Legendre’s Geometry. 299 
conceiving clearly the situation of the various planes used, 
and their projections. The niecciee is treated in a Sine 
and rigorous style. 
The second section of par If. treats of polyedrons Fo of 
their measure. . We have before suggested, . that we feel 
ourselves unable pinogvey 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 has made 
a distinction between two different 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 pens been 
made in elementary books ; and for want of havin; 
é 
as 
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 of 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 figures, ‘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- 
* Traus. Note L, p. 202. 
