SOURCE: "Euclid's Elements" in An Introduction to the Foundations and Fundamental Concepts of Mathematics, Holt, Rinehart and Winston, 1958, pp. 30-57.

[In the following excerpt, Eves and Newsom review the formal nature and significance of Elements, arguing that the work offers the earliest extensive development of the axiomatic method, and that the impact of this form of analysis on the development of mathematics has been tremendous.]

The Importance and Formal Nature of Euclid's Elements

The earliest extensively developed example of the use of the axiomatic method that has come down to us is the very remarkable and historically important Elements of Euclid. The production of this treatise is generally regarded as the first great landmark in the history of mathematical thought and organization, and its subsequent influence on scientific thinking can hardly be overstated.

Of Euclid himself, however, disappointingly little is known. It is from Proclus' Commentary on Euclid, Book I, that we obtain our most satisfying information about Euclid. He writes,

Euclid, who put together the Elements, collected many of the theorems of Eudoxus. He perfected many of the theorems of Theaetetus, and also brought to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors. This man lived in the time of the first Ptolemy, for Archimedes, who came immediately after the first Ptolemy, makes mention of Euclid, and furthermore, it is said that Ptolemy once asked him if there was in geometry any shorter way than that of the Elements, and Euclid answered that there was no royal road to geometry. It is evident, then, that Euclid came after the time of Plato, but preceded Eratosthenes and Archimedes. [The quotations from Proclus and Aristotle which appear in this and the next chapter are adapted, by permission, from T. L. Heath, The Thirteen Books of Euclid's Elements (New York: Cambridge University Press, 1926), pp. 1, 115, 116, 117-118, 119, 121-122, 153-155, 202-203, 241-242.]

This statement would imply that Euclid lived about 300 B.C. Also, from other evidence, it seems quite certain that Euclid was the first professor of mathematics at the famous University of Alexandria, [For an interesting exposition upon Alexandria, see R. E. Langer (Bibliography).] and that he was the founder of the distinguished and long-lived Alexandrian School of Mathematics. Even his birthplace is not known, but there is some reason to believe that he received his mathematical training in the Platonic School at Athens.

Although Euclid wrote at least ten treatises on mathematics, posterity has come to know him chiefly through his Elements, a monumental work written in thirteen books, or parts. This extraordinary work so quickly and so completely superseded all previous works of the same nature that now no copies remain of the earlier efforts. Apparently from its very first appearance it was accorded the highest respect, and the mere citation of Euclid's book and proposition numbers has been regarded ever since as sufficient to identify a particular theorem or construction. With the single exception of the Bible, no work has been more widely studied or edited. For more than two millennia it has dominated all teaching of geometry, and over a thousand editions of it have appeared since the first one printed in 1482. And, as the prototype of the axiomatic or postulational method, its impact on the development of mathematics has been enormous.

Proclus has clarified for us the meaning of the term "elements." It seems that the elements of any demonstrative study are to be regarded as the leading, or key, theorems which are of wide and general use in the subject. Their function has been compared to that of the letters of the alphabet in relation to language; as a matter of fact, letters are called by the same name in Greek. The selection of the theorems to be taken as the elements of the subject requires the exercise of considerable judgment. As Proclus says,

Now it is difficult, in each science, both to select and arrange in due order the elements from which all the rest is resolved. And of those who have made the attempt some were able to put together more and some less; some used shorter proofs; some extended their investigations to an indefinite length; some avoided the method of reductio ad absurdum; some avoided proportion; some contrived preliminary steps directed against those who reject the principles; and, in a word, many different methods have been invented by various writers of elements.

It is essential that such a treatise should be rid of everything superfluous (for this is an obstacle to the acquisition of knowledge); it should select everything that embraces the subject and brings it to a point (for this is of supreme service to science); it must have great regard at once to clearness and conciseness (for their opposites trouble our understanding); it must aim at the embracing of theorems in general terms (for the piecemeal division of instruction into the more partial makes knowledge difficult to grasp). In all these ways Euclid's system of elements will be found to be superior to the rest.

And, elsewhere, Proclus says,

Starting from these elements, we shall be able to acquire knowledge of the other parts of this science as well, while without them it is impossible for us to get a grasp of so complex a subject, and knowledge of the rest is unattainable. As it is, the theorems which are most of the nature of principles, most simple, and most akin to the first hypotheses are here collected, in their appropriate order; and the proofs of all other propositions use these theorems as thoroughly known, and start from them. Thus Archimedes in the books on the sphere and cylinder, Apollonius, and all other geometers, clearly use the theorems proved in this very treatise as constituting admitted principles.

Aristotle, in his Metaphysics, speaks of "elements" in the same sense when he says, "Among geometrical propositions we call those 'elements' the proofs of which are contained in the proofs of all or most of such propositions."

It is no reflection upon the brilliance of Euclid's work that there had been other Elements anterior to his own. According to the Eudemian Summary, Hippocrates of Chios made the first effort along this line, and the next attempt was that of Leon, who in age fell somewhere between Plato and Eudoxus. It is said that Leon's work contained a more careful selection of propositions than did that of Hippocrates, and that these propositions were more numerous and more serviceable. The textbook of Plato's Academy was written by Theudius of Magnesia and was praised as an admirable collection of elements. The geometry of Theudius seems to have been the immediate precursor of Euclid's work and was undoubtedly available to Euclid, especially if he studied in the Platonic School. Euclid was acquainted also with the important work of Theaetetus and Eudoxus. Thus it is probable that Euclid's Elements is, for the most part, a highly successful compilation and systematic arrangement of works of earlier writers. No doubt Euclid had to supply a number of the proofs and to perfect many others, but the chief merit of his work lies in the skillful selection of the propositions and in their arrangement into a logical sequence presumably following from a small handful of initial assumptions.

In the thirteen books that comprise Euclid's Elements there is a total of 465 propositions. Contrary to popular impression, many of these propositions are concerned, not with geometry, but with number theory and with elementary (geometric) algebra. Book I contains the necessary preliminary material, together with theorems on congruence, parallel lines, and rectilinear figures. Book II is devoted to geometric algebra, Book III to circles, and Book IV to the construction of regular polygons. Books V and VI contain the Eudoxian theory of proportion and its application to geometry. Books VII, VIII, and IX, containing a total of 102 propositions, deal with elementary number theory. Book X is devoted to the study of irrationals, much of the material probably from Theaetetus. The remaining three books are concerned with solid geometry. The material of Books I, II, and IV was, in all likelihood, developed by the early Pythagoreans. The material found in current American high school plane and solid geometry texts is largely that found in Euclid's Books I, III, IV, VI, XI, and XII.

Certainly there is a good deal in the contents of Euclid's Elements which is of considerable interest, but in the present study our concern is with the formal nature of the Elements rather than with its mathematical contents. In fact, the various consequences of the formal character of this great work will constitute some of our chief avenues of investigation. At the moment, we are especially interested in Euclid's conception of the axiomatic method and in the precise manner in which he applied the method to the development of his Elements. We consider these matters in the two following sections.

Aristotle and Proclus on the Axiomatic Method

It is a misfortune that no copy of Euclid's Elements has been found which actually dates from the author's own time. Modern editions of the work are based upon a revision that was prepared by the Greek commentator Theon of Alexandria, who lived almost 700 years after the time of Euclid. Theon's revision was, until the early nineteenth century, the oldest edition of the Elements known to us. In 1808, however, when Napoleon ordered valuable manuscripts to be taken from Italian libraries and to be sent to Paris, F. Peyrard found, in the Vatican library, a tenth-century copy of an edition of Euclid's Elements which predates Theon's recension. A study of this older edition and a careful sifting of citations and remarks made by early commentators indicate that the introductory material of Euclid's original treatise undoubtedly underwent some editing in the subsequent revisions, but that the propositions and their proofs, except for minor additions and deletions, have remained essentially as Euclid wrote them.

Because of our lack of a copy of Euclid's original treatise, and because of the changes and additions made by later editors, it is not certain precisely what statements Euclid assumed at the start of his work, nor even how many such statements he had. Also, unfortunately, there is no known commentary by Euclid himself upon the nature of the deductive organization used so successfully in his mathematical studies. It would be valuable to have Euclid's own point of view upon the meaning of proof, or upon the significance which he attached to such terms as "definition," "axiom," and "postulate." Even partially to understand Euclid, therefore, we must study the ideas held by Euclid's contemporaries. Aristotle, in particular, is an important source of information. Since Aristotle studied at Plato's Academy, his scholastic background may have been quite similar to that of Euclid.

A student of mathematics would do well to study Aristotle's Analytica posteriora. The following passage from that work is particularly full and enlightening:

By the first principles of a subject I mean those the truth of which it is not possible to prove. What is denoted by the first terms and those derived from them is assumed; but, as regards their existence, this must be assumed for the principles but proved for the rest. Thus what a unit is, what a straight line is, or what a triangle is, must be assumed; and the existence of the unit and of magnitude must also be assumed, but the existence of the rest must be proved. Now of the premises used in demonstrative sciences some are peculiar to each science and others common to all, the latter being common by analogy, for of course they are actually useful insofar as they are applied to the subject-matter included under the particular science. Instances of first principles peculiar to a science are the assumptions that a line is of such and such a character, and similarly for a straight line; whereas it is a common principle, for instance, that if equals be subtracted from equals, the remainders are equal. But it is enough that each of the common principles is true as regards the particular subject-matter; in geometry, for instance, the effect will be the same even if the common principles be assumed to be true, not of everything, but only of magnitudes, and, in arithmetic, of numbers.

Now the things peculiar to the science, the existence of which must be assumed, are the things with reference to which the science investigates the essential attributes, for example arithmetic with reference to units, and geometry with reference to points and lines. With these things it is assumed that they exist and that they are of such and such a nature. But, with regard to their essential properties, what is assumed is only the meaning of each term employed; thus arithmetic assumes the answer to the question what is meant by "odd" or "even," "a square" or "a cube," and geometry to the question what is meant by "the irrational," or "deflection," or the so-called "verging" to a point; but that there are such things is proved by means of the common principles and of what has already been demonstrated. It is similar with astronomy. For every demonstrative science has to do with three things, (1) the things which are assumed to exist, namely the subject-matter in each case, the essential properties of which the science investigates, (2) the so-called common axioms, which are the primary source of demonstration, and (3) the properties, with regard to which all that is assumed is the meaning of the respective terms used.

This remarkable passage is almost modern in its point of view. It says that a demonstrative science must start from a set of assumptions, known as the first principles of the subject. These first principles constitute a sort of platform of initial agreement from which the rest of the discourse can be launched by purely deductive procedures. Of these principles, according to Aristotle, some are common to all sciences and others are peculiar to the particular science being studied. The first principles common to all sciences are called axioms (illustrated by, "if equals be subtracted from equals, the remainders are equal"). Among the first principles, or initial assumptions, peculiar to the science being studied, we have, first of all, statements of the existence of the subject matter and of the fundamental things whose properties the science intends to investigate (for example, in geometry, we must assume the existence of "magnitude," of "points," and of "lines"). Also among the first principles peculiar to the science being studied we have the connotation of the technical terms employed in the discourse. That is, we must accept certain definitions concerning manifestations or attributes of our subject matter (for example, in geometry, we must assume what is meant by "triangle" and by "irrational"). These definitions, however, say nothing of the existence of the things defined, but must be merely understood. The existence of only the subject matter and the fundamental things is assumed; the existence of all other things defined must be proved.

In addition to the definitions, one might expect to find among the first principles which are peculiar to the particular science being studied some statements concerning properties or relationships of the technical terms of the discourse. Certainly, since we cannot prove all the statements of our discourse, we anticipate the need for some such assumed statements for the purpose of getting started. About such assumptions Aristotle, again in his Analytica posteriora, has the following to say:

Now anything that the teacher assumes, though it is matter of proof, without proving it himself, is a hypothesis if the thing assumed is believed by the learner, and it is moreover a hypothesis, not absolutely, but relatively to the particular pupil; but if the same thing is assumed when the learner either has no opinion on the subject or is of contrary opinion, it is a postulate. This is the difference between a hypothesis and a postulate; for a postulate is that which is rather contrary than otherwise to the opinion of the learner, or whatever is assumed and used without being proved, although matter for demonstration. Now definitions are not hypotheses, for they do not assert the existence or non-existence of anything, while hypotheses are among propositions. Definitions only require to be understood; a definition is therefore not a hypothesis, unless indeed it be asserted that any audible speech is a hypothesis. A hypothesis is that from the truth of which, if assumed, a conclusion can be established.

It must be admitted that Aristotle's notion of a postulate and of the role that a postulate plays in a demonstrative science is not too clear. His remarks imply that a postulate represents the assumption of a thing which is properly a subject of demonstration, and that the assumption is made without, perhaps, the assent of the student. In other words, a postulate may not appeal to a person's sense of what is right, but it has been adopted as basic in order that the work may proceed. From this point of view, then, a postulate is a first principle. In contradistinction to this, a hypothesis is an assumption believed in by the learner, and thus is introduced apparently in order to continue an argument. For example, once a theorem has been established, and hence is acceptable to the learner, that theorem may be taken as a hypothesis from which to deduce some later theorem. If we read further in the works of Aristotle we find other passages which are of special significance in comprehending the organization of Euclid's Elements. In several places we find that Aristotle regards an axiom as a universal assumption that is so self-evident that no sane person would question it; also he considers an axiom to be too fundamental ever to be regarded as matter for demonstration. We thus seem to have, according to Aristotle, the following four distinctions between an axiom and a postulate. An axiom is common to all sciences, whereas a postulate is related to a particular science; an axiom is self-evident, whereas a postulate is not; an axiom cannot be regarded as a subject for demonstration, whereas a postulate is properly such a subject; an axiom is assumed with the ready assent of the learner, whereas a postulate is assumed without, perhaps, the assent of the learner. Some of Aristotle's statements appear somewhat contradictory, but the interpretations just given seem especially appropriate in any attempt to understand Euclid's work.

Aristotle's characterizations of definitions, axioms, and postulates are further clarified by the following account given by Proclus in his Commentary on Euclid, Book I. [We have everywhere corrected a confusion that exists in the original statement caused by Proclus' consistent misuse of the term "hypothesis" for the term "definition."]

The compiler of elements in geometry must give separately the principles of the science, and, after that, the conclusions from those principles, not giving any account of the principles but only of their consequences. No science proves its own principles, or even discourses about them; they are treated as self-evident…. Thus the first essential was to distinguish the principles from their consequences. Euclid carries out this plan practically in every book and, as a preliminary to the whole enquiry, sets out the common principles of this science. Then he divides the common principles themselves into definitions, postulates, and axioms. For all these are different from one another; an axiom, a postulate, and a definition are not the same thing, as the inspired Aristotle has somewhere pointed out. Whenever that which is assumed and ranked as a principle is both known to the learner and convincing in itself, such a thing is an axiom, for example the statement that things which are equal to the same thing are also equal to one another. When, on the other hand, the pupil has not the notion of what is told him which carries conviction in itself, but nevertheless lays it down and assents to its being assumed, such an assumption is a definition. Thus we do not preconceive by virtue of a common notion, and without being taught, that the circle is such and such a figure, but, when we are told so, we assent without demonstration. When, again, what is asserted is both unknown and assumed even without the assent of the learner, then, he says, we call this a postulate, for example that all right angles are equal. This view of a postulate is clearly implied by those who have made a special and systematic attempt to show, with regard to one of the postulates, that it cannot be assented to by any one straight off. According then to the teaching of Aristotle, an axiom, a postulate, and a definition are thus distinguished.

That there was no unanimity of opinion, even among the early Greek mathematicians themselves, concerning the precise nature of, and the difference between, an axiom and a postulate is borne out by remarks made by Proclus. Proclus points out the following three distinctions advocated by various parties: (1) An axiom is a self-evident assumed statement about something, and a postulate is a self-evident assumed construction of something; thus axioms and postulates bear a relation to one another much like that which exists between theorems and construction problems. (2) An axiom is an assumption common to all sciences, whereas a postulate is an assumption peculiar to the particular science being studied. (3) An axiom is an assumption of something that is both obvious and acceptable to the learner; a postulate is an assumption of something that is neither necessarily obvious nor necessarily acceptable to the learner. (This last is essentially the Aristotelian distinction.) Further confusion is indicated by Proclus when he points out that some preferred to call them all postulates.

In summary, then, according to the Greek conception of the axiomatic method, every demonstrable science must start from assumed first principles. These first principles consist of definitions, axioms (or common notions), and postulates. The definitions describe the technical terms used in the discourse, and, except in the case of a few fundamental terms, are not meant to imply the existence of the entities described. The axioms and the postulates are initial statements that must be assumed in order that the discourse may proceed. Just which of these statements should be called axioms and which postulates was a matter of varying opinion.

Euclid's Definitions, Axioms, and Postulates

Adhering to the Greek conception of the axiomatic method, we find, at the very start of Book I of Euclid's Elements, a list of the definitions, postulates, and common notions which are to serve as the first principles of the work. Some of the succeeding books of the work commence with additional lists of definitions. It is presumed by the author that all of the 465 propositions included in the treatise are logically deduced from these principles. For reference, we now give here the complete set of first principles for Book I essentially as furnished by T. L. Heath [T. L. Heath [5], 1, 153-155.] in his translation of the distinguished Heiberg text of Euclid's Elements.

Definitions

1. A point is that which has no part.

2. A line is length without breadth.

3. The extremities of a line are points.

4. A straight line is a line which lies evenly with the points on itself.

5. A surface is that which has only length and breadth.

6. The extremities of a surface are lines.

7. A plane surface is a surface which lies evenly with the straight lines on itself.

8. A plane angle is the inclination to one another of two lines in a plane if the lines meet and do not lie in a straight line.

9. When the lines containing the angle are straight lines, the angle is called a rectilinear angle.

10. When a straight line erected on a straight line makes the adjacent angles equal to one another, each of the equal angles is called a right angle, and the straight line standing on the other is called a perpendicular to that on which it stands.

11. An obtuse angle is an angle greater than a right angle.

12. An acute angle is an angle less than a right angle.

13. A boundary is that which is an extremity of anything.

14. A figure is that which is contained by any boundary or boundaries.

15. A circle is a plane figure contained by one line such that all the straight lines falling upon it from one particular point among those lying within the figure are equal.

16. The particular point (of Definition 15) is called the center of the circle.

17. A diameter of a circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle. Such a straight line also bisects the circle.

18. A semicircle is the figure contained by a diameter and the circumference cut off by it. The center of the semicircle is the same as that of the circle.

19. Rectilinear figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.

20. Of the trilateral figures, an equilateral triangle is one which has its three sides equal, an isosceles triangle has two of its sides equal, and a scalene triangle has its three sides unequal.

21. Furthermore, of the trilateral figures, a right-angled triangle is one which has a right angle, an obtuse-angled triangle has an obtuse angle, and an acute-angled triangle has its three angles acute.

22. Of the quadrilateral figures, a square is one which is both equilateral and right-angled; an oblong is right-angled but not equilateral; a rhombus is equilateral but not right-angled; and a rhomboid has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. Quadrilaterals other than these are called trapezia.

23. Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

Postulates

Let the following be postulated:

1. A straight line can be drawn from any point to any point.

2. A finite straight line can be produced continuously in a straight line.

3. A circle may be described with any center and distance.

4. All right angles are equal to one another.

5. If a straight line falling on two straight lines makes the interior angles on the same side together less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are together less than two right angles.

Common notions

1. Things which are equal to the same thing are also equal to one another.

2. If equals be added to equals, the wholes are equal.

3. If equals be subtracted from equals, the remainders are equal.

4. Things which coincide with one another are equal to one another.

5. The whole is greater than the part.

We observe that the first principles of Euclid's Elements fit quite well the Aristotelian account of definitions, postulates, and axioms as given in Section 2.2. It would also seem that Euclid strove to keep his list of postulates and axioms to an irreducible minimum. This economy, too, is in keeping with Aristotle's views, for in his Analytica posteriora he says, "other things being equal, that proof is the better which proceeds from the fewer postulates, or hypotheses, or propositions."

We shall pass over Euclid's definitions without much comment. Most of them probably were taken from earlier works, which would account for the fact that some terms, like oblong, rhombus, and rhomboid, are included but are never used anywhere in the work. It is curious that after having defined parallel lines Euclid does not give a formal definition of parallelogram. The existence of a parallelogram is established in I 33, [I 33 means Proposition 33 of Book I.] and in I 34 it is referred to as a parallelogramic area; then in I 35 this latter expression is shortened to parallelogram. We note that to the definition of a diameter of a circle (Definition 17) is appended the statement, "Such a straight line also bisects the circle." This addition is, of course, really a theorem (one of those attributed by Proclus in the Eudemian Summary to Thales), but its statement in Definition 17 is necessary in order to justify the definition of a semicircle which immediately follows. There are indications for believing that the definitions of a straight line and of a plane (Definitions 4 and 7) were original with Euclid. These definitions are not easy to understand, but can be comprehended, at least partially, if we appeal to sight by considering an eye placed at an extremity of the line or the plane and looking, respectively, along the line or the plane. Other interpretations of these definitions have been given. A number of Euclid's definitions are vague and virtually meaningless; we shall return to this in the next section. The work of Heath previously referred to contains a full and valuable commentary on Euclid's definitions.

Some aspects of Euclid's postulates are of especial interest. The first three are postulates of construction, for they assert what we are permitted to draw. Since these postulates restrict constructions to only those that can be made in a permissible manner with straightedge and compasses, these instruments, so limited, have become known as Euclidean tools, although their use under these restrictions certainly predates Euclid. The construction of figures with only straightedge and compasses, viewed as a game played according to the rules set down in Euclid's first three postulates, has proved to be one of the most fascinating and absorbing games ever devised. One is surprised at the really intricate constructions that can be accomplished in the allowed manner, and accordingly it is hard to believe that certain seemingly simple construction problems, like that of trisecting a given arbitrary angle, for example, cannot also be so accomplished. The energetic efforts of early Greek geometers to solve legitimately some of the construction problems which are now known to be beyond the use of Euclidean tools profoundly influenced the development of much of the content of early geometry. For example, the invention of the conic sections, of many cubic and quartic curves, and of several transcendental curves resulted from this work. A later outgrowth was the development, in modern times, of portions of the theory of equations, of the theory of algebraic numbers, and of group theory. This whole line of mathematical development, so intimately tied to Euclid's first three postulates, has little connection with our present line of investigation, and so will not be further considered here….

Postulates 1 and 3 refer to existence. In other words, the existence of a straight line joining any two given points is assumed, as is the existence of a circle having any given center and radius. From applications that Euclid makes of Postulates 1 and 2, it appears that these postulates are meant also to imply that the straight line segment joining two points in the one case, and the produced portion in the other case, are unique, although it must be admitted that the postulates do not explicitly say as much. Postulate 3 may be construed as implying something in regard to the continuity and extent of the space under consideration, since the radius of the circle may be as small or as large as one desires.

Postulates 4 and 5 are quite different from the first three postulates. The meaning of Postulate 4 is certainly evident, but there has been much debate on whether it is properly classified when placed among the postulates. If it should be classified as a theorem its proof would have to be accomplished by applying one pair of adjacent right angles to another such pair, but Euclid preferred to shun, as much as possible, such proofs by superposition. In any event, Euclid had to place Postulate 4 before his Postulate 5, since the condition in Postulate 5 that a certain pair of interior angles be together less than two right angles would be useless unless it were first made clear that all right angles are equal.

Postulate 5, known as Euclid's parallel postulate, has become, as we shall see, one of the most famous statements in mathematical history. There is more evidence for the origin of this postulate with Euclid than for the origin of any of the other four, because Aristotle alludes to a petitio principii, or a circularity in reasoning, that was involved in the theory of parallels current in his time. It is a mark of Euclid's mathematical acumen that he perceived that the only way out of the difficulty was to lay down some postulate as a basis for the theory of parallels that is so essential to the development of his geometry. The postulate that he formulated serves this purpose admirably and also, at the same time, furnishes a criterion for determining whether two straight lines in a figure will or will not meet if extended. This fact is an advantage of Euclid's postulate over the substitutes that were later suggested to take its place, and this advantage is actually employed in the Elements as early as I 44. The consequences of investigations carried on in connection with Euclid's fifth postulate proved to be very far-reaching. Not only did these investigations supply the stimulus for the development of much of the mathematics which we characterize as modern, but they led to a far deeper examination, and consequent perfection, of the axiomatic method….

Of the common notions, or axioms, there is reason to believe that the first three were given by Euclid, but that the last two may have been added at a later time. Axiom 4 has been criticized on the ground that its subject matter is special rather than general, and that it ought therefore to be listed as a postulate instead of as an axiom. Objections which can be raised to the method of superposition, used by Euclid with apparent reluctance to establish some of his early congruence theorems, can be at least partially met by Axiom 4. Again the student is referred to the excellent commentary given by Heath.

In conclusion, we may summarize Euclid's conception and use of the axiomatic method somewhat as follows: Every deductive system requires assumptions from which the deduction may proceed. Therefore, as initial premises, Euclid puts down five postulates, or assumed statements about his subject matter. In addition to the five postulates, Euclid lists five axioms, or common notions, which he also needs for his proofs. These axioms are not peculiar to his subject matter but are general principles valid in any field of study. Now in the postulates a number of terms occur, such as point, straight line, right angle, and circle, of which it is not certain that the reader has a precise notion. Hence some definitions are also given. These definitions are not, like the postulates, assumptions about the nature of the subject matter but are merely explanations of the meanings of the terms. Definition 10, for example, tells what a right angle is, and how an angle may be identified as a right angle, but it says nothing about the existence of right angles, nor does it state what is assumed about such angles. These latter functions are left to the postulates and to deduced propositions. Thus Postulate 4 informs us that all right angles are equal, and Proposition I 11 proves that right angles exist. On the other hand, Postulate 4 gives no clue regarding the nature of a right angle, nor does it tell how the term is to be employed; it merely states a fundamental assumption about such angles. Finally, the natural order for presenting the postulates, axioms, and definitions to the student is, first, the definitions explaining the meanings of the technical terms of the discourse, next, the postulates which are so closely related to the definitions, and lastly, the axioms or common notions.

Some Logical Shortcomings of Euclid's Elements

It would be very surprising indeed if Euclid's Elements, because it is such an early and extensive application of the axiomatic method, should be free of logical blemishes. Therefore it is no great discredit to the work that critical investigations have revealed a number of defects in its logical structure. Probably the gravest of these defects are certain tacit assumptions that are employed later in the deductions and are not granted by the first principles of the work. This danger exists in any deductive study when the subject matter is overly familiar to the author. Usually a thorough grasp of the subject matter in a field of human endeavor is regarded as an indispensable prerequisite to serious work, but in developing a deductive system such knowledge can be a definite disadvantage unless proper precautions are taken.

A deductive system differs from a mere collection of statements in that it is organized in a very special way. The key to the organization lies in the fact that all statements of the system other than the original assumptions must be deducible from these initial hypotheses, and that if any additional assumptions should creep into the work the desired organization is not realized. Now anyone formulating a deductive system knows more about his subject matter than just the initial assumptions he wishes to employ. He has before him a set of statements belonging to his subject matter, some of which he selects for postulates and the rest of which he presumably deduces from his postulates as theorems. But, with a large body of information before one, it is very easy to employ in the proofs some piece of this information which is not embodied in the postulates. Any piece of information used in this way may be so apparently obvious or so seemingly elementary that it is assumed unconsciously. Such a tacit assumption, of course, spoils the rigidity of the organization of the deductive system. Morever, should that piece of information involve some misconception, its introduction may lead to results which not only do not strictly follow from the postulates but which may actually contradict some previously established theorem. Herein, then, lies the pitfall of too great a familiarity with the subject matter of the discourse; at all times in building up a deductive system one must proceed with the appearance of being completely ignorant of the developing material. This does not mean that in building up a deductive system one refrains from making any use of one's intuitive appreciation of the significance of the axioms and of possible interpretations of the primitive terms. On the contrary one makes full use of these things, but only to conjecture possible theorems and possible avenues of investigation. In the actual establishment of these theorems and in the actual development of these avenues of investigation, one must be careful to proceed only in terms of the accepted assumptions….

In short, the truth of the matter is that Euclid's first principles are simply not sufficient for the derivation of all of the 465 propositions of the Elements. In particular, the set of postulates needs to be considerably amplified. The work of perfecting Euclid's initial assumptions, so that all of his geometry can rigorously follow, occupied mathematicians for more than two thousand years. Not until the end of the nineteenth century and the early part of the twentieth century, after the foundations of geometry had been subjected to an intensive study, were satisfactory sets of postulates supplied for Euclidean plane and solid geometry. The history of this struggle is of major concern in our present study, and in following it we shall encounter the device that mathematicians contrived for avoiding the pitfall, into which Euclid so often fell, of overfamiliarity with the subject matter.

Not only is Euclid's work marred by numerous tacit assumptions, but some of his preliminary definitions are also open to criticism. Euclid, following the Greek pattern of material axiomatics, makes some sort of attempt to define, or at least to explain, all the terms of his discourse. Now, actually, it is as impossible to define explicitly all of the terms of a discourse as it is to prove all of the statements of the discourse, for a term must be defined by means of other terms, and these other terms by means of still others, and so on. In order to get started, and to avoid circularity of definition where term x is defined by means of term y, and then later term y by means of term x, one is forced to set down at the very start of the discourse a collection of primitive, or basic, terms whose meanings are not to be questioned. All subsequent terms of the discourse must be defined, ultimately, by means of these initial primitive ones. The postulates of the discourse are, then, in final analysis, assumed statements about the primitive terms. From this point of view, the primitive terms may be regarded as defined implicitly, in the sense that they are any things or concepts which satisfy the postulates, and this implicit definition is the only kind of definition that the primitive terms can receive.

In Euclid's development of geometry the terms point and line, for example, could well have been included in a set of primitive terms for the discourse. At any rate, Euclid's definition of a point as "that which has no part" and of a line as "length without breadth" are easily seen to be circular and therefore, from a logical viewpoint, woefully inadequate. One distinction between the Greek conception and the modern conception of the axiomatic method lies in this matter of primitive terms; in the Greek conception there is no listing of the primitive terms. The excuse for the Greeks is that to them geometry was not just an abstract study, but was an attempted logical analysis of idealized physical space. Points and lines were, to the Greeks, idealizations of very small particles and of very thin threads. It is this idealization that Euclid attempts to express in his two initial definitions….

The End of the Greek Period and the Transition to Modern Times

[This section is largely skimmed from the appropriate places in H. Eves.] Very little in the further development of the axiomatic method took place after Euclid until relatively modern times. We must mention, however, the brilliant exploitation of the method by Archimedes (287-212 B.C.), one of the greatest mathematicians of all time, and certainly the greatest of antiquity. Although Archimedes lived most of his long life in the Greek city of Syracuse, on the island of Sicily, it seems that he studied for a time at the University of Alexandria. He was thoroughly schooled in the Euclidean tradition, and he left deep imprints on both geometry and mechanics. Archimedes' works are masterpieces of mathematical exposition and resemble to a remarkable extent, because of their high finish, economy of presentation, and rigor in demonstration, the articles found in present-day research journals. It is interesting that Archimedes employed the axiomatic method in his writings on theoretical mechanics, as well as in his purely geometrical studies, always laying down the first principles of the work and then deducing a sequence of propositions. Thus, in his treatise On Plane Equilibriums, Archimedes establishes 25 theorems of mechanics on the basis of three simple postulates suggested by common experience. The postulates are as follows: (1) Equal weights at equal distances balance; equal weights at unequal distances do not balance but incline toward the weight which is at the greater distance. (2) If, when weights at certain distances balance, something be added to one of the weights, equilibrium will not be maintained, but there will be inclination on the side of the weight to which the addition was made; similarly, if anything be taken away from one of the weights, there will be inclination on the side of that weight from which nothing was taken. (3) When equal and similar plane figures coincide if placed on one another, their centroids similarly coincide; and in figures which are unequal but similar the centroids will be similarly situated. From these simple postulates Archimedes locates, for example, the centroid of any parabolic segment and of any portion of a parabola lying between two parallel chords. Problems of this sort would today be worked out by means of the integral calculus.

Again, in his work On Floating Bodies, Archimedes rests the establishment of the 19 propositions of the work on two fundamental postulates. This treatise is the first recorded application of mathematics to hydrostatics, and it begins by developing those familiar laws of hydrostatics which nowadays are encountered in an elementary physics course. The treatise then goes on to consider several rather difficult problems, culminating with a remarkable investigation of the positions of rest and of stability of a right segment of a paraboloid of revolution floating in a fluid. Not until the sixteenth-century work of Simon Stevin did the science of statics and the theory of hydro-dynamics appreciably advance beyond the points reached by Archimedes. It is worthy of note that these early researches in theoretical physics were developed by the use of the axiomatic method.

There is a geometrical assumption explicitly stated by Archimedes in his work On the Sphere and Cylinder which deserves special mention; it is one of the five postulates assumed at the start of Book I of the work and it has become known as the postulate of Archimedes. A simple statement of the postulate is as follows: Given two unequal linear segments, there is always some finite multiple of the shorter one which is longer than the other. In some modern treatments of geometry this postulate serves as part of the postulational basis for introducing the concept of continuity. It is a matter of interest that in the nineteenth and twentieth centuries geometric systems were constructed which denied the Archimedean postulate, thus giving rise to so-called non-Archimedean geometries. Although named after Archimedes, this postulate had been considered earlier by Eudoxus.

There were other able Greek mathematicians in ancient times after Euclid besides Archimedes—for example, Apollonius, Eratosthenes, Menelaus, Claudius Ptolemy, Heron, Diophantus, and Pappus—but these men did little to advance the development of the axiomatic method and so have slight connection with our present study. After Pappus, who flourished toward the end of the third century A.D., Greek mathematics practically ceased as a living study, and thenceforth merely its memory was perpetuated by minor writers and commentators, such as Theon and Proclus. This closing period of ancient times was dominated by Rome. One Greek center after another had fallen before the power of the Roman armies; in 146 B.C. Greece had become a province of the Roman Empire, although Mesopotamia was not conquered until 65 B.C., and Egypt held out until 30 B.C. The economic structure of the empire was based essentially on agriculture and an increasing use of slave labor. Conditions proved more and more stifling to original scientific work, and a gradual decline in creative thinking set in. The eventual collapse of the slave market, with its disastrous effect on Roman economy, found science reduced to a mediocre level. The famous Alexandrian school gradually faded with the breakup of ancient society, and finally, in 641 A.D., Alexandria was taken by the Arabs, who put the torch to what the Christians had left. The long and glorious era of Greek mathematics was over.

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 bequeathed by the ancient world were forgotten. Only the monks of the Catholic monasteries, and a few cultured laymen, preserved a slender thread of Greek and Latin learning. The period was marked by great physical violence and intense religious faith. The old social order gave way, and society became feudal and ecclesiastical.

The Romans had never taken to abstract mathematics but had contented themselves with merely a few practical aspects of the subject that were associated with commerce and civil engineering. With the fall of the Roman Empire and the subsequent closing of much of east-west trade and the abandonment of state engineering projects, even these interests waned, and it is no exaggeration to say that very little in mathematics, beyond the development of the Christian calendar, was accomplished in the West during the whole of the half millennium covered by the Dark Ages.

During this bleak period of learning the people of the east, especially the Hindus and the Arabs, became the major custodians of mathematics. However, the Greek concept of rigorous thinking—in fact, the very idea of proof—seemed distasteful to the Hindu way of doing things. Although the Hindus excelled in computation, contributed to the devices of algebra, and played an important role in developing our present positional numeral system, they produced nothing of importance so far as basic methodology is concerned. Hindu mathematics of this period is largely empirical and lacks those outstanding Greek characteristics of clarity and logicality in presentation and of insistence on rigorous demonstration.

The spectacular episode of the rise and decline of the Arabian empire occurred during the period of Europe's Dark Ages. Within a decade following Mohammed's flight from Mecca to Medina in 622 A.D., the scattered and disunited tribes of the Arabian peninsula were consolidated by a strong religious fervor into a powerful nation. Within a century, force of arms had extended the Moslem rule and influence over a territory reaching from India, through Persia, Mesopotamia, and northern Africa, clear into Spain. Of considerable importance for the preservation of much of world culture was the manner in which the Arabs seized upon Greek and Hindu erudition. The Bagdad caliphs not only governed wisely and well but many became patrons of learning and invited distinguished scholars to their courts. Numerous Hindu and Greek works in astronomy, medicine, and mathematics were industriously translated into the Arabic tongue and thus were saved until later European scholars were able to retranslate them into Latin and other languages. But for the work of the Arabian scholars a great part of Greek and Hindu science would have been irretrievably lost over the long period of the Dark Ages.

Not until the latter part of the eleventh century did Greek classics in science and mathematics begin once again to filter into Europe. There followed a period of transmission during which the ancient learning preserved by Moslem culture was passed on to the western Europeans through Latin translations made by Christian scholars traveling to Moslem centers of learning, and through the opening of western European commercial relations with the Levant and the Arabian world. The loss of Toledo by the Moors to the Christians in 1085 was followed by an influx of Christian scholars to that city to acquire Moslem learning. Other Moorish centers in Spain were infiltrated, and the twelfth century became, in the history of mathematics, a century of translators. One of the most industrious translators of the period was Gherardo of Cremona, who translated into Latin more than 90 Arabian works, among which were Ptolemy's Almagest and Euclid's Elements. At the same time Italian merchants came in close contact with eastern civilization, thereby picking up useful arithmetical and algebraical information. These merchants played an important part in the European dissemination of the Hindu-Arabic system of numeration.

The thirteenth century saw the rise of the universities at Paris, Oxford, Cambridge, Padua, and Naples. Universities were to become potent factors in the development of mathematics, since many mathematicians associated themselves with one or more such institutions. During this century Campanus made a Latin translation of Euclid's Elements which later, in 1482, became the first printed version of Euclid's great work.

The fourteenth century was a mathematically barren one. It was the century of the Black Death, which swept away more than a third of the population of Europe, and during this century the Hundred Years War, with its political and economic upheavals in northern Europe, got well under way.

The fifteenth century witnessed the beginning of the European Renaissance in art and learning. With the collapse of the Byzantine Empire, culminating in the fall of Constantinople to the Turks in 1453, refugees flowed into Italy, bringing with them treasures of Greek civilization. Many Greek classics, up to that time known only through the often inadequate Arabic translations, could now be studied from original sources. Also, about the middle of the century, occurred the invention of printing, which revolutionized the book trade and enabled knowledge to be disseminated at an unprecedented rate. Mathematical activity in this century was largely centered in the Italian cities and in the central European cities of Nuremberg, Vienna, and Prague, and it concentrated on arithmetic, algebra, and trigonometry, under the practical influence of trade, navigation, astronomy, and surveying.

In the sixteenth century the development of arithmetic and algebra continued, the most spectacular mathematical achievement of the century being the discovery, by Italian mathematicians, of the algebraic solution of cubic and quartic equations. In 1572 Commandino made a very important Latin translation of Euclid's Elements from the Greek. This translation served as a basis for many subsequent translations, including a very influential work by Robert Simson, from which, in turn, so many English editions were derived.

The seventeenth century proved to be particularly outstanding in the history of mathematics. Early in the century Napier revealed his invention of logarithms, Harriot and Oughtred contributed to the notation and codification of algebra, Galileo founded the science of dynamics, and Kepler announced his laws of planetary motion. Later in the century Desargues and Pascal opened a new field of pure geometry, Descartes launched modern analytic geometry, Fermat laid the foundations of modern number theory, and Huygens made distinguished contributions to the theory of probability and other fields. Then, toward the end of the century, after many mathematicians had prepared the way, the epoch-making creation of the calculus was made by Newton and Leibniz. Thus, during the seventeenth century, many new and vast fields were opened for mathematical investigation. The dawn of modem mathematics was at hand, and it was perhaps inevitable that sooner or later some aspect of the axiomatic method itself should once again claim the attention of researchers.

Bibliography

… Eves, H., An Introduction to the History of Mathematics, revised edition. New York: Holt, Rinehart and Winston, Inc., 1964….

[Heath, T. L.], The Thirteen Books of Euclid's Elements. 3 vols., 2nd ed. New York: Dover Publications, Inc., 1956….

Langer, R. E., "Alexandria—shrine of mathematics," American Mathematical Monthly, 48 (1941), 109-125….