Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

sides. And in the following numbers, all the species of superpartient
ratio will be found in an orderly progression.
The manner however in which each of them is produced
ad infinitum, is as follows: If each of the terms that form the
superbipartient habitude is doubled, superbipartient ratio will
always be generated. Thus by doubling 3 and 5,6 and 10 will
be produced, and these numbers will form a superbipartient
ratio. And if these again are doubled, the same order of ratio
will arise. By thus proceeding also ad infinitum, the states of
the former habitude will not be changed.
Again, in order to find supertripartient habitudes, the first
supertripartients 7 and 4 must be tripled, and numbers of this
kind will be generated. And if those that are produced from
these are also tripled, the same ratio will be formed.
Thus, too, in order to produce superquadripartient habitudes
ad infinitum, the first roots of them, i.e. 9 and 5, must be rnultiplied
by 4, and the products of this multiplication must be
also quadruplicated, and the same ratio will present itself to
the view. The other species likewise will be generated by
causing the roots always to increase by one multiplication. But
the numbers in the above table are called roots, because all the
before-mentioned habitudes are derived from them. In the
superbipartient ratio also, because the greater contains the less,
and two thirds of the less, the habitude is called superbipartient-tertian.
Thus, too, the supertripartient ratio is denominated
supertripartient-quartan, because the greater contains
the less, and three fourths of it besides. Thus again, the superquadripartient
is denominated superquadripartient-quintan.
And after a similar manner in the rest. Hence the ratio which
is called superbipartient, may also be called superbitertian.
That which is denominated supertripartient, may also be called
supertriquartan. And that which has the appellation of