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Thomas Jefferson

Currency:USD Category:Collectibles / Autographs Start Price:NA Estimated At:15,000.00 - 20,000.00 USD
Thomas Jefferson

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Auction Date:2019 Sep 12 @ 18:00 (UTC-5 : EST/CDT)
Location:15th Floor WeWork, Boston, Massachusetts, 02108, United States
ALS - Autograph Letter Signed
ANS - Autograph Note Signed
AQS - Autograph Quotation Signed
AMQS - Autograph Musical Quotation Signed
DS - Document Signed
FDC - First Day Cover
Inscribed - “Personalized”
ISP - Inscribed Signed Photograph
LS - Letter Signed
SP - Signed Photograph
TLS - Typed Letter Signed
Extraordinary manuscript written in the hand of Thomas Jefferson, unsigned, one page, 4.75 x 7.75, no date but circa March 1814. The manuscript is comprised of Jefferson’s explanation of Napier’s rule on spherical triangles, a branch of geometry crucial to astronomy, navigation, architecture, and geodesy, which is the science of accurately measuring and understanding Earth's geometric shape, orientation in space, and gravitational field. The content of this writing matches neatly with that of a letter addressed by Jefferson to Louis H. Girardin, a professor at the College of William and Mary. In the letter to Girardin, Jefferson introduces his explanation of Napier's ‘Catholic rule’ with a discussion of the many English and French mathematical texts that omit it or consider it too difficult or ‘too artificial to be applied by young computists.’

A transcription of the offered manuscript, headed “L’d Nepier’s Catholic rule for solving Spherical r’t angled triangles,” reads as follows:

“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. x 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.

L’d 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 given ?, and if the sides, or angles adjacent to the base be of the same character, i.e. both of 90° 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 90° 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 ? 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 uncompromised ?. 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.” In fine condition.

John Napier (1550–1617) was a Scottish nobleman and mathematician best known for inventing logarithms and pioneering the use of decimal points. He contributed to geometry, spherical trigonometry, physics, and astronomy, along with interests in millennial theology and the occult. 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), published in English as A Description of the Admirable Table of Logarithmes (London, 1616; trans. Edward Wright). Originating in ancient Greek scholarship, the field of spherical and hyperbolic triangles leapt forward in early modern Europe with developments by Napier and French astronomer and mathematician Jean Baptiste Joseph Delambre. The discipline was essentially complete by 1859 with the publication of Isaac Todhunter’s book Spherical Trigonometry. A polymath of considerable degree, Jefferson’s expertise ranged across a broad array of disciplines, with his extensive library at Monticello offering him a near unlimited wealth of literature. As this manuscript plainly reveals, Jefferson’s genius extended far beyond the realms of legation, agriculture, and government.