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Let ABC be a spherical triangle right-angled at C; with B as pole describe a great circle DEFG, and with A as pole describe a great circle HFKL, and produce the sides of the original triangle ABC to meet these great circles. Then since B is a pole of DEFG the angles at D and G are right angles, and since A is a pole of HFKL the angles at H and I are right angles. Hence the five triangles BAC, AED, EFH, FKG, KBL are all right-angled ; and moreover it will be found on examination that, although the elements of these triangles are different, yet their circular parts are

We will consider, for example, the triangle AED; the angle EAD is equal to the angle BAC; the side AD is the complement of AB; as the angles at C and G are right angles E is a pole of GC (Art. 13), therefore EA is the complement of AC; as B is a pole of DE the angle BED is a right angle, therefore the angle AED is the complement of the angle BEC, that is, the angle AED is the complement of the side BC (Art. 12); and similarly the side DE is equal to the angle DBE, and is therefore the complement of the angle ABC. Hence, if we denote the elements of the triangle ABC as usual by a, b, c, A, B, we have in the angle AED the hypotenuse equal to a - b, the angles equal to

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same as those of ABC. Similarly the remaining three of the five right-angled triangles may be shewn to have the same circular parts as the triangle ABC has.

Now take troo of the theorems in Art. 65, for example (1) and (3); then the truth of the ten cases comprised in Napier's Rules will be found to follow from applying the two theorems in succession to the five triangles formed in the preceding figure. Thus this method of considering Napier's Rules regards each Rule, not as the statement of dissimilar properties of one triangle, but as the statement of similar properties of five allied triangles.

69. In Napier's work a figure is given of which that in the preceding Article is a copy, except that different letters are used; Napier briefly intimates that the truth of the Rules can be easily seen by means of this figure, as well as by the method of induction from consideration of all the cases which can occur. The late T. S. Davies, in his edition of Dr Hutton's Course of Mathematics, drew attention to Napier's own views and expanded the demonstration by a systematic examination of the figure of the preceding Article.

It is however easy to evade the necessity of examining the whole figure; all that is wanted is to observe the connexion between the triangle AED and the triangle BAC. For let a, a,, az, az, az represent the elements of the triangle BAC taken in order, beginning with the hypotenuse and omitting the right angle; then the elements of the triangle AED taken in order, beginning with the hypotenuse and omitting the right angle, are 2

az 2

If, therefore, to characterise

2 the former we introduce a new set of quantities Pie Px P 59 such that a, + P2 = a, + P2 = 0; + P5

and 2

P= a. then the original triangle being characterised by P, Pg Po Po P the second triangle will be similarly characterised by P.x, Pa, Ps, P P As the second triangle can give rise to a third in like manner, and so on, we see that every right-angled triangle is one

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of a system of five such triangles which are all characterised by the quantities Pie Pos Pixo Po Ps, always taken in order, each quantity in its turn standing first.

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The late R. L. Ellis pointed out this connexion between the five triangles, and thus gave the true significance of Napier's Rules. The memoir containing Mr Ellis's investigations, which was unpublished when the first edition of the present work appeared, will be found in pages 328...335 of The Mathematical and other writings of Robert Leslie Ellis... Cambridge, 1863.

Napier's own method of considering his Rules was neglected by writers on the subject until the late T. S. Davies drew attention to it. Hence, as we have already remarked in Art. 68, an erroneous statement was made respecting the Rules. For instance, Woodhouse says, in his Trigonometry: “There is no separate and independent proof of these rules;...” Airy says, in the treatise on Trigonometry in the Encyclopædia Metropolitana : “ These rules are proved to be true only by showing that they comprehend all the equations which we have just found.”

70. Opinions have differed with respect to the utility of Napier's Rules in practice. Thus Woodhouse says, “In the whole compass of mathematical science there cannot be found, perhaps, rules which more completely attain that which is the proper object of rules, namely, facility and brevity of computation.” (Trigonometry, chap. x.) On the other hand may be set the following sentence from Airy's Trigonometry (Encyclopædia Metropolitana): “In the opinion of Delambre (and no one was better qualified by experience to give an opinion) these theorems are best recollected by the practical calculator in their unconnected form.” See Delambre's Astronomie, vol. 1. p. 205. Professor De Morgan

, strongly objects to Napier's Rules, and says (Spherical Trigono metry, Art. 17): “There are certain mnemonical formulæ called Napier's Rules of Circular Parts, which are generally explained. We do not give them, because we are convinced that they only create confusion instead of assisting the memory."

71. We shall now proceed to apply the formulæ of Art. 62 to the solution of right-angled triangles. We shall assume that the given quantities are subject to the limitations which are stated in Arts. 22 and 23, that is, a given side must be less than the semicircumference of a great circle, and a given angle less than two right angles. There will be six cases to consider.

72. Having given the hypotenuse c and an angle A. Here we have from (3), (5) and (2) of Art. 62, tan b = tan c cos A, cot B= cos c tan A, sin a = sin c sin A.

Thus b and B are determined immediately without ambiguity; and as a must be of the same affection as A (Art. 64), a also is determined without ambiguity.

It is obvious from the formulæ of solution, that in this case the triangle is always possible.

If c and A are both right angles, a is a right angle, and b and B are indeterminate.

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73. Having given a side b and the adjacent angle A.
Here we have from (3), (4) and (6) of Art. 62,

tan 6
tan c=

tan a =
tan A sin b,

= cos b sin A.

cos B

COS A

a

Herec, a, B are determined without ambiguity, and the triangle is always possible.

74. Having given the two sides a and b.
Here we have from (1) and (4) of Art. 62,

cos c = cos a cos b, cot A cot a sin b, cot B = cot b sin an

Here C, A, B are determined without ambiguity, and the triangle is always possible.

75. Having given the hypotenuse c and a side a.
Here we have from (1), (3) and (2) of Art. 62,

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sin a

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sin c

Here b, B, A are determined without ambiguity, since A must be of the same affection as a. It will be seen from these formulæ that there are limitations of the data in order to insure a possible triangle ; in fact, c must lie between a and a -a in order that the values found for cos b, cos B, and sin A may be numerically not greater than unity.

If c and a are right angles, A is a right angle, and b and B are indeterminate.

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Here c, a, b are determined without ambiguity. There are limitations of the data in order to insure a possible triangle. First suppose A less than then B must lie between 5-A and+A;

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77. Having given a side a and the opposite angle A. Here we have from (2), (4) and (6) of Art. 62,

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Here there is an ambiguity, as the parts are determined from their sines. If sin a be less than sin A, there are two values admissible for c; corresponding to each of these there will be in general only one admissible value of b, since we must have cos c = cos a cos b, and only one admissible value of B, since we must have cos c = cot A cot B. Thus if one triangle exists with the given parts, there will be in general two, and only two, triangles with the given parts. We say in general in the preceding sen

. tences, because if a = A there will be only one triangle, unless a

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