new Axiom, or
rather "Quasi-Axiom"--for it's _self-evident_ character is open
to dispute. This Axiom is as follows:--

In any Circle the inscribed equilateral Tetragon (Hexagon in
editions 1st and 2nd) is greater than any one of the
Segments which lie outside it.

Assuming the truth of this Axiom, Mr. Dodgson 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 Appendix III. he propounds the question, "How
should Parallels be defined?"

Appendix IV., which deals with the theory of Parallels as it stands
to-day, concludes with the following words:--

I am inclined to believe that if ever Euclid I. 32 is proved
without a new Axiom, it will be by some new and ampler
definition of the _Right Line_--some definition which
shall connote that mysterious property, which it must
somehow possess, which causes Euclid I. 32 to be true. Try
_that_ track, my gentle reader! It is not much trodden
as yet. And may success attend your search!

In the Introduction, which, as is frequently the case, ought to be
read _last_ in order to be appreciated properly, he relates his
experiences with two of those "misguided visionaries," the
circle-squarers. One of them had selected 3.2 as the value for
"_pi_," and the other proved, to his own satisfaction at least,
that it is correctly represented by 3! The Rev. Watson Hagger, to
whose kindness, as I have alre