First occurrence of hyperboloid paraboloid

Question

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?

Answer 1

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.