473 



were, in fact, stuilicil by the Greek geometers,* ami, along 

 with the ohlate splieroid anil the cone, they make up all the 

 surfaces of the second order with which the ancients were ac- 

 quainted. Except the cone, the surfaces considered by 

 them are all of revolution ; and there is only one surface 

 of revolution, the hyperboloid of one sheet, which was not 

 noticed until modern times. This surface is mentioned (un- 

 der the name of the hyperbolic cylindroid) by Wren,f who re- 

 marks that it can be generated by the revolution of a right 

 line round another right line not in the same i)lane. As to the 

 general conception of surfaces of the second order, the sug- 

 gestion of it was reserved for the algebraic geometry of Des- 

 cartes. In that geometry the curves previously known as sec- 

 tions of the cone are all expressed by the general equation of 

 the second degree between two coordinates ; and hence it 

 occurred to Eulerj about a century ago, to examine and 

 classify the different kinds of surfaces comprised in the 

 general equation of the second degree among three coordinates . 

 The new and more general forms thus brought to light have 

 since engaged a large share of the attention of geometers ; 

 but the want of some other than an algebraic principle of con- 

 nexion has prevented any great progress from being made 

 in the investigation of such of their properties as do not im- 

 mediately depend on transformations of coordinates. This 

 want the modular method of generation perfectly supplies, 

 by evolving the different forms from a simple geometrical 

 conception, at the same time that it brings them within the 

 range of ideas familiar to the ancient geometry, and places 

 their relation to the conic sections in a striking point of view. 



* The hyperboloid of two sheets, and the paraboloid of revolution, were 

 known by the name of conoids. Archimedes has left a treatise on Conoids and 

 Spheroids, as well as a treatise on the Sphere and Cylinder. 



t In the Philosophical Transactions for the year 1669, p. 961. 



X See his Introductio in Analysiii IiiJinitoium,p. 373 ; Lausanne, 1748. 



2s 2 



