our school work is now organized. Where mathematics is required of all students, there are sure to be many for whom such advanced work would be meaningless. With less required mathematics in the high school, those who elect it in the higher courses would be in a position to use analytics and the calculus. Where mathematics is taught with reference to its applications in science, as Professor Perry would have it, there would be opportunity for the practical applications of certain phases of advanced mathematics. This seems wise provided it is used as a tool, and students are not required to master its logical associations. Such an addition to the subject-matter of high school mathematics adds nothing to the field of synthetic geometry per se.

5. The enrichment of the subject-matter of geometry by the introduction of modern geometry.—In recent years the text-books on geometry have added some phases of modern synthetic geometry. Regarding the development of modern geometry, Professor David Eugene Smith says:

"The nineteenth century has seen a notable increase of interest in the geometry of the circle and the straight edge, a geometry which can, however, hardly be called elementary in the ordinary sense. France has been the leader in this phase of the subject, with England and Germany following. Carrying out the suggestion made by Desargues in the seventeenth century, Chasles, about the middle of the nineteenth century, developed the theory of anharmonic ratio, making this the basis of what may be designated modern geometry. Brocard, Lemoine, and Neuberg have been largely instrumental in creating a geometry of the circle and the triangle, with special reference to certain interesting angles and points. How much of all this will find its way into elementary text-books of the next generation, replacing, as it might safely do, some of the work which we now give, it is impossible to say. The teacher who wishes to become familiar with the elements of this modern advance could hardly do better than read Casey's Sequel to Euclid (London, fifth edition, 1888). .

"Among the improvements which affect the teaching of elementary geometry to-day, a few deserve brief mention. Among these is the contribution of Möbius on the opposite senses of lines, angles, surfaces, and solids; the principle of duality as

given by Gergonne and Poncelet; the contributions of De Morgan to the logic of the subject; the theory of transversals as worked out by Monge, Brianchon, Servois, Carnot, Chasles, and others; the theory of the radical axis, a property discovered by the Arabs, but introduced as a definite concept by Gaultier (1813) and used by Steiner under the name of 'line of equal power'; the researches of Gauss concerning inscriptible polygons, adding the 17- and 257-gon to the list below the 1000-gon; . . . and the researches of Muir on stellar polygons. .

"1

Our best texts to-day have included some little of the subjectmatter of modern synthetic geometry, but usually in the form of isolated propositions. The theorems of Menelaus, Carnot, and Pascal are of sufficient importance to be brought into the elementary work. We also usually find some exercises on the radical axis, and the "nine-point circle" is assigned as an original. A glance at Casey's "A Sequel to Euclid" will show what a large number of theorems and exercises we teach to-day which are not found in Euclid.2 All this of course is not classed as "Modern Geometry." To what extent any systematized treatment of modern geometry should enter the school work, no definite answer can yet be given.

3

Two assumptions in Euclidean geometry mark points of departure for other systems of geometry. One of these is the postulate that two intersecting straight lines cannot inclose a space. If this be denied it is possible to work out a logical geometry of the plane which has some characteristics in common with the geometry of the sphere. Such a system has been called elliptic geometry. The parallel axiom, which is equivalent to assuming that two intersecting straight lines cannot both be

1

4

5

Smith, The Teaching of Elementary Mathematics, pp. 231-232. For a more complete treatment, see Smith, History of Modern Mathematics.

2 For a history of the "nine-point circle," see Mackay, History of the Nine-point Circle, Edinb. M. S. Proc., XI, pp. 19-57. For the history of other familiar problems and theorems not given by Euclid, see articles by Professor Mackay in Edinb. M. S. Proc., V, VIII, IX, and XX.

3 Euclid himself did not employ this, but tacitly assumed it.

* Also called Riemannian geometry on account of its great development by Riemann.

5 The equivalent of this appears as Euclid's 5th postulate. It is sometimes called his 11th (or 12th) axiom.

parallel to a given straight line, marks another point of departure from our ordinary geometry. The denial of this axiom leads to the non-Euclidean geometry of Lobachevski and Bolyai (cir. 1825). Saccheri (1733), however, was the first to suggest investigating the subject in a scientific way.1

Should non-Euclidean geometry find a place in the elementary teaching? The teacher should be familiar with the subject for the sake of its bearing on Euclidean geometry and for the interest which it might arouse in the regular work, but young minds are not interested in the foundations of mathematics.

6. The nature of the text-book.-Texts of elementary geometry which were essentially academic began to appear in the eighteenth century. The nineteenth century has produced a great number of these, while the last ten years has seen an enormous production of texts, the character of which shows that there is at hand a general awakening to the need of better teaching.

Up to the last century, the geometries still copied Euclid in one respect. All of the propositions were explained, thus leaving little chance for the originality of the pupil. In recent times, the originals ("riders" of the English) have occupied a prominent place in the books. Recent geometries place these, to a great extent, as exercises among the proved propositions, but we still have a few texts that put them at the ends of the different books. Some texts even have placed as exercises, to be worked out by the students, propositions fully explained by Euclid. Why should not the simple theorem, "If two lines are parallel to a third line, they are parallel to each other," be left to the student for proof? On the other hand the student might well have "hints" on the more difficult originals.

Euclid first set us the example of giving demonstrations in full. According to De Morgan, we have gone beyond the "Elements" in this respect. In the sixteenth century the idea was current that Euclid gave only the enunciations and that his commentators later supplied the proofs. Many of the books of that century copied what they thought was the real plan of

1 Smith, History of Modern Mathematics, p. 566. Also Loria, Della varia fortuna di Euclide. For a more complete historical account, see Stäckel und Engel, Die Theorie der Parallelinien von Euclid bis auf Gauss. The works of Lobachevski and Bolyai have been made accessible in English by Professor George B. Halsted. See his Lobachevski's Non-Euclidean Geometry and Bolyai's Science Absolute of Space.

Euclid. The French texts perhaps come nearer to the “Elements" in their easy, essay style. The opposite to this essay style is seen in the form of demonstration adopted in some of our American texts, where the proof is outlined under various "steps." Some texts have adopted a "developmental" method whereby the student supplies the answers to a series of questions. As Professor Smith points out, "the printed questions usually admit of but a single answer each, and hence they merely disguise the usual formal proof." Provision should be made, however, for developmental work. Frequently a suggestion may be made regarding the nature of the proof, the student completing it. There should be completed proofs also to serve as models, especially as phases of new work enter.

The modern text-book should put the teacher and pupil not only in touch with the best methods, but should bring into relief other improvements that affect the teaching to-day. This will necessitate some attention to certain important theorems not found in Euclid and a recognition of such conceptions as the one-to-one correspondence between algebra and geometry, the law of converse, and reciprocal theorems. Regarding the elements which should be included in a text-book on geometry, Professor Smith says there should be "(1) A sequence of propositions which is not only logical, but psychological; not merely one which will work theoretically, but one in which the arrangement is adapted to the mind of the pupil; (2) Exactness of statement, avoiding slip-shod expressions as, 'A circle is a polygon of an infinite number of sides,' 'Similar figures are those with proportional sides and equal angles,' without other explanation; (3) Proofs given in a form which shall be a model of excellence for the pupil to pattern after; (4) Abundant exercises from the beginning, with practical suggestions as to methods of attacking them; (5) Propædeutic work in the form of questions or exercises, inserted long enough before the propositions concerned to demand thought—that is, not immediately preceding the author's proof.' In a small way only can the text-book bring about fundamental reforms in the teaching of geometry. It can simplify the logic, it can bring out the significant relations between the parts of the subject-matter, it can introduce new

12

1 Smith, The Teaching of Elementary Mathematics, p. 255.

2 Ibid., p. 255.

material, but it is the trained teacher only who can bring, the materials of the science into vital connection with every-day experiences. At the most the text is a thing of logic. Under the touch of the teacher it may either remain a mere machine or become a living reality to the student.1

7. The question of method.-The methods of attack used in elementary geometry to-day follow the lines laid down by the Greeks, who invented and used the methods of analysis, reductio ad absurdum, and exhaustions. To them is also due the conception of geometric locus, which was used by them rather sparingly in the solution of problems. Modern practice puts more emphasis on the employment of this effective method. It is of particular value in that it is a method of discovery. Many theorems in geometry are proved when the student knows full well that they are true. That such work is necessary is not to be denied, but ample provision should be made for the solution of geometric problems, in particular where the intersection of loci can be employed. Time should not be put on the proving of the necessary theorems on loci to the exclusion of their applications. Above all the discussion should not be omitted in such work. Here the student becomes an investigator in the best sense of the word. For example, in the problem to draw a circle with a given radius tangent to a given circle and a given straight line, the discussion, based on the intersection of loci, is exceedingly rich in material for this kind of work.

cance.

We shall now consider the term method in its broader signifiEver since the time of Socrates, the method of question and answer has found service in the school room. Such a method gives opportunity for a developmental process. At the same time it may be utilized in an extreme dogmatic sense. In the early secondary schools this dogmatism found favor, where rules were valued more than logic.

It is perhaps safe to say that more attention is given to-day to the demonstrative teaching of geometry than ever before. When geometry was first taught, it was in the higher classes of the different institutions. As has been shown, it has been gradually taught to younger students, and pretty much in the

1

1 For mention of worthy present-day texts in geometry, both for the elementary school and the secondary school, see the chapters on geometryin Smith's The Teaching of Elementary Mathematics.