EQUATIONS, construction of, in alge bra, is the finding the roots or unknown quantities of an equation, by geometrical construction of right lines or curves, or the reducing given equations into geo metrical figures. And this is effected by lines or curves, according to the order or rank of the equation. The roots of any equation may be determined, that is, the equation may be constructed, by the intersections of a straight line with ano ther line or curve of the same dimensions as the equation to be constructed : for the roots of the equation are the ordinates of the curve at the points of intersection with the right line ; and it is well known that a curve may be cut by a right line in as many points as its dimensions amount to. Thus, then, a simple equation will be constructed by the intersection of one right line with another ; a quadratic equation, or an affected equation of the second rank, by the intersections of a right line with a circle, or any of the co nic sections, which are all lines of the se cond order; and which may be cut by the right line in two points, thereby giving the two roots of the quadratic equation. A cubic equation may be constructed by the intersection of the right line with a line of the third order, and so on. But if, instead of the right line, some other line of a higher order be used, then the second line, whose intersections with the former are to determine the roots of the equation, may be taken as many dimen sions lower as the former is taken high er. And, in general, an equation of any height will be constructed by the inter section of two lines, whose dimensions multiplied together produce the dimen sion of the given equation. Thus, the in tersections of a circle with the conic sec tions, or of these with each other, will construct the biquadratic equations, or those of the fourth power, because 2 X 2 = and the intersections of the circle, or conic sections, with a line of the third order, will construct the equations of the fifth and sixth power, and so on.—For example : To construct a simple equation. This is done by resolving the given simple equa tion into a proportion, or finding a third or fourth proportional, &c. Thus, 1. It the equation be ax=bc; then a:bnc:x b c = —, the fourth proportional to a,b,c. 2. a b' If a x=b; then a:b::b:x a third proportional to a and b. 3. If a x = c'; then, since b'— =6+ c Xb—o, it will be a: b+ c::b—c:x= a fourth proportional to a, b±c, and b—c.
4. If a x = 6' + c' ; then construct the right-angled triangle ABC (Plate V. Mis eel. fig. 5.) whose base is b, and perpen dicular is c, so shall the square of the hy pothenuse be P.-1-c", which call h' ; then the equation is ax--=1L', and a third proportional to a and h.
To construct a quadratic equation. 1./f it be a simple quadratic, it may be reduced to this form, x. = a ; and hence a ,x x : 6, or x =via b, a mean proportional between a and b. Therefore upon a straight line take 9. B = a, and B C = 6; then upon the diameter A C describe a semicircle, and raise the perpendicular B D to meet it in D ; so shall B D be x, the mean proportional sought between A B and B C, or between a and b. 2. If the quadratic be affected, let it first be x' -I- 2 ax = 6'; then form the right-angled triangle, whose base A B is a, and dicular B C is 6 ; and with the centre A and radius A C describe the semicircle C E; so shall D B and B E be the two roots of the given quadratic equation 2 ax = b". 3. If the quadratic be x' — 2 a x =1,', then the construction will be the very same as of the preceding one x' + 2 a x = b.. 4. But if the form be 2 a x x' b', form a right-angled triangle (fig. 1.) whose hypothenuse F G is a, and per pendicular G H is b ; then with the radius F G and centre F describe a semicircle I G K; so shall I II and H K be the two roots of the given equation-2 a x — 6', or x' — 2 a x — 6'.
To construct cubic and biquadratic equa. tions. These are constructed by the inter sections of two conic sections; for the equation will rise to four dimensions, by which are determined the ordinates from the four points in which these conic sec tions may cut one another ; and the conic sections may be assumed in such a manner as to make this equation co incide with any proposed biquadratic ; so that the ordinates from these four intersections will be equal to the roots of the proposed biquadratic. When one of the intersections of the conic section falls upon the axis, then one of the or dinates vanishes, and the equation by which these ordinates are determined will then be of three dimensions only, or a cubic to which any proposed cubic equation may be accommodated ; so that the three remaining ordinates will be the roots of that proposed cubic. The conic sections for this purpose should be such as are most easily described ; the circle may be one, and the para bola is usually assumed for the other. See Simpson's and Itlaclaurin's Algebra.