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On Napier’s Rules of the Circular Parts. 25 
condly, we may employ the two rules and the circular parts of 
Napier. The two methods are fundamentally the same, and 
differ from one another only in form. 
In the preceding investigation only three related triangles have 
been mentioned, whereas the author of the rules employs five. 
It is to be observed that each of the two complementary trian- 
gles has itself a pair of complementary triangles; and as the given 
triangle is one of each pair, there are no more than two new tri- 
angles found in this manner, and these complete the five of Na- 
pier. All the five triangles will be exhibited on the surface of 
the sphere, if each of the two oblique angles of the given trian- 
gle be made the pole of a great circle; for the intersections of 
the two great circles and the three sides of the triangle will form 
five different right-angled triangles, the hypothenuses of which 
inclose a pentagonal figure. Every one of the five triangles 
has its two complementary triangles next it on either hand. The 
real principles of Napier’s Theory consist in these two things: 
First, all the five related triangles agree in having the same cir- 
cular parts ; secondly, if we take the circular parts of all the tri- 
angles, making a similar part always occupy the middle place, 
we shall obtain all the arrangements of which they are suscep- 
tible. Wherefore, since there is the same relation between every 
triangle and its circular parts, when the two rules are proved, by 
means of the proper triangle, to be true in any one arrangement, 
it follows that they must be universally true in every arrange- 
ment. The words of Napier, at the close of the demonstration 
of his rules by induction, are as follows, viz. 
‘¢ Preter hanc probationem per inductionem omnium casuum, 
qui occurrere possunt, potest idem theorema lucidé perspici ex 
19 et 20, precedentibus, in quorum schemate, homologa circu- 
larium partium constitutio earundem analogie similitudinem 
arguit: ita ut quod de und intermedia et ejus extremis circum- 
positis, aut oppositis, vere enuntiatur, de ceteris quatuor inter- 
mediis et earum extremis respective circumpositis, aut oppositis, 
negari non possit.”’ 
The rules of Napier were therefore investigated by means of 
' properties belonging to right-angled triangles. They are a de- 
duction from a theory of considerable subtilty, bearing marks of 
the same deep and original thinking and profound research, to 
which we are indebted for the invention of logarithms. 
J. Ivory. 
Kk 2 LVI, 4 Re- 
