GEOMETRICAL

, something that has a relation to Geometry, or done after the manner, or by the means of Geometry. As, a Geometrical construction, a Geometrical curve, a Geometrical demonstration, genius, line, method, Geometrical strictness, &c.

Geometrical Consiruction, of an equation, is the drawing of lines and figures, so as to express by them the same general property and relation, as are denoted by the algebraical equation. See Construction of Equations.

Geometrical Curve or Line, called also an Algebraical one, is that in which the relations between the abscisses and ordinates may be expressed by a finite algebraical equation. See Algebraical Curves.

Geometrical Lines, as observed by Newton, are distinguished into classes, orders, or genera, according to the number of the dimensions of the equation that expresses the relation between the ordinates and abscisses; or, which comes to the same thing, according to the number of points in which they may be cut by a right line.

Thus, a line of the first order, is a right line, since it can be only once cut by another right line, and is expressed by the simple equation : those of the 2d, or quadratic order, will be the circle, and the conic sections, since all of these may be cut in two points by a right line, and expressed by the equation : those of the 3d or cubic order, will be such as may be cut in 3 points by a right line, whose most general equation is ; as the cubical and Neilian parabola, the cissoid, &c. And a line of an infinite order, is that which a right line may cut in infinite points; as the spiral, the cycloid, the quadratrix, and every line that is generated by the infinite revolutions of a radius, or circle, or wheel, &c.

In each of those equations, x is the absciss, y its corresponding ordinate, making any given angle with it; and a, b, c, &c, are given or constant quantities, affected with their signs + and -, of which one or more may vanish, be wanting or equal to nothing, provided that by such defect the line or equation does not become one of an inferior order.

It is to be noted that a curve of any kind is denominated by a number next less than the line of the same kind: thus, a curve of the 1st order, (because the right line cannot be reckoned among curves) is the same with a line of the 2d order; and a curve of the 2d kind, the same with a line of the 3d order; &c.

It is to be observed also, that it is not so much the equation, as the construction or description, that makes any curve, geometrical, or not. Thus, the circle is a geometrical line, not because it may be expressed by an equation, but because its description is a postulate: and it is not the simplicity of the equation, but the easiness of the description, that is to determine the choice of the lines for the construction of a problem. The equation that expresses a parabola, is more simple than that which expresses a circle; and yet the circle, by reason of its more simple construction, is admitted before it. Again, the circle and the conic sections, with respect to the dimensions of the equations, are of the same order; and yet the circle is not numbered with them in the construction of problems, but by reason of its simple description is depressed to a lower order, viz, that of a right line; so that it is not improper to express that by a circle, which may be expressed by a right line; but it is a fault to construct that by the conic sections, which may be constructed by a circle.

Either, therefore, the law must be taken from the dimensions of equations, as observed in a circle, and so the distinction be taken away between plane and solid problems: or the law must be allowed not to be strictly observed in lines of superior kinds, but that some, by reason of their more simple description, may be preferred to others of the same order, and be numbered with lines of inferior orders.

In constructions that are equally Geometrical, the most simple are always to be preferred: and this law is so universal as to be without exception. But algebraical expressions add nothing to the simplicity of the construction; the bare descriptions of the lines here are only to be considered; and these alone were considered by those geometricians who joined a circle with a right| line. And as these are easy or hard, the construction becomes easy or hard: and therefore it is foreign to the nature of the thing, from any other circumstance to establish laws relating to constructions.

Either, therefore, with the ancients, we must exclude all lines beside the circle, and perhaps the conic sections, out of geometry; or admit all, according to the simplicity of the description. If the trochoid were admitted into geometry, by means of it we might divide an angle in any given ratio; would it be right therefore to blame those who would make use of this line to divide an angle in the ratio of one number to another; and contend, that you must make use only of such lines as are defined by equations, and therefore not of this line, which is not so defined? If, when an angle is proposed to be divided, for instance, into 10001 parts, we should be obliged to bring a curve defined by an equation of more than 100 dimensions to do the business; which nobody could describe, much less understand; and should prefer this to the trochoid, which is a line well known, and easily described by the motion of a wheel, or circle: who would not see the absurdity?

Either therefore the trochoid is not to be admitted at all in geometry; or else, in the construction of problems, it is to be preferred to all lines of a more difficult description; and the reason is the same for other curves. Hence the trisections of an angle by a conchoid, which Archimedes in his Lemmas, and Pappus in his Collections, have preferred to the inventions of all others in this case, must be allowed as good; because we must either exclude all lines, beside the circle and right line, out of geometry, or admit them according to the simplicity of their descriptions; in which case the conchoid yields to none, except the circle. Equations are expressions of arithmetical computation, and properly have no place in geometry, excepting so far as quantities truly Geometrical (that is, lines, surfaces, solids, and proportions) may be said to be some equal to others. Multiplications, divisions, and such like computations, are newly received into Geometry, and that unwarily, and contrary to the first design of this science. For whoever considers the construction of problems by a right line and a circle, found out by the first geometricians, will easily perceive that geometry was introduced, that by drawing lines, we might easily avoid the tediousness of computation. For which reason the two sciences ought not to be confounded together: the ancients so carefully distinguished between them, that they never introduced arithmetical terms into geometry; and the moderns, by confounding them, have lost the simplicity in which all the elegance of geometry consists. In short, that is arithmetically more simple, which is determined by the more simple equations; but that is Geometrically more simple, which is determined by the more simple drawing of lines; and in geometry that ought to be reckoned best, which is geometrically most simple. Newton's Arith. Univers. appendix. See Curves.

Geometrical Locus, or Place, called also simply Locus, is the path or track of some certain Geometrical determination, in which it always falls. See Locus.

Geometrical Medium. See Medium.

Geometrical Method of the Ancients. The ancients established the higher parts of their geometry on the same principles as the elements of that science, by demonstrations of the same kind: and they were careful not to suppose any thing done, till by a previous problem they had shewn that it could be done by actually performing it. Much less did they suppose any thing to be done that cannot be conceived; such as a line or series to be actually continued to infinity, or a magnitude diminished till it become infinitely less than what it is. The elements into which they resolved magnitudes were sinite, and such as might be conceived to be real. Unbounded liberties have of late been introduced; by which geometry, which ought to be perfectly clear, is filled with mysteries. Maclaurin's Fluxions, Introd. p. 39.

Geometrical Pace, is a measure of 5 feet long.

Geometrical Plan, in Architecture. See Plan.

Geometrical Plane. See Plane.

Geometrical Progression, a progression in which the terms have all successively the same ratio: as 1, 2, 4, 8, 16, &c, where the common ratio is 2.

The general and common property of a Geometrical progression is, that the product of any two terms, or the square of any one single term, is equal to the product of every other two terms that are taken at an equal distance on both sides from the former. So of these terms, 1, 2, 4, 8, 16, 32, 64, &c, .

In any Geometrical Progression, if a denote the least term, z the greatest term, r the common ratio, n the number of the terms, s the sum of the series, or all the terms; then any of these quantities may be found from the others, by means of these general values, or equations, viz, . When the series is infinite, then the least term a is nothing, and the sum . See also PROGRESSION.

Geometrical Proportion, called also simply Proportion, is the similitude or equality of ratios.

Thus, if , or , the terms a, b, c, d are in Geometrical Proportion; also 6, 3, 14, 7, are in Geometrical Proportion, because , or .|

In a Geometrical Proportion, the product of the extremes, or 1st and 4th terms, is equal to the product of the means, or the 2d and 3d terms: so ad = bc, and . See Proportion.

Geometrical Solution, of a problem, is when the problem is directly resolved according to the strict rules and principles of geometry, and by lines that are truly Geometrical. This expression is used in contradistinction to an arithmetical, or a mechanical, or instrumental solution; the problem being resolved only by a ruler and compasses.

The same term is likewise used in opposition to all indirect and inadequate kinds of solutions, as by approximation, infinite series, &c. So, we have no Geometrical way of finding the quadrature of the circle, the duplicature of the cube, or two mean proportionals; though there are mechanical ways, and others, by infinite series, &c.

Pappus informs us, that the ancients endeavoured in vain to trisect an angle, and to find out two mean proportionals, by means of the right line and circle. Afterwards they began to consider the properties of several other lines; as the conchoid, the cissoid, and the conic sections; and by some of these they endeavoured to resolve some of those problems. At length, having more thoroughly examined the matter, and the conic sections being received into geometry, they distinguished Geometrical problems and solutions into three kinds; viz,

1. Plane ones, which, deriving their origin from lines on a plane, may be properly resolved by a right line and a circle.—2. Solid ones, which are resolved by lines deriving their original from the consideration of a solid; that is, of a cone.—3. Linear ones, to the solution of which are required lines more compounded.

According to this distinction, we are not to resolve solid problems by other lines than the conic sections; especially if no other lines beside the right line, circle, and the conic sections, must be received into geometry.

But the moderns, advancing much farther, have received into geometry all lines that can be expressed by equations; and have distinguished, according to the dimensions of the equations, those lines into classes or orders; and have laid it down as a law, not to construct a problem by a line of a higher order, that may be constructed by one of a lower.