To Louis H. Girardin
Monticello Mar. 18. 14.
According to your request of the other day, I send you my formula and explanation of Ld Napier’s theorem for the solution of right angled Spherical triangles. with you I think it strange that the French mathematicians have not used, or noticed, this method more than they have done. Montucla, in his account of Lord Napier’s inventions, expresses a like surprise at this fact, and does justice to the ingenuity, the elegance, and convenience of the theorem, which, by a single rule, easily preserved in the memory, supplies the whole Table of Cases, given in the books of Spherical trigonometry. yet he does not state the rule; but refers for it to Wolf’s Cours de Mathmatiques. I have not the larger work of Wolf’s and in the French translation of his abridgment (by some member of the congregation of St Maur) the branch of Spherical trigonometry is entirely omitted. Potter, one of the English authors of Courses of mathematics, has given the Catholic proposition, as it is called, but in terms unintelligible, and leading to error, until, by repeated trials, we have ascertained the meaning of some of his equivocal expressions. in Robert Simson’s1 Euclid we have the theorem, with it’s demonstrations, but, less aptly for the memory, divided into two rules; and these are extended, as the original was, only to the cases of right angled triangles. Hutton, in his Course of Mathematics, declines giving the rules as ‘too artificial to be applied by young computists.’ but I do not think this. it is true that, when we use them, their demonstration is not always present to the mind: but neither is this the case generally in using mathematical theorems, or in the various steps of an Algebraical process. we act on them however mechanically, & with confidence as truths of which we have heretofore been satisfied by demonstration, altho we do not at the moment retrace the processes which establish them. Hutton however in his Mathematical Dictionary, under the terms ‘Circular parts’ & ‘Extremes’ has given us the rules, and in all their extensions to oblique spherical, & to plane triangles. I have endeavored to reduce them to a form best adapted to my own frail memory, by couching them in the fewest words possible, & such as cannot, I think, mislead, or be misunderstood. my formula, with the explanation which may be necessary for your2 pupils, is as follows.3
Ld Napier noted first the parts, or elements of a triangle, to wit, the sides & angles: and, expunging from these the right angle, as if it were a non-existence, he considered the other five parts, to wit, the 3. sides and 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, and the Complements of the hypothenuse & of the 2. oblique angles. if the 3. of these, given and required. were all adjacent, he called it the case of Conjunct parts, the middle element the Middle part, and 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. Extremes Conjunct, and to that of the Cosines of the 2. Extremes Disjunct.’
And to aid our recollection in which case the Tangents, and in which the Cosines are to be used, preserving the original designations of the inventor, we may observe that the tangent belongs to the Conjunct case, terms of sufficient affinity to be associated in the memory; and the Sine-Complement remains of course for the Disjunct case: and further, if you please, that the initials of Radius and Sine, which are to be used together, are Alphabetical consecutives.
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 2. 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, and opposite to a given 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, and the Supplements of the angles as Circular parts in these cases; and, dropping a perpendicular from any angle from which it would fall internally on the opposite side, they assumed that angle, or that side, as the Middle part, and the other angles, or other sides, as the Opposite or4 Extreme parts, disjunct in both cases. then ‘the rectangle under the tangents of ½ the sum, and ½ the difference of the Segments of the Middle part is equal to the rectangle under the tangents of ½ the sums & ½ the difference of the Opposite parts.’
And, since every plane triangle may be considered as described on the surface of a sphere of an infinite radius, these 2. rules may be applied to plane right angled triangles, and, through them, to the Oblique. but as Ld Napier’s rule gives a direct solution only in the case of 2. sides and an uncomprised angle, 1. 2. or 3. operations, with this combination of parts, may be necessary to get at that required.
You likewise requested,5 for the use of your school, an explanation of a method of platting the courses of a survey which I mentioned to you as of my own practice. this is so obvious and simple, that as it occurred to myself, so I presume it has to others, altho’ I have not seen it stated in any of the books. for drawing parallel lines, I use the triangular rule, the hypothenusal side of which being applied to the side of a common strait rule, the triangle slides on that, as thus always parallel to itself. instead of drawing meridians on his paper let the pupil draw a parallel of latitude, or East and West line, and note in that a point for his 1st station. then applying to it his protractor, lay off the 1st course and distance in the usual way to ascertain his 2d station.
—for the 2d course, lay the triangular rule to the E. and W. line, or 1st parallel, holding the strait- or guide-rule6 firmly against it’s hypothenusal side. then slide up the triangle (for a Northernly course) to the point of his 2d station, and pressing it firmly there, lay the protractor to that, and mark off the 2d course and distance as before, for the 3d station.
—then lay the triangle to the 1st parallel again, and sliding it as before to the point of the 3d station, there apply to it the protractor for the 3d course & distance, which gives the 4th station: and so on. Where a course is Southwardly, lay the protractor as before to the Northern edge of the triangle but prick it’s reversed course, which reversed again in drawing, gives the true course. when the station has got so far from the 1st parallel as to be out of the reach of the parallel rule sliding on it’s hypothenuse7 another parallel must be drawn8 by laying the edge, or longer leg of the triangle to the 1st parallel as before, applying the guide-rule to the end, or short leg (instead of the hypothenuse) as in the margin, & sliding the triangle up to the point for the new parallel.—I have found this in practice the quickest and most correct method of platting which I have ever tried, and the neatest also, because it disfigures the paper with the fewest unnecessary lines.
If these Mathematical trifles can give any facilities to your pupils, they may in their hands become matters of use, as in mine they have been of amusement only. ever and respectfully yours
PoC (DLC); at foot of first page: “Mr Girardin.” Tr (ViU: GT); posthumous copy.
The edition of Christian von Wolff’s Cours de Mathématique owned by TJ was translated by Antoine Joseph Pernety and Jean François de Brézillac, both members of the congregation of st maur (Paris, 1747; Sowerby, description begins E. Millicent Sowerby, comp., Catalogue of the Library of Thomas Jefferson, 1952–59, 5 vols. description ends no. 3682). TJ also cited John potter, A System of Practical Mathematics (London, 1759; Sowerby, description begins E. Millicent Sowerby, comp., Catalogue of the Library of Thomas Jefferson, 1952–59, 5 vols. description ends no. 3667); Robert simson’s The Elements of Euclid (Glasgow, 1756; Sowerby, description begins E. Millicent Sowerby, comp., Catalogue of the Library of Thomas Jefferson, 1952–59, 5 vols. description ends no. 3702); and Charles hutton, A Course of Mathematics (New York, 1812; Sowerby, description begins E. Millicent Sowerby, comp., Catalogue of the Library of Thomas Jefferson, 1952–59, 5 vols. description ends no. 3683) and A Mathematical and Philosophical Dictionary (London, 1795; Sowerby, description begins E. Millicent Sowerby, comp., Catalogue of the Library of Thomas Jefferson, 1952–59, 5 vols. description ends no. 3684). David Steuart Erskine, 11th Earl of buchan, and Walter minto discussed their analogous rule in An Account of the Life, Writings, and Inventions of John Napier, of Merchiston (Perth, 1787), 97–103.
1. Tr: “Simons’s.”
2. Tr: “our.”
3. Letter from this point to end of paragraph concluding with “to get at that required” is either based on or the basis for TJ’s Notes on Napier’s Theorem, [ca. 18 Mar. 1814].
4. Preceding two words interlined.
5. Tr: “request.”
6. Reworked from “strait rule.”
7. Preceding four words interlined.
8. Remainder of sentence interlined.
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