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.
00000SeſquialteraThe Quadruple.0000Seſquitertia00000000doubledadding 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.
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
Quadruple Proportion.
000000000doubledThe Quadruple000000000a third added000000000000a third addedtake 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
Quadruple Proportion.
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.
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.
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
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