Hipparchus had a problem. The second-century-BC astronomer wanted to quantify his models for the motions of the heavenly bodies—especially the Sun and Moon—so that he could accurately predict their positions and hence eclipses. To be able to say that the Sun would be in a certain place at a certain time, he needed a trigonometry for bodies that move not in planes but on spherical surfaces.

Scholars do not agree who invented spherical trigonometry. It might have been Hipparchus himself, who is credited with inventing planar trigonometry. Possibly it was Menelaus two centuries later, or even Ptolemy a half century after that. In any case, Ptolemy’s astronomical classic Syntaxis mathematica contains a fully realized spherical trigonometry, which differed from ours in several ways. For example, its basic function was the chord of a circular arc rather than the sine, which would be invented centuries later in India. Arabic astronomers who deemed the book “majestic” gave the work its better-known name, Almagest, long after its AD 140 appearance.

Triangles in curved space Section: Choose Top of page ABSTRACT Triangles in curved space << From the heavens to Earth Supplemental material Additional resources figure Imagine you and a friend are at the north pole of a sphere. You each head off on a trek to the equator, departing along paths at right angles to each other. When you reach the equator, you both turn toward each other and walk until you meet. As panel a in theshows, the two of you have now formed a spherical triangle with three right angles, two on the equator and one at the pole; the angle sum is not 180° as in a plane but 270°. And that sum is not the same for all triangles. For a tiny spherical triangle, which would be nearly flat, the angle sum is just over 180°. For a large one, the angle sum can approach 540°. Examine the sides of the triangle in panel a and you’ll see that they are all 90° arcs of great circles. And you’ve probably made a crucial mental transition: On a sphere, side lengths are measured in degrees just as angles are. Let’s use that understanding to solve an astronomical problem as relevant today as it was to Hipparchus more than two millennia ago. Panel b shows the geometry and defines the terms we’ll use. Bold white arcs are on the surface of the celestial sphere, taken to have unit radius and center A; thin colored lines lie within the sphere. The Sun, F, travels along the ecliptic, which intersects the celestial equator (the same plane as Earth’s equator) at an angle ε = 23.4°, the same angle that describes the tilt of Earth’s axis with respect to its orbital plane. The arcs GH and GC are both 90°, so CH also equals ε. The Sun’s longitude, λ = GF, determined by the time of year, should give us enough information to calculate the Sun’s altitude above the equator, the declination δ = FJ. But how? Here is a sketch of the proof; additional details are provided in the online version of this Quick Study. Considering slice HAC, we see that the angle made by the blue lines at A is equal to arc CH, which is ε. Since AC = 1, BC = sin ε. Likewise, looking at slice GAF yields DF = sin λ, and slice JAF similarly yields EF = sin δ. But triangles ABC and DEF are similar. Thus, sin δ/sin λ = sin ε/1, which is just another way to express the standard declination formula sin δ = sin λ · sin ε. Nowadays any pocket calculator can handle the declination formula with ease, but before 20th-century technology, astronomers worked well into the day doing declination calculations by hand. In 1614 John Napier announced a solution to the difficulty. With the logarithms he introduced in his Description of the Miraculous Table of Logarithms, the laborious multiplication of sines in the declination formula can be expressed in terms of a more manageable sum. Additional information about Napier’s book is included online.

From the heavens to Earth Section: Choose Top of page ABSTRACT Triangles in curved space From the heavens to Earth << Supplemental material Additional resources Trigonometry, both plane and spherical, was intended for astronomers; 15th-century astronomer Regiomontanus called it “the foot of the ladder to the stars.” But spherical trigonometry has also influenced actions of those with Earth-bound concerns. The earliest occasions were provided by medieval Islamic scholars, who often needed astronomy to resolve demands required by ritual. Examples included predicting the beginning of the sacred month of Ramadan, which is defined by the emergence of the lunar crescent from the glare of the Sun at the time of the new moon, and determining the times of the five daily prayers, some of which required knowledge of the Sun’s altitude. Moreover, to pray, a worshipper needed to face toward Mecca, a requirement that was also a problem in spherical trigonometry: Determine the position in the sky of the point in the celestial sphere directly above Mecca. Drop that point to the horizon and face in that direction. One of the most dramatic stories in the history of mathematics is the 1837 adventure of Thomas Hubbard Sumner. He set off from South Carolina, and three weeks later he needed to sail through St George’s Channel between Wales and Ireland. However, miserable weather and obscured skies made him unsure of his position, and potentially fatal rocks awaited along the shores. The clouds parted momentarily, which gave him just enough of a window to measure the Sun’s altitude, 12° 10′ above the horizon. Then, with his creativity perhaps sharpened by the stakes of survival, he reasoned as follows: The collection of places on Earth’s surface where the Sun is at a given altitude forms a circle whose center is the Sun’s geographic position—that is, the point on the surface directly below the Sun at a given time (see panel c). Sumner knew he had to be somewhere on that circle, whose location he could calculate. Such a circle is called a small circle, not because it is small but because it isn’t a great circle. In Sumner’s vicinity off the south coast of Ireland, his circle was actually extremely large, very nearly a straight line; navigators call it a line of position. By great fortune, Sumner’s line of position passed through the sea in a northeasterly direction and very nearly contacted Smalls Lighthouse off the coast of Wales in a well-charted region. Although Sumner didn’t know where on the line he was, all he had to do was to keep traveling along it. He would be assured of eventually spotting Smalls Lighthouse, and from there he could navigate to safety. A variation of Sumner’s ingenious reasoning allows you to pinpoint your location on Earth, no matter where you are. If measuring the altitude of a star determines your location on a small circle on Earth’s surface, measuring the altitude of two celestial bodies places you at the intersection of a pair of circles. The two circles intersect at two points, one of which is your location. Almost always, those two points are far away from each other, and if you cannot tell whether you’re off the south coast of Ireland or the south coast of India, you have bigger problems than navigation can solve. Nautical almanacs tabulate the positions of several dozen reference stars, the Sun, and several planets, so you have plenty of celestial bodies to choose from. In practice, you can determine a ship’s location to well within a mile and can increase the reliability of the method by observing the altitudes of more than two bodies. Modern technologies such as GPS have rendered traditional practices of spherical trigonometry obsolete other than for hobbyists. But they are making at least a small comeback. At the US Naval Academy in Annapolis, Maryland, one of the last institutions to give up teaching spherical trigonometry back in the 1960s, officers in training are now being instructed in celestial navigation. The potential for GPS systems to be jammed by enemies at times of conflict may cause threatened sailors to turn their eyes to the heavens, not to cry for help from divine powers, but to apply ancient ingenuity to save themselves and their shipmates.

Supplemental material Section: Choose Top of page ABSTRACT Triangles in curved space From the heavens to Earth Supplemental material << Additional resources Download Original Video (163.1 MB) (QuickTime is needed to play this video in Firefox. To download the QuickTime player plugin, click here .) https://aip-prod-streaming.literatumonline.com/journals/content/pto/2017/pto.2017.70.issue-12/pt.3.3798/20171207/media/pt.3.3798.mm.original.v1.,3840,2560,1920,1605,965,773,645,.mov.m3u8?b92b4ad1b4f274c70877518b1cabb28bd7f063d0ea1e59d65b52a178c3222e73a96e5d2cf9ef940daf1a39c1f7bf21f2977cec26b2ce789f04c248c2e1171efea1033c63da03e512795f8e1d0ac0fc91c95750e6cfcc4c18e72cec1e7797882b52fe8c9b3fe914696a9e6b082d491b5f006466f392691c16b303e52cba40fb9afa4da2879550144cbdd46b7f3b3dd50fa4b75a31425d923cb38c03f3a77b31d99bc4b9d9b7233319dc34740e8109c439402d4f1cfeeccacd3ad7b167 Similar triangles in panel b figure ε, the angle needed to rotate the line AH around the perpendicular axis AG onto the line AC. Accepting for the moment that GDF is also a right angle and given that GDE is right by construction, the angle EDF is also ε, the angle needed to rotate DE around the perpendicular axis AG onto the line DF. The right-angled triangles ABC and DEF are therefore similar. One way to see that GDF is indeed a right angle is to look at the tetrahedron ADEF, three of whose faces are manifestly right triangles. With respect to the fourth face ADF, AF2 = AE2 + EF2 = (AD2 + DE2) + (DF2 − DE2) = AD2 + DF2. That is, triangle ADF satisfies the Pythagorean theorem, so it also must be a right-angled triangle with DF perpendicular to AG. In the proof of the declination formula, I stated that triangles ABC and DEF in panel b of theare similar. Left unstated was that fact that angle ADF is a right angle. Here I justify those assertions. First note that both B and E are defined by dropping perpendiculars from a point on the ecliptic to the plane of the celestial equator. As a result, triangles ABC and DEF are right-angled. Since AGJH and AGFC are quadrants, the angles GAH and GAC are right angles. Thus the angle BAC is, the angle needed to rotate the line AH around the perpendicular axis AG onto the line AC. Accepting for the moment that GDF is also a right angle and given that GDE is right by construction, the angle EDF is also, the angle needed to rotate DE around the perpendicular axis AG onto the line DF. The right-angled triangles ABC and DEF are therefore similar. One way to see that GDF is indeed a right angle is to look at the tetrahedron ADEF, three of whose faces are manifestly right triangles. With respect to the fourth face ADF, AF= AE+ EF= (AD+ DE) + (DF− DE) = AD+ DF. That is, triangle ADF satisfies the Pythagorean theorem, so it also must be a right-angled triangle with DF perpendicular to AG. Logarithms matter δ = sin λ ⋅ sin ε relates the declination of the Sun δ to its longitude λ and the tilt of Earth’s axis ε. The product sin λ ⋅ sin ε involves two quantities, each of which are usually long decimal fractions. Scottish nobleman John Napier, realizing how tedious it would be to have to multiply such numbers repeatedly, announced a solution in his 1614 Description of the Miraculous Table of Logarithms. (The figure δ) = log (sin λ) + log (sin ε). Instead of multiplying the sines, all astronomers had to do was to add their logarithms—which is a whole lot less time- and soul-consuming. The declination formula sin= sin⋅ sinrelates the declination of the Sunto its longitudeand the tilt of Earth’s axis. The product sin⋅ sininvolves two quantities, each of which are usually long decimal fractions. Scottish nobleman John Napier, realizing how tedious it would be to have to multiply such numbers repeatedly, announced a solution in his 1614. (Theshows the cover of the book, courtesy of Special Collections, Lehigh University Libraries, Bethlehem, Pennsylvania.) With logarithms, the declination formula transforms to log (sin) = log (sin) + log (sin). Instead of multiplying the sines, all astronomers had to do was to add their logarithms—which is a whole lot less time- and soul-consuming. LEHIGH UNIVERSITY LIBRARIES But Napier did more than invent logarithms. He includes in his book a systematic compilation of all 10 of the fundamental formulas of a spherical right triangle. They are given below, with the vertices denoted by A, B, and C and the sides opposite those vertices by a, b, and c respectively, with C reserved for the right angle and c for the hypotenuse. Table sin b = tan a ⋅ cot A sin a = sin A ⋅ sin c cos c = cot A ⋅ cot B cos A = sin B ⋅ cos a sin a = cot B ⋅ tan b cos B = cos b ⋅ sin A cos A = tan b ⋅ cot c sin b = sin c ⋅ sin B cos B = cot c ⋅ tan a cos c = cos a ⋅ cos b Many of those formulas were already well known. For instance, the declination formula appears in disguise at the top of the right column. The formula at the bottom of that column is the spherical equivalent of the Pythagorean theorem. Several great 16th-century mathematicians, including Georg Rheticus and François Viète, knew all 10. But Napier recognized the many patterns in the list. Can you spot any? Here’s one. Pick any occurrence of the letter b, read downward, and loop back to the top of the column if you reach the bottom. Every time, you will read the pattern b, c, a, A, B. Additional patterns helped 17th-century educators design various devices to aid in the memorization of the 10 formulas.

I am grateful to Joel Silverberg and my students at Quest University and the summer camp MathPath for reading this article critically and providing helpful comments. I thank Quest student Tala Schlossberg for creating a video based on this work and for making it available with the online version of the Quick Study.