CONSTRUCTION
, in Geometry, the art or manner of drawing or describing a figure, scheme, the lines of a problem, or such like.
Construction of Equations, in Algebra, 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 geometrical 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 another 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 2d rank, by the intersections of a right line with a circle, or any of the conic sections, which are all lines of the 2d 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 3d order: and so on.
But if, instead of the right line, some other line of a higher order be used; then the 2d line, whose intersections with the former are to determine the roots of the equation, may be taken as many dimensions lower, as the former is taken higher. And, in general, an equation of any height will be constructed by the intersections of two lines whose dimensions, multiplied together, produce the dimension of the given equation. Thus, the intersections of a circle with the conic sections, or of these with each other, will construct the biquadratic equations, or those of the 4th power, because ; and the intersections of the circle or conic sections with a line of the 3d order, will construct the equations of the 5th and 6th power; and so on.—For example,
To construct a Simple Equation. This is done by resolving the given simple equation into a proportion, or finding a third or 4th proportional, &c. Thus, 1. If the equation be ax = bc; then , the fourth proportional to a, b, c.
2. If ax = b2; then , a third proportional to a and b.
3. If ; then, since , it will be , a fourth proportional to a, b + c and b - c.
4. If ; then construct theright-angled triangle ABC, whose base is b, and perpendicular is c, so shall the square of the hypothenuse be b2 + c2, which call h2; then the equation is ax = b2, and x = h2/a a third proportional to a and h.
1. If it be a simple quadratic, it may be reduced to this form x2 = ab; and hence a : x :: x : b, or x = √ab a mean proportional between a and b. Therefore upon a straight line take AB = a, and BC = b; then upon the diameter AC describe a semicircle, and raise the perpendicular BD to meet it in D; so shall BD be = x the mean proportional sought between AB and BC, or between a and b.
2. If the quadratic be affected, let it first be ; then form the right-angled triangle whose base AB is a, and perpendicular BC is b; and with the centre A and radius AC describe the semi- circle DCE; so shall DB and BE be the two roots of the given quadratic equation .
3. If the quadratic be , then the construction will be the very same as of the preceding one .
4. But if the form be : form a rightangled triangle whose hypothenuse FG is a, and perpendicular GH is b; then with the radius FG and centre F describe a semi-circle IGK; so shall IH and HK be the two roots of the given equation , or . See Maclaurin's Algebra, part 3, cap. 2, and Simpson's Algebra, pa. 267.
To construct Cubic and Biquadratic Equations.— These are constructed by the intersections of two conic sections; for the equation will rise to 4 dimensions, by which are determined the ordinates from the 4 points in which these conic sections may cut one another; and the conic sections may be assumed in such a manner, as to make this equation coincide with any proposed biquadratic: so that the ordinates from these 4 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 ordinates vanishes, and the equation, by which these ordinates are determined, will then be of 3 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 parabola is usually assumed for the other.
Vieta, in his Canonica Recensione Effectionum Geometricarum, and Ghetaldus, in his Opus Posthumum de Resolutione & Compositione Mathematica, as also Des Cartes, in his Geometria, have shewn how to construct simple and quadratic Equations. Des Cartes has also shewn how to construct cubic and biquadratic equations, by the intersection of a circle and a parabola: And the same has been done more generally by Baker in his Clavis Geometrica, or Geometrical Key. But the genuine foundation of all these constructions was first laid and explained by Slusius in his Mesolabium, part 2. This doctrine is also pretty well handled by De la Hire, in a small treatise, called La Construction des Equations Analytiques, annexed to his Conic Sections. Newton, at the end of his Algebra, has given the construction of cubic and biquadratic equations mechanically; as also by the conchoid and cissoid, as well as the conic sections. See also Dr. Halley's Construction of Cubic and Biquadratic Equations; Colson's, in the Philos. Trans.; the Marquis de l'Hospital's Traité Analytique des Sections Coniques; Maclaurin's Algebra, part 3, c. 3 &c.
