84 Prof. Cayley. On the [June 19, 



cos (A + B)=0, and thence A + B=a right angle, or A + B-f C=two 

 right angles. 



Hence also in any triangle ABC, drawing a perpendicular, say AD 

 from A to the side BC, and so dividing the triangle into two right- 

 angled triangles, we prove that the snm A+B + C of the angles is 

 = two right angles, and we further establish the relations 



6=ccos A + acosC, c=acosB + 6cos A, 



which are the fundamental formulae of plane trigonometry ; that is, 

 we derive the metrical geometry or trigonometry of the plane from 

 the two original equations a=ccos B, 6=c sin B. 



5. Supposing the plane bent in any manner, converted that is into a 

 developable surface or torse, and using the term straight line to 

 denote a geodesic of the surface, then the straight line of the surface 

 is in fact the form assumed, in consequence of the bending, by a 

 straight line of the plane. The sides and angles of the rectilinear 

 triangle ABC on the surface are equal to those of the rectilinear 

 triangle ABC on the plane, and the metrical relations hold good 

 without variation. But it is not simpliciter true that the descriptive 

 properties of the torse are identical with those of the plane. This 

 will be the case if the points of the plane and torse have with each 

 other a (1, 1) correspondence, but not otherwise. For instance, 

 consider a plane curve (such as the parabola or one branch of the 

 hyperbola) extending from infinity to infinity, and let the torse be 

 the cylinder having this curve for a plane section ; then to each point 

 of the plane their corresponds a single point of the cylinder ; and 

 conversely to each point of the cylinder there corresponds a single 

 point of the plane ; and the descriptive geometries are identical. In 

 particular two straight lines (geodesies) on the cylinder cannot inclose 

 a space ; and Euclid's twelfth axiom holds good in regard to the 

 straight lines (geodesies) of the cylinder. But take the plane curve 

 to be a closed curve, or (to fix the ideas) a circle ; the infinite plane is 

 bent into a cylinder considered as composed of an infinity of 

 convolutions ; to each point of the plane there corresponds a single 

 point of the cylinder, but to each point of the cylinder an infinity of 

 points of the plane; and the descriptive properties are in this case 

 altered ; the straight lines (geodesies) of the cylinder are helices ; and 

 we can through two given points of the cylinder draw, not only 01 

 but an infinity of helices; any two of these will inclose a spi 

 And even if instead of the geodesies we consider only the shorl 

 lines, or helices of greatest inclination ; yet even here for a pair 

 points on opposite generating lines of the cylinder, there are 

 helices of equal inclination, that is, two shortest lines inclosing 

 space. We have in what precedes an illustration in regard to 

 descriptive geometry of the pseudosphere ; this is not identical wil 





