called, from the use which he first made of them, Napier's circular parts, are the five parts of a right-angled or a quadrantial spherical triangle; they are the two legs, the complement of the hypothenuse, and the complements of the two oblique angles.

Concerning these circular parts, Napier gave a general rule in his Logarithmorum Canonis Descriptio, which is this: "The rectangle under the radius and the fine of the middle part is equal to the rectangle under the tangents of the adjacent parts, and to the rectangle under the cotangents of the opposite parts. The right angle or quadrantial side being neglected, the two sides and the complements of the other three natural parts are called the circular parts, as they follow each other as it were in a circular order. Of these, any one being fixed upon as the middle part, those next it are the adjacent, and those farther from it the opposite parts."

This rule contains within itself all the particular rules for the solution of right-angled spherical triangles, and they were thus brought into one general comprehensive theorem, for the sake of the memory; as thus, by charging the memory with this one rule alone: All the cases of right angled spherical triangles may be reduced, and those of oblique ones also, by letting fall a perpendicular, excepting the two cases in which there are given either the three sides, or the three angles. And for these a similar expedient has been devised by Lord Buchan and Dr. Minto, which may be thus expressed:

"Of the circular parts of an oblique spherical triangle, the rectangle under the tangents of half the sum and half the difference of the segments at the middle part (formed by a perpendicular drawn from an angle to the opposite side), is equal to the rectangle under the tangents of half the sum and half the difference of the opposite parts." By the circular parts of an oblique spherical triangle are meant its three sides and the supplements of its three angles. Any of these six being assumed as a middle part, the opposite parts are those two of the same denomination with it, that is, if the middle part is one of the sides, the opposite parts are the other two, and, if the middle part is the supplement of one of the angles, the opposite parts are the supplements of the other two. Since every plane triangle may be considered as described on the surface of a sphere of an infinite radius, these two rules may be applied to plane triangles, provided the middle part be restricted to a side.

Thus it appears that two simple rules suffice for the solution of all the possible cases of plane and spherical triangles. These rules, from their neatness, and the manner in which they are expressed, cannot fail of engraving themselves deeply on the memory of every one who is a little versed in trigonometry. It is a circumstance worthy of notice, that a person of a very weak memory may carry the whole art of trigonometry in his head.