To accomplish 10. To convert 19. To propose

THE HISTORY OF GEOMETRY

The story of the history of geometry, like that of many other growing and changing subjects, is composed of two intertwined strands. One strand nar­rates the growing content of the subject and the other the changing nature of the subject. The following is a brief outline of the birth and the development of geometry.

Subconscious Geometry

The first geometrical considerations of man are unquestionably very ancient. They had their origin in simple observation stemming from human ability to recognize physical form and to compare shapes and sizes. The notion of distance was undoubtedly one of the first geometric concepts developed Many observations in the daily life of early man led to the notion of simple geometric concepts such as rectangles, squares, triangles, curves, surfaces and solids. Such geometry may, for want of a better name, be called '‘subconscious geometry'’.

Scientific Geometry

In the beginning, “man considered only concrete geometrical problems, which present themselves individually and with no observed interconnec­tions. Later (but still before the dawn of recorded history), human intelli­gence evolved to the point where it was able to extract from a number of observations certain general properties and relationships. This introduced the advantage of ordering practical geometrical problems into sets such that the problems in a set can be solved by the same general procedure. One thus arrives at the notion of a geometrical law or rule. This higher stage of geom­etry may be called “scientific geometry”, in view of the fact that induction, trial and error, and empirical procedures were the tools of discovery. Geometry became a collection of general rule-of-thumb and laboratory results, concerning areas, volumes, and relationships of various figures sug­gested by physical objects. No evidence permits us to estimate the number of centuries that passed before man was able to raise geometry to the status of a science, but the writers of antiquity unanimously agree upon the Nile Valley of ancient Egypt and Babylonia as the place where subconscious geometry first became scientific geometry. Geometry remained of this type until the great Greek period of antiquity.

Demonstrative Geometry (Early Greek Geometry)

The economic and political changes of the last centuries of the second millennium B.C. caused the power of Egypt and Babylonia to wane. New peoples came to the fore, and it happened that the further development of geometry passed over to the Greeks, who transformed the subject into something vastly different from the set of empirical conclusions worked out by their predecessors. The Greeks insisted that geometric fact must be established not by empirical procedures, but by deductive reasoning; geo­metrical truth was to be attained in the classroom rather than in laborato­ry. In short, the Greeks transformed the empirical or scientific geometry of the ancient Egyptians and Babylonians into what we may call “systematic” or “demonstrative” geometry. Greek geometry started in an essential way with the work of Thales of Miletus in the first half of the sixth century B.C. This versatile genius, one of the “seven wise men” of antiquity was a wor­thy founder of demonstrative geometry. He is the first known individual with whom the use of deductive methods in geometry is associated. He is credited with a number of very elementary geometrical results the value of which is not to be measured by their content but rather by the belief that he supplied them with a certain amount of logical reasoning instead of intuition and experiment. The next outstanding Greek geometer is Pythagoras who continued the systematization of geometry begun some fifty years earlier by Thales.

Later Greek Geometry

The three most outstanding Greek geometers of antiquity are Euclid (c. 300 B.C.). Archimedes (287-212 B.C.) and Apollonius (c. 225 B.C.) and it is no exaggeration to say that almost every subsequent significant geometrical development, right up to and including the present time, finds its seeds of origin in some work of these three great scholars.

With the passing of Apollonius the golden age of Greek geometry came to an end The geometers who followed did little more than fill in details and perhaps independently develop certain theories the germs of which were already contained in the works of the three great predecessors. Among these later geometers special mention should be made of Heron (or Hero) of Alexandria (c. A.D. 75), Menetaus (c. 100) and Claudius Ptolemy (c. 85-c. 165). In ancient Greek geometry both in its form and its content, we find the fountamhead of the subject.

Middle Ages

The closing period of ancient times comes when in 146 B.C. Greece became a province of the Roman Empire and a gradual decline in creative thinking set in. The period starting with the fall of the Roman Empire in the middle of the fifth century and extending into the eleventh century is known as Europe’s Dark Ages, for during this period civilization in western Europe reached a very low ebb. Schooling became almost nonexistent, Greek learn­ing all but disappeared, and many of the arts and crafts were forgotten. During this period of learning, the peoples of the East, especially the Hindus and the Arabs, became the major custodians of maths. Although the Hindus excelled in computation, contributed to the devices of algebra, and played an important role in the development of our present positional numeral system, they produced almost nothing of importance in geometry or in basic math methodology.

It was not until the latter part of the eleventh century that Greek classics in science and maths began once again to filter into Europe. The fifteenth century, the early period of the Renaissance, witnessed the rebirth of art and learning in Europe. Many Greek classics, known up to that time only through Arabic translations, often quite inadequate, could now be studied from original sources. Math activity in this century was largely centered in the Italian cities and in the central European cities of Nuremberg, Vienna and Prague. It concentrated on arithmetic, algebra, and trigonometry, under the practical influence ot trade, navigation, astronomy, and surveying.

Projective GeometryIn an effort to produce more realistic pictures, many of the Renaissance artists and architects became deeply interested in discovering the formal laws controlling the constructions of objects on a screen, and as early as the fif­teenth century a number of these men created the elements of an underlying geometrical theory of perspective... Some aspects of this subject which con­cerns a way of representing and analyzing three-dimensional objects by means of their projections on certain planes had their origin in the design of fortifications.

Analytic Geometry Projective geometry was overshadowed by the more supple analytic geom­etry introduced by Rene Descartes and Pierre de Fermat. There is a funda­mental distinction between the two studies, for the former is a branch of geometry whereas the latter is a method of geometry. Analytic geometry is often described as the “royal road” in geometry that Euclid thought did not

Differential GeometryMany new and extensive fields of math investigation were opened up in the seventeenth century, making that era an outstandingly productive one in the development of maths. Unquestionably, the most remarkable math achieve­ment of the period was the invention of the calculus by Isaac Newton and Gottfried Wilhelm von Leibnitz. A fair share of its remarkable applicability lies in the field of geometry and there is an exceedingly vast body of geometry wherein one studies properties of curves and surfaces, and their generaliza­tions, by means of the calculus. This body of geometry is known as “differ­ential geometry”. For the most part, differential geometry investigates curves and surfaces only in the immediate neighbourhood of any one of their points. This aspect of differential geometry is known as “local differential geometry” or “differential geometry in the small”. However, sometimes properties of the total structure of a geometric figure are implied by certain local properties of the figure that hold at every point of the figure. This leads to what is known as “integral geometry” or “global differential geometry”, or “differential geometry in the large”. It is probably quite correct to say that differential geometry, at least in its modern dress, started in the early part of the eighteenth century with the interapplications of the calculus and analyt­ic geometry. Karl Friedrich Gauss (1777-1855) introduced the fruitful method of studying the differential geometry of curves and surfaces by means of parametric representation of these objects. Bernhard Riemann introduced an improved notation and a procedure independent of any par­ticular coordinate system employed. The tensor calculus was accordingly devised and developed. Here we find an assertion of the tendency of maths in recent times to strive for the greatest possible generalization.

Generalized differential geometries, known as Riemannian geometries were explored intensively, and this in turn led to non-Riemannian, and other, geometries. Much of this material finds significant application in relativity theory and other parts of modern physics.

Non-Euclidean GeometryThere is evidence that a logical development of the theory of parallels gave •he early Greeks a lot of trouble. Euclid met the difficulties by defining par- •Mel lines as coplanar straight lines that do not meet one another however far they nuy be produced in cither di rev turn, and by uditpt itiy m *n imtwi assun^tion his now famous parallel postulate:’‘If a straight line inier^u two straight lines so asto make the interior angles on oneside of it togethc i(ess than two right angles, thetwo straight lines will intersect,if indefinitely produced, on the side on which are the angles whtch aretogether i?** two right angles’*. Actually, the postulate is the converse of Proportion } 7 of Euclid’s Book IIand it seemed more like a proposition than » postulate Itwas natural to ask if the postulate was really needed at all. or perhaps it couldbe derived as a theorem, or, at least, it could be replaced by si more accept able equivalent. The attempts to devise substitutes and to derive it as a thenrem from the rest of Euclid’s postulates occupied geometers for over two thousand years and culminated in the most far-reaching developmentof modern maths - non - Euclidean geometry.

Topology started as a branch of geometry, but during the second quarter of the twentieth centur. it underwent such generalization and became involved with so many other branches of maths that it is now more properly consid­ered, along with geometry, algebra, and analysis, a fundamental division of maths. Today topology may roughly be defined as the math study of contin­uity, though it still reflects its geometric origin. Topology is the study of those properties of geometric figures which remain invariant under so-called topo­logical transformations, that is. under single-valued continuous mapping possessing single-valued continuous inverses.

The Erlanger Program

In the middle of the nineteenth century a number of different geometries came into existence, and the time was ripe for some sort of codification, syn­thesis and classification to give a sense of order to these geometries. Such a scheme was announcedin 1871 by Felix Klein, in his inaugural address upon appointment to the Philosophical Faculty and the Senate of the University of Erlanger This address, based on work he and Sophus Lie did in group the ory, set forth aremarkable definition of “a geometry”, one that served to codify essentially all the existing geometries of the time and pointed the way to new fields of research in geometry. This address with the program of geo metrical study advocated by it is known as Erlanger Program. Somewhat oversimply stated, the Erlanger Program claims that geometry is the investi­gation of those properties of figures which remain unchanged when the fig­ures are subjected to a group of transformations. It advocates the classifica­tion of existing geometries and the creation and study of new geometries, according to this scheme. In particular one should study the geometries characterized by the various proper subgroups of the transformation group of a given geometry, thereby obtaining geometries that embrace others.

For plane Euclidean metric geometry, the group of transformations is the set of all rotations and translations in the plane; for plane projective geome­try, the group of transformations is the set of all so-called planar projective transformations; for topology, the group of transformations is the set of all topological transformations. Each geometry has its underlying controlling transformation group. In building up a geometry, then, one is at liberty to choose, first of all, the fundamental elements (point, line, etc.), next the man­ifold of these elements (plane of points, ordinary space of points, spherical surface of points, plane of lines, pencil of circles, etc.), and, finally, the group of transformations to which the manifold of elements is to be subjected.

Abstract Spaces

In the twentieth century the study of “abstract spaces” was inaugurated and some very general geometries came into being. A “space” became merely a set of objects, for convenience called “points” together with a set of relations in which these points are involved, and a geometry becomes simply the the­ory of such a space. The set of relations to which the points are subjected is called the “structure” of the space, and this structure may or may not be explainable in terms of the invariant theory of a transformation group. Through set theory geometry received a further generalization or metamor­phosis. These new geometries find invaluable application in the modem development of analysis. Important among abstract spaces are the so-called metric spaces, Hausdorf spaces, topological spaces, Hilbert’s spaces, and vector spaces.

Hilbert's Formal Axiomatics

The discovery by Lobachevsky, Bolyai and Gauss of a self-consistent geometry different from the geometry of Euclid liberated geometry from its traditional mold. A deep-rooted and centuries-old conviction that there can be only one possible geometry is shattered and the way is opened for the cre­ation of many different systems of geometry. With the possibility of creating such purely “artificial” geometries, it becomes apparent that geometry is not necessarily tied to actual physical space. The postulates of geometry become, for the mathematician, mere hypotheses whose physical truth or falsity need not concern him. The mathematician may take his postulates a suit he pleas­es, provided they are consistent with one another. Whereas it is customary, in Euclidean geometry, to think of the objects that represent the primitive terms of ail axiomatic discourse as being known prior to the postulates, now the postulates become regarded as prior to the specification of primitive terms. This new point of the axiomatic method is known as “formal axiomatics” in contrast to the earlier “material axiomatics”. In a formal axiomatic treatment the primitive terms have no meaning whatever except that implied by the potu- lates, and the postulates have nothing to do with “self-evidence” or “truth” - they are merely assumed statements about the undefined primitive terms.

Many mathematicians now regard any discourse conducted by formal axiomatics as a “branch of pure maths’*. If for the primitive terms in such a postulation discourse we substitute terms of definite meaning which convert the postulates into true statements about those terms, then we have an “inter­pretation” of the postulate system. Such an interpretation may also, if the reasoning is valid, convert the derived statements of the discourse into true statements. Such an evaluation of a branch of pure maths is called a “branch of applied maths”. Clearly, a given branch of pure maths may possess many interpretations and may thus lead to many branches of applied maths. From this point of view, we see that material axiomatics is the independent axiomat­ic development of some branch of applied axiomatics. In a formal axiomatic treatment one strips the discourse of all concrete content and goes to the abstract development that lies behind any specific application.

Formal axiomatics was first systematically developed by David Hilbert in his famous book The Foundations of Geometry in 1899. This little work, which ran through nine editions, is today a classic in its field. Next to Euclid’s Elements it may be regarded as perhaps the most influential work so far writ­ten in the field of geometry. Backed by the author’s great math authority, the work firmly implants the postulation method of formal axiomatics not only in the field of geometry but also in nearly every branch of maths of the twen­tieth century. The book offers a completely acceptable postulate set for Euclidean geometry, and it can be read by any intelligent person.

ACTIVE VOCABULARY

to accomplish 10. to convert 19. to propose

2.to approach 11. to deserve 20. to realize

3.to assert 12. to display2J. to substitute

4.to assume 13.to doubt 22. to survey

5.to conceive 14.to emerge 23. to survive

6.to concern 15. to famish 24. to trace

7.to confide 16.to inaugurate 25. to unify

8.to confine 17.to infer 26. to yield