Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

equired to insert all these middles. Let the arithmetical middle
then be first inserted. And this will be effected, if 25 is
placed between these numbers; for 10, 25, and 40, are in arithmetical
proportion. Again, if between 10 and 40, the number
20 is inserted, the geometric middle, with all its properties, will
be immediately produced; for as 10 is to 20, so is 20 to 40. But
if 16 is inserted between 10 and 40, the harmonic middle will
be produced; for as 10 is to 40, so is the difference between 16
and 10, i.e. 6, to the difference between 40 and 16, i.e. 24.
If odd numbers, however, are proposed as the two extremes,
such as are 5 and 45, it will be found that 25 will constitute
the arithmetical, 15 the geometrical, and 9 the harmonic middle.
It is necessary, however, to show how these middles may be
found. Two terms being given, if it is required to constitute
an arithmetical middle, the two extremes must be conjoined,
and the sum arising from the addition of them must be divided
by 2, and the quotient will be the arithmetical mean that was to
be found. Thus 10+40=50, and 50 divided by 2 is equal to
25. This, therefore will be the middle term, according to
arithmetical proportion. Or if the number by which the
greater surpasses the less term, is divided by 2, and the quotient
is added to the less term; the sum thence arising, will be the
arithmetical mean required. Thus the difference between 40
and 10 is 30, and if this is divided by 2 the quotient will be 15.
But if 15 is added to 10, the sum will be 25, the arithmetical
mean between 10 and 40. Again, in order to find the geometrical
mean, the two extremes must he multiplied together, and
the square root of the product will be the required mean. Thus
10X 40=400, and the square root of 400 is 20. Hence 20
will be the geometrical mean between 10 and 40. Or if the
ratio which the given terms have to each other is divided by 2,
the quotient will be the middle that was to be found. For 40