9

14

We shall see, in due time, what those five
premisses are. Suffice it for the present to say
that there are only five. Now if each of them
could be efficient and that in only one way, as one
would have a right to infer from the account of
reasoning given in the text-books, it would
necessarily follow that there could not be but
32 theorems of the theory of numbers in all; whereas
of highly interesting theorems already known there
are hundreds. Euclid's Elements, which was never
designed to be more than an introduction to
geometry and algebra (or that theory which with the
Greeks served the purpose of our algebra) has only
5 postulates, which are the main premisses, together
with 9 axioms and 132 definitions. From these
Euclid deduces 369 theorems, 96 problems,

14

We shall see, in due time, what those five
premisses are. Suffice it for the present to say
that there are only five. Now if each of them
could be ifficient and that in only one way, as one
would have a right to infer from the account of
reasoning given in the text-books, it would
necessarily follow that there could not be but
32 theorems of the theory of numbers in all; whereas
of highly interesting theorems already known there
are hundreds. Euclid's Elements, which was never
designed to be more than an introduction to
geometry and algebra (or that theory which with the
Greeks served the purpose of our algebra) has only
5 postulates, which are the main premisses, together
with 9 axioms and 132 definitions. From these
Euclid deduces 369 theorems, 96 problems,