Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

for instance, 12 is a number longer in the other part, the greater
side of which is 4, but the less 3. Hence the former has to
the latter a sesquitertian ratio, which is less than a sesquialter
ratio, because is less than f. Euclid never distinguishes numbers
longer in the other part from such as are oblong, and
comprehends all the species of them under the general name
of planes; for which he is reprehended by the acute Jamblichus
as follows: "This again Euclid not perceiving, confounds the
diversity and variety of explanation. For he thought that the
number which is longer in the other part, is simply that
which is produced by the multiplication of two unequal numbers,
and does not distinguish it from the oblong number. If
any one however should grant him this, it would happen that
contraries which are naturally incapable of subsisting together,
would be found in the same subject. For his definition comprehends
both square numbers and such as are longer in the
other part."
P. 131. Perfect numbers.-As every perfect is a triangular
number, the side of which is the prime number from which
the perfect number is formed ; if that prime be squared, it will
be equal to the sum of the perfect number itself, and the triangular
number immediately preceding it. Thus 7 )< 7=49,
and 49=21+28. Thus too 31 )(31=961, and 961=465+496.
And so of the rest.
P. 125. In the series 1 +~+ii-i$~+&, &c.-It
is re-
markable likewise, that the sum of any finite number of terms
in the series f + 4 + + &, kc. may be obtained by multiplying
the last term by the denominator of the term next to
the last. Thus the sum of f f is equal to 3 X -1.-8-1
6 - 0 2 '
Thus too the sum of $ + $ + is equal to 6 X = A. And
thus also the sum of f + % 4-
of the rest.
FINIS.
+,A
is equal to
And so