DEAR SIR,--According to your request of the other day, I send you my formula and explanation of Lord Napier's theorem, for the solution of right-angled spherical triangles. With you I think it strange that the French mathematicians have not used or noticed this method more than they have done. Montucla, in his account of Lord Napier's inventions, expresses a like surprise at this fact, and does justice to the ingenuity, the elegance, and convenience of the theorem, which, by a single rule easily preserved in the memory, supplies the whole table of cases given in the books of spherical trigonometry. Yet he does not state the rule, but refers for it to Wolf, Cours de Mathematiques. I have not the larger work of Wolf; and in the French translation of his abridgement, (by some member of the congregation of St. Maur,) the branch of spherical trigonometry is entirely omitted. Potter, one of the English authors of Courses of Mathematics, has given the Catholic proposition, as it is called, but in terms unintelligible, and leading to error, until, by repeated trials, we have ascertained the meaning of some of his equivocal expressions. In Robert Simson's Euclid we have the theorem with its demonstrations, but less aptly for the memory, divided into two rules, and these are extended as the original was, only to the cases of right-angled triangles. Hutton, in his Course of Mathematics, declines giving the rules, as "too artificial to be applied by young computists." But I do not think this. It is true that when we use them, their demonstration is not always present to the mind; but neither is this the case generally in using mathematical theorems, or in the various steps of an algebraical process. We act on them, however, mechanically, and with confidence, as truths of which we have heretofore been satisfied by demonstration, although we do not at the moment retrace the processes which establish them. Hutton, however, in his Mathematical Dictionary, under the terms "circular parts," and "extremes," has given us the rules, and in all their extensions to oblique spherical, and to plane triangles. I have endeavored to reduce them to a form best adapted to my own frail memory, by couching them in the fewest words possible, and such as cannot, I think, mislead, or be misunderstood. My formula, with the explanation which may be necessary for your pupils, is as follows: