[ 255 j 
LVI. On NaprEr’s Rules of the Circular Parts. By James 
Ivory, M.A. F.R.S. 
Tue illustrious inventor of logarithms applied the whole force 
of his mind to the shortening of mathematical calculations. Be- 
sides his great discovery, he bequeathed to posterity some other 
contrivances well adapted to the end he had in view. Among 
these, his two Rules of the Cireular Parts, which alone are suf- 
ficient for solving all the cases of right-angled spherical triangles, 
are the most distinguished in point of usefulness. 
In our treatises of trigonometry the rules of Napier are repre- 
sented as enunciations so contrived that, by a particular classifi- 
cation and nomenclature of the parts of a triangle, they include 
all the propositions necessary for solving every case. They are 
held up as one of the happiest examples of artificial memory 
that can any where be found. Rules, entirely dependent on dex- 
terity of arrangement, cannot, it is said, admit of a separate de- 
monstration: they canbe proved to be just in no other way than 
by showing that they comprehend every result, and thus fulfil 
the purpose for which they were contrived. 
In the original tract* in which the rules were first published, 
the author no doubt demonstrates them by an induction of every 
possible case. But this mode of proof he was led to adopt, be- 
cause, not composing an express treatise on trigonometry, it be- 
came necessary to show the agreement of the rules with the 
writings of others. At the close of this demonstration he im- 
mediately indicates another and a more general one, which ex- 
hibits the whole theory in one view, amounting to this proposi- 
tion: That two theorems, applied either to the triangle ori- 
ginally proposed, or to one or other of four triangles reiated 
to it, comprehend the whole doctrine of right-angled spherical 
trigonometry. The invention of the circular parts merely en- 
ables the author to enunciate the two theorems with reference 
to the given triangle alone. 
It appears therefore that the rules are suggested by real pro- 
perties of right-angled triangles. The purpose of their inventor 
seems to have been to reduce trigonometry to the least number 
of necessary principles, rather than to collect a variety of un- 
connected particulars into a compendium commodious to the 
memory. The views of Napier may be applied to abridge the 
theory of trigonometry, as well as to exhibit its practi ical pre- 
cepts in ashort abstract. That this is really the case will better 
appear from what follows, 
1. Ifa great circle of the sphere be described about either of 
* Mirifici Logarithmorwn Canonis Descriptio. 
the 
