2

age (early in the third century B.C.) or later could not define
mathematics as the science of that which the abstract [no?]
quantity expresses. A line, however, was classed as a quantity,
or quantum, by Aristotle and his followers; so that even perspective
(which deals wholly with intersections and projections, not at all with
lengths) could be said to be a science of quantities, "quantity" being
taken in the concrete sense. That this was what was originally
meant by the definition 'Mathematics is the science of quan-
tity' is sufficiently shown by the circumstance that these
writers who first enunciate it, about A.D. 500, that
is, Ammonius Hermiae and Boethius, make astronomy and
music branches of mathematics; and it is confirmed by the
reasons they give for doing so*.^ That Aristotle did not regard
mathematics as the science of quantity, ^in the modern abstract sense, is evidented in various
ways. The subjects of mathematics are, according to him, the
how much and the continuous. (See Metahph.
K.iii.1061a33.) He referred the continuous to the category of
quantum, and therefore he did make quantum, in a broad sense, the
one [??] of mathematics.

[notes on the side:]
Even Philo of Alexandria, who defined mathe
matics as the science of ideas furnished by
sensation and reflection in respect to their
necessary consequences, since he includes
under mathematics, besides its more essential parts,
the theory of numbers and geometry, also the
practical arithmetic of the Greeks, geodesy, mechanics,
optics (or projective geometry), magic and astronomy, must [??]

*I regret I have
not noted the passage of Am-
monius to which
I refer. It is pro-
bably one of the [??] given
by [B?]. My MS.
[no?] states that
he gives reasons
showing [?] to be
hos meaning.