200
these other two in a just Relation or proporti­
onate Interval, which Interval is the equal re­
lative Distance which this Number stands from
the other two. Of the three Methods most
approved by the Philosophers for finding this
Mean, that which is called the arithmetical is
the most easy, and is as follows. Taking the
two extreme Numbers, as for Instance, eight
for the greatest, and four for the least, you add
them together, which produce twelve, which
twelve being divided in two equal Parts, gives
us six.

 8 4 12 6

THIS Number six the Arithmeticians say, is
the Mean, which standing between four and
eight, is at an equal Distance from each of
them.

 8 6 4

THE next Mean is that which is called the
Geometrical, and is taken thus. Let the small­
est Number, for Example, four, be multiplied
by the greatest, which we shall suppose to be
nine; the Multiplication will produce 36:
The Root of which Sum as it is called, or the
Number of its Side being multiplied by itself
must also produce 36. The Root therefore
will be six, which multiplied by itself is 36,
and this Number six, is the Mean.

 4 Times 9 36 6 Times 6 36

THIS geometrical Mean is very difficult to
find by Numbers, but it is very clear by Lines;
but of those it is not my Business to speak
here. The third Mean, which is called the
Musical, is somewhat more difficult to work
than the Arithmetical; but, however, may be
very well performed by Numbers. In this the
Proportion between the least Term and the
greatest, must be the same as the Distance be­
tween the least and the Mean, and between the
Mean and the greatest, as in the following Ex­
ample. Of the two given Numbers, let the
least be thirty, and the greatest sixty, which is
just the Double of the other. I take such
Numbers as cannot be less to be double, and
these are one, for the least, and two, for the
greatest, which added together make three. I
then divide the whole Interval which was be­
tween the greatest Number, which was sixty,
and the least, which was thirty, into three
Parts, each of which Parts therefore will be
ten, and one of these three Parts I add to the
least Number, which will make it forty; and
this will be the musical Mean desired.

 30 60 1 2 3 3 30 10 30 10 30 40 60

AND this mean Number forty will be dis­
tant from the greatest Number just double the
Interval which the Number of the Mean is
distant from the least Number; and the Con­
dition was, that the greatest Number should
bear that Portion to the least. By the Help of
these Mediocrites the Architects have discover­
ed many excellent Things, as well with Rela­
tion to the whole Structure, as to its several
Parts; which we have not Time here to par­
ticularize. But the most common Use they
have made of these Mediocrities, has been how­
ever for their Elevations.

CHAP. VII.

Of the Invention of Columns, their Dimensions and Collocation.

It will not be unpleasant to consider some
further Particulars relating to the three
Sorts of Columns which the Ancients invent­
ed, in three different Points of Time: And it
is not at all improbable, that they borrowed the
Proportions of their Columns from that of the
Members of the human Body. Thus they
found that from one Side of a Man to the
other was a sixth Part of his Height, and that
from the Navel to the Reins was a tenth. From
this Observation the Interpreters of our sacred
Books, are of Opinion, that Noah's Ark for
the Flood was built according to the Propor­
tions of the human Body. By the same Pro­
portions we may reasonably conjecture, that the
Ancients erected their Columns, making the
Height in some six Times, and in others ten
Times, the Diameter of the Bottom of the