He makes three hypotheses which were recognized later to correspond to the elliptic, Euclidean and hyperbolic geometries. But at the end of his work, in order to exhibit a contradiction when Euclid's postulate is denied, he is forced to make use of a somewhat vague and unacceptable assumption about "the nature of a straight line.' Gauss's activity in connection with the parallel postulate is of especial interest because of its psychologic aspect. It is difficult for us to picture a mathematician hesitating to publish a discovery for fear of the outcry that its publication might produce-perhaps not many would be displeased to awaken an echo; yet this is believed by some to have been the attitude of Gauss. Though he was keenly interested and thought deeply on the subject of parallels, he published nothing; he feared, as he said, "the clamor of the Boeotians." When forced to write a letter on the subject, he begs his correspondent to keep silence as to the information imparted. In 1831 he writes in a letter: "In the last few weeks I have begun to put down a few of my Meditations [on parallels] which are already to some extent forty years old. These I had never put in writing, so that I have been compelled three or four times to go over the whole matter afresh in my head. Also I wished that it should not perish with me." It is only when we call to mind the unrivalled place of honor held by Euclidean geometry among branches of human knowledge-a respect no doubt enhanced by the prominence given it in Kant's Critique of Pure Reason-that we realize the uncomfortable position of one who even appeared to attack its validity. Gauss's meditations were leading him through tedious and painstaking labors to the conclusion that Euclid's fifth postulate was not deducible from his other postulates. The minds of those not conversant with the intricacies of the problem might easily rush to the conclusion that Euclid's geometry was therefore untrue, and feel the whole structure of human learning crashing about their ears. Between 1820 and 1830, the conclusion toward which Gauss tended was finally made sure by the invention of the hyperbolic non-Euclidean geometry by Lobachevsky and Johann Bolyai, working simultaneously and independently. The question, Is Euclid's fifth postulate logically deducible from his other postulates? is answered by showing that the denial of this postulate while all the others are retained leads to a geometry as consistent as Euclid's own. The method, we recall, was that used by Saccheri, whose intellectual conservatism alone prevented his reaching the same result. The famous postulate is only one of three mutually exclusive hypotheses which are logically on the same footing. Thus was Euclid "vindicated" in an unexpected manner. Knowingly or not, the wise Greek had stated the case correctly, and only his followers had been at fault in their efforts for improvement. To quote Heath: "We cannot but admire the genius of the man who concluded that such an hypothesis, which he found necessary to the validity of his whole system, was really indemonstrable." Thus in some sense the problem of the parallel postulate was laid to rest, but its spirit marches on. If the fifth postulate could without logical error be replaced by its contradictory, could the other postulates be similarly treated? What is the nature of a postulate or axiom? What requirements should a satisfactory system of axioms fulfill? Are we sure that accepted proofs will bear as keen scrutiny as that to which proofs of the postulate have been subjected? The facing of these questions has brought us to the modern critical study of the foundations of geometry. It has been realized that if geometry is to continue to enjoy its reputation for logical perfection, it should at least try to deserve it. The edge of criticism, sharpened on the parallel postulate, is turned against the whole structure. Out of this movement has grown the critical examination of the foundations of algebra, of projective geometry, of mechanics, of logic itself; and the end is not in sight. One obvious result of this critical study is that geometrical axioms are not necessary truths, but merely presuppositions: they are the hypotheses on which the whole body of theorems rests. It is essential that a system of axioms should be consistent with each other, and desirable that they be non-redundant, and complete. No one has found Euclid's system inconsistent,1 and redundancy would be a crime against elegance rather than against logic. But on the score of completeness Euclid is far from giving satisfaction. He not infrequently states conclusions which could be arrived at only by looking at a figure, i.e., by space intuition; but we are all familiar with cases where space intuition misleads (for example in the fallacious proof that all triangles are isosceles), and if we accept it as a guide how can we be sure that our intuitions will always agree? In constructing an equilateral triangle Euclid says, "From point С where the two circles meet, draw...." Perhaps they do meet-but not on the basis of anything previously stated. In dropping a perpendicular from a point to a line, he says a certain circle will meet the line twice. Why should it not cut thrice or not at all? In another proof he says that a certain line will lie within a certain angle. I see that it does, but I do not see it proved. We are told in the midst of a proof to bisect an angle of a triangle and produce the bisector to meet the opposite side. How do we know it will meet? Because it is not a parallel. And probably it is not parallel because it is inside the triangle. How do we know it is inside; or, being inside, that it must get outside? When have these terms been defined? You may answer: It is not necessary to define them because everyone with common sense knows inside from outside without being told. "Who is so dull as not to perceive...?" says Simson, one of Euclid's apologists. This may be granted. But it must be pointed out that common sense knows that two straight lines cannot enclose space, yet this is given prominence as an axiom; or that a straight line is the shortest distance between two points, yet this is proved as a proposition. To state in words what distinguishes the inside from the outside of a polygon is not easy. The word "between" is likewise difficult of definition. But the modern geometer imbued with the critical spirit feels it necessary to define such terms, and what is more, he finds a way of doing it. 1 It is in fact possible to show that the system is consistent, provided we agree to accept the axioms of arithmetic as consistent. While this is only a transformation of the problem, it is logically important to recognize the possibility of such a transformation.-EDITOR. Hilbert's Grundlagen der Geometrie, published in 1899, is a classic product of this movement. It presupposes no space concepts at all, but only such general logical terms as "corresponds to," "associated with," "determined by." Contrary to tradition, it does not begin by defining terms. The first sentence is: "I think of three systems of things which I call points, lines and planes." Note the unadorned simplicity of the concept things. The axioms serve as definitions. They state, in non-spatial terms, relations between these "things"; that is to say, the points, lines and planes are such things as have such and such relations. "That is all ye know on earth, and all ye need to know." Twenty-one axioms are found necessary, as against Euclid's meager five. The whole work could be read and comprehended by a being with no space intuitions whatever. We could substitute the names of colors or sounds for points, lines and planes, and get on equally well. The ideal of making a thing "so plain that a blind man could see it" is literally realized. And the age-old ideal of a body of proved propositions, close-knit together by unassailable logic, is immeasurably nearer realization. Although to the best of my knowledge no one has yet had the hardihood to invite a child of fourteen to consider "three systems of things," the modern critical movement is not without bearing on problems of teaching. I wish, with proper humility, to put forward a few ideas on this subject. If it is true that our traditional formal geometry, taken directly or indirectly from Euclid, is not the logically perfect thing we had imagined, and if its modern perfected descendant is so abstract that not even the most rationalistic of us would venture to force it on beginners, why not acknowledge these facts and bravely face anew the question of how we can best make the study of elementary geometry serve its proclaimed purpose of training the mind? I would suggest two lines along which progress might be made. First, by sacrificing the ideal of nonredundancy in our underlying assumptions we could save time and stimulate interest by arriving more quickly at propositions whose truth is not immediately evident and which could be presented as subjects for investigation. Must we, because Euclid did it, prove that the base angles of an isosceles triangle are equal? A child that has cut the triangle out of paper and folded it over knows as much as any proof can teach him. If to treat the proposition in this way is repugnant to the teacher's logical conscience, let him privately label it "axiom" or "postulate," and proceed, even though this proposition could have been proved. The place to begin producing arguments is the place where the truth of the proposition is even momentarily in doubt. One statement which presents itself with a question mark and is found after investigation to be true or to be false is worth ten obviously true statements proved with all the paraphernalia of hypothesis, conclusion, step one and step two, with references. The only apparent reason for proving in the traditional way the theorems on the isosceles triangle and congruent triangles is in order to familiarize the student with the above-mentioned paraphernalia. This brings me to my second suggestion. Formal geometry has been looked upon as a complete and perfect thing to 1 It is possible that Halsted's Rational Geometry has a somewhat similar purpose.—EDITOR. which the learner can with profit play the sedulous ape. Yet I sometimes think that by emphasizing too early the traditional form of presentation of geometrical argument, and paying too little attention to the psychology of the learner, we may have corrupted some very good minds. "I wish to prove . . .," says the student; meaning, I wish to prove something stated and accepted as true in advance of argument. Should we not prefer to have our students say, "I wish to examine . . ., to understand, to find out whether . to discover a relation between to invent a means of doing . . ."? What better slogan could prejudice desire than "I wish to prove"? The conscienceless way in which college debaters collect and enumerate arguments regardless of the issues involved is another aspect of the same evil. A student said not long ago, "The study of mathematics would be good fun if we did not have to learn proofs." It had never been brought home to her that mathematical reasoning is not a thing to be acquired, like a knowledge of Latin verbs, but a thing to be participated in like any other form of exercise. Another said, "I cannot apply my geometry because all we did in school was to learn the proofs and pass the examinations." In the midst of a proof the student hesitates and says, "I am sure this is the next step, but I cannot recall the reason for it." The step and its reason would occur simultaneously to a mind that had faced the proposition as a problem and thought it through. I should like to see in every text an occasional page of exercises to prove or disprove. And if formal proofs must be printed in full, by all means let some of the proofs be wrong. When the student has thus halted with one foot in the air in this progress from step to step down the printed page, on what does the ability to proceed depend? On the ability to quote something: to quote, usually, a single statement-compact, authoritative, triumphantly produced. Surely it is bad training that leads the mind to expect to find support for its surmises in a form so simple; and the temptation to substitute ability to quote in place of the labor of finding out the truth may be a real danger. What wonder if the mind so trained quotes Washington's Farewell Address or the Monroe Doctrine and feels that its work is done? I do not mean that formal proof should never be given. It has its place as an exercise in literary composition; for it deals with the form in which thought is expressed. We should, however, take every possible precaution to see that the thought is first there to be expressed, lest the form be mistaken for the substance. Just how this is to be brought about I am not prepared to discuss, although I suspect that drawing and measurement in the early stages of study, problems of construction and investigation, and the total absence of complete proofs from the printed page, would help. I wish merely to state my belief that only in so far as we succeed in these aims shall we succeed in making geometry really train the mind. It can be done, said the butcher, I think; It must be done, I am sure. One point further. Perhaps we are a little too modest about the importance of having our students retain something of the subject-matter of the courses we teach. Evidently it is here that memory, based on understanding, should rightly be used. I sometimes think we might in some way collectively take out insurance against a student's arriving at the junior year in college in the belief that two triangles are similar whenever they have a side in common. RECENT PUBLICATIONS. REVIEW. Introductory Mathematical Analysis. By W. PAUL WEBBER and LOUIS C. PLANT. New York, Wiley, 1919. 13+ 304 pages. Price $2.00. This combination book for freshmen covers plane trigonometry and topics from advanced algebra, analytic geometry, and the calculus, which are treated in the order named. The first seven chapters (70 pages) may be said to form an introduction containing a review of algebra and geometry, and such general topics as computations, including logarithms and the slide rule, rectangular coördinates with graphical representations of statistics and equations, classes of numbers, variables, functions, and limits. Interpolation in tables involving two independent variables is given, and simple problems to determine the form of an equation representing a given empirical table are solved. In a later chapter some problems of the latter sort are solved by using logarithmic and semi-logarithmic paper. Chapter VIII (50 pages) gives the usual course in plane trigonometry in condensed form. In defining the six functions a tabular form is used in which a row is given to each function and a column to an angle in each of the four quadrants, a point P1(x1,y1) being chosen on the terminal line of an angle a in the first quadrant, a point P2(x2, y2) on the terminal line of an angle a2 in the second quadrant, etc. The table includes, in parentheses, the sign of each function for each quadrant. A radian "is defined by the equation Arc AB = r·0, where r is the radius of the arc AB and 0 is the angle in radians." Portions of a traverse table and of a table of haversines, and some problems in indirect fire, appear among the applications. The chapter on the circular functions is followed by one on polar coördinates, complex numbers, and vectors, which connects trigonometry with algebra. Vector and scalar products are given considerable attention, with application to equilibrium of particles and rigid bodies. A short chapter on equations gives a few of the theorems from the theory of equations, but the opportunity to tie it to the preceding chapter by the theorem that complex roots occur in conjugate pairs was not improved. The reviewer holds no brief for this theorem, especially as he thinks other material more suitable for and valuable to freshmen than complex numbers, but the omission is noticeable in a text which considers the derivative of u" for a complex value of n, which treats series with complex terms, and which expresses the sine and cosine in the well-known exponential forms in order to |