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EOMETRY treats both of the mag
nitude and position of extension, and their connections.
Algebra treats only of magnitude. Therefore, of the relations which subsist in
geometrical figures, those of magnitude only can be immediately expressed by algebra.
The opposite position of straight lines may indeed be expressed simply by the
signs + and - But, in order to express the various other positions of geometrical figures by algebra, from the principles of geometry, some relations of magnitude muft be found, which depend upon these positions, and which can be exhibited by equations: And conversely, by the same principles may the positions of figures be inferred from the equations denoting such relations of their parts.
Though this application of algebra appears to be indirect, yet such is the fimplicity of the operations, and the general nature of its theorems, that investigations, especially in the higher parts of
geometry, are generally easier and more expeditious by the algebraical method, though less elegant than by what is purely geometrical. The connections also, and analogies of the two sciences established by this application, have given rise to many curious fpeculations
; geometry has been rendered far more extensive and useful, and algebra itself has received considerable improve
1. Of the Algebraical Expression of Geoine
trical Magnitudes. A line, whether known or unknown, is represented by a single letter ; a rectangle is properly expressed by the product of the two letters representing its fides ; and a rectangular parallelopiped by the product of three letters, two of which represent the sides of any of its rectangular bases, and the third the altitude.
These are the most simple expressions of geometrical magnitudes, and any other which has a known proportion to these, may, in like manner, be expressed algebraically. Conversely, the geometrical magnitudes, represented by such algebraical quantities, may be found, only the algebraical dimensions above the third, not having any corresponding geometrical dimensions, must be expressed by proportionals *.
* All algebraical dimensions above the third must be expreffed by inferior geometrical dimensions; and, tho’ any algebraical quantities, of two and three dimensions, may be immediately expreffed by surfaces and solids re, fpectively, yet it is generally necessary to express them, and all superior dimensions, by lines.
The opposite position of straight lines, it has been remarked, may be expressed by the signs + and
Thus, If, in any geometrical investigation by algebra, each line is expreffed by a single letter, and each surface or folid by an algebraical quantity of two or three dimenfions respectively, then whatever legitimate operations are performed with regard to them, the terms in any equation derived will, when properly reduced, be all of the same dimension; and any such equation may be easily expressed geometrically by means of proportionals, as in the following example.
Thus, if the algebraical equation a4+14=c4--d4, is to be expressed geometrically, a, b, c, and d, being suppos fed to represent straight lines ; let à :b:e:f:g, in continued proportion, then 24: 64::a:g and at : a++b+::a :a+g; then let a:c:h:k:), and at:c*::a:l; also, let c:d:m:n :p, and ct :d4::c:p, or c4:04--04::c:
By combining the two former proportions, (Chap. 2. L'art 1.) 04:2*+b::1:a+g, and combining the latter with this last found, c4d4:a4 +64 :: 0-P *1::cxa+5; therefore c—pxl=cxa+g, and c:c-p ::1:a+g. known line is assumed as I, as its powers
do not appear, the terms of an equation, including any of them, may be of very different dimensions; and before it can be properly expressed by geometrical magnitudes, the deficient dimenfions must be supplied by powers of
When an equation has been derived from geometrical relations, the line denoting 1 is known; and
Thus, let a point A be given in the line
AP, any segment AP taken to the right hand, being considered as positive, a segment Ap to the left is properly represented by a negative quantity. If a and b
reprefent two lines ; and if, upon the line AB from the point A, AP be taken towards the right equal to a, it may be expressed by + a; then PM taken to the left and equal to b, will be properly represented by -b, for AM is equal to amb: If a=b, then M will fall upon A, and a-bro: By the same notation, if b is greater than a, M will fall to the left of A ; and in this case, if 2a=b, and if Pp be taken equal to b, then a-b=ma will represent Ap, which is equal to a, and situated to the left of A. This use of the signs, however, in particu
when an assumed equation is to be expressed by the relations of geometrical magnitudes, the 1 is to be assumed.
In this manner may any single power be exprefled by a line. If it is x5, then to 1, x find four quantities in continued proportion, so that 1:*:m:n:p:9, then 1:9::15: *5, or q=x5, and so of others.