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Jefferson’s Notes Explaining Napier’s Rule on Spherical Triangles, a Branch of Geometry Crucial to Astronomy, Geodesy, Navigation, & Architecture
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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.

THOMAS JEFFERSON. Autograph Manuscript. Notes on Napier’s Theorem. [Monticello, Va.], [ca. March 18, 1814].

Inventory #23358       Price: $35,000

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.”


Collected 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. 


[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 given . 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.

John Napier (1550-1617) a Scottish nobleman and mathematician, is best known for inventing logarithms and pioneering the use of decimal points. He also 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).

Louis Hue Girardin, a Frenchman, was appointed professor of modern languages at the College of William and Mary in 1803. He is best known for assembling volume IV of Burke’s History of Virginia. He had earlier served as the Librarian for Louis XVI.  In addition to modern languages, Girardin also taught history and geography, the latter of which perhaps explains his interest in Napier’s formula.

Spherical and hyperbolic triangles

Originating in ancient Greek scholarship, the field leapt forward in early modern Europe with developments by Scotsman John 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.

Spherical polygons are determined by the intersection of a number of great circles—circles that can be drawn around a sphere dividing it completely in half. The shapes created by these intersecting lines are, in effect, laid out on a spherical surface. Spherical triangles are among the most useful, especially in determining size and distance.

Unlike two-dimensional (planar) triangles, whose 3 angles always add up to 180°, the sum of a spherical triangle adds up to more than 180°. 

Consider three great circles on the sphere of the Earth. The first is the Equator. The second is the Prime Meridian, 0° longitude, and its counterpart on the other side of the world, 180° longitude. The third is a great circle made up of 90° East and 90° West longitude. All three circles perfectly bisect the globe.

The Equator and the Prime Meridian intersect at a right angle. The circle made up of 90° E and W longitude also intersects the Equator at a right angle. And the Prime Meridian and 90° E and W longitude intersect at right angles. The sum of the three right angles is 270°—the result of a triangle drawn on a sphere instead of a flat surface.

Following similar reasoning, a hyperbolic triangle, one drawn on a concave surface, adds up to less than 180°.

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