HISTORY OF GEOMETRY The history of geometry may be considered under separate heads as follows : Ancient empirical geometry, ancient demonstra tive geometry, sporadic developments during mediaeval times and the Renaissance, analytical geometry, modern synthetic geometry and, finally, the foundations of geometry.
Early Babylonian records disclose the use in auguries of triangles, quadrangles and parallel lines; drawings of Babylonian carriages indicate the division of a circumference into four and six equal parts. For angular measure ment the circumference was divided into 36o equal parts or de grees, each degree into 6o minutes, and each minute into 6o seconds. Like the Hebrews (I. Kings vii. 23), the Babylonians took 7r
One tablet reveals the approximate computation of the diagonal of a rectangle, expressed in the sexagesimal notation. More attention to geometry was given in Egypt. Greek writers state that land surveying was practised in Egypt because frequent overflows of the Nile obliterated landmarks. More reliable in formation is obtained from papyri and from inscriptions on the walls of temples. The Ahmes papyrus written about 155o B.C. or, according to others, about 170o B.C., is the most important ancient mathematical manuscript known. In this more stress is laid upon arithmetic and algebra than upon geometry. Ahmes calculates the areas of squares and rectangles; he approximates the area of an isosceles triangle by multiplying the slanting side by half the base, and of an isosceles trapezoid by multiplying half the sum of the two bases by the slant height. Remarkable is the rule for finding the area of a circle : From the diameter subtract one-ninth of it, and square the remainder. This rule implies that 71- nearly equals 3.16. Examples on the measurement of pyramids in Ahmes are of doubtful interpretation, but in another nearly contemporaneous papyrus (see Ancient Egypt, 1917), a calculation is given yielding the exact volume of the frustum of a rectangular pyramid—an astounding achievement in early Egyptian times.
The Greek geometry of Thales and his school, about the 7th century B.e., was only just emerging from the empirical stage; it dealt with such cases as the equality of vertical angles, the equal ity of the base angles of an isosceles triangle and the bisection of a circle by a diameter. This was a geometry of lines; Egyptian geometry was mainly one of areas and volumes. Thales's measure ments of the heights of the pyramids by their shadows and of the distance of ships at sea presupposes the use of similar triangles.
The transition from empirical to rigorous demonstrative geometry was necessarily very slow. The discovery by the Pythagoreans in Italy of the existence of incommensurable magnitudes (such as the side and diagonal of a square) marks a long step toward reasoned conclusions. The story of the slaughter of a hecatomb of oxen in celebration of the discovery of a proof of the "Pythagorean theorem" of the right triangle is probably a myth, but it indicates a high appreciation of an intellectual achievement in pure geometry. Another indication of progress was the proposing, by the Sophists of Athens, of the three famous problems of construction—the squaring of the circle, the trisection of any given angle, and the duplication of a cube. These are now known to be impossible under the restrictions im posed by the Greeks—the use of a pair of compasses and an un graduated or unmarked ruler in a finite number of steps of con struction. The Greeks themselves found solutions of these prob lems when the above restrictions were abandoned. Hippias of Elis invented a curve, the "quadratrix," by which angles could be tri sected and the circle could be squared. But the drawing of the quadratrix involved theoretically an infinite number of steps. Hip pocrates of Chios, evidently hoping eventually to achieve the quadrature of the circle, successfully "squared" certain Tunes and thus furnished the earliest example of a curvilinear area for which under the Greek restrictions an exactly equal area could be con structed in a plane which was bounded by straight lines. The duplication of the cube which, algebraically expressed, means con structing X= 2S, where S is the side of a given cube, was solved "mechanically" by the aid of the conchoid of Nicomedes and by other means. Progress in giving definitions of fundamental concepts like "point," "line," "surface" and in giving explicit ex pression to axioms, was made by Plato and his pupils. The theory of proportion as related to magnitudes was developed by Eudoxus and Theaetetus. Eudoxus is credited also with the "method of ex haustion" which is not the same as the modern theory of limits, even though it involves the concept of a variable and a constant. The procedure as found in Euclid's Elements, bk. xii., Prop. 2, involves a part of our modern process of showing that a constant is the limit of a variable, but the Greeks did not actually pass, as we do, from the variable to its limit, but resorted, instead, to a process of reductio ad absurdum.
A most interesting phase in the development of the Greek phil osophy of mathematics is seen in Zeno of Elea's arguments on motion. As explained in Aristotle's Physics, Zeno tried to prove that motion is impossible. Swift-footed Achilles could not catch a tortoise, the arrow in its flight is at any moment at rest, etc. For centuries Zeno was branded as a paralogist, but such recent writers as Paul Tannery and Bertrand Russell (Principles of Mathemat ics, 1903) advance the view that Zeno was misunderstood and that his arguments were sound and involved profound questions which have been successfully resolved only by the theory of the con tinuum as developed in recent mathematics. The paradoxes of Zeno, as well as Antiphon's attempt to square the circle by in scribing a series of regular polygons of an increasingly greater number of sides, the ultimate polygon coinciding with the circle, convinced Greek mathematicians that a clear and logical science of geometry could not be attained, except by eliminating the seemingly mystic concepts of infinity and of fixed infinitesimals. And thus we find Euclid excluding the infinitely little from his Elements by a definition (bk. v., def. 4) : "Magnitudes are said to have a ratio to one another, when the less can be multiplied so as to exceed the other"; if the less were infinitely small, any finite multiple of it would still be infinitely small and could not satisfy this definition. Likewise, Archimedes, in the preface to his Quad rature of the Parabola, gives a postulate which he attributes to Eudoxus: If a and b are magnitudes, such that a < b, it shall be possible to find a finite number n, such that na> b. Thus in the classical writings of Euclid, Archimedes, and Apollonius of Perga, the infinitely small is carefully avoided.
Euclid was the author of several works, the most important of which is his Elements, written between 33o and 32o B.C. It con tained 13 books, of which the first six and the last three were de voted to geometry (plane and solid), the seventh, eighth and ninth to arithmetic, and the tenth to irrationals. The proof of the theorem of Pythagoras on the right triangle is the only part of the geometry which Greek commentators definitely ascribe to Euclid. The geometrical theorems and the methods of proof (the method of exhaustion and the definition of proportion of magnitudes) ap pear to be due to earlier investigators. The great achievement of Euclid was the arrangement of the material handed down to him into a coherent, logical system. He ranks as a great systematizer. It is one of the marvels in the history of mathematics that the Elements, written in the 4th century B.c., should have established and maintained itself as a text-book in geometry for over 2,000 years. In England it was the authorized text down to the opening of the present century.
Euclid eschewed all practical applications of geometry. This attitude was abandoned by Archimedes who found the relation, 34>7r >3.41, needed in computing the area of a circle, and who discovered theorems on the areas of the surface of a sphere and of a cylinder, as well as on the volumes of a sphere and cylinder.
Besides the development of ordinary elementary geometry, the Greeks must be credited with the study of the conic sections (the ellipse, parabola, and hyperbola). The beginnings were made in the time of Plato ; the culmination was reached in the work called the Conic Sections, written by Apollonius of Perga.
With Euclid, Archimedes and Apollonius, geometry reached the highest development during ancient times. Later Greek writers discovered certain curves (the conchoid of Nicomedes and the cissoid of Diocles) ; Pappus reached certain theorems and view points which were more fully developed in modern times. But these were the afterglow following the sunset of Greek geometry. Mediaeval Times and the Renaissance.—The Hindus of this period did not excel in geometry. Their creative work is limited to theorems on the area and diagonals of a quadrilateral inscribed in a circle, theorems developed mainly by Brahmagupta (c. 628). The outstanding Arabic achievement was the geometric solution of cubic equations, by the method of intersecting conics, a process which had been foreshadowed by Archimedes. The fullest Arabic exposition of this topic was given by the poet Omar Khayyam. In Europe creative work began with Johann Kepler, who made use of the concepts of infinitely small and infinitely great quantities, which Euclid and Archimedes had carefully avoided in their classical writings. Kepler looked upon a circle as a polygon having an infinite number of sides, and upon a sphere as consisting of an infinite number of pyramids. He arrived at the areas and volumes of figures generated by curves revolving about a line as axis. The Italian, Bonaventura Cavalieri, a pupil of Galileo, developed the Geometry of Indivisibles, and succeeded in solving many of the problems on volumes which had been proposed by Kepler for solution. Researches which foreshadowed the great achievements of a more modern period are found in the work of Evangelista Torricelli, Vincenzo Viviani, Gilles P. de Roberval, and especially of Gerard Desargues and Blaise Pascal on modern synthetic geometry.
Analytic geometry was created by two Frenchmen, Rene Descartes and Pierre de Fermat. The chief credit is rightly awarded to Descartes, who promptly published his results in his La geometrie, 1637; Fermat's treatise Ad locos pianos et solidos isagoge appeared posthumously in 1679. The two main ideas involved in analytical geometry are the location of points in a figure by the use of co-ordinates and the algebraic representation of a curve or surface by an equation involving two or three variables. Of these, only the latter was new in the I7th century; co-ordinate representation was practised in ancient times by Apollonius and others. Descartes's La geometrie does not contain a systematic development of analytical geometry in the manner found in modern texts. The method must be constructed from isolated statements occurring in different parts of the treatise. Nevertheless, it is a work of genius occupying a conspicuous place in the history of geometry. The words "abscissa" and "ordinate" were not due to Descartes. In the technical sense of analytical geometry they were first used by Leibniz in 1692, in the Acta Eruditorum. An important example solved by Descartes in his La geometrie was the "Problem of Pappus" : Given several straight lines in a plane find the locus of a point such that the perpendicular drawn from the point to the given lines, shall satisfy the condition that the product of certain of them shall be in a given ratio to the product of the others. This problem afforded an excellent example of the power of the analytical method, a power which Boltzmann more recently described by saying that the formula appears at times cleverer than the man who invented it.
The cultivators of analytical geometry in the 18th century were Jean Paul de Gua de Malves, Gabriel Cramer, Leonhard Euler and, in general, the mathematicians who developed the differential and integral calculus. Thus Newton in 1704 published a classifi cation of cubic curves. The calculus offered a general and expedi tious method of finding tangents at any point of a continuous curve having derivatives.
During the 19th century, new principles were introduced into analytic geometry which afforded greater power and generality to the science. Thus the principle of duality was applied by Julius Plucker to equations of lines and curves. The duality consisted in a double interpretation of one and the same equation so that ux + vy + I =0, for example, could be interpreted as having two variables x, y representing co-ordinates of points, u and v being constants, or the equation could be interpreted as having two variables u, v representing lines, x and y being constants. In the first case the equation represents a straight line, in the second case, a point. By this duality, one and the same process would yield two theorems. But for the full analytic application of duality, Augustus F. Mobius and Plucker found it necessary to abandon the ordinary Cartesian co-ordinates and to introduce the more general homogeneous co-ordinates. Plucker studied the singularities of plane curves and developed four equations (the "Plucker equations") expressing the relations between the number of double points, double tangents, stationary points, and stationary tangents of a curve of a given degree and class. The discovery of these relations Arthur Cayley considered "the most important one beyond all comparison in the entire subject of modern geometry." Etienne Bobillier and Plucker introduced an "abridged nota tion." J. J. Sylvester and Otto Hesse showed how processes of elimination could be simplified by the use of determinants. Particularly prominent in elaborating the higher fields of the science were Alfred Clebsch, Henri Halphen, and Jean Gaston Darboux. Curves and surfaces of higher order afford fields of never-ending research. (See CURVES ; CURVES, SPECIAL.) Modern Synthetic Geometry.—This was cultivated in the 19th century simultaneously with analytic geometry. The two movements occupied the same field of study, but differed in method of exposition. Rivalry existed between the followers of the two methods, which was usually but not always friendly. The continual direct viewing of figures as existing in space adds exceptional charm to the study of synthetic geometry, but the equation of the analytic method may outrun thought itself and constitutes a powerful tool in research. Jean Victor Poncelet and others used both the synthetic and the analytic methods; Jakob Steiner used only the former, Plucker only the latter. Modern synthetic geometry was first cultivated by Gaspard Monge, L. N. M. Carnot, and J. V. Poncelet in France, and by Mobius and Steiner in Germany and Switzerland, and was developed to still higher perfection by Michel Chasles in France and von Staudt in Germany. Monge in 1795 was the first to stress the "descriptive geometry" used in engineering. The principle of j duality was advanced by J. D. Gergonne and Poncelet for the study of descriptive properties without reference to the analytic processes elaborated by Plucker. The use of the anharmonic or cross ratio was stressed by Steiner and Chasles.
The parallel postulate of Euclid (according to which two lines in a plane meet if the sum of the two interior angles on the same side of a transversal is less than two right angles) seemed unsatisfactory even to some of the ancient mathematicians. Proofs of it were attempted on the assumption of the other Euclidean axioms, but were always found invalid. After many failures, certain investigators tried to build up a geometry in which the postulate does not hold, in which, in other words, the angle sum in question may be less than two right angles and yet the lines may not meet, no matter how far they are produced. The result was a non-Euclidean geometry, perfectly consistent with its assumptions, developed independently by the Russian, Nicolai I. Lobachevski (1829) and the Hungarian mathematician, Janos Bolyai (1834). So novel were these creations that they failed to secure general attention for many years. With reference to the assumptions about parallel lines, there are now recognized three principal geometries—the "parabolic" or Euclidean based on Euclid's parallel postulate, the "hyperbolic" or Lobachevskian based on the denial of that postulate, and the "elliptic" or Riemannian geometry in which parallel lines do not exist at all. Felix Klein considered two forms of "elliptic" geometry. Two-dimensional geometric figures in the first of these geometries are visualized when drawn in a plane; those in the second geometry are partly visualized on a saddle shaped surface like the pseudosphere ; those in the third geometry are visualized on a sphere. Recently, these geometries have assumed importance in cosmological speculations. According to Albert Einstein, the universe is finite and its geometry is "elliptic." The question of the number of dimensions has agitated mathe maticians and philosophers since the time of the Greeks. The first to assume definitely the existence of a fourth dimension of space was the Platonist, Henry More, of Cambridge, England, a contemporary of Isaac Newton. But not till the 1 gth century did mathematicians enter upon an extensive study of geometries of higher dimensions. As a rule it was not claimed that these higher dimensions had real existence in our physical space; they were ideal creations of the human mind. However, as early as the 18th century, D'Alembert and Lagrange looked upon time, which appeared as a fourth variable in mechanics, as a fourth dimension. This idea was developed in more recent time by Hermann Minkowski and Einstein in a manner leading to a fourth dimen sional world, a "fusion of geometry and physics." Said Minkowski (1908) : "Nobody has ever noticed a place except at a time, or a time except at a place." The foundations of geometry are the last part of the geometric structure to be firmly established. Italian and German mathe maticians (Giuseppe Peano, 188o; Moritz Pasch, 1882; Mario Pieri, 1899) were the first to enter upon a minute study of independent, consistent and complete sets of axioms enabling the different geometries to be built up without borrowing anything from intuition. It was recognized that Euclid, who for centuries had been admired for the rigour of his demonstrations, does de pend here and there upon facts not deduced from the axioms but obtained from visual inspection of the figures. Thus, in Euclid's Elements, in the very first proposition, it is assumed without proof that two circles drawn in the figure intersect each other. The study of the foundations of different geometries was con tinued in Germany, by David Hilbert, in France by Henri Poin care and in the United States mainly by Oswald Veblen.
historical details consult the general histories of Bibliography.--For historical details consult the general histories of mathematics; also such special works as G. Loria, Die hauqtsdch lichst en Theorien der Geometrie (i 888) ; G. Loria, Ebene Kurven, Theorie and Geschichte (1911) ; F. Gomes Teixeira, Traite des courbes speciales remarquables (19o8) ; R. Bonola, Non-Euclidean Geometry (1912) ; E. Kotter, Entwickelung der synthetischen Geometrie (19oI).
(F. CA.)
