Abstract
In the late 18th century, both Anders Lexell (1740–1784) and his mentor Leonhard Euler (1707–1783) published a proof of a result in spherical geometry that is now known as Lexell’s Theorem. In this paper, I discuss both proofs and provide some background to the theorem. I will also outline some of the later proofs of the theorem as well as discuss some applications. In this connection, I argue that some of the work of Euler’s disciples Fuss and Schubert may be connected to Lexell’s investigations. In conclusion, I touch upon the role of Lexell’s Theorem in the history of hyperbolic geometry.
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Notes
- 1.
On Lexell’s life and work, see Stén (2014).
- 2.
- 3.
On this, see Delambre (1827) for a quick overview.
- 4.
- 5.
On this, see Footnote 10.
- 6.
Strictly speaking, unless we want to bring in the notion of oriented area, the locus consists of two lines on opposite sides of AB as C can lie on either side of AB. Euclid avoids this issue by only talking about triangles on the same side of the fixed base.
- 7.
From the Greek “ana” (again) and “logos” (ratio), i.e., equalities between ratios.
- 8.
Indeed, let V ′ be any point not on the circle with center P and passing through O and N, but on the same side of AB as P. Then, there is a unique circle through V ′, O, and N with its center P′ on the perpendicular bisector of ON. If the area of ABV ′ equals 2δ′, it follows that ∠P′ON equals \(\frac{1} {2}\pi -\delta '\). Since P and P′ will be distinct, so will δ and δ′. In other words, V ′ does not have the desired property.
- 9.
See Calinger (2016), p. 515.
- 10.
Euler does not actually indicate how he knew about Lexell’s result. He just refers to it as a theorem “proposed” by Lexell. If this is a reference to the presentation of Lexell’s paper, it would place the presentation before January 1778 (assuming that the pertinent part of Euler’s paper is not a later insertion).
- 11.
- 12.
The theorem was already included in the first edition of the book, but its very short proof was based on the unverified assumption that the maximum area is reached for exactly one configuration of the two given sides. Starting with the third edition, a much longer proof replaces the original (but incomplete) proof. See Legendre (1800), pp. 254–257.
- 13.
Actually, this construction is only possible as long as the antipode B′ of B lies “outside” the circle with center A, i.e., as long as the sum of the given sides is less than two right angles. If the sum of the given sides is greater or equal to two right angles, the area of the triangle increases as the angle between the given line segments increases until the triangle degenerates into three arcs on the same great circle. Steiner does not distinguish between these two cases, although Legendre does.
- 14.
Note that the latter characterization implies in its turn that the chordal (rectilinear) triangle ABC has to be right-angled at A. This would allow for an explicit construction of the spherical triangle ABC of maximal area, but clearly it is not a property intrinsic to spherical geometry as the study of the geometry of the surface of a sphere. In terms of spherical trigonometry, given that A has to equal B + C, it follows from the Napier analogy used by Lexell (see above) that \(\cos (A) = -\tan (\frac{1} {2}b)\tan (\frac{1} {2}c)\), where b and c are the given lengths of the triangle. Since b + c < π, A is well-defined.
- 15.
Among other things, it established that any spherical figure bounded by great circles can be decomposed into any other such spherical figures of the same area. The corresponding result for the plane, also known as the Bolyai-Gerwien Dissection Theorem, was proved independently by Wolfgang Bolyai and Paul Gerwien in the 1830s, although the British mathematician William Wallace is said to have proved the same result in 1807.
- 16.
One exception is Catalan (1843), pp. 272–273, which follows Steiner’s argument very closely.
- 17.
- 18.
- 19.
See Engel and Stäckel (1898), p. 321. It seems unlikely that at the time Gauss was aware of these proofs. On the other hand, a summary of Lobachevsky’s work on non-Euclidean geometry had been published in German in 1840. Even if the exact chronology is not entirely clear, it is tempting to think that Gauss’ request was inspired by this summary and his own interest in non-Euclidean geometry.
- 20.
In 1844, very similar diagrams are used a number of times in Meikle (1844) to argue that the sum of the angles of a triangle is a measure for the area of that triangle. Likewise, Hilbert uses the same diagram in his proof of Legendre’s Second Theorem (“If in some triangle the sum of its angles is equal to two right angles, then the same is true in every triangle”) in the later editions of his Grundlagen der Geometrie (see Hilbert (1971), p. 38).
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Atzema, E.J. (2017). “A Most Elegant Property”: On the Early History of Lexell’s Theorem. In: Zack, M., Schlimm, D. (eds) Research in History and Philosophy of Mathematics. CSHPM 2016. Proceedings of the Canadian Society for History and Philosophy of Mathematics/La Société Canadienne d’Histoire et de Philosophie des Mathématiques. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-64551-3_8
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