Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

terms should accord with its proper principle." We shall likewise
find this property to be of the greatest advantage in ascertaining
those numbers which are called superfluous, dirninished,"rnd
perfect.
This also must not be passed over in silence, that in this
series when the number of terms is even, the rectangle under
the extremes is equal to the rectangle under the two means;
for when the series is even the media are two. Thus in that
disposition of evenly-even numbers in which the last term is
128, the two means are 8 and 16, which multiplied by each
other produce 128, equal to 1 X 128. The numbers likewise
which are above these, if they are multiplied, will produce the
same number. For 4x32 is equal to 1 X 128. But if the number
of terms is odd, one middle term is found, and this multiplied
into itself will be equal to the product of the two extremes.
Thus in that series of terms, in which the extreme is
64, one middle term alone is found, and this is 8, which multiplied
into itself is equal to l X64. The terms also which are
above this medium will, when multiplied into each other,
give the same product; for 4X 16164. Thus also 32 multiplied
by 2 and 1 multiplied by 64, produce the same number without
any variation.
+ It occurred to me that a similar property might be found in the series whose
terms are in a triple, quadruple, quintuple hc. ratio. And I discovered that if
in the series whose terms are in a triple ratio, each of the terms is doubled, the sum
of all the terms but the last will be equal to the last term less by 1. Thus in the
series 1. 3. 9. 27. 81. 243. 729, kc. if each term is doubled, the series wilI be 2.
6. 18. 54. 162. 486, kc. And 2+6=8, which is less than 9 by 1. Again, 2+6+
18~26, which is less than 27 by 1. And 2+6+18+54=80, which is less than 81
by 1. And so in other instances. But in a series whose terms arc in a quadruple
ratio, each of the terms must be tripled. In a quintuple series, the terms must be
made quadruple; in a sextuple, quintuple, and the same property will take place.
+ It is evident that each of the terms in the duple series is diminished or defective,
for the sum of its parts is less than the whole. And from what we have
shown, this is evident a fortiori in the triple, quadruple kc. series.