Notes on Napier’s Theorem
[ca. 18 Mar. 1814]
Ld Nepier’s Catholic rule for solving Spherical rt angled triangles.
He noted first the parts, or elements of a triangle, to wit, the sides and angles, and, expunging from these the right angle, as if it were a non existence, he considered the other 5. parts, to wit, the 3. sides, & 2. oblique angles, as arranged in a circle, and therefore called them the Circular parts; but chose (for simplifying the result) instead of the hypothenuse, & 2. oblique angles themselves, to substitute their complements: so that his 5. circular parts are the 2. legs themselves, & the Complements of the hypothenuse, & of the 2. oblique angles. if the 3. of these, given & required, were all adjacent, he called it the case of Conjunct parts, the middle element the Middle part, & the 2. others the Extremes conjunct with the middle, or Extremes Conjunct: but if one of the parts employed was separated from the others by the intervention of the parts unemployed, he called it the case of Disjunct parts, the insulated, or opposite part, the Middle part, and the 2. others the Extremes Disjunct from the middle, or Extremes Disjunct. he then laid down his Catholic rule, to wit, ‘the rectangle of the Radius, & Sine of the Middle part, is equal to the rectangle of the Tangents of the 2. [adjacent parts/Extremes Conjunct] and to that of the Cosines of the 2. [opposite parts/Extremes Disjunct.’] or R. × Si. Mid. part = □ Tang. of the 2 [adjacent parts/Extr. Conj.] = □ of Cos. of 2. [opposite parts/Extr. Disjunct.]
In applying the Catholic rule, instead of using literally the Sine of a Complement, seek at once the Cosine; for the Tangent of a Complement, seek the Cotangent, and for the Cos. of a complement, use the Sine of the same side or angle.
And to fix this rule artificially in the memory, it is observable that the 1st letter of Adjacent parts is the 2d of the word Tangents to be used with them; & that the 1st letter of Opposite parts is the 2d of Cosines, to be used with them: and further, that the initials of Rad. and Sine, which are to be used together, are consecutive in the alphabetical order.
Ld Napier’s rule may also be used for the solution of Oblique spherical triangles. for this purpose a perpendicular must be let fall from an angle of the given triangle, internally, on the base, forming it into two right angled triangles, one of which may contain 2. of the data. or, if this cannot be done, then letting it fall externally on the prolongation of the base, so as to form a right angled triangle, comprehending the oblique one, wherein 2. of the data will be common to both. to secure 2. of the data from mutilation this perpendicular must always be let fall from the end of a given side, & opposite to a given1 ∠. and if the sides, or angles adjacent to the base be of the same character, i.e. both of 90o or of less, or more, it will fall on the base internally: if otherwise, externally.
The sides and angles are of the same, or different characters under the following circumstances. 1. in a rt angled triangle the angles adjacent to the hypoth. are of the same character each as it’s opposite leg. 2. in a rt angled △ if the hypoth. is of less than 90o the legs & angles will be of the same character; if of more, different. 3. in a rt angled △ if a leg or angle be of less than 90o the other & the hypoth. are of the same character; if more, different. 4. in every spherical △, the longest side & greatest angle are opposite: & the shortest side and least angle.
But there will remain yet 2 cases wherein Ld Napier’s rule cannot be used, to wit, where all the sides, or all the angles alone are given. to meet these 2 cases, Ld Buchan & Dr Minto devised an analogous rule. they considered the sides themselves, & the supplements of the angles as Circular parts in these cases, & dropping a perpendicular from any ∠ from which it would fall internally on the opposite side, they assumed that ∠ or that side as the middle part, & the other ∠s or other sides as the opposite, or Extreme parts, disjunct in both cases. then the rectangle under the Tangents of ½ the Sum, & ½ the Difference of the segments of the middle part, = the □ under the Tangents of ½ the sums, & ½ the difference of the Opposite parts.
Corollary. since every plane △ may be considered as described on the surface of a sphere of an infinite radius, these 2. rules may be applied to plane rt angled △s & thro’ them to the Oblique: but as Ld Napier’s rule gives a direct solution only in the case of 2. sides & an uncomprised ∠. 1. 2. or 3. operations, with this combination of parts, may be necessary to get at that required.
In using the analogous rule, when unknown segments of an ∠ or base are to be subtracted the one from the other, the greatest segment is that adjacent to the longest side, or to the least angle at the base.
MS (NjVHi); entirely in TJ’s hand; undated, with conjectural date based on identical or similar language in TJ to Louis H. Girardin, 18 Mar. 1814; all brackets beneath dateline in original.
John Napier of Merchiston (1550–1617), mathematician, was a minor Scottish nobleman with a deep interest in millennialism. He wrote important arithmetical and algebraic treatises, developed concrete aids to calculations, and won lasting fame as the inventor of logarithms (DSB description begins Charles C. Gillispie, ed., Dictionary of Scientific Biography, 1970–80, 16 vols. description ends ; ODNB description begins H. C. G. Matthew and Brian Harrison, eds., Oxford Dictionary of National Biography, 2004, 60 vols. description ends ). Napier developed his analogies for the solution of right-angled spherical triangles in book 2, chapter 4 of his Mirifici Logarithmorum Canonis descriptio (Edinburgh, 1614), 30–9, published in English as A Description of the Admirable Table of Logarithmes (London, 1616; trans. Edward Wright), 43–57.
1. Manuscript: “gven.”
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