Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

BOOK Two 97
are opposite to those above mentioned. The philosophers however,
posterior to Plato and Aristotle, on account of the perfection
of the decad according to Pythagoras, added four other
middles, that in these proportionalities a decad might be formed.
It is indeed in conformity ta this number, that the former
five habitudes and comparisons, which we have discussed, are
described; where to the five greater proportions, which we
called leaders, we adapted other less terms which we called
attendants. Hence it is evident, that in the description of the
ten predicaments by Aristotle, and prior to him by Archytas
(though it is considered as dubious by some, whether these
predicaments were invented by Archytas), the Pythagoric decad
is to be found.
Let us now, however, direct our attention to proportionalities
and middles. And, in the first place, let us discuss that
middle which preserves the habitudes of terms according to
equality of quantity, neglecting similitude of ratio. But with
these quantities that middle is conversant in which there is an
equal difference of the terms from each other. What the difference
of terms is, however, has been before defined. And
that this middle is arithmetical, the ratio itself of numbers will
evince, because its proportion consists in the quantity of number.
What then is the reason that the arithmetical habitude
is prior to all other proportionalities ? In the first place, it is
because the natural series of numbers in which there is the
same difference of the terms, comprehends this middle. For
the terms differ from each other by unity, by the fecundity of
which the nature of number is first unfolded. In the second
place, it is because, as was observed in the first chapter of the
first book, arithmetic is prior both to geometry and music; the
two latter cointroducing at the same time the former; and the
former when subverted, subverting the two latter. The discussion,
therefore, will proceed in order, if we first begin from