Legendre's Gtomelry. 299 



conceiving clearly the situation of the various planes used, 

 and their projections. The subject is treated in a clear 

 and rigorous style. 



The second section of part II. treats ofpolyedrons and of 

 their measure. We have before suggested, that we feel 

 ourselves unable to convey an adequate view of the merit of 

 this part of Legendre's work. Those who are only acquaint- 

 ed with the geometry of solids or volumes as given by the 

 older writers, we are sure, will be surprised and delighted 

 at the luminous and novel manner, in which this part of el- 

 ementary geometry is exhibited- On volumes he iias made 

 a distinction between two difParent kinds of equality anal- 

 ogous to that which we before noticed with respect to the 

 comparison of surfaces. " Two solids, two solid angles, 

 two spherical triangles, or two spherical polygons, may be 

 equal in all their constituent parts without coinciding when 

 applied. It does not appear that this observation has been 

 made in elementary books ; and for want of having regard 

 to it, certain demonstrations founded upon the coincidence 

 of figures, are not exact. Such are the demonstrations, by 

 which several authors pretend to prove the equality of 

 spherical triangles, in the same cases and in the same man- 

 ner, as they do that of plane triangles. We are furnished 

 with a striking example of this by Robert Simson, who, in 

 attacking the demonstration ot" Euclid, B. XI. prop. 28, 

 fell himself into the error of founding his demonstration 

 upon a coincidence which does not exist. We have thought 

 it proper, therefore, to give a particular name to this kind 

 of equality, which does not admit of coincidence ; we have 

 called it equality by symmetry ^ and the figures which are 

 thus related, we call symmetrical figures."* We think, as in 

 the case of surfaces, that the defects in the usual language 

 would have been better supplied by calling those figures 

 which would coincide, coincident tigures, that is, figures 

 equal by coincidence ; and those figures which will not 

 coincide, symmetrical figures, that is, figures equal by sym- 

 metry. 



In the propositions relating to polyedrons, as well as in 

 those relating to polygons and solid angles, those having re- 

 entering angles are excluded as not belonging to the ele- 



* Trans. Note l.p,202. 



