An introductory text-book of logic - Mellone, Sydney - Rare Books at ...

24$ THE GENERAL NATURE OF INDUCTION.
angle, simply because it is isosceles, must agree with our
specimen in all the qualities necessary for the proof.
The length of any of the sides, or the size of any of the
angles, points in which any triangle may differ from
any other, are not included in the definition of the
triangle, and they are not the points on which the proof
depended.
The universality of the result depends on our being
absolutely certain of what are the essentials of the kind
of triangle in question ; and we can be certain of these
because in geometry definitions have not to be discovered.
The geometrician can frame his own definitions, and
change them, if necessary.
(b) Let us next consider an algebraical formula which
is true universally /.<?., true whatever quantities the letters
may represent.
It may easily be proved that
(a + &) (a-b) = a?-P.
Having proved this in the single case, we know that the
result is of absolutely universal validity, whatever the
quantities may be, provided
that a and b are different
quantities. How do we know this ? Because the proof
depended only on the definition and rules of algebraical
"
multiplication," on the
"
essential qualities," so to
speak, of this operation, and not on any quantity that
the terms a and b might represent. And the definition
and rules of the operation have not to be discovered ;
the algebraist, like the geometrician, frames his own
definitions.
Mathe
(c) There is a process technically termed
"
matical Induction," which reaches a universal conclusion
from two or three instances. It illustrates the same
principle as the previous inductions ; but it is specially
applicable to terms which may be arranged in a regular
series whose order of progression is known. The follow-