practice, cannot fail to have the six theorems themselves fixed in his memory, be a fair judge of the value of rules, which, to him at least, must necessarily be useless. Where it is remarkable, that the very same kind of trigonometrical functions of the angles obtain, in both branches; and that, in the spherical forms, the functions of the sides are of the same kind as the functions of the angles. The last spherical form is easily deduced from that which precedes it; and by equating the two numerators, and also the two denominators, there result two of Naper's four analogies; and the two analogies thus obtained may readily be translated into the remaining two, by means of the polar triangle. If the second spherical form be, likewise, applied to the parts of the polar triangle, there will result the only additional form which is wanted, for the solution of oblique-angled spherical triangles; namely, |