The subject which I propose to discuss in this paper is the value of the method of demonstration by superposition. I am satisfied that it has been much under-rated, and in some cases misunderstood. It may be stated, that the essential characteristic of this method of demonstration, is the mental comparison of two magnitudes, by placing one of them upon the other. Euclid's axiom of equality (which, perhaps, is rather a definition) is this: “Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal.” Accordingly, in his first four books, Euclid never regards two magnitudes as equal, except under circumstances wherein it can be shown that this condition is satisfied. Only in one proposition has he avoided the labour which a strict attention to this requirement necessarily imposes; and perhaps, even in that case, it is hypercritical to object to what he has done. All that he assumes is this: it being admitted that when A fills the same space as B, A is equal to B; it must therefore be admitted, that when A and C together fill the same space as B and C together, A is also equal to B.