Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

from the triangle above it, and thrice the antecedent triangle,
as 15 from 6 and thrice 3. And so on, the number of the preceding
triangle being increased by unity. The pentagon 12 is
composed from the square 9 and the preceding square 4, the
first triangle unity being subtracted from the sum. The hexagon
15 consists of the pentagon 12 placed above it, and the
antecedent pentagon 5, the two triangles of unity being sub
tracted from the sum. The heptagon 18 consists of the hexagon
15 placed above it, and the preceding hexagon 6, the three
triangles of unity being subtracted from the sum. The square
16 consists of the triangle 10 and 6. The pentagon 22 consists
of the square 16 and the triangle 6. The hexagon 28, of the
pentagon 22, and the triangle 6. The heptagon 34, of the hexagon
28, and the triangle 6. The square 16 consists of the triangles
10 and 6, precisely and entirely. But the pentagon 22
consists of the squares 16 and 9, less by once the first triangle
in energy 3. The hexagon 28, consists of the pentagons 22 and
12, less by twice the first triangle in energy, i.e. less by 6. And
the heptagon 34, consists of the hexagons 28 and 15, less by
thrice the first triangle in energy, i.e. less by 9.
P. 79. The numbers called Aetevomekeis, &c.-Because these
numbers depart from that equality which is in squares, the
Pythagoreans arranged them in the cwrdination of things
evil; but they arranged squares in the co-ordination of things
good. The reason however of this arrangement is, that all
square numbers or figures are similar to themselves, and have
equal sides, which are in the same ratio to each other. Hence
they are at rest, and do not depart from unity. But numbers
and figures longer in the other part, are dissimilar to each
other, and have also a dissimilar ratio of their sides. For as in
the natural series of numbers 1,2,3,4,5,6, &c. the ratio of the
last to the next antecedent term, becomes by proceeding less
and less, so in numbers longer in the other part, the more they
increase the less is the ratio of the greater side to the less. Thus