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A1A

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Originally appearing in Volume V11, Page 713 of the 1911 Encyclopedia Britannica.
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A1A =AoA2, B1B = BoB2. Then rabatted. This line corresponds therefore to the plan of a"—that r Aa ;BQ y AB will give the length required. is, to the axis xy, corresponding points on these lines being those The construction might have which lie on a perpendicular to a'. A been performed in the elevation We have thus one pair of corresponding lines and can now find b y m a king A2A =AoA1 and for any point B1 in the plan the corresponding point B in the rabatted B2B = BoB1 on lines perpendicular plane. We draw a line through B1, say B1P1, cutting a' in C. To it to A2B2. Of course AB must have corresponds the line CP, and the point where this is cut by the project- A B, the same length in both cases. ing ray through B1, perpendicular to a', is the required point B. B This figure may be turned into Similarly any figure in the rabatted plane can be found when the a model. Cut the paper along plan is known; but this is usually found in a different manner piece AIABBI over along A1BI till As this method and the reasoning employed for it have their peculiar it stands upright at right angles to the horizontal plane. The points advantages, we give it also. A, B will then be in their true position in space relative to In . Simi- Supposing the planes n1 and 2r2 to be in their positions in space larly if B2BAA2 be cut out and turned along A2B2 through a right perpendicular to each other, we take a section of the whole figure angle we shall get AB in its true position relative to the plane by a plane perpendicular to the trace a' about which we are going 1-2. Lastly we fold the whole plane of the paper along the axis x to rabatt the plane a. Let this section pass through the point Q in till the plane I"2 is at right angles to in. In this position the two a'. Its traces will then be the lines QP1 and PIP2 (fig. 9). These sets of points AB will coincide if the drawing has been accurate. will be at right angles, and will therefore, together with the section Models of this kind can be made in many cases and their con- QP2 of the plane a, form a right-angled triangle QP1P2 with the struction cannot be too highly recommended in order to realize right angle at P1, and having the sides P1Q and PIP2 which both orthographic projection. are given in their true lengths. This triangle we rabatt about its § 14. To find the angle between two given lines a, b of which the base P1Q, making P1R=PIP2. The line QR will then give the true projections a1, b1 and a2, b2 are given. length of the line QP in space. If now the plane a be turned about Solution.—Let a1, b1 (fig. 44) meet in PI, a2, b2 in T, then if the line a' the point P will describe a circle about Q as centre with radius PIT is not perpendicular to the axis the two lines will not meet. In QP=QR, in a plane perpendicular to the trace a'. Hence when the this case we draw a line parallel plane a has been rabatted into the horizontal plane the point P will to b to meet the line a. This is lie in the perpendicular PIQ to a', so that QP=QR. easiest done by drawing first the If AI is the plan of a point A in the plane a, and if AI lies in QP1, line PIP2 perpendicular to the then the point A will lie vertically above Al in the line QP. On axis to meet a2 in P2, and then turning down the triangle QP1P2, the point A will come to Ao, the drawing through P2 a line c2 line A1Ao being perpendicular to QP1. Hence A will be a point in parallel to b2; then b1, c2 will be QP such that QA=QAo. the projections of a line c which If B1 is the plan of another point, but such that A1B1 is parallel is parallel to b and meets a in P. to a', then the corresponding line AB will also be parallel to a'. The plane a which these two Hence, if. through A a line AB be drawn parallel to a', and B1 B lines determine we rabatt to the perpendicular to a', then their intersection gives the point B. Thus plan. We determine the traces of any point given in plan the real position in the plane a, when a' and c' of the lines a and c; rabatted, can be found by this second method. This is the one then a'c' is the trace a' of their most generally given in books on geometrical drawing. The first plane. On rabatting the point method explained is, however, in most cases preferable as it gives P comes to a point S on the line the draughtsman a greater variety of constructions. It requires a P1Q perpendicular to a'c', so somewhat greater amount of theoretical knowledge. that QS=QP. But QP is the hypotenuse of a triangle PP1Q with If instead of our knowing the plan of a figure the latter is itself a right angle PI. This we construct by making QR = PoP2; then given, then the process of finding the plan is the reverse of the P1R=PQ. The lines a'S and c'S will therefore include angles equal above and needs little explanation. We give an example. to those made by the given lines. It is to be remembered that two § 16. It is required to draw the plan and elevation of a polygon of lines include two angles which are supplementary. Which of these which the real shape and position in a given plane a are known. is to be taken in any special case depends upon the circumstances. We first rabatt the plane a (fig. 46) as before so that PI comes to To determine the angle between a line and a plane, we draw through P, hence OPI to OP. Let the given polygon in a be the figure any point in the line a perpendicular to the plane (§ 12) and determine ABCDE. We project, not the vertices, but the sides. To project the angle between it and the given line. The complement of this the line AB, we produce it to cut a' in F and OP in G, and draw project angle. is the required one. perpendicular to a'; then G1 corresponds to G, therefore FG1 to FG. To determine the angle between two planes, we draw through any In the same manner we might project all the other sides, at least II those which cut OF and OP in convenient points. It will be best, however, first to produce all the sides to cut OP and a' and then to draw all the projecting rays through A, B, C . . perpendicular to a', and in the same direction the lines G, G1, &c. By drawing FG we get the points Al, Bl on the projecting ray through A and B. We then join B to the point NI where BC produced meets the trace a'. This gives C1. So we go on till we have found El. The line Al El must then meet AE in a', and this gives a check. If one of the sides cuts a' or OP beyond the drawing paper this method fails, but then we may easily find the projection of some other line, say of a diagonal, or directly the projection of a point, by the former methods. The also serve to check the drawing, for two corresponding diagonals must meet in the trace a'. Having got the plan we easily find the elevation. The elevation of G is above G, in a", and that of F is at F2 in the axis. This gives the elevation F2G2 of FG and in it we get A2132 .in the verticals through Al and B1. As a check we have OG=OG2. Similarly the elevation of the other sides and vertices are found. § 17. We proceed to give some applications of the above principles to the representation of solids and of the solution of problems connected with them. Of a pyramid are given its base, the length of the perpendicular from the vertex to the base, and the point where this perpendicular cuts the base; it is required first to develop the whole surface of the pyramid into one plane, and second to determine its section by a plane which cuts the plane of the base in a given line and makes a given angle with it. 1. As the planes of projection are not given we can take them as we like, and we select them in such a manner that the solution becomes as simple as possible. We take the plane of the base as the horizontal plane and the vertical plane perpendicular to the plane of the section. Let then (fig. 47) ABCD be the base of the pyramid, Vi the plan of the vertex, then the elevations of A, B, C, D will be in the axis at A2, B2, C2, D2, and the vertex at some point V2 above V, at a known distance from the axis. The lines V1A, V1B, &c., will be the plans and the lines V2A2, V2B2, &c., the elevations of the edges of the pyramid, of which thus plan and elevation are known. We develop the surface into the plane of the base by turning each lateral face about its lower edge into the horizontal plane by the method used in § 14. If one face has been turned down, say ABV to ABP, then the point Q to which the vertex of the next face BCV comes can be got more simply by finding on the line V1Q perpendicular to BC the point Q such that BQ =BP, for these lines represent the same edge BV of the pyramid. Next R is found by making CR=CQ, and so on till we have got the last vertex —in this case S. The fact that AS must equal AP gives a convenient check. 2. The plane a whose section we have to determine has its horizontal trace given perpendicular to the axis, and its vertical trace makes the given angle with the axis. This determines it. To find the section of the pyramid by this plane there are two methods applicable: we find the sections of the plane either with the faces or with the edges of the pyramid. We use the latter. As the plane a is perpendicular to the vertical plane, the trace a' contains the projection of every figure in it; the points E2, F2t G2, H2 where this trace cuts the elevations of the edges will therefore , be the elevations of the points where the edges cut a. From these we find the plans E1, Fl, G1, H1, and by joining them the plan of the section. If from El, Fl lines be drawn perpendicular to AB, these will determine the points E, F on the developed face in which the plane a cuts it; hence also the line EF. Similarly on the other faces. Of course BF must 'be the same length on BP and on BQ. If the plane a be rabatted to the plan, we get the real shape of the section as shown in the figure in EFGH. This is done easily bymaking FoF=OF2, &c. If the figure representing the development of the pyramid, or better a copy of it, is cut out, and if the lateral faces be bent along the lines AB, BC, &c., we get a model of the pyramid with the section marked on its faces. This may be placed on its plan ABCD and the plane of elevation bent about the axis x. The pyramid stands then in front of its elevations. If next the plane a with a hole cut out representing the true section be bent along the trace a' till its edge coincides with a", the edges of the hole ought to coincide with the lines EF, FG, &c., on the faces. § i8. Polyhedra like the pyramid in § 17 are represented by the projections of their edges and vertices. But solids bounded by curved surfaces, or surfaces themselves, cannot be thus represented. For a surface we may use, as in case of the plane, its traces—that is, the curves in which it cuts the planes of projection. We may also project points and curves on the surface. A ray cuts the surface generally in more than one point; hence it will happen that some of the rays touch the surface, if two of these points coincide. The points of contact of these rays will form some curve on the surface, and this will appear from the centre of projection as the boundary of the surface or of part of the surface. The outlines of all surfaces of solids which we see about us are formed by the points at which rays through our eye touch the surface. The projections of these contours are therefore best adapted to give an idea of the shape of a surface. Thus the tangents drawn from any finite centre to a sphere form a right circular cone, and this will be cut by any plane in a conic. V2 It is often called the projection of a sphere, but it is better called the contour-line of the sphere, as it is the boundary of the projections of all points on the sphere. If the centre is at infinity the tangent cone becomes a right circular cylinder touching the sphere along a great circle, and if the projection is, as in our case, orthographic, then the section of this cone by a plane of projection will be a circle equal to the great circle of the sphere. We get such a circle in the plan and another in the elevation, their centres being plan and elevation of the centre of the sphere. Similarly the rays touching a cone of the second order will lie in two planes which pass through the vertex of the cone, the contour-line of the projection of the cone consists therefore of two lines meeting in the projection of the vertex. These may, however, be invisible if no real tangent rays can be drawn from the centre of projection; and this happens when the ray projecting the centre of the vertex lies within the cone. In this case the traces of the cone are of importance. Thus in representing a cone of revolution with a vertical axis we get in the plan a circular trace of the surface whose centre is the plan of the vertex of the cone, and in the elevation the contour, consisting of a pair of lines intersecting in the elevation of the vertex of the cone. The circle in the plan and the pair of lines in the elevation do not determine the surface, for an infinite number of surfaces might be conceived which pass through the circular trace and touch two planes through the contour lines in the vertical plane. The surface becomes only completely defined if we write down to the figure that it shall represent a cone. The same holds for all surfaces. Even a plane is fully represented by its traces only under the silent understanding tl'at the traces are those of a plane. § 19. Some of the simpler problems connected with the representation of surfaces are the determination of plane sections and of the curves of intersection of two such surfaces. The former is constantly used in nearly all problems concerning surfaces. Its solution depends of course on the nature of the surface. To determine the curve of intersection of two surfaces, we take a plane and determine its section with each of the two surfaces, rabatting this plane if necessary. This gives two curves which lie in the same plane and whose intersections will give us points on both surfaces. It must here be remembered that two curves in space do not necessarily intersect, hence that the points in which their projections intersect are not necessarily the projections of points common to the two curves. This will, however, be the case if the two curves lie in a common plane. By taking then a number of plane sections of the surfaces we can get as many points on their curve of intersection as we like. These planes have, of course, to be selected in such a way that the sections are curves as simple as the case permits of, and such that they can be easily and accurately drawn. Thus when possible the sections should be straight lines or circles. This not only saves time in drawing but determines all points on the sections, and therefore also the points where the two curves meet, with equal accuracy. § 20. We give a few examples how these sections have to be selected. A cone is cut by every plane through the vertex in lines, and if it is a cone of revolution by planes perpendicular to the axis in circles. A cylinder is cut by every plane parallel to the axis in lines, and if it is a cylinder of revolution by planes perpendicular to the axis in circles. A sphere is cut by every plane in a circle. Hence in case of two cones situated anywhere in space we take sections through both vertices. These will cut both cones in lines. Similarly in case of two cylinders we may take sections parallel to the axis of both. In case of a sphere and a cone of revolution with vertical axis, horizontal sections will cut both surfaces in circles whose plans are circles and whose elevations are lines, whilst vertical sections through the vertex of the cone cut the latter in lines and the sphere in circles. To avoid drawing the projections of these circles, which would in general be ellipses, we rabatt the plane and then draw the circles in their real shape. And so on in other cases. Special attention should in all cases be paid to those points in which the tangents to the projection of the curve of intersection are parallel or perpendicular to the axis x, or where these projections touch the contour of one of the surfaces. (O. H.) IV. ANALYTICAL GEOMETRY r. In the name geometry there is a lasting record that the science had its origin in the knowledge that two distances may be compared by measurement, and in the idea that measurement must be effectual in the dissociation of different directions as well as in the comparison of distances in the same direction. The distance from an observer's eye of an object seen would be specified as soon as it was ascertained that a rod, straight to the eye and of length taken as known, could be given the direction of the line of vision, and had to be moved along it a certain number of times through lengths equal to its own in order to reach the object from the eye. Moreover, if a field had for two of its boundaries lines straight to the eye, one running from south to north and the other from west to east, the position of a point in the field would be specified if the rod, when directed west, had to be shifted from the point one observed number of times westward to meet the former boundary, and also, when directed south, had to be shifted another observed number of times southward to meet the latter. Comparison by measurement, the beginning of geometry, involved counting, the basis of arithmetic; and the science of number was marked out from the first as of geometrical importance. But the arithmetic of the ancients was inadequate as a science of number. Though a length might be recognized as known when measurement certified that it was so many times a standard length, it was not every length which could be thus specified in terms of the same standard length, even by an arithmetic enriched with the notion of fractional number. The idea of ,possible incommensurability of lengths was introduced into Europe by Pythagoras; and the corresponding idea of irrationality of number was absent from a crude arithmetic, while there were great practical difficulties in the way of its introduction. Hence perhaps it arose that, till comparatively modern times, appeal to arithmetical aid in geometrical reasoning was in allpossible ways restrained. Geometry figured rather as the helper of the more difficult science of arithmetic. 2. It was reserved for algebra to remove the disabilities of arithmetic, and to restore the earliest ideas of the land-measurer to the position of controlling ideas in geometrical investigation. This unified science of pure number made comparatively little headway in the hands of the ancients, but began to receive due attention shortly after the revival of learning. It expresses whole classes of arithmetical facts in single statements, gives to arithmetical laws the form of equations involving symbols which may mean any known or sought numbers, and provides processes which enable us to analyse the information given by an equation and derive from that equation other equations, which express laws that are in effect consequences or causes of a law started from, but differ greatly from it in form. Above all, for present purposes, it deals not only with integral and fractional number, but with number regarded as capable of continuous growth, just as distance is capable of continuous growth. The difficulty of the arithmetical expression of irrational number, a difficulty considered by the modern school of analysts to have been at length surmounted (see FUNCTION), is not vital to it. It can call the ratio of the diagonal of a square to a side, for instance, or that of the circumference of a circle to a diameter, a number, and let a or x denote that number, just as properly as it may allow either letter to denote any rational number which may be greater or less than the ratio in question by a difference less than any minute one we choose to assign. Counting only, and not the counting of objects, is of the essence of arithmetic, and of algebra. But it is lawful to count objects, and in particular to count equal lengths by measure. The widened idea is that even when a or x is an irrational number we may speak of a or x unit lengths by measure. We may give concrete interpretation to an algebraical equation by allowing its terms all to mean numbers of times the same unit length, or the same unit area, or &c. and in any equation lawfully derived from the first by algebraical processes we may do the same. Descartes in his Geometrie (1637) was the first to systematize the application of this principle to the inherent first notions of geometry; and the methods which he instituted have become the most potent methods of all in geometrical research. It is hardly too much to say that, when known facts as to a geometrical figure have once been expressed in algebraical terms, all strictly consequential facts as to the figure can be deduced by almost mechanical processes. Some may well be unexpected consequences; and in obtaining those of which there has been suggestion beforehand the often bewildering labour of constant attention to the figure is obviated. These are the methods of what is now called analytical, or sometimes algebraical, geometry. 3. The modern use of the term " analytical " in geometry has obscured, but not made obsolete, an earlier use, one as old as Plato. There is nothing algebraical in this analysis, as distinguished from synthesis, of the Greeks, and of the expositors of pure geometry. It has reference to an order of ideas in demonstration, or, more frequently, in discovering means to effect the geometrical construction of a figure with an assigned special property. We have to suppose hypothetically that the construction has been performed, drawing a rough figure which exhibits it as nearly as is practicable. We then analyse or critically examine the figure, treated as correct, and ascertain other properties which it can only possess in association with the one in question. Presently one of these properties will often be found which is of such a character that the construction of a figure possessing it is simple. The means of effecting synthetic-ally a construction such as was desired is thus brought to light by what Plato called analysis. Or again, being asked to prove a theorem A, we ascertain that it must be true if another theorem B is, that B must be if C is, and so on, thus eventually finding that the theorem A is the consequence, through a chain of intermediaries, of a theorem Z of which the establishment is easy. This geometrical analysis is not the subject of the present article; but in the reasoning from form to form of an equation or system of equations, with the object of basing the algebraical proof of a geometrical fact on other facts of a more obvious character, the same logic is utilized, and the name " analytical geometry " is thus in part explained. 4. In algebra real, positive number was alone at first dealt with, and in geometry actual signless distance. But in algebra it became of importance to say that every equation of the first degree has a root, and the notion of negative number was introduced. The negative unit had to be defined as what can be added to the positive unit and produce the sum zero. The corresponding notion was readily at hand in geometry, where it was clear that a unit distance can be measured to the left or down from the farther end of a unit distance already measured to the right or up from a point 0, with the result of reaching 0 again. Thus, to give full interpretation in geometry to the algebraically negative, it was only necessary to associate distinctness of sign with oppositeness of direction. Later it was discovered that algebraical reasoning would be much facilitated, and that conclusions as to the real would retain all their soundness, if a pair of imaginary units Al -1 of what might be called number were allowed to be contemplated, the pair being defined, though not separately, by the two properties of having the real sum o and the real product i. Only in these two real combinations do they enter in conclusions as to the real. An advantage gained was that every quadratic equation, and not some quadratics only, could be spoken of as having two roots. These admissions of new units into algebra were final, as it admitted of proof that all equations of degrees higher than two have the full numbers of roots possible for their respective degrees in any case, and that every root has a value included in the form a+b J — i, with a, b, real. The corresponding enrichment could be given to geometry, with corresponding advantages and the same absence of danger, and this was done. On a line of measurement of distance we contemplate as existing, not only an infinite continuum of points at real distances from an origin of measurement 0, but a doubly infinite continuum of points, all but the singly infinite continuum of real ones imaginary, and imaginary in conjugate pairs, a conjugate pair being at imaginary distances from 0, which have a real arithmetic and a real geometric mean. To geometry enriched with this conception all algebra has its application. 5. Actual geometry is one, two or three-dimensional, i.e. lineal, plane or solid. In one-dimensional geometry positions and measurements in a single line only are admitted. Now descriptive constructions for points in a line are impossible without going out of the line. It has therefore been held that there is a sense in which no science of geometry strictly confined to one dimension exists. But an algebra of one variable can be applied to the study of distances along a line measured from a chosen point on it, so that the idea of construction as distinct from measurement is not essential to a one-dimensional geometry aided by algebra. In geometry of two dimensions, the flat of the land-measurer, the passage from one point 0 to any other point, can be effected by two successive marches, one east or west and one north or south, and, as will be seen, an algebra of two variables suffices for geometrical exploitation. In geometry of three dimensions, that of space, any point can be reached from a chosen one by three marches, one east or west, one north or south, and one up or down; and we shall see that an algebra of three variables is all that is necessary. With three dimensions actual geometry stops; but algebra can supply any number of variables. Four or more variables have been used in ways analogous to those in which one, two and three variables are used for the purposes of one, two and three-dimensional geometry, and the results have been expressed in quasi-geometrical language on the supposition that a highez, space can be conceived of, though not realized, in which four independent directions exist, such that no succession of marches along three of them can effect the same displacement of a point as a march along the fourth; and similarly for higher numbers than four. Thus analytical, though not actual, geometries exist for four and more dimensions. They are in fact algebras furnished with nomenclature of a geometrical cast, suggested by convenient forms of expression which actual geometry has, in return for benefits received, conferred on algebras of one, two and three variables. We will confine ourselves to the dimensions of actual geometry, and will devote no space to the one-dimensional, except incident-ally as existing within the two-dimensional. The analytical method will now be explained for the cases of two and three dimensions in succession. The form of it originated by Descartes, and thence known as Cartesian, will alone be considered in much detail. I. Plane Analytical Geometry. 6. Coordinates.—It is assumed that the points, lines and figures considered lie in one and the same plane, which plane therefore need not be in any way referred to. In the plane a point 0, and two lines x'Ox, y'Oy, intersecting in 0, are taken once for all, and regarded as fixed. 0 is called the origin, and x'Ox, y'Oy the axes of x and y respectively. Other positions in the plane are specified in relation to this fixed origin and these fixed axes. From any point P we y Fie. 48. FIG. 49. suppose PM drawn parallel to the axis of y to meet the axis of x in M, and may also suppose PN drawn parallel to the axis of x to meet the axis of y in N, so that OMPN is a parallelogram. The position of P is determined when we know OM (=NP) and MP ~ON). If OM is x times the unit of a scale of measurement chosen at pleasure, and MP is y times the unit, so that x and y have numerical values, we call x and y the (Cartesian) coordinates of P. To distinguish them we often speak of y as the ordinate, and of x as the abscissa. It is necessary to attend to signs; x has one sign or the other according as the point P is on one side or the other of the axis of y, and y one sign or the other according as P is on one side or the other of the axis of x. Using the letters N, E, S, W, as in a map, and considering the plane as divided into four quadrants by the axes, the signs are usually taken to be: x y For quadrant + + N E + — S E + N W — S W A point is referred to as the point (a, b), when its coordinates are x =a, y =b. A point may be fixed, or it may be variable, i.e. be regarded for the time being as free to move in the plane. The coordinates (x, y) of a variable point are algebraic variables, and are said to be " current coordinates." The axes of x and y are usually (as in fig. 48) taken at right angles to one another, and we then speak of them as rectangular axes, and of x and y as " rectangular coordinates " of a point P; OMPN is then a rectangle. Sometimes, however, it is convenient to use axes which are oblique to one another, so that (as in fig. 49) the angle xOy between their positive directions is some known angle co distinct from a right angle, and OMPN is always an oblique parallelogram with given angles; and we then speak of x and y as " oblique coordinates." The coordinates are as a rule taken to be rectangular in what follows. 7. Equations and loci. If (x, y) is the point P, and if we are given that x=o, we are told that, in fig. 48 or fig. 49, the point M lies at 0, whatever value y may have, i.e. we are told the one fact that P lies on the axis of y. Conversely, if P lies anywhere on the axis of y, we have always OM =o, i.e. x=o. Thus the equation x = o is one satisfied by the coordinates (x, y) of every point in the axis of y, and not by those of any other point. We say that x=o is the equation of the axis of y, and that the axis of y is the locus represented by the equation x =o. Similarly y =o is the equation of the axis of x. An equation x=a, where a is a constant, expresses that P lies on a parallel to the axis of y through a point M on the axis of x such that OM =a. Every line parallel to the axis of y has an equation of this form. Similarly, every line parallel to the axis of x has an equation of the form y =b, where b is some definite constant. These are simple cases of the fact that a single equation in the current coordinates of a variable point (x, y) imposes one limitation on the freedom of that point to vary. The coordinates of a point N P se, O M taken at random in the plane will, as a rule, not satisfy the equation, but infinitely many points, and in most cases infinitely many real ones, have coordinates which do satisfy it, and these points are exactly those which lie upon some locus of one dimension, a straight line or more frequently a curve, which is said to be represented by the equation. Take, for instance, the equation y=mx, where in is a given constant. It is satisfied by the coordinates of every point P, which is such that, in fig. 48, the distance MP, with its proper sign, is m times the distance OM, with its proper sign, i.e. by the co-ordinates of every point in the straight line through 0 which we arrive at by making a line, originally coincident with x'Ox, revolve about 0 in the direction opposite to that of the hands of a watch through an angle of which m is the tangent, and by those of no other points. That line is the locus which it represents. Take, more generally, the equation y =0(x), where 0(x) is any given non-ambiguous function of x. Choosing any point M on x'Ox in fig. i, and giving to x the value of the numerical measure of OM, the equation determines a single corresponding y, and so determines a single point P on the line through M parallel to y'Oy. 'This is one point whose coordinates satisfy the equation. Now let M move from the extreme left to the extreme right of the line x'Ox, regarded as extended both ways as far as we like, i.e. let x take all real values from — so to co. With every value goes a point P, as above, on the parallel to y'Oy through the corresponding M ; and we thus find that there is a path from the extreme left to the extreme right of the figure, all points P along which are distinguished from other points by the exceptional property of satisfying the equation by their coordinates. This path is a locus; and the equation y=4(x) represents it. More generally still, take an equation f(x, ) =o which involves both x and y under a functional form. Any particular value given to x in it produces from it an equation for-the determination of a value or values of y, which go with that value of x in specifying a point or points (x, y), of which the coordinates satisfy the equation f(x, y) =o. Here again, as x takes all values, the point or points describe a path or paths, which constitute a locus represented b the equation. Except when y enters to the first degree only in f(x, y), it is not to be expected that all the values of y, determined as going with a chosen value of x, will be necessarily real; indeed it is not uncommon for all to be imaginary for some ranges of values of x. The locus may largely consist of continua of imaginary Koints; but the real parts of it constitute a real curve or real curves. cite that we have to allow x to admit of all imaginary, as well as of all real, values, in order to obtain all imaginary parts of the locus. A locus or curve may be algebraically specified in another way ; viz. we may be given two equations x=f(0), y=F(0), which express the coordinates of any point of it as two functions of the same variable parameter 0 to which all values are open. As 0 takes all values in turn, the point (x, y) traverses the curve. It is a good exercise to trace a number of curves, taken as defined by the equations which represent them. This, in simple cases, can be done approximately by plotting the values of y given by the equation of a curve as going with a considerable number of values of x, and connecting the various points (x, y) thus obtained. But methods exist for diminishing the labour of this tentative process. Another problem, which will be more attended to here, is that of determining the equations of curves of known interest, taken as defined by geometrical properties. It is not a matter for surprise that the curves which have been most and longest studied geometrically are among those represented by equations of the simplest character. 8. The Straight Line.—This is the simplest type of locus. Also the simplest type of equation in x and y is Ax+By+C =o, one of the first degree; Here the coefficients A, B, C are constants. They are, like the current coordinates, x, y, numerical. But, in giving interpretation to such an equation, we must of course refer to numbers Ax, By, C of unit magnitudes of the same kind, of units of counting for instance, or unit lengths or unit squares. It will now be seen that every straight line has an equation of the first degree, and that every equation of the first degree represents a straight line. It has been seen (§ 7) that lines parallel to the axes have equa- tions of the first degree, free from one of the variables. Take now a straight line ABC inclined to both axes.. Let it make a given angle a with the positive direction of the axis of x, i.e. in fig. 5o let this be the angle through which Ax must be re- volved counter-clockwise about A in order to be made coin- cident with the line. Let C, of coordinates (h, k), be a fixed point on the line, and P (x, y) any other point upon it. Draw the ordinates CD, PM of C .and P, and let the parallel to the axis of x through C meet PM, produced if necessary, in R. The right-angled triangle
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