1theſe 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.
84126onate 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.
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 Mean, which ſtanding between four and
eight, is at an equal Diſtance from each of
them.
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 9366 Times 636Geometrical, 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.
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.
3060123330103010304060find 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.
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.
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
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