Alberti, Leone Battista
,
Architecture
,
1755
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<
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>THIS
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Quadruple
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may be alſo formed by
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adding a
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Seſquialtera
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and a
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Seſquitertia
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to the
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Duple;
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and how this is done, is manifeſt by
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what we have ſaid above: But for its clearer
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Explanation, we ſhall give a further Inſtance
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of it here. </
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<
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>The Number two, for Example,
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by Means of a
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Seſquialtera
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is made three, which
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by a
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Seſquitertia
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becomes four, which four
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being doubled makes eight.
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>00</
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<
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>000</
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<
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Seſquialtera
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<
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<
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>The
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Quadruple.
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<
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<
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>0000</
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<
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>
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Seſquitertia
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</
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<
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<
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<
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>00000000</
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<
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>doubled</
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<
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>OR rather in the following Manner. </
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<
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>Let us
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take the Number three; this being doubled
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makes ſix, to which adding another three, we
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have nine, and adding to this a third of itſelf,
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it produces twelve, which anſwers to three in a
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Quadruple
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Proportion.
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<
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<
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>000</
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</
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<
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<
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<
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>000000</
cell
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<
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>doubled</
cell
>
</
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>
<
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<
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>The
<
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Quadruple
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</
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<
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</
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<
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<
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<
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>000000000</
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<
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>a third added</
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</
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<
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<
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>000000000000</
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<
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>a third added</
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<
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>THE Architects make uſe of all the ſeveral
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Proportions here ſet down, not confuſedly and
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indiſtinctly, but in ſuch Manner as to be con
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ſtantly and every way agreeable to Harmony:
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As, for Inſtance, in the Elevation of a Room
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which is twice as long as broad, they make
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uſe, not of thoſe Numbers which compoſe the
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Triple, but of thoſe only which form the
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Duple; and the ſame in a Room whoſe Length
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is three Times its Breadth, employing only its
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own proper Proportions, and no foreign ones,
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that is to ſay, taking ſuch of the triple Pro
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greſſions above ſet down, as is moſt agreeable
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to the Circumſtances of their Structure. </
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<
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>There
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are ſome other natural Proportions for the Uſe
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of Structures, which are not borrowed from
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Numbers, but from the Roots and Powers of
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Squares. </
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<
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>The Roots are the Sides of ſquare
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Numbers: The Powers are the Areas of thoſe
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Squares: The Multiplication of the Areas
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produce the Cubes. </
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<
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>The firſt of all Cubes,
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whoſe Root is one, is conſecrated to the Deity,
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becauſe, as it is derived from One, So it is
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One every Way; to which we may add, that
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it is the moſt ſtable and conſtant of all Fi
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gures, and the very Baſis of all the reſt. </
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<
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>But
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if, as ſome affirm, the Unite be no Number,
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but only the Source of all others, we may then
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ſuppoſe the firſt Number to be the Number
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two. </
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<
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>Taking this Number two for the Root,
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the Areas will be four, which being raiſed up
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to a Height equal to its Root, will produce a
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Cube of eight; and from this Cube we may
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gather the Rules for our Proportions; for here
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in the firſt Place, we may conſider the Side of
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the Cube, which is called the Cube Root,
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whoſe Area will in Numbers be ſour, and the
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compleat or entire Cube be as eight. </
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>
<
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>In the
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next Place we may conſider the Line drawn
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from one Angle of the Cube to that which is
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directly oppoſite to it, ſo as to divide the Area
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of the Square into two equal Parts, and this is
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called the Diagonal. </
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<
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>What this amounts to
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in Numbers is not known: Only it appears
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to be the Root of an Area, which is as Eight
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on every Side; beſides which it is the Diago
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nal of a Cube which is on every Side, as twelve,
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Fig.
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1.</
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*</
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<
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>LASTLY, In a Triangle whoſe two ſhorteſt
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Sides form a Right Angle, and one of them
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the Root of an Area, which is every Way as
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four, and the other of one, which is as twelve,
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the longſt Side ſubtended oppoſite to that
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Right Angle, will be the Root of an Area,
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will be the Root of an Area, which is as ſix
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teen
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Fig.
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2.</
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*</
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<
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>THESE ſeveral Rules which we have here
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ſet down for the determining of Proportions,
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are the natural and proper Relations of Num
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bers and Quantities, and the general Method
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for the Practice of them all is, that the ſhorteſt
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Line be taken for the Breadth of the Area,
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the longeſt for the Length, and the middle
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Line for the Height, tho' ſometimes ſor the
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Convenience of the Structure, they are inter
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changed. </
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<
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>We are now to ſay ſomething of
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the Rules of thoſe Proportions, which are not
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derived from Harmony or the natural Pro
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portions of Bodies, but are borrowed elſewhere
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for determining the three Relations of an
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Apartment; and in order to this we are to
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obſerve, that there are very uſeful Conſidera
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tions in Practice to be drawn from the Muſi
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cians, Geometers, and even the Arithmeticians,
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of each of which we are now to ſpeak. </
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<
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>Theſe
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the Philoſophers call
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Mediocrates,
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or
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Means,
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and the Rules for them are many and various;
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but there are three particularly which are the
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moſt eſteemed; of all which the Purpoſe is,
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that the two Extreams being given, the middle
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Mean or Number may correſpond with them
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in a certain detemined Manner, or to uſe
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ſuch an Expreſſion, with a regular Affinity.
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</
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<
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>Our Buſineſs, in this Enquiry, is to conſider
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three Terms, whereof the two moſt remote
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are one the greateſt, and the other the leaſt;
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the third or mean Number muſt anſwer to
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