equation of fractions, solve a quadratic equation, plot the graph of an equation, etc., as a test of their ability to state in good English the way to get results. Our students learn a certain rigmarole which kills the true spirit needed in any mathematical endeavor. Teaching mathematics by rigmarole is the easy way. It develops no power in people for clear, concise thinking. This kind of teaching makes pupils as well as the general public opponents of our high school mathematics.

Teachers of mathematics need to realize that their problem is a very hard one. They need to give thought to it. More concentrated effort in our teaching will lead to better results, will remove many of the objections lodged against the science into whose rooms every student should look, many of whom will want to enter and contribute to its development.

THE DICTIONARY OF GEOMETRY.

BY CHARLES H. SAMPSON

Boston, Mass.

The dictionary of geometry is sadly neglected both by teachers and pupils. It should not be. One can hardly learn to write if the dictionary of the language that he speaks is neglected. One can hardly be expected to think intelligently in geometry if the fundamental truths (the dictionary) are not clearly understood.

A page of definitions or rules is more or less uninteresting reading for the average pupil. There is little to create or promote a lively interest there. And yet, no pages in the text need emphasis more than these. They are the dictionary of the subject.

Proofs or theorems occupy, and rightly so, a most important place in the study of the subject of geometry. A proof, if it be complete, requires statements of fact for which reasons must be given. There is always a "why" to be answered. A proof without reason, is worthless. Correct reasoning depends very largely upon a sound and broad knowledge of the geometrical dictionary.

"The radius perpendicular to a chord bisects it and the arc which it subtends." Here is a short theorem but even here, in spite of the brevity of it, several definitions must be understood if the proof is to be worked out successfully. What is a radius? What is the meaning of perpendicular? What is the meaning of chord? To bisect a chord is to do what to it? What now, does that word "subtends" indicate? Always there is an oppor

tunity to test one's knowledge of the geometrical dictionary

A good place-in fact, the best place to start the geometrical ball rolling is in the last two years of the grammar grades (the first two years of the junior high school). Children of the age. represented here can visualize and understand the construction of such things as squares, rectangles, triangles, circles, tangents, etc. They can also understand and become proficient in applying the formulas that are used for finding the areas and perimeters of these common figures. To do this at this time is opening the pages of the geometrical dictionary to them. They will enjoy the new knowledge acquired and later on they will profit greatly by it.

It is true, of course, that all of this work will have to be reviewed at a later period but if the foundation is built at a time when the mind is more receptive and keen to learn than is perhaps the truth later on, the review will be easier and of greater value.

There are many ways in which to learn the dictionary of geometry, depending very largely on the way that the subject is taught. Here is a chance to make a teaching problem an interesting one.

The geometrical terms should not be learned as one might learn a poem. No laborious and tiresome memory work should be necessary in the sense generally thought of. Let there always be an illustration at hand. If a rectangle is a plane figure bounded by four sides all the angles of which are right angles, let there be a cardboard figure to illustrate. Children absorb much visually. Let the eye see the definition. The right angles should be pointed out; the fact that one side is longer than the other should be indicated. Explain why this figure is called a rectangle.

Where areas are to be found and perimeters determined there ought to be a definite illustration to tie the mental thought and the visual thought together. The young student is slowly but surely learning his geometrical dictionary.

It is a practical impossibility to teach geometry successfully if the illustrations are neglected. A geometry class room without a blackboard in it is a good way from being ideal. One of the most important and necessary features of such a room would have been neglected if such an aid to teaching were missing. The teacher should use a lot of chalk when teaching rules and definitions. Illustrations are needed if the geometrical dictionary is to be mastered.

TANGENT LINES AMONG THE GREEKS.
BY FLORIAN CAJORI

University of California, Berkeley, Calif.

Some time in the future I may discuss Professor G. A. Miller's numerous and interesting criticisms of my histories of mathematics, which have appeared in this journal, with the view of determining which of his criticisms are correct and which are false. In the present note I wish simply to enlarge upon his comment on the fundamental question of tangent lines. Professor Miller refers to my History of Mathematics, 1919, and says:1 "One would probably not regard a statement appearing on page 163 of this history as reasonably accurate. It is here stated that Roberval 'broke off from the ancient definition of a tangent as a straight line having only one point in common with a curve.' It is evident that the ancients might have thought that a tangent line at the point of tangency had only one point in common with the curve, and that they might have concluded that a necessary and sufficient condition that a stright line is tangent to a circle or to an ellipse is that it has one and only one point in common with such a curve but they must have noticed that such a condition is not sufficient to insure that a straight line is tangent to such a curve as the cissoid of Diocles, for instance."

The claim made here that my history is "probably not reasonably accurate" is based on a mere guess on the part of the critic as to what the Greeks might, perhaps, have thought. The critic refers to the cissoid, but he is unable to point out that the Greeks ever constructed, or tried to construct, tangents to the cissoid or that they considered the drawing of tangents at cuspidal points. There is no evidence to show that the drawing of tangents at singular points of curves and the special precautions necessary in treating that subject received the attention. of antiquity. Really, Professor Miller's comment is an artistic piece of fiction. What we do know is that the Greeks constructed tangents to circles, to conic sections and to certain other curves having no singular points. For these curves the point of view stated in my history is necessary and sufficient. If one examines the proof of Euclid III, 16, it is evident, not that Euclid "might have thought that a tangent line at a point of tangency had only one point in common with the curve," but that Euclid actually and truly established this to be the case for the circle. If further evidence is desired on this point, it will be found in

This journal, March, 1922, p. 278.

Euclid III, 2: "If any two points be taken in the circumference of a circle, the straight line which joins them shall fall within the circle." Professor Miller's criticism is wholly unjustifiable.

The concept that a tangent touches a curve at only one point was with Euclid a logical necessity. For suppose the tangent touched the curve at two distinct fixed points; these two points could not be consecutive points for the reason that no such points exist on a continuous curve. Between any two points there exist other points on such a curve. If the two points are not consecutive, how was Euclid to distinguish between a tangent line and a secant? A secant may cut a curve in non-consecutive points.

The statement quoted from my History, was made in order to contrast the tangent concept of Roberval and some of his contemporaries with that of the Greeks. Roberval's assumption of two points of contact rested on a crude intuition resembling that which regarded a curve as a polygon. More recently, logical demands have led to the abandonment of Roberval's concept and to a return to that of the ancients, according to which a tangent cannot have more than one point in common with a curve at a point of tangency. Most of the modern processes of determining tangents rest on the theory of limits.

A SCORE CARD FOR JUDGING THE VALUE OF GENERAL SCIENCE TEXT-BOOKS.

BY ALLAN PETERSON,

East High School, Des Moines, Iowa.

During the present year it became necessary in the Des Moines schools to adopt a number of new texts in various subjects. To accomplish this Superintendent J. W. Studebaker and his administrative assistants devised a plan of procedure somewhat more systematic and professional than the ordinary.

At the head of the entire work of text-book selection was a General Administrative Committee composed of the superintendent director of research, supervisors and high school principals. This committee outlined the general plan of procedure and formulated the main divisions to be used by all committees on their score cards together with their definite values. These values were derived experimentally by securing the average judgments of eighty teachers, principals and supervisors in the Des Moines schools.

For each subject in which a text was to be selected there was a special Text Book Committee composed of four teachers familiar with that work. These committees first prepared their score cards which they submitted to the General Committee, or in some cases to special Curriculum Committees, for approval. They then examined the various texts submitted, scored them and prepared a final report and summary which was submitted to the General Committee, and through them to the Board of Education.

Publishing houses had previously been notified of the plan and dates were arranged for their representatives to appear before the various committees to explain the merits of their books. No member of a committee was interviewed personally.

Each committee member then scored each text, after which the committee met, compared scores and agreed upon a committee score on each item for each text. A summary of the total committee score for each text was made as well as any recommendations the committee felt necessary. This report together with all individual score sheets was filed with the General Committee.

The General Science Committee was composed of Miss Margaret Brick, of West High, Mr. S. L. Thomas of North High, Mr. C. B. Houser, of East High, with the writer as chair

man.

The score card below is not submitted with the idea that it even approaches perfection, but with the hope that it may be suggestive and helpful to others who may be faced with the problem of text book selection. As intimated above the five general divisions and their assigned values were fixed by the General Committee. This was also true for all of Topic V which was the same for all texts. The sub-topics of the first four divisions was the work of the General Science Text Book Committee.

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