Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

tween them, is multiplied by 2, and added to the sum of the
two squares, the aggregate will be a square number. Thus 1
+4=5, and 5 added to twice 2 is equal to 9. Thus too 4+9
=13, and 13 added to twice 6, is equal to 25; and so of the
rest. But if the process is converted, and between the two first
terms longer in the other part, the second square is placed,
which is in order the second, but in energy the first; and if this
square is doubled, and is added to the sum of the two other
terms, the aggregate will again be a square. Thus 2 added to
6, added to twice 4, is equal to the square 16. Thus too, if
the second and third terms longer in the other part are added
together, and to their sum the double of the third square is
added, viz. if to 12+6=18 the double of 9 is added, the
aggregate will be the square 36. And thus also, 12+20=32,
added to twice 16, is equal to the square 64; and so of the rest.
This too, is no less admirable, that when a number longer in
the other part is placed as a medium between two squares, and
a square by the above process is produced from the three terms,
the square so produced has always an odd number for its side.
Thus from 1+4 and twice 2 the square 9 is produced, the side
of which is the odd number 3. And the square 25 which is
produced from 4+9, and twice 6, has for its side the odd number
5. Thus likewise, the square 49, arising from the addition
of 9, 16, and twice 12, has for its side the odd number 7. And
the like will be found to take place in the rest.
But when one square subsists between two numbers longer
in the other part, all the squares produced from them will have
even numbers for their sides. Thus the square 16, produced
from the addition of 2,6, and twice 4, has for its side the even
number 4. Thus too, the square 36, arising from the addition
of 6, 12, and twice 9, has for its side the even number 6. And
thus also the square 64, produced by the addition of 12, 20,