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Jefferson’s Notes Explaining Napier’s Rule on Spherical Triangles, a Branch of Geometry Crucial to Astronomy, Geodesy, Navigation, & Architecture
Monticello, VA 1814 - Autograph Manuscript. Notes on Napier's Theorem. [Monticello, Va.], [ca. March 18, 1814]. John Napier, who is also credited with inventing logarithms and pioneering the use of the decimal point, first published his rule in 1614. While spherical trigonometry was the foundation for many scientific pursuits including astronomy, celestial navigation, geodesy (the measurement and mathematical representation of the Earth), architecture, and other disciplines, Napier's Theorum remained largely unknown in America because of its complexity. Since it was so important to his own scholarly pursuits, Jefferson, the Sage of Monticello, was the perfect person to school a professor friend on this important, but complicated mathematical formula.For instance, a navigator's distance and position can be determined by "solving" spherical triangles with latitude and longitude lines-essentially very large triangles laid out on a curved surface. Astronomers apply similar principles; stargazers imagine the sky to be a vast dome of stars, with triangles laid out on curved (in this case concave) surface. The distance of stars can be calculated by the viewer, who is considered to be standing at the center (the Earth) and looking up at stars and planets as if they were hung on the inside surface of the sphere. In architecture, spherical triangles fill the corner spaces between a dome that sits on foursquare arches-called a dome on pendentives. The Papers of Thomas Jefferson assign the date based on nearly identical language found in a letter of March 18, 1814 from 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" (meaning all inclusive or universal) with a discussion of the many English and French mathematical texts that omit it or consider it too difficult for "young computists."ProvenanceCollected in the late 19th or early 20th century and donated to a historical society in New Jersey. The only time this manuscript has ever been publicly offered was in 1979. However, when it appeared then in a Charles Hamilton auction, the Papers of Thomas Jefferson noticed it with suspicion and checked with its owner to make sure the sale was authorized. They were right to question it: the document had been stolen. It was withdrawn before the auction, and returned to its rightful owner, from whom we recently bought it. Transcript[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 par. (See website for full description)
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