Elements of the philosophy of the human mind. By Dugald Stewart. Rev. and abridged, with critical and explanatory notes, for the use of colleges and schools. By Francis Bowen ...

RPEASON'ING AND I)EDUCTIVE EVIDENCE, 459
This remarkable, and indeed singular coincidence of propositions purely hypothetical, with facts which fall under the
examination of our senses, is owing, as I already hinted, to the
peculiar nature of the objects about which mathematics is conversant; and to the opportunity which we have (in consequence
of that mensurability,* which belongs to all of them) of adjusting, with a degree of accuracy approximating nearly to the
truth, the data from which we are to reason in our practical
operations, to those which are assumed in our theory. The
only affections of matter which these objects comprehend are
extension and figure; affections which matter possesses in common with space, and which may therefore be separated in fact,
as well as abstracted in thought, from all its other sensible qualities. In examining, accordingly, the relations of quantity conmany of the sublimest heights of mathematical speculation; that the lat
ter had found out the area of the circle, and calculated its circumference
to more than a hundred places of decimals, when the former could hardly
divide an arc into minutes of a degree; and that many excellent treatises
had been written on the properties of curve lines, before a straight line
of considerable length had ever been carefully drawn, or exactly mcasured on the surface of the earth."
* In an Essay on Quantity, by Dr. Reid, published in the transactions
of the Royal Society of London, for the year 1748, mathematics is very
correctly defined to be "the doctrine of measure." " The object of this
science," the author observes, " is commonly said to be quantity; in which
case, quantity ought to be defined, what may be measured. Those who
have defined quantity to be whatever is capable of more or less, have given
too wide a notion of it, which has led some persons to apply mathematical
reasoning to subjects that do not admit of it." The appropriate objects of
this science are therefore such things alone as admit, not only of being increased and diminished, but of being multiplied and divided. In other
words, the common quality which characterizes all of them is their msensumrability.
In the same essay, Dr. Reid has illustrated, with much ingenuity, a dis.
tinction (hinted at by Aristotle) of quantity into proper and improper.
"I call that," says he, " proper quzantity, which is measured by its owzn kind;
or which, of its own nature, is capable of being doubled or trebled, without taking in any quantity of a different kind as a measure of it. Thus a
line is measured by known lines, as inches, feet, or miles; and the length
of a foot being known, there can be no question about the length of two