Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

And after this manner, if all the numbers in the duple series
arc multiplied by those in the upper series, the products wil!
be found to be unevenly-even numbers.
This also is admirable in this species of numbers, that if the
disposition and description of them according to breadth is regarded,
the property of the evenly-odd numbers will present
itself to the view, but the property of the evenly-even, if the
disposition of them is regarded according to length. For according
to breadth, the two extremes are equal to the two
media, or if there is but one medium, the double of it is equal
to the extremes. But according to length, the property of the
evenly-even number will be discovered; for here the product
of the extremes is equal to that of the two media, or if there is
but one medium the square of it is equal to the product of the
extremes. The description of them however, according to
length and breadth, is as follows; The products arising from
the multiplication of evenly-even numbers in an orderly series
by 3, are to be placed in the first row. Again, the products
arising from the multiplication of the same evenly-even numbers
by 5, are to be placed in the second row. Those arising
from the multiplication by 7, are to be placed in the third row,
and so of the rest.
3 5 7 9 etc. Odd numbers.
4 8 16 32 etc. Evenly-even numbers.
Length.
12 24 48 %
20 40 80 160
Breadth,
28 56 112 224
36 72 144 288
Length.
Breadth,
Here in the breadth, if three terms are taken, as for instance