Geometry

GEOMETRY, one of the three principal branches of mathe matics (the other two being algebra and analysis), may be de scribed as the branch which deals with the properties of space. Its most elementary part is known to every schoolboy under the name of plane and solid geometry, the former dealing with the properties of figures in a plane, the latter with the properties of figures in space (of three dimensions). These two subjects form, however, only a small part of geometry as the term is now under stood. The present article attempts, first, to describe briefly these two elementary topics in geometry, and secondly, to give the reader some conception of the content of geometry as a whole with references to other articles where more detailed information on the various topics may be found. The article is then intended, in part, to orient the reader in a very large subject, sending him to other articles for more elaborate discussion of the special fields.

Like most other branches of knowledge, geometry arose origi nally in response to man's practical needs. The word `geometry" (Gr. earth and measure) means "earth measurement." Indeed, the subject seems to have had its birth in ancient Egypt, where the periodic inundations of the Nile made the surveying of the land for the re-establishment of boundary lines a necessity. This early empirical geometry consisted merely of a number of crude rules for the mensuration of various simple geometric figures; for the laying out of angles, especially right angles, etc. (See below for details as to history.) The ancient Greeks developed this crude beginning into the science which is now studied in the schools under the name of demonstrative geometry, the plane and solid geometry already mentioned. This form of geometry depends on the observation that the propositions of geometry are logically inter-related; i.e., that, if certain propositions are granted, certain others can be proved as logical consequences of those assumed. This suggests the possibility of arranging all the propositions in a sequence such that every proposition in the list, after a certain one, is a logical consequence of some or all the propositions that precede it. The first comprehensive and systematic attempt to exhibit the prop ositions of geometry in such a sequence, which has come down to us, is one of the most famous works in all literature, the Elements of Euclid (q.v.) of Alexandria (c. 30o B.e.). This work consists of 13 books, the first six and the last three of which are devoted to plane and solid geometry respectively. More or less literal trans lations of this ancient work were used to within a generation ago as textbooks in the public schools of England. The textbooks of the present time in all countries are adaptations of Euclid's Elements designed to meet the pedagogical needs of young pupils ; they may claim pedagogical advantages, but at the sacrifice of some logical rigour and comprehensiveness.

The Foundations of Geometry.—If the propositions of geometry have been arranged in a strictly logical sequence, as above indicated, it is evident that a certain number at the be ginning of the list are not logical consequences of the preceding ones. The first proposition is, of course, not a logical conse quence of a preceding one; nor is it likely that the second is a logi cal consequence of the first. The question then, naturally, arises as to the logical status of these unproved propositions on which all the others depend. Moreover, these propositions involve cer tain terms, such as point, straight line, circle, etc. What meaning attaches to such terms? A definition defines a term in terms of certain others, the meaning of which is supposed known. In order to avoid defining in a circle some terms must remain un defined. The foundations of geometry must then consist, from the purely logical point of view, of a set of undefined terms and a set of unproved propositions concerning them, such that every new term can be defined in terms of the undefined, and such that every new proposition can be proved a consequence of the un proved. The unproved propositions are usually called axioms (q.v.) or postulates. Are these to be regarded as self-evident truths? Are they imposed on our minds a priori, as Kant (q.v.) taught, and is it impossible to think logically without granting them? Or are they, in accordance with the teaching of John Stuart Mill (q.v.) of experimental origin? Do the undefined terms de note primitive notions, the meaning of which is clear without definition to everybody? Prevailing opinion regards a geometric theorem as true beyond possibility of doubt by a reasonable being. Will a critical inspection bear out this opinion? The answer is in the negative. Indeed, Bertrand Russell has said: "Mathematics may be defined as the science in which we never know what we are talking about nor whether what we say is true." ("Recent work on the Principles of Mathematics," The Interna tional Monthly, 1901.) Many a reader, in looking back on his school days, may heartily agree with this definition. Modern work on the foundations has shown that Kant was wrong and that Mill was only partly right. Logically considered, the axioms and postulates are mere assumptions. A certain writer has con sidered the dethroning of the "self-evident" as analogous to the change from an absolute monarchy to a democracy. The "self evident truth," which ruled by the Divine right of the alleged inconceivability of the opposite, has been replaced by the "as sumption," which is elected for its qualifications to serve (the reference is obviously to an ideal democracy).

Greek Geometry.

The more elementary part of Greek geometry is too well known to warrant a detailed exposition here. The reader will recall with pleasure or pain, but without diffi culty, the theorems on the equality or congruence (to use the more modern term) of triangles, on parallel lines, on the circle, on the measurement of angles by various circular arcs, on the similarity of triangles and other figures, on regular polygons, on proportion, on perimeters and areas, etc., in plane geometry; and the various properties of planes and lines, dihedral, trihedral, and polyhedral angles, the mensuration of pyramids, prisms, cones, cylinders, and the sphere, etc., in solid geometry.

Among the high spots of this material we may mention the famous so called Pythagorean proposition (the square on the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides) of which many proofs have been pub lished. Among them the visual proofs are of special interest, one of which is reproduced herewith. A mere comparison of the ad joined figures establishes the theorem. The theorem is of interest also for the problems to which it gave rise. Even in ancient times the problem of finding right triangles with integral sides, i.e., to find three whole numbers x, y, and z, such that engaged the attention of mathematicians and was completely solved by them. The generalization to the solution in whole numbers of the equation xn+y11= zn is still one of the unsolved problems in the theory of numbers (see THEORY OF NUMBERS), no solutions exist ing for any value of it greater than two and less than ioo. Certain other famous problems deserve mention on account of their influence on the development of geometry and of mathematics in general.

The Duplication of the Cube.—The Athenians, so the story goes, appealed to the oracle at Delos to know how to stay the plague which visited their city in 43o B.C. The oracle replied that they must double in size the altar of Apollo without changing its shape. The altar being in the form of a cube, the problem was to find the size of a cube whose volume was twice that of a given cube. In modern notation, if a is the side of the given cube, the problem is to construct x such that = It must be remem bered that, to the Greeks, "construct" meant construct with ruler and compasses only. With this limitation on the means to be employed the problem is now known to be impossible. But the at tempts to solve it led to the invention of numerous new curves (see CURVE; CURVES, SPECIAL) and was a powerful stimulus to mathematicians through many centuries.

The Trisection of an Angle.—To construct the bisector of any angle is one of the easiest problems in plane geometry. It was inevitable that a construction for dividing any angle into three (or more) equal parts should be sought. Again, if the construc tion is to apply to any angle (certain special angles such as a right angle are, of course, readily trisected) and if ruler and compasses only may be used, the problem is impossible ; i.e., there is no such construction. Many new curves, however, have been invented for the purpose of solution. Among these we may mention the curve that was later known as the quadratrix, because it furnished a solution also for the problem of squaring the circle (see below). In the figure shown above OX rotates at a uniform rate through one right angle from the position OB to the position OA, in the same time that the line MN, always parallel to OA, moves at a uniform rate from the position BC to the position OA. The intersection P of the line OX and MN then traces the curve in question. The use of this curve in trisecting an angle is almost trivially simple. Let XOB be the angle to be trisected. Divide BM into three equal parts and draw lines through the points of division parallel to OA. These lines meet the quadratrix in points which if joined to 0 yield the required trisectors.

The Squaring of the Circle.—This problem, perhaps the most famous one of all, consists in constructing a square equal in area to the area of a given circle. It, too, is impossible if ruler and compasses are the only instruments permitted. The use of the quadratrix in the solution of this problem depends on the following relation. In the preceding figure it may be proved that the length of the quadrant BXA satisfies the following proportion : This makes it possible to construct a line equal in length to the circumference of the circle, from which a square equal in area to that of the circle is easily constructed. In spite of the fact that the solution of these problems by means of ruler and compasses alone has been known to be impossible for over ioo years, angle trisecters and circle-squarers continue to appear. The best that can be said of these deluded individuals is that their enthusiasm has outstripped their scholarship. A later section of this article is devoted to the modern aspect of construction problems, and another section to developments arising from Euclid's parallel postulate.

The Conic Sections.—Partly under the stimulation of the problems just mentioned, the ancient Greeks investigated a class of curves known as the conic sections (q.v.), or more briefly the conics, and developed a large number of their properties. These curves arise as the plane sections of a right circular cone and have three distinct forms, the ellipse, the parabola, and the hyper bola. They have played a fundamentally important role in the development, not only of pure mathematics, but also in the ap plications. The reader may find further information in the article referred to (see also PROJECTIVE GEOMETRY). Suffice it to say that, had the conic sections not been previously studied, Kepler could not have discovered his famous laws concerning the motion of the planets, nor would we to-day have the benefit of search lights with their parabolic reflectors. With the work of Apollonius on conics and the work of Archimedes on certain spirals and his remarkable determination of certain areas—he succeeded in find ing the areas of an ellipse and of a parabolic segment—we reach the limits of ancient Greek geometry. No essential progress was made in this subject for over i,000 years. We may, however, at this point say a few words about a more modern development, which is largely in the spirit of ancient geometry.

The Geometry of the Triangle and Circle.—This develop ment relates to a detailed study of the triangle (q.v.) and circle (q.v.). It consists largely of the discovery of numerous points and lines connected with a triangle or circle and the discussion of their properties. Some of the latter are very remarkable. As an example, we may mention the nine-point circle related to a tri angle, so-called because it passes through (a) the three mid points of the sides, (b) the feet of the perpendiculars drawn from the vertices to the opposite sides, and (c) the mid-points of the lines joining the intersection of the three altitudes to the three vertices. This circle is tangent to the inscribed circle of the tri angle, and also to the three circles which are each tangent to one side and the other two sides produced.

Descartes and the Invention of Analytic Geometry.—A new stimulus came to the development of geometry by the intro duction through Descartes (1637, q.v.) of the so-called analytic methods. By representing a point in a plane by means of two numbers (co-ordinates, q.v.), giving the distance and direction of the point from two intersecting lines (axes) of the plane, it was found possible to translate any geometric situation into an alge braic situation, whereby the powerful methods of algebra became available as a means of geometric investigation. The resulting analytic geometry (q.v.) is distinguished from the older or syn thetic geometry by its method rather than by its content. Analytic and synthetic geometry do not then constitute two different branches of geometry; they denote, rather, two distinct methods of studying geometry. There seems to be no need, therefore, to give further details as to this method at this point ; the reader should consult the article just mentioned. We may, however, try to characterize briefly the effect on the development of geometry of the introduction of these new methods.

In the first place, it provided a systematic plan for further progress. A curve in the plane is represented by an equation in the variable co-ordinates (x, y) of a point on the curve. The straight lines are represented by equations of the first degree. Equations of the second degree turn out to represent the conic sections (see above). It was therefore natural to study next the curves represented by equations of the third degree (cubic curves) ; then those of the fourth degree (quartic curves) ; and so on. It was possible even to develop a general theory of curves of degree n. Some curves, such as the quadratrix previously men tioned, lead to equations that are not algebraic but transcendental (see EQUATION) . Similar remarks apply to the geometry of space (of three dimensions). A point in space is represented by three co-ordinates (x, y, z) ; an equation in these three variables represents a surface ; and surfaces may then be classified accord ing to their degree (those of the first degree being the planes), and then systematically studied.

Construction Problems.

In the second place, and closely related to the preceding, the new method gave a means of classi fying the construction problems inherited from the ancients, as well as new ones. A construction problem, when formulated analytically, is found to be equivalent to the solution of an equa tion or to that of a system of simultaneous equations. A con struction with ruler and compasses is possible only when the cor responding equations can be solved by means of the rational operations (addition, subtraction, multiplication, division) and the extraction of square roots. The equations underlying the problem of duplicating the cube and that of trisecting an angle cannot be solved without introducing irrational operations of higher order than square roots (in both these cases cube roots suffice). These problems are, therefore, impossible with ruler and compasses, as previously stated. Another problem that had a great influence on the development of algebra was that of constructing a regular polygon of a given number of sides. The ancient Greeks were able, with ruler and compasses, to construct regular polygons of three, four, five, six, eight, ten, ... sides; but failed in their at tempts to construct one of seven sides. The determination of those values of n for which a regular polygon of n sides can be constructed with ruler and compasses led to a detailed study of the so-called cyclotomic equations, x'i — i = o. As a result it was shown that the problem is possible only when n has the form n= 2kplp2p3 . . ., where each p represents a prime number of the form Since seven is not of this form, the problem is im possible when n= 7. It will be noted that the first prime of the required form greater than 5 is 17 ; the next 257. The general result given is due to K. F. Gauss (1796) (q.v.).

Spaces of More Than Three Dimensions.

In the third place the analytic formulation of geometry led naturally, almost inevit ably, to a vast extension of the domain of geometry. Since the points of a plane could be represented by pairs of co-ordinates (x, y), the points of a space of three dimensions by triples (x, y, z) of co-ordinates, the question arises as to the geometric inter pretation of a set of four independent co-ordinates (x, y, z, u) of a set of five, . . . ; in general, of a set of n such co-ordinates. This led to the conception of a space of four, five, ... , or in general of n dimensions. The precise formulation of the idea of dimension (q v.), as here used, need not concern us at this place. We may note, however, that the notion of a space of four or more dimen sions is not, as is generally supposed, necessarily beyond our powers of concrete representation. True, a space of points of more than three dimensions is an abstract generalization to vis ualize which is beyond the present powers of our imagination. But, if we use other elements than points with which to build our space, no such difficulty arises. Thus the totality of straight lines in ordinary space (of three dimensions) is four-dimensional; i.e., constitutes a four-dimensional space (of lines). The totality of all spheres in ordinary space is a four-dimensional space of spheres. The corresponding geometries (line geometry and sphere geometry, see below) are just as concrete as the ordinary point geometries in the plane or in ordinary space. Nevertheless, point spaces of more than three dimensions are the ones usually thought of when n-dimensional spaces are being considered. Strangely enough, they find a large part of their interest in the field of applied mathematics. The four dimensional space-time manifold of Einstein-Minkowski is one that has recently aroused considerable popular interest in connection with the theory of relativity (q.v.). In the field of applied mathematics the concep tion of a space of more than three dimensions is generally of value in that it makes possible the application of geometric lan guage to a problem that is essentially analytic. This language is of great convenience and is often suggestive of the relations sought. As an example we may cite the dynamical theory of gases. Suppose a gas consists of a (very large) number, N, of molecules. The dynamical condition of each of these molecules is represented by six co-ordinates, three to specify its position in space and three more to give the three components of its velocity. To describe completely the state of the gas at any instant 6N co-ordinates would be necessary. Jeans, in his Dynamical Theory of Gases, says : "We can suppose this dynamical system repre sented in a generalized space of 6N dimensions." A space of 6N dimensions, where N is the num ber of molecules in a certain vol ume of a gas ! After this, it will cause no surprise to learn that spaces of an infinite number of dimensions have also been defined and their properties investigated. Indeed, there are several differ ent kinds of such spaces. Before leaving this topic, it should per haps be noted that, while we have chosen to introduce the con cept of an n-dimensional space on an analytic basis, the same can be done equally well on a purely synthetic basis.

Imaginary Elements.

In the fourth place, the introduction of analytic methods led to the conception of imaginary points, lines, etc., a conception which proved very fruitful. Not only did it make possible a greater generality in the statement of theorems and give a deeper insight into geometric relations, as, for exam ple, the characterization of a circle as any conic which passes through certain two imaginary points at infinity, but it also opened up a new field of enquiry. A complex variable x = x'+ix" = — I) depends on two real variables x', x". A complex line has then two real dimensions; a complex plane of points (x, y), where x and y each depend on two real variables, is a space of four real dimensions. The discussions of the one-dimensional loci on the complex line, and the one-, two-, and three-dimensional spreads in the complex plane constitutes a new world for the geometer to explore.

Finally, we should mention the fact that the introduction of the analytic method contributed largely to the idea of the unity of mathematics. There have been several instances in the preceding paragraphs not merely illustrating the possibility of regarding every geometric situation from an analytic point of view, which is, of course, the very essence of analytic geometry, but also ex emplifying the fact that analytic situations may be given a geo metric interpretation. It becomes increasingly difficult to dis tinguish analysis from geometry; these two branches of mathe matics appear rather as different aspects of the same thing.

Differential Geometry.

The introduction of the differen tial and integral calculus (Newton and Leibniz, qq.v.) only a half century after the introduction of analytic geometry, greatly in creased the power of the latter. The application of the funda mental ideas of the calculus to geometry introduced many new and valuable concepts. The resulting discipline is known as differ ential geometry (q.v.). Among the more elementary concepts referred to we may cite, by way of example, the curvature of a plane curve. The curvature at a point on a curve is a number which measures the sharpness of bending at that point. More precisely, if the curve is thought of as traced by a point moving with uniform speed along the curve, and this speed is taken as unity, the curvature at a point is proportional to the rate at which the tangent to the curve is turning at the instant when the moving point is at the given point. The curvature of a straight line is zero, the curvature of a circle is the same at every point and is equal to the reciprocal of the radius of the circle. This conception leads to a very beautiful theorem in the general theory of surfaces. The line drawn through a point on a surface perpen dicular to the tangent plane at that point is called the normal to the surface. Let us consider the plane curves obtained as sections of our surface by planes passing through this normal—the so-called normal sections. Each of these plane curves has a curvature at the point. Among all these normal sections there will, in general, be one for which the curvature is a maximum, and one for which the curvature is a minimum. (The exception is when all the curvatures are the same, in which case the surface is a plane or a sphere, or the point is some sort of exceptional point. We shall consider only "general" points, i.e., points that are not exceptional.) The theorem referred to states that these two normal sections are always at right angles to each other. (The theorem is true, as stated, for all respectable surfaces other than planes or spheres; the exceptions may be disregarded in a survey of this kind.) Further, this theorem makes possible the definition of the so called total curvature of a surface at a point as the product of the maximum and minimum curvatures of the normal sections just referred to. The investigation of surfaces of constant total curva ture leads to the result that the only such surfaces are the plane whose total curvature at every point is zero ; the sphere whose total curvature at every point is a positive constant (equal to the reciprocal of the radius squared) ; and the so-called pseudo sphere whose total curvature at every point is a negative con stant. (A negative total curvature indicates a saddle-shaped sur face.) The pseudo-sphere is obtained by revolving about the line AB the curve (known as the tractrix) in the adjoined figure. These surfaces of constant curvature are of interest in connection with non-Euclidean geometry. (See below.) Projective Geometry.—Early in the r 7th century Desargues (1593-1662) proved the theorem that, if the vertices of two tri angles lie on three lines meeting in a point, then their sides meet in three points lying on a line (see fig. 4). This theorem, it will be observed, has nothing to do with measurement, while the theorems of the older g e o m e t r y are metric. Such theorems occur singly in the 17th century; they were not recognized as forming part of a new kind of geometry until about the beginning of the i9th century, when the so-called projective geometry (q.v.) was de veloped as a self-contained body of doctrine. If we take a photograph of a straight railroad track, with equally spaced ties, the parallel lines of the track appear in the photograph as con verging, the equal distances between the ties appear unequal in the photograph, getting shorter and shorter as they recede in the picture. The right angles between the ties and the track appear as acute angles. If the landscape contains a circular pond it appears in the photograph as an ellipse. And yet the picture represents the scene ; something of the actual relationships of the scene must be preserved, even though distances and angles are distorted, and parallel lines are no longer parallel. The properties of the scene that are preserved in the photograph, i.e., the prop ' erties that are unchanged ("invariant" is the technical term), under the transformation produced by taking the photograph, are the properties studied by projective geometry. If the mechanism of this transformation is analysed, it is readily seen to consist of the following : Given any plane geometric figure (we confine our selves to plane figures for the sake of simplicity), let 0 be any point not in the plane of the figure and think of straight lines drawn from 0 to every point of the figure. Let this set of lines be cut by any other plane not passing through 0. This plane section of the lines through 0 gives a new figure in the cutting plane which is said to have been obtained from the original figure by the process of projection and section. It will be noted that in the case of our photograph the original scene is our first figure, the photographic plate is the cutting plane, while the point 0 is the point in the camera in which the light rays from the scene converge. Corresponding to every point of the original figure there is by this process determined a definite point in the new figure ; corresponding to every straight line in the original there is a straight line in the new ; if a point P in the original lies on a line 1, the corresponding point P' in the new figure will lie on the line l' corresponding to 1. A projective transformation consists of such a transformation by projection and section or of the result of a sequence of any number of such projections and sections. A projective transformation then transforms points into points, and straight lines into straight lines, and preserves the property of the incidence of points and lines. It does not preserve distances or angles; the latter, therefore, have no place in projective geom etry as such. Its nature is to some extent characterized by calling it the geometry of position (Geometrie der Lage). It is, as to structure, a very simple geometry and an extraordinarily sym metric one. Further details must be sought in the article re ferred to above. Projective geometry, moreover, furnishes a scientific basis for descriptive geometry (q.v.) which is a branch of applied rather than of pure geometry.

The set of all projective transformations (in a plane or in space) form what is known as a group of transformations by virtue of the fact that the result of performing any two of the operations of the set in succession is equivalent to a third trans formation of the set. Projective geometry is characterized com pletely by the fact that it studies those properties of figures which are invariant under the group of all projective transformations. Similarly, the set of all rigid motions in space form a group (see GROUPS). Ordinary elementary metric Euclidean geometry is then the geometry which studies those properties of figures that remain invariant under the group of all such motions. These are special cases of a fundamental principle, first enunciated by Felix Klein in 1872, to the effect that corresponding to every group of transformations in space there is a geometry consisting of those properties of space which are invariant under the given group. This principle provides at once a general classification of geome tries; it also provides a systematic method of procedure in study ing geometry as a whole, by the systematic investigation of all possible groups of transformations. In this project Sophus Lie (q.v.) laid the foundations in his theory of continuous groups. The group of motions just referred to is a sub-group of the general projective groups; other sub-groups are equivalent to the non-Euclidean (hyperbolic or elliptic) displacements; so that ordinary Euclidean and the two forms of non-Euclidean geometry are all implicitly contained in projective geometry, from which they are obtained by specialization. Furthermore, non-projective transformations may be defined by means of the projective, so that Arthur Cayley (q.v.) was led to exclaim: "Projective geom etry is all geometry." Inversion Geometry.—Among other geometries that have received extensive study, and in which the point is still the primary element of space, we may mention the inversion geometry in which the fundamental transformations are the so-called inversions with respect to a circle (plane inversion geometry), or with respect to a sphere (inversion geometry in space), or with respect to a hyper-sphere (in spaces of more than three dimensions). Analysis Situs.—The set of all possible continuous formations, i.e., roughly speaking, the set of all twistings, ings, stretchings or contractions without tearing anything apart, also form a group. The corresponding geometry is known as analysis situs (q.v.). It must consider, by what has been said, those properties of figures that remain invariant under any tinuous transformation whatever. Are there such properties? A simple closed curve in a plane divides the plane into two regions, an inside and an outside. This is a theorem of analysis situs. A surface may be either sided or two-sided (a simple one-sided surface may be tained by taking a strip of paper and fastening together the two ends, having pr e v i o u s l y turned one of the ends over through an angle of i80° ; see fig. 5) and this property is variant under any continuous transformation. If a map is drawn on any simple closed surface, say a sphere, by any set of tersecting lines on the surface dividing the surface into a set of regions (countries), and if the number of regions is denoted by r, the number of sides (the portions of the lines between two points of intersection) by s and the number of vertices (the points of intersection of the lines) by v, then the relation v+r=s+2 ways holds. This, too, is a theorem of analysis situs (due to L. Euler, q.v.) . It has been stated that, given any such map, four different colours are sufficient to colour the countries in such a way that no two countries with a boundary line in common shall have the same colour. The proof of this apparently simple theorem, however, presents serious difficulties; it is still one of the un solved problems of analysis situs.

Primitive Elements of Space.

We have hitherto assumed the point as the primitive element of space, except that once or twice we have hinted at the fart that this is not essential. In fact, the idea that some other geometric entity might be used as the primitive element, immediately opens up a new and vast field of geometric investigation. If, for example, we think of ordinary space as made up of all the straight lines in it, and study the properties of these lines and of systems of these lines, we enter the field of line geometry (q.v.), which, as has been said, is four dimensional. If the sphere is taken as the primitive element of space, we obtain the so-called sphere geometry, and so on.

There are other aspects of geometry which it might be thought desirable to include in this general survey. For example, some may seek here a reference to such matters as curves without tangents, or to the so-called "crinkly" or surface-filling curves. But these are essentially problems of higher analysis in geomet ric garb. Others may look in vain for a discussion of the very important concept of elements at infinity in the various geome tries ; such a discussion will be found in the articles devoted to the geometries in question. Enough has been said, it is hoped, to give the reader some notion of the scope of geometry as a whole, and some idea of its fundamental conceptions and of the type of problem with which it deals.

BIBLIOGRAPHY.-Any

one of the numerous school texts on plane or Bibliography.-Any one of the numerous school texts on plane or solid geometry will serve to refresh the reader's memory on these subjects. The standard edition of Euclid's Elements in English, is that by T. L. Heath in 3 vols. (Cambridge, 1908). For bibliographies of the various branches of geometry the reader should consult the articles referred to in this article. On certain special topics the following may be helpful: J. W. Young, The Fundamental Concepts of Algebra and Geometry (New York, 1911), for brief discussions of non-Euclidean geometry, construction problems, foundations of geometry. n-dimensional space, etc. This book presupposes no mathematical preparation beyond elementary algebra and geometry. Somewhat more mathematics is presupposed in Monographs on Topics of Modern Mathematics, edit. by J. W. A. Young (New York, 19"). It contains chapters on The Foundations of Geometry by O. Veblen; Modern Pure Geometry by T. F. Holgate ; Non-euclidean Geometry, by F. S. Woods; Constructions With Ruler and Compasses, by L. E. Dickson. For imaginary elements in geometry see J. L. Coolidge, Geometry in the Complex Domain (Oxford, 1924). (J. W. Y.)