ELEMENTS

OF

ALGEBRA.

PART· III.

Of the Application of Algebra to Geometry.

G

CHAP. I. /

General Principles,

EOMETRY treats both of the magnitude and position of extenfion, and their connections.

Algebra treats only of magnitude. Therefore, of the relations which subsist in geometrical figures, thofe of magnitude only can be immediately expreffed by algebra.

The oppofite pofition of ftraight lines may indeed be expreffed fimply by the

figns

figns and But, in order to exprefs the various other pofitions of geometrical figures by algebra, from the principles of geometry, fome relations of magnitude muft be found, which depend upon these pofitions, and which can be exhibited by equations: And converfely, by the fame principles may the pofitions of figures be inferred from the equations denoting fuch relations of their parts.

Though this application of algebra appears to be indirect, yet fuch is the fimplicity of the operations, and the general nature of its theorems, that investigations, efpecially in the higher parts of geometry, are generally eafier and more expeditious by the algebraical method, though less ele gant than by what is purely geometrical. The connections alfo, and analogies of the two sciences established by this application, have given rife to many curious fpeculations ; geometry has been rendered far more extenfive and ufeful, and algebra itfelf has received confiderable improve

ments.

Of the Algebraical Expression of Geometrical Magnitudes.

A line, whether known or unknown, is represented by a single letter'; a rectangle is properly expreffed by the product of the. two letters reprefenting its fides; and a rectangular parallelopiped by the product of three letters, two of which represent the fides of any of its rectangular bases, and the third the altitude.

These are the moft fimple expreffions of geometrical magnitudes, and any other which has a known proportion to these, may, in like manner, be expreffed algebraically. Conversely, the geometrical magnitudes, represented by such algebraical quantities, may be found, only the algebraical dimenfions above the third, not having any corresponding geometrical dimensions, must be expreffed by proportionals *.

The

* All algebraical dimenfions above the third must be expreffed by inferior geometrical dimenfions; and, tho' any algebraical quantities, of two and three dimensions, may be immediately expreffed by furfaces and folids refpectively, yet it is generally neceffary to express them, and all fuperior dimenfions, by lines.

If,

The oppofite position of straight lines, it has been remarked, may be expreffed by the figns 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 refpectively, then whatever legitimate operations are performed with regard to them, the terms in any equation derived will, when properly reduced, be all of the fame dimenfion; and any fuch equation may be cafily expreffed geometrically by means of proportionals, as in the following example.

:

:

:

Thus, if the algebraical equation a4b4c4d4, is to be expreffed geometrically, a, b, c, and d, being suppofed to represent straight lines; let a:b:e:f:g, in continued proportion, then a bag and a1; a++b4:: a a+g; then let a:c:h:k:l, and a* : c^ :: a : 1; alfo, let cd:m:n: p, and c: d4::c:p, or c4 : c4-d4::c: cp. By combining the two former proportions, (Chap. 2. Part 1.) c4 : a++b+ ::/:a+g, and combining the latter with this laft found, c4-d4 : at +b4 :: c-p xlcxa+g; therefore c-pxlcxa+g, and e: e-p ::/:a+g.

If any known line is affumed as 1, as its powers do not appear, the terms of an equation, including any of them, may be of very different dimenfions; and before it can be properly expreffed by geometrical magnitudes, the deficient dimenfions must be fupplied by powers of the I. When an equation has been derived from geometrical relations, the line denoting I is known; and

when

Thus, let a point A be given in the line

P

A M Р

B

AP, any segment AP taken to the right hand, being confidered as pofitive, a fegment 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 expreffed by + a; then PM taken to the left and equal to b, will be properly reprefented by -b, for AM is equal to a-b: If a=b, then M will fall upon A, and a b=0: By the fame notation, if b is greater than a, M will fall to the left of A; and in this cafe, if 2a=b, and if Pp be taken equal to b, then a-ba will reprefent Ap, which is equal to a, and fituated to the left of A. This use of the figns, however, in particu

lar

when an affumed equation is to be expreffed by the relations of geometrical magnitudes, the 1 is to be affumed.

In this manner may any fingle power be expreffed by a line. If it is x5, then to 1, x find four quantities in continued proportion, fo that 1: x : m : n :p : q, then 1:915x5, or q=x5, and fo of others.