Proof of Napier's Rules

Abstract

I AM greatly obliged to “J. J. W.” for pointing out the objection of a want of generality in the construction of the figure contained in my former letter (in NATURE, No. 106), for the proof of Napier's Rules; which the more general construction now described by “J. J. W.” most simply and most effectually removes. To illustrate his more perfect general construction with a figure—D is the centre, and B12B′ a part of the circumference of a circular piece of cardboard, upon which the ares B1, 12 are taken equal to the sides of the right-angled spherical triangle which it is required to represent. If we join DB, D1, D2, and draw BC, CA perpendicular to D1, D2, the latter perpendicular prolonged meeting the circle of the circumference in B′, and join DB′; and on AB′ as diameter describe the semicircle AC′B′; and with the centre A, and radius AC, another circle, meeting the semicircle in C′, so that the straight line AC′is equal to AC; and join B′C′. Then it is easily shown that if AC CB are the two sides, AB′ is the hypothenuse of a right-angled triangle, which, when the four triangles are closed together so as to form a solid figure, will coincide with the triangle AC′B′. As BC (or B′C′) will then be perpendicular both to CD and to CA (or C′A), it will be perpendicular to the plane DCA; and the are Br, which is in the same plane with it, will be at right angles to the are 12. The third are 2B′ will therefore be the hypothenuse of a right-angled spherical triangle, of which B1, 12, are the two sides. Calling these ares or the angles of the faces resented by them, a,b,c, and the angles opposite to them in the spherical triangle, A,B,C, the proof of Napier's Rules, with this solid figure, proceeds by the same direct steps as those already described, with a special example of the figure in my former letter. As the construction there described is confined to the representation of a particular kind of right-angled spherical triangle, and is therefore inapplicable to illustrate the proof of Napier's Rules experimentally in every given case, the general construction supplied by “J. J. W.,” which is limited by no such restrictions, and which is at least equally convenient, will evidently serve more effectively the same practically useful and instructive purpose.