is induction: but the difficulties in sciences
often lie (as, e.g. in geometry, where the inductions are the simple
ones of which the axioms and a few definitions are the formulae) not at
all in the inductions, but only in the formation of trains of reasoning
to prove the minors; that is, in so combining a few simple inductions as
to bring a new case, by means of one induction within which it evidently
falls, within others in which it cannot be directly seen to be included.
In proportion as this is more or less completely effected (that is, in
proportion as we are able to discover marks of marks), a science, though
always remaining inductive, tends to become also _deductive_, and, to
the same extent, to cease to be one of the _experimental_ sciences, in
which, as still in chemistry, though no longer in mechanics, optics,
hydrostatics, acoustics, thermology, and astronomy, each generalisation
rests on a special induction, and the reasonings consist but of one step
each.

An experimental science may become deductive by the mere progress of
experiment. The mere connecting together of a few detached
generalisations, or even the discovery of a great generalisation working
only in a limited sphere, as, e.g. the doctrine of chemical equivalents,
does not make a science deductive as a whole; but a science is thus
transformed when some comprehensive induction is discovered connecting
hosts of formerly isolated inductions, as, e.g. when Newton showed that
the motions of all the bodies in the solar system (though each motion
had been separately inferred and from separate marks) are all marks of
one like movement. Sciences have become deductive usually through its
being shown, either by deduction or by direct experiment, that the
varieties of some phenomenon in them uniformly attend upon those of a
better known phenomenon, e.g. every variety of sound, on a distinct
variety of oscillatory motion. The science of number has been the grand
agent in thus making sciences deductive. The truths of numbers are,
indeed, affirmable of all things only in respect of their quantity; but
since the variations of _quality_ in various classes of phenomena have
(e.g. in mechanics and in astronomy) been found to correspond regularly
to variations of _quantity_ in the same or some other phenomena, every
mathematical formula applicable to quantities so varying becomes a mark
of a corresponding general truth respecting the accompanying variations
in qual