1 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 extension, 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 position of ftraight lines may indeed be expreffed fimply by the figns and. figns 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 positions of figures be inferred from the equations denoting fuch relations of their parts. But, in order to express 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, especially in the higher parts of geometry, are generally cafier and more expeditious by the algebraical method, though less elegant 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. I. İ. Of the Algebraical Expreffion of Geométrical Magnitudes. A line, whether known or unknown, is represented by a single letter a fingle letter; a rectangle is properly expreffed 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 fides of any of its rectangular bafes, and the third the altitude. These are the moft fimple expreffions of geometrical magnitudes, and any other which has a known proportion to thefe, may, in like manner, be expreffed algebraically. Conversely, the geometrical magnitudes, represented by fuch algebraical quantities, may be found, only the algebraical dimensions above the third, not having any correfponding 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 pofition of straight lines, it has been remarked, may be expressed by the figns and —. Thus, If, in any geometrical investigation by algebra, each line is expreffed by a fingle letter, and each furface 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 eafily expreffed geometrically by means of proportionals, as in the following example. : Thus, if the algebraical equation a4+b4c4d4, is to be expreffed geometrically, a, b, c, and d, being fuppo fed to reprefent ftraight lines; let a:b:e:f:g, in continued proportion, then a bag and a+ : a++b+::a a+g; then let a:c:h:k:l, and atca: 1; alfo, let cd:mn: p, and ct: d4::c:p, or c4c4d4::c: c-p. By combining the two former proportions, (Chap. 2. Part 1.) c4; a++b+ ::/:a+g, and combining the latter with this last found, c4—d+ : aa +b4 :: c-p xlcxa+g; therefore c-pxl=cxa+g, and c:c-p 1: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 supplied 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 P 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 represented by —b, for AM is equal to a-b: If a=b, then M will fall upon A, and a-b=o: 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-b-a will reprefent Ap, which is equal to a, and fituated to the left of A. This ufe of the figns, however, in particu I 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 1915 x5, or q=x5, and fo of others. |