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Analytic Geometry

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ANALYTIC GEOMETRY. When Descartes first came upon the idea of co-ordinates, a link was made between the sciences of geometry and analysis. Now the connection has grown so close that the results of either can be passed over to the other by definite laws of interpretation. In what follows, a few leading ideas are set out in terms of plane geometry, most of which can be extended to three or more dimensions.

Cartesian co-ordinates Co-ordinates (q.v.) are numbers measuring certain geometrical quantities and used to fix the position of a point, e.g., latitude and longitude fix a place on a map. It is possible to build up a geometry, which is not analytical, on the ideas of order and in cidence alone, without introducing measurement; but in ana lytic geometry we assume that we can measure such things as lengths and angles, and compare two different things of the same kind by measuring them in terms of a common unit. The value of a co-ordinate may depend on the choice of units; or it may be independent of them if it is defined as a ratio of two quantities of the same kind.

The simplest plane co-ordinates are rectangular Cartesian. Through a fixed point (origin) 0, we take two lines (axes) OX, OY in fixed directions at right angles to each other. (In oblique carte sian co-ordinates, the axes are not rectangular, but inclined at a given angle a; the conditions of a problem often make this an advantage.) We reach any point P of the plane by going from 0 a certain distance x =OM along OX, and then another distance y =MP parallel to OY. The pair of numbers (x, y) are the co ordinates of P. If x and y are positive, this procedure leads only to points in the first quadrant of the plane, bounded by the half lines OX, OY. Points in the other three quadrants are reached by taking one or both of the steps OM, MP in the opposite direction; to these reversed steps we assign negative co-ordinates. A zero co-ordinate belongs to a point on one or other axis, and the co-ordinates of the origin are (o, o). Every point of the plane has a unique pair of real numbers as co-ordinates, and every pair of real numbers are the co-ordinates of a unique point.

Thus the point P, and its pair of co-ordinates (x, y) are equiva lent data. Any length, or other geometrical quantity, that is fixed when the point P and the axes are given, is fixed when x, y are given; thus, = and tan POM = y/x. (If, instead of x, y, we are given r = OP = and 0 = L POM = then, since x =r cos 0, y = r sin 0, both x and y are determined; thus r, 0, are data equivalent to x, y, and can serve just as well in determining P; they are called its polar co-ordinates.) With any system of co-ordinates, the formula for the distance d between two given points P2 is of great importance. Let the co-ordinates of P2 have the suffixes 1, a. Then, with rectangular cartesian co-ordinates, oblique cartesian co-ordinates, cos a; polar co-ordinates, cos (02-01).

The simplest analytic geometry of all is of one dimension; e.g., when we deal with only one line and the points on it. Any point P can be specified by its distance x =OP, measured from a fixed origin 0 in terms of a fixed unit, which can be shown by marking the point A at unit distance from 0, so that OA= 1. Then x is the one co-ordinate of P.

By far the most important assumption is that of a single point with the co-ordinate oo , at an infinite distance from 0. This is required, for example, as the homologue of 0 in ordinary in version, in which the point whose co-ordinate is x corresponds to the point whose co-ordinate is 1/x.

Equations.

If we interpret the variables as the co-ordinates of a variable point P, an equation in x, y is a statement of the position of P. For example, the equation = tells us that P is at a distance a from 0, that is, P is some point on the circle centre 0, radius a. This does not fix P, for one equation does not determine both x and y, but it relates x and y, and it limits P to lie on the curve. The equation holds between the co-ordinates of any point on the circle, and of no other point ; it is called the equation of the curve.

Any geometrical property of the circle is a statement about P, true if, and only if, P lies on the circle ; it has a corresponding equation in x, y which follows from the equation = Suppose we want to prove that, if OX meets the circle in A, A', then = A'M • MA. The expression of this desired result in terms of the co-ordi nates is (a+x) (a—x), which is a sim ple algebraic deduction from = Thus, in general, any equation in x, y represents a curve, whose properties, when expressed in terms of the co-ordinates of a gen eral point lying on it, are precisely the analytical consequences of this equation.

Since there is, as a general rule, a very much more certain and obvious way of verifying an analytical deduction than of devising a geometrical proof, the translation into equations introduces method and helps to turn geometry from an art into a science. One reason is that an algebraic expression has the property of form, which is more easily recognized than its geometrical equiva lent. By the help of the conventions of the notation, the expression is analysed, almost at first sight, into the components of its structure : variables and constants, terms, factors and indices.

The application of analysis to geometry saves effort by showing the direct way to success for soluble problems, and no less by showing certain others to be insoluble; it enables us to prove a negative. A particular figure may suggest wrongly that a certain property is true in general. In such a case, it is often quite hard to prove by pure geometry that the surmise does not always hold good. But when equations are written down expressing all the properties, both those given by hypothesis and those guessed from the figure, we can find out whether the second set can be deduced from the first or not. If they can, we may return more hopefully, and usually with some definite clue, to the search for a pure proof; if they are not deducible, we cease to waste our time on the matter. The classic example is the squaring of the circle by an Euclidean construction. For centuries, geometers were convinced that only a little more skill or luck was wanted for success, and no one was then in a position to assert that every conceivable attempt must fail. By the aid of analysis, that assertion is now proved. For this kind of special theorem, the analytical method is one of verification rather than discovery.

It is sheer folly to keep the two methods apart. There are occasions for using either alone, but we gain most power by using pure, algebraic and infinitesimal ideas and notations alongside, keeping ever present the relation between them, and the inter pretation of each in terms of the others. "Every equation has its little meaning" is a better motto than "shut your eyes and write down the equations." For example, analytic geometry could hardly get going at all without relying on the theorem of Pythagoras, as was done tacitly earlier in this article in writing down = Algebraic Curves.—The most important elementary prop erty of Cartesian co-ordinates is that an equation of the first degree in x, y represents a line, and conversely every line has an equation of the first degree. Thus : y = constant is a line parallel to OX, x = constant is a line parallel to 0Y, ax+by =o is a line through 0, ax+by+c = o is a line parallel to the last, but does notpass through 0 if o, x/a+y/b = i is the line cutting off lengths a, b from theaxes, y — b = m(x — a) is the line through the point A whose co ordinates are (a, b), making with OX the angle whose tangent is m. For different values of m, this can represent any one of the set of lines through A. The co-ordinates of the point of inter section of two given lines satisfy both their equations. Two inde pendent linear equations in x, y always have one and only one solution, provided we admit infinite values; e.g., with the pair ax+by=o, The corresponding geometrical con vention is that two parallel lines intersect in one point at in finity.

A quadratic equation represents a conic (see CoNIC the simplest being the circle + It illustrates the unity of mathematics that the same curves should be the first to present themselves naturally along the two lines of approach. The co ordinates of the intersection of a line and conic satisfy a linear and a quadratic equation ; their determination depends on the solu tion of the eliminant, a quadratic equation in one variable. Now a line may meet a conic in two different real points, or touch it at one point or pass clear of it. The conditions are the same as that the corresponding quadratic equation may have real and distinct, or equal or imaginary roots. It is therefore convenient to speak of the intersections of the line and conic as always two in number, being real or coincident or imaginary.

If the equation of a curve in Cartesian co-ordinates is algebraic and not transcendental, its degree, in the co-ordinates jointly, is called the degree of the curve. The co-ordinates of the inter sections of a curve of degree n and any line depend on an elimi nant of degree ii; there are n such intersections, real or imaginary, finite or coinciding with the one point at infinity on the line. The fundamental theorem about such curves is that two al gebraic curves of degrees n2 have points of intersection; as we have just seen in the case when one is a line, i. Thus two conics meet in four points.

A common application of co-ordinates in subjects such as en gineering or statistics is to show the variation of two connected quantities by means of a graph. This is not really analytic geom etry, but rather its converse ; not the use of analysis to increase our geometrical knowledge, but the use of geometry to illustrate some analytical knowledge. We can draw a graph whenever we can calculate one co-ordinate directly in terms of the other, and its equation has the form y = f (x) .

co-ordinates, equation, line, terms and co-ordinate