Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy
Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy
Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy
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4, and terminate in 20, are compared to these, the double, i.e.<br />
the first species of multiplicity will be exhibited. Hence, the<br />
first will surpass the first by unity alone, as 2 surpasses 1. The<br />
second will exceed the second by 2, as 4 exceeds 2. The third<br />
will exceed the third by 3, as 6 exceeds 3 by 3. The fourth<br />
will surpass the fourth by 4, and in this way 8 surpasses 4.<br />
And after a similar manner in the rest. But if the third angle<br />
is considered, which begins from 9, and which extends both in<br />
length and breadth as far as to the number 30, and if this is<br />
compared with the first length and breadth, the triple species<br />
of multiplicity will present itself to the view, so that the comparison<br />
will take place through the black angle. And these<br />
numbers will surpass each other according to the natural progression<br />
of the even number. For the first number will surpass<br />
the first by 2, as 3 surpasses 1. The second surpasses the second<br />
by 4, as 6 surpasses 2. The third surpasses the third by 6, as 9<br />
surpasses 3. And after the same mode of progression the rest<br />
are increased. If again, the boundary of the fourth angle is<br />
considered, which is distinguished by the quantity of the number<br />
16, and which terminates its length and breadth in the<br />
number 40, here also a similar comparison being made with<br />
the preceding, will unfold the quadruple species of multiplicity.<br />
Hence, the first will surpass the first by 3, as 4 surpasses<br />
unity. The second will surpass the second by 6, as 8 surpasses<br />
2. The third will exceed the third by 9, as 12 exceeds 3. And<br />
after a similar manner in all the following numbers. If the<br />
remaining angles likewise are considered, the same thing will<br />
take place through all the species of multiplicity, as far as to<br />
the decuple species.<br />
If however, in this description, the superparticular species<br />
are required, they may be found by the following method.<br />
For if we direct our attention to the second angle, the begirlning<br />
of which is 4, and above which is 2, and if the row in
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