Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

Unequal differences.( 1. 2. 2. 3. 3. 4. 4. 5. 5. 6,
-- -
Numbers longer in
the other part, be- } 1. 2.
tween squares. J
4. 6. 9. 12. 16. 20. 25. 30. 36,
Equal differences.
I
2. 2. 3. 3. 4. 4. 5. 5. 6. 6,
Squares between
numbers longer in 2. 4. 6. 9. 12. 16. 20. 25. 30. 36. 42,
the other oar$.
In the first of these series it is evident that the numbers longer
in the other part, accord in ratio with the squares between
which they are inserted; but are discordant in differences.
Thus 2 is to 1 as 4 to 2. And 6 is to 4 as 9 to 6. And 12 is
to 9 as 16 to 12. And so of the rest. But the difference between
1 and 2 is 1; but between 2 and 4 is 2. The difference
between 4 and 6 is 2; but between 6 and 9 is 3. The difference
between 9 and 12 is 3; but between 12 and 16 is 4.
And so of the rest. But in the second of these series the
differences are equal; but the ratios are discordant. For 4 is
not to 2 as 6 to 4. Nor is 9 to 6 as 12 to 9. Nor is 16 to
12 as 20 to 16. Hence it is evident that squares when they
come between numbers longer in the other part, preserve an
arithmetical mean; but that numbers longer in the other part,
when they come between squares preserve a geometrical mean.
Thus 4 is an arithmetical mean between 2 and 6. Thus too 9
is an arithmetical mean between 6 and 12. And this is the
case with 16 between 12 and 20. And so of the rest. But
in the other series 2 is a geometrical mean between 1 and 4;
6 between 4 and 9; 12 between 9 and 16. And so on.
In the first series also, the differences are unequal in quantity,
but equal in denomination. Thus the difference between
1 and 2 is 1, but between 2 and 4, the difference is 2. In quantity,
however, 1 and 2 are unequal, but in appellation equal. For
each is the whole of the less, and the half of the greater quant-