199

THIS Quadruple may be also formed by
adding a Sesquialtera and a Sesquitertia to the
Duple; and how this is done, is manifest by
what we have said above: But for its clearer
Explanation, we shall give a further Instance
of it here. The Number two, for Example,
by Means of a Sesquialtera is made three, which
by a Sesquitertia becomes four, which four
being doubled makes eight.

 00 000 Sesquialtera The Quadruple. 0000 Sesquitertia 00000000 doubled

OR rather in the following Manner. Let us
take the Number three; this being doubled
makes six, to which adding another three, we
have nine, and adding to this a third of itself,
it produces twelve, which answers to three in a

THE Architects make use of all the several
Proportions here set down, not confusedly and
indistinctly, but in such Manner as to be con­
stantly and every way agreeable to Harmony:
As, for Instance, in the Elevation of a Room
which is twice as long as broad, they make
use, not of those Numbers which compose the
Triple, but of those only which form the
Duple; and the same in a Room whose Length
is three Times its Breadth, employing only its
own proper Proportions, and no foreign ones,
that is to say, taking such of the triple Pro­
gressions above set down, as is most agreeable
to the Circumstances of their Structure. There
are some other natural Proportions for the Use
of Structures, which are not borrowed from
Numbers, but from the Roots and Powers of
Squares. The Roots are the Sides of square
Numbers: The Powers are the Areas of those
Squares: The Multiplication of the Areas
produce the Cubes. The first of all Cubes,
whose Root is one, is consecrated to the Deity,
because, as it is derived from One, So it is
One every Way; to which we may add, that
it is the most stable and constant of all Fi­
gures, and the very Basis of all the rest. But
if, as some affirm, the Unite be no Number,
but only the Source of all others, we may then
suppose the first Number to be the Number
two. Taking this Number two for the Root,
the Areas will be four, which being raised up
to a Height equal to its Root, will produce a
Cube of eight; and from this Cube we may
gather the Rules for our Proportions; for here
in the first Place, we may consider the Side of
the Cube, which is called the Cube Root,
whose Area will in Numbers be sour, and the
compleat or entire Cube be as eight. In the
next Place we may consider the Line drawn
from one Angle of the Cube to that which is
directly opposite to it, so as to divide the Area
of the Square into two equal Parts, and this is
called the Diagonal. What this amounts to
in Numbers is not known: Only it appears
to be the Root of an Area, which is as Eight
on every Side; besides which it is the Diago­
nal of a Cube which is on every Side, as twelve,

Fig. 1.

*

LASTLY, In a Triangle whose two shortest
Sides form a Right Angle, and one of them
the Root of an Area, which is every Way as
four, and the other of one, which is as twelve,
the longst Side subtended opposite to that
Right Angle, will be the Root of an Area,
will be the Root of an Area, which is as six­

teen Fig. 2.

*

THESE several Rules which we have here
set down for the determining of Proportions,
are the natural and proper Relations of Num­
bers and Quantities, and the general Method
for the Practice of them all is, that the shortest
Line be taken for the Breadth of the Area,
the longest for the Length, and the middle
Line for the Height, tho' sometimes sor the
Convenience of the Structure, they are inter­
changed. We are now to say something of
the Rules of those Proportions, which are not
derived from Harmony or the natural Pro­
portions of Bodies, but are borrowed elsewhere
for determining the three Relations of an
Apartment; and in order to this we are to
observe, that there are very useful Considera­
tions in Practice to be drawn from the Musi­
cians, Geometers, and even the Arithmeticians,
of each of which we are now to speak. These
the Philosophers call Mediocrates, or Means,
and the Rules for them are many and various;
but there are three particularly which are the
most esteemed; of all which the Purpose is,
that the two Extreams being given, the middle
Mean or Number may correspond with them
in a certain detemined Manner, or to use
such an Expression, with a regular Affinity.
Our Business, in this Enquiry, is to consider
three Terms, whereof the two most remote
are one the greatest, and the other the least;
the third or mean Number must answer to