Proof of Napier's Rules

Abstract

SUCH a structure in cardboard as that described by Prof. A, S. Herschel in NATURE, No. 106, may be found very useful in facilitating the study of the proof of “Napier's Rules,” but the ingenious learner might object that the demonstration was confined to one particular species of triangle—the isosceles right-angled with a perimeter equal to a quadrant; for Mr. Herschel's angles a and b are plainly equal, and together with c make up a right angle. The corresponding construction for any case would be as follow:—Take a circular piece of cardboard with centre D (referring to Mr. Herschel's diagram), and on the circumference, in the same direction, take any two arcs B1, 12. Let a perpendicular from B or D₁ meet it in D, and a second from C or D₂ meet it in A, and be produced to reach the circumference in B′. Finally, a semicircle on A B' as diameter and another with centre A and radius A C will determine by their intersection the.point C'. To a construction thus generalised all that Prof. Herschel adds would apply.