The economic arid political changes of the last centuries of the secon 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 (287212 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), Menelaus (c. 100) and Claudius Ptolemy (c. 85 c. 165). In ancient Greek geometry both in its form and its content, we find the fountainhead 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 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 mathematics. 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 mathematical methodology.

It was not until the latter part of the eleventh century that Greek classics in science and mathematics 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. Mathematical 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 of trade, navigation, astronomy, and surveying.

Formal Axiomatics

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 mathematical authority, the work firmly implants the postulation method of formal axiomatics not only in the field of geometry-bat also in nearly every branch of mathematics 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

1. to accomplish 14. to emerge

2. to approach 15. to furnish

3. to assert 16. to inaugurate

4. to assume 17. to infer

5. to conceive 18. to proceed

6. to concern 19. to propose

7. to confide 20. to realize

8. to confine 21. to substitute

9. to contradict 22. to survey

10. to convert 23. to survive

11. to deserve 24. to trace

12. to display 25. to unify

13. to doubt 26. to yield

3.1.8. Non-Euclidian Geometry

The man who deserves the honour for the creation of non-Euclidean geometry is the distinguished Russian mathematician N. Lobachevsky.Lobachevsky challenged the parallel axiom and substituted another: "Through a point outside a line L there are an infinite number of lines parallel to L". He built a new geometry on the basis of a parallel axiom contradicting Euclid's, it is a logically consistent geometry, one in which there are not contradictions. The most unbelievable theorems to which he was led did not discourage him and he came to the conclusion: "There are geometries different from Euclid's and equally valid".

Lobachevsky succeeded in creating a new geometry with many surprising theorems. The most unexpected is the theorem that the sum of the angles of any triangle is always less than 180°. Moreover of two triangles the one with a larger area has a smaller angle sum, i.e., two similar triangles must be also congruent. As a final example is the following: the distance between two parallel lines approaches zero in one direction along the lines and becomes infinite in the other direction.

Which then is correct geometry? Which is the correct theory of the universe? Which is the most convenient theory? Which fits the observed data best and involves the least computation and the simplest mathematics? We may think that the new geometry cannot be applied to the physical world because, for example, it asserts that similar triangles must be congruent. The surprising revelation that emerged from all the attempts to decide which of the two geometries fits physical space is that bothfit equally well.

The creation of non-Euclidean geometry brought into clear light a distinction between mathematical and physical spaces. The axioms of Euclidean geometry are true of physical space. With the creation of non-Euclidean geometry mathematicians appreciated the fact that systems of thought based on statements about physical space might be different from that physical space.

If both Euclidean and non-Euclidean geometry can represent physical space equally well, which one is the truth about space arid figures in space? One cannot say. In fact, the choice may not be limited to just these two. Geometry is not the truth about physical space but the study of possible spaces. Several of these mathematically constructed spaces, differing sharply from one another, can fit physical space equally well as far as experience can decide.

We must give due credit also to other mathematicians who contributed much to the creation of non-Euclidean geometry. J.Bolyai,aHungarian mathematician worked out the notion of a non-Euclidean geometry simultaneously with Lobachevsky, but independently. Since Lobachevsky's publications precede Bolyai's it is customary to name Lobachevsky as the discoverer of the concept of non-Euclidean geometry.

K. F. Gauss, the great mathematical giant of the ninetieth century, discovered the same results as Lobachevsky and Bolyai before either and lacked the courage to publish facts so startling. After carefully considering the parallel axiom, Gauss gave a criterion for determining the truth of Euclidean geometry: measuring the angles of a triangle must decide which geometry fits the physical world in the particular case.

These radical departures from Euclid followed by Riemann's geometry with many striking theorems. The German mathematician Riemann(18261866) postulated no parallels.In other words, he substituted for Euclid's parallel postulate the assumption that: 'Through a point P outside a line L there is no line parallel to it; that is, every pair of lines in a plane must intersect". In his geometry all the perpendiculars to a straight line meet in a point, the sum of the angles of any triangle is greater than 180°, etc.

To test the "truth" of all these theorems revealed in Gauss's criterion, mathematicians once tried to measure a huge triangle with vertices on three peaks in Germany. But all experiments failed to bring about a decisive conclusion. The sum of the angles found was always so close to 180° that the excess or deficit in each case could be made by the unavoidable imperfection in the measuring techniques. Even if the three theories fit experimental facts equally well, they are not equal in convenience of computation. For ordinary everyday purposes, the Euclidean system is the simplest and hence we use it not because of the "absolute" and only truth, but because it makes our work easier. Riemann's system is the simplest for use in Einstein theory.

When the term non-Euclidean geometry is used in mathematical literature, the geometries of Lobachevsky and Riemann are always meant, although the term may well be applied generically to any geometry that denies one or more axioms of Euclid. After the days of Lobachevsky and Riemann it became the fashion to challenge axioms. To correct the defects in Euclid's "Elements" many axiom (postulate) systems were suggested and developed. Among these systems are those of Pasch(1882), Peano(1889), Veblen(1904), Hilbert(1909) and Birkhoffand Beatley(1940). Each is different; some have certain advantages over the others. Hilbert's system, perhaps because he is known as one of the outstanding mathematicians of the twentieth century, had the most profound effect. Perhaps too, this is because his system, as compared to the others, is most similar to Euclid's. Whatever the reasons, Hilbert's system was so widely used, revised, and refined over the years that many variations of his system changes in statements and phraseology are now in existence.

Euclidean geometry is only one applied science furnishing an interpretation of Hilbert's pure science; spherical geometry is another. There are an infinite number of others besides some vital, many trivial, but-all possible interpretations. It took unusual imagination to entertain the possibility of geometry different from Euclid's, for the human mind was for two millennia bound by the prejudice of tradition to the firm belief that Euclid's system was most certainly the only way geometrically to describe physical space, and that any contrary geometric system simply could not be consistent. One may ask today whether a geometry is based on a set of consistent postulates, whether these postulates are independent of one another, or whether this geometry serves better than another geometry for a given application. But the question of whether a geometry is "true" has no place in pure science.

3.1.9. Euclid's Elements

When most people describe the Greeks' contribution to modern civilization they talk in terms of art, literature and philosophy. No doubt the Greeks deserve the highest praise in all these fields. Nevertheless, the contribution of the Greeks that determines most the character of present-day civilization was their mathematics!

In a relatively brief period (from about 600 till 300 B. C.) great intellects such as Thales, Pythagoras, Euclid, Eudoxus,Archimedes and Apolloniuscreated an amazing amount of first-class mathematics. It is disappointing that unlike the situation with the ancient Egyptian and Babylonian geometry, there exist virtually no primary sources for the study of very early Greek geometry. We are forced to rely on manuscripts and accounts that are dated several hundred years after the birth of the original treatment.

Our principal source of information concerning very early Greek geometry is the so-called "Eudemian Summary" of Proclus.This summary constitutes several pages of Proclus's "Commentary on Euclid, Book I" and is a brief outline of the development of Greek geometry from the earliest times up to Euclid. Although Proclus lived in the fifth century A. D., a good thousand years after the inception of Greek geometry he still had access to a number of historical and critical works that are now lost to us except for the fragments and allusions preserved by him and others.

Although much of the information on plane geometry was known to the Babylonians of earlier times, thedeductive aspectofgeometrywas introduced for the first time by the Pythagoreans.Chains of propositions in which successive propositions are derived from earlier ones began to emerge in the works of Thales. As the chains lengthen and some are tied to others, the bold idea of developing all of geometry in one long chain suggests itself.

During the first three hundred years of Greek mathematics there developed the Greek notion of a logical discourseas a sequence of statements obtained by deductive reasoning from the accepted set of initial statements. Now both the initial and the derived statements of the discourse were statements about the technical matter of the discourse and hence involved special or technical terms. The meanings of these terms must be clear to the reader, and so for the Greeks the discourse must start with a list of explanations and definitions of these technical terms. After these explanations and definitions the initial statements, called "axioms" or "postulates" of the discourse, were to be listed. These initial statements, according to the Greeks, should be so carefully chosen that their truths were quite acceptable to the reader in the light of the explanations and definitions already cited. Certainly, the most outstanding contribution of the early Greeks to mathematics was the formulation of the mathematicalmethod (400 B. C) for this method is the very core of modern mathematics. Unfortunately, we do not know with whom the mathematical method originated personally, but H evolved with the Pythagoreans as a natural outgrowth and refinement of the early application of deductive procedures to mathematics.

It is claimed in Proclus's "Summary" that a Pythagorean, Hippocrates of Chios, attempted, with at least partial success, a logical presensation of geometry in the form of a single chain of propositions based on a few initial definitions and assumptions. There followed other writers' attempts and then, about 300 B. C., Euclid produced his epoch-making effort, the "Elements", a single deductive chain of 465 propositions neatly and beautifully comprising plane and solid geometry, number theory, and Greek geometrical algebra.

From its very first appearance this work was accorded the highest respect, and it so quickly and so completely superseded all previous efforts of the same nature that now no trace remained of the earlier systems. Euclid unified the work of many schools and isolated individuals in this most famous textbook on geometry. Euclid deduced all the most important results of the Greek masters of the classical period and therefore the "Elements" constituted the mathematical history of the age as well as the logical presentation of Geometry. The effect of this single book en the future development of geometry was enormous and is difficult to overstate.

The plan of Euclid's "Elements" is as follows. It begins with a list of definitionsof such notions as point and line; for example, a line is defined as "length without breadth". Next appear various statements some of which are labeled axiomsand others postulates.It appears that the axioms are intended to be principles of reasoning which are valid in any science (for example, one axiom asserts that "equals to the same thing are equal to each other"), while the postulates are intended to be assertions about the subject matter, that is geometry (for example, one postulate asserts that "it is possible to draw a line joining any two distinct points").

From a modern viewpoint it may be said that Euclid treats point and line essentially as primitive or undefined notions, subject only to the restrictions stated in the postulates, and that his definitions of these notions offer merely an intuitive description which helps one in thinking about formal properties of points and lines. Concerning the postulates, he probably believed that they were true statements on the basis of the meaning suggested by his definitions of the terms involved and the proofs acquired status of "self-evident truths".

The axioms chosen by Euclid state properties of points, lines and other geometric figures that are possessed by their physical counterparts. The properties in question are so obviously true of these physical objects that all mathematicians agreed on them as a basis for further reasoning. In the selection of axioms Euclid displayed great insight and judgement. Euclid chose a very limited number of axioms, twelvein ail (later generations reduced this number to ten),and constructed the whole system of geometry.

From this starting point of definitions, axioms and postulates, Euclid proceeds to derive propositions (theorems)and at appropriate places to introduce further definitions (for example, an obtuse angle is defined as an angle which is greater than a right angle). His methodof proof is strictly deductive that is, his theorems are proved by several deductive arguments, each employs unquestionable premises and yields an unquestionable conclusion.

A discourse developed according to the above plan is referred to as "material axiomiatics".Certainly, the most outstanding contribution of the early Greeks to mathematics was the formulation of the pattern of material axiomatics and the insistence that geometry should be systematized according to this pattern. Euclid's Elements is the earliest extensively developed example of this use of the pattern available to us. In recent years, this pattern was significantly generalized to yield a more abstract form of discourse known as "formal axiomatics".

The creation of Euclidean geometry is more than the contribution of numerous useful theorems. It reveals the power of reason. No other human creation demonstrates how much knowledge can be derived by reasoning alone as have the hundreds of proofs in Euclid's "Elements". The necessity for accurate and exact definitions, for clearly stated assumptions and for rigorous proof became evident in Euclid's "Elements".

We know much of the material of Euclid's "Elements" through our high-school studies. By studying Euclid hundreds of generations from Greek times learned how to reason, how perfect logical reasoning must proceed, how to master the procedure, how to distinguish exact reasoning from vague pretence of proof. Even nowadays this masterpiece of Euclid serves as a logical exercise and as a model of reasoning and the art of the mind.

3.1.10. A Modern View of Geometry

For a long time geometry was intimately tied to physical space, actually beginning as a gradual accumulation of subconscious notions about physical space and about forms, content and spatial relations of specific objects in that space. We call this very early geometry "subconscious geometry". Later human intelligence evolved to the point where it became possible to consolidate some of the early geometrical notions into a collection of somewhat general laws or rules. We call this laboratory phase in the development of geometry "scientific geometry". About 600 B. C. the Greeks began to inject deduction into geometry giving rise to what we call "demonstrative geometry".

In time demonstrative geometry becomes a material-axiomatic study of idealized physical space and of the shapes, sizes, and relations of idealized physical objects in that space. The Greeks had only one space and one geometry; these were absolute concepts. The space was not thought of as a collection of points but rather as a realm or locus, in which objects could be freely moved about and compared with one another. From this point of view, the basic relation in geometry was that of congruence or superposability.

With the elaboration of analytic geometry in the first half of the seventeenth century, space came to be regarded as a collection of points; and with the invention about two hundred years later of the classical non-Euclidean geometries, mathematicians accepted the situation that there is more than one conceivable space and hence more than one geometry. But space was still regarded as a locus in which figures could be compared with one another. The central idea was to consider a group of congruent transformations of a space into itself and geometry came to be regarded as the study of those properties of configurations of points which remain unchanged when the enclosing space is subjected to these transformations, and geometry is defined as the invariant theory of a transformation group. Geometry came to be rather far removed from its former intimate connection with physical space, and it became a relatively simple matter to invent new and even bizarre geometries.

At the end of the last century Hilbert and others formulated the concept of formal axiomaticsand there developed the idea of the branch of mathematics as an abstract body of theorems deduced from a set of postulates. Each geometry became, from this point of view, a particular branch of mathematics. Postulate sets for a large variety of geometries were studied.

In the twentieth century the study of abstract spaces was inaugurated and some very general studies came into being. A space became merely a set of objects together with a set of relations in which the objects are involved, and geometry became the theory of such a space. It must be confessed that this latter notion of geometry is so embracive that the boundary lines between geometry and other areas of mathematics became very blurred, if not entirely obliterated. It is essentially only the terminology and the mode of thinking involved that makes the subject geometric.

There are many areas of mathematics where the introduction of geometrical terminology and procedure greatly simplifies both the understanding and the presentation of some concept or development. This becomes increasingly evident in so much of mathematics that some mathematicians of the second half of the twentieth century feel that perhaps the best way to describe geometry today is not as some separate and prescribed body of knowledge but as a point of view a particular way of looking at a subject. Not only is the language of geometry often much simpler and more elegant that the language of algebra and analysis, but it is at times possible to carry through rigorous trains of reasoning in geometrical terms without translating them into algebra or analysis. There results a considerable economy both in thought and in communication of thought. Moreover, and perhaps most important, the suggested geometrical imagery frequently leads to further results and studies, thus furnishing us with a powerful tool of inductive or creative reasoning. A great deal of modern analysis becomes singularly compact and unified through the employment of geometrical language and imagery.