Page:Collingwood - Life and Letters of Lewis Carroll.djvu/306

 son proves a series of Propositions, which lead up to and enable him to accomplish the feat referred to above.

At the end of Book II. he places a proof (so far as finite magnitudes are concerned) of Euclid's Axiom, preceded by and dependent on the Axiom that "If two homogeneous magnitudes be both of them finite, the lesser may be so multiplied by a finite number as to exceed the greater." This Axiom, he says, he believes to be assumed by every writer who has attempted to prove Euclid's 12th Axiom. The proof itself is borrowed, with slight alterations, from Cuthbertson's "Euclidean Geometry."

In Appendix I. there is an alternative Axiom which may be substituted for that which introduces Book II., and which will probably commend itself to many minds as being more truly axiomatic. To substitute this, however, involves some additions and alterations, which the author appends.

Appendix II. is headed by the somewhat startling question, "Is Euclid's Axiom true?" and though true for finite magnitudes—the sense in which, no doubt, Euclid meant it to be taken—it is shown to be not universally true. In