# First occurrence of hyperboloid paraboloid

The ancient greeks considered surfaces such as cones, but did they study the hyperbolic paraboloid? What is the first occurrence of such surface in history?

• At storyofmathematics.com/hyperbolic-paraboloid they claim that "Jean-Baptiste Listing and Carl Friedrich Gauss significantly influenced the study of curved surfaces and the growth of differential geometry" but provide no further details. Oct 19, 2023 at 13:33
• Check A history of the conic sections and quadric surfaces. Coolidge, Julian Lowell, 1873-1954 [1968] (the original edition seems to have been published in 1945). Oct 19, 2023 at 14:03
• Thanks for your reference. Coolidge asserts that only quadrics of revolution were considered by the greeks and then cites Fermat, Brianchon and Dupin as the next mathematicians working on quadrics. A wiki transcription of the work of Fermat tells us that Fermat did not discover the hyperbolic paraboloid, see fr.wikisource.org/wiki/%C5%92uvres_de_Fermat/I/Lieux_en_surface. So the mystery remains. Maybe Descartes? Monge proved a few theorems about the hyperbolic paraboloid but this is much later. Oct 19, 2023 at 15:33

I suspect that Coolidge goes from Fermat to Brianchon, and only mentions Monge in passing, because his focus is on synthetic geometry. He remarks:

"It is certainly surprising that the synthetic geometers of the seventeenth and eighteenth centuries who were so much interested in the conic sections should have shown no particular interest in the quadric surfaces."

But the main work on quadrics in the 18th century was done analytically, first by Clairaut, in connection with his investigations of the shape of the Earth, and then by Euler, Lagrange and Monge. In the appendix on surfaces to the second volume of his Introductio in analysin infinitorum (1748), Euler gave a complete geometric classification of quadrics, and it is there that we meet the hyperbolic paraboloid for the first time. Some historical commentary can be found in Kline's Mathematical Thought from Ancient to Modern Times, v.2, ch.23:

"The quadric surfaces, e.g. sphere, cylinder, paraboloid, hyperboloid of two sheets, and ellipsoid, were of course known geometrically before 1700; in fact, some of them appear in Archimedes' work. Clairaut in his book of 1731 gave the equations of some of these surfaces... Though Euler had done some earlier work on the equations of surfaces, it is in Chapter 5 of the Appendix to the second volume of his Introductio (1748) that he systematically takes up three-dimensional coordinate geometry.

He presents much of what had already been done and then studies the general second degree equation in three variables [denoted (1)]. He now seeks to use change of axes to reduce this equation to the forms that result from having the principal axes of the quadric surfaces represented by (1) as the coordinate axes... Euler uses this transformation to reduce (1) to canonical forms and obtains six distinct cases: cone, cylinder, ellipsoid, hyperboloid of one and two sheets, hyperbolic paraboloid (which he discovered), and parabolic cylinder.

Monge later showed that hyperboloid of one sheet and hyperbolic paraboloid are doubly ruled surfaces, i.e. can be "assembled" from two families of straight lines, which precipitated their popularity in engineering and architecture. Euler went on to develop differential geometry of general surfaces in Recherches sur la courbure des surfaces (1760) and study developable surfaces in De Solidis Quorum Superficiem in Plan um Explicare Licet (1771). Their study was also later taken up by Monge.

• It would be good to have a more reliable reference than Morris Kline. Oct 22, 2023 at 8:24
• @MikhailKatz Introductio in analysin infinitorum is available on Euler Archive, classification of quadrics is on pp. 373ff. Aside from modernized notation, Kline's description follows the source. Oct 22, 2023 at 8:38
• Kline is not a reliable secondary source (I can provide some details if you are interested; they are manifold). Do you have a reliable secondary source? Oct 22, 2023 at 8:43
• @MikhailKatz I do not know of one that goes into the details of the appendix, and most users here are unlikely to read Latin. Oct 22, 2023 at 8:46
• I find this odd, because Kline typically copies ("synthesizes") things from author authors - often Boyer. Does this happen to be in Boyer by any chance (not that he is much more reliable than Kline)? Oct 22, 2023 at 8:50