scientific centre-to study at Alexandria the healing art, anatomy, mathematical science, geography, and astronomy. Neither Athens, Rome, Carthage, nor any other city of the ancient world can boast similar distinction as a home of science.

EUCLID. Three centuries after Thales had introduced the rudiments of Egyptian mathematics into Greece, the focus of mathematical activity was again transferred to that ancient land, but its spirit and aims remained there still for centuries essentially Greek. Continuing the ancient register, Proclus writes: —

Not much younger than these (the Aristotelians) is Euclid, who brought the elements together, arranged much of the work of Eudoxus in complete form, and brought much which had been begun by Theætetus to completion. Besides he supported what had been only partially proved by his predecessors with irrefragable proofs. . . . It is related that King Ptolemy asked him once if there were not in geometrical matters a shorter way than through the Elements: to which he replied that in geometry there is no straight path for kings.

As a recent writer has well said: "There are royal roads in science; but those who first tread them are men of genius and not kings."

Euclid's period of activity was about 300 B.C.; his place of birth and even his race are unknown; he is said to have been of a mild and benevolent disposition, and to have appreciated fully the scientific merits of his predecessors. While we know next to nothing of his life and personality, his writings have had an influence and a prolonged vitality almost, if not quite, unparalleled.

EUCLID'S "ELEMENTS."- Scientifically, Euclid is attached to the Platonic philosophy. Thus he makes the goal of his Elements the construction of the so-called "Platonic bodies" i.e., the five regular polyhedrons. This treatise, which served as the basis of practically all elementary instruction for the following 2000 years, is naturally his best-known work, and appears to have been accepted in the Greek world after many previous attempts as a

finality. It consisted of thirteen books, of which only the first six are ordinarily included in modern editions. The whole is essentially a systematic introduction to Greek mathematics, consisting mainly of a comparative study of the properties and relations of those geometrical figures, both plane and solid, which can be constructed with ruler and compass. The comparison of unequal figures leads to arithmetical discussion, including the consideration of irrational numbers corresponding to incommensurable lines. The contents may be briefly summarized as follows: Book I deals with triangles and the theory of parallels: Book II with applications of the Pythagorean theorem, many of the propositions being equivalent to algebraic identities, or solutions of quadratic equations, which seem to us more simple and obvious than to the Greeks. It should be noted however that the geometrical treatment is relatively advantageous for oral presentation. Book III deals with the circle, Book IV with inscribed and circumscribed polygons. These first four books thus contain a general treatment of the simpler geometrical figures, together with an elementary arithmetic and algebra of geometrical magnitudes. In Book V, for lack of an independent Greek arithmetical analysis, a theory of proportion (which has thus far been avoided) is worked out, with the various possible forms of the equation The results are applied in Book VI to the comparison of similar figures. This contains the first known problem in maxima and minima, the square is the greatest rectangle of given perimeter, - also geometrical equivalents of the solution of quadratic equations. The next three Books are devoted to the theory of numbers, including for example the study of prime and composite numbers, of numbers in proportion, and the determination of the greatest common divisor. He shows how to find the sum of a geometrical progression, and proves that the number of prime numbers is infinite.

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b d

If there were a largest prime number n then the product 1 x 2 x 3... n increased by 1 would always leave a remainder 1 when divided by n or by any smaller number. It would thus either be prime

itself, or a product of prime factors greater than n, either of which suppositions is contrary to the hypothesis that n itself is the greatest prime number.

Book X deals with the incommensurable on the basis of the theorem: If two unequal magnitudes are given, and if one takes from the greater more than its half, and from the remainder more than its half and so on, one arrives sooner or later at a remainder which is less than the smaller given magnitude. Books XI, XII, and XIII are devoted to solid geometry, leading up to our familiar theorems on the volume of prism, pyramid, cylinder, cone, and sphere, but in every case without computation, emphasizing the habitual distinction between geometry and geodesy or mensuration... a distinction expressed by Aristotle in the form: "One cannot prove anything by starting from another species, for example, anything geometrical by means of arithmetic. Where the objects are so different as arithmetic and geometry one cannot apply the arithmetical method to that which belongs to magnitudes in general, unless the magnitudes are numbers, which can happen only in certain cases." Book XIII passes from the regular polygons to the regular polyhedrons, remarking in conclusion that only the known five are possible.

The extent to which Euclid's Elements represent original work rather than compilation of that of earlier writers cannot be determined. It would appear, for example, that much of Books I and II is due to Pythagoras, of III to Hippocrates, of V to Eudoxus, and of IV, VI, XI, and XII, to later Greek writers; but the work as a whole constitutes an immense advance over previous similar attempts.

Proclus (410-485 A.D.) is the earliest extant source of information about Euclid. Theon of Alexandria edited the Elements nearly 700 years after Euclid, and until comparatively recent times modern editions have been based upon his.

Like other Greek learning, Euclid has come down to later times through Arab channels. There is a doubtful tradition that an English monk, Adelhard of Bath, surreptitiously made a Latin

translation of the Elements at a Moorish university in Spain in 1120. Another dates from 1185, printed copies from 1482 onward, and an English version from 1570. After Newton's time it found its way from the universities into the lower schools.

Different versions vary widely as to the axioms and postulates on which the work as a whole is based. It is believed that Euclid originally wrote five postulates, of which the fourth and fifth are now known as Axioms 11 and 12,-"All right angles are equal"; and the famous parallel axiom :-"If a straight line meets two straight lines, so as to make the two interior angles on the same side of it together less than two right angles, these straight lines will meet if produced on that side." The necessarily unsuccessful attempts which have since been made to prove this as a proposition rather than a postulate constitute an important chapter in the history of mathematics, leading in the last century to the invention of the generalized geometry known as nonEuclidean, in which this axiom is no longer valid.

INFLUENCE OF EUCLID. The Elements of Euclid have exerted an immense influence on the development of mathematics, and particularly of mathematical pedagogy. Aside from their substance of geometrical facts, they are characterized by a strict conformity to a definite logical form, the formulation of what is to be proved, the hypothesis, the construction, the progressive reasoning leading from the known to the unknown, ending with the familiar Q.E.D. There is a careful avoidance of whatever is not geometrical. No attempt is made to develop initiative or invention on the part of the student; the manner in which the results have been discovered is rarely evident and is even sometimes concealed; each proposition has a degree of completeness in itself. This treatise translated into the languages of modern Europe has been a remarkable means of disciplinary training in its special form of logic. No other science has had any such single permanently authoritative treatise.

CRITICISM OF EUCLID. On the other hand, its narrowness of aim, its deliberate exclusion of the concrete, its laborious methods of dealing with such matters as infinity, the incommensurable or

irrational, its imperfect substitutes for algebra, as in the theory of proportion, have diminished its usefulness, and have in comparatively recent times (in English-speaking countries) led to the substitution of modernized texts. Still, no other mathematical treatise has had even approximately the deservedly far-reaching influence of Euclid. Its subject-matter is so nearly complete that its author's name is still a current synonym for elementary geometry.

His elements are particularly admired for the order which controls them, for the choice of theorems and problems selected as fundamental (for he has by no means inserted all which he might give, but only those which are really fundamental), and for the varied argumentation, producing conviction now by starting from causes, now by going back to facts, but always irrefutable, exact and of most scientific character. ... Shall we mention the constantly maintained invention, economy and orderliness, the force with which he establishes every point? If one adds to or takes from it, one will recognize that he departs thereby from science, tending towards error or ignorance.

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Elsewhere Proclus:

It is difficult in every science to choose and dispose in suitable order the elements from which all the rest may be derived. Of those who have attempted this some have increased their collection, others have diminished it; some have employed abridged demonstrations, others have expanded their presentation indefinitely, etc.

In such a treatise it is necessary to avoid everything superfluous ... to combine all that is essential, to consider principally and equally clearness and brevity, to give theorems their most general form, for the detail of teaching particular cases only makes the acquisition of knowledge more difficult. From all these points of view, Euclid's Elements will be found superior to every other.

In a recent interesting discussion of Euclid's Elements, F. Klein (Elementar-Mathematik vom Höheren Standpunkt aus. II) says in substance: "A false estimation of the Elements finds its source in the general misunderstanding of Greek genius which long pre