analysis" (Scientific Method, F. W. Westaway, p. 412). Deduction is the method for presenting final results where the false starts, the wrong suggestions, the study of special cases, and the unsuccessful trials of discovery have been left out; in presenting proofs in deductive form all the scaffolding of discovery has been removed. If students see only polished deductive proofs in the textbooks, and neither originate a proof by themselves nor witness the teacher discover a proof, there should be no surprise at their lack of ability in the discovery of proofs.

As suggested at the beginning of this paper, the first great difficulty in beginning demonstrative geometry is to make the students see the purpose of it, to make them feel what we are driving toward. The poorest possible beginning is in attempting to prove theorems which seem obviously true to the students. We should early make them see that we are trying to give valid reasons for our beliefs about geometric things. Simply stated, a proof consists in the production of facts as sufficient reasons to support some conclusion, implicitly challenging a denial of either the truth or the relevance of these reasons. The first step is to get the students to make some statements which they believe to be true about some figures they have drawn; these can be brought out by good questions from the teacher. The second step is to call for reasons for their belief, and the third step is for the teacher to attack and show insufficient some reason given. In the beginning, obvious reasons should be accepted without discussion, meanwhile watching for an incorrect reason to attack. Probably the first wrong inference will be that some lines are equal or parallel because they look equal or parallel. That appearances are often deceptive may well be shown by certain optical illusions, which the teacher keeps at hand for such purpose. (See 1908 Proceedings of the Central Association of Science and Mathematics Teachers, p. 177, and WentworthSmith Plane Geometry, p. 15, for useful optical illusions.) Likewise measurement always appeals to beginners as a good reason for the truth of equalities in geometry. Although often using measurements to suggest truths, yet students must early be convinced of the difficulty of making accurate drawings, that measurements are at best only approximations, and that more simple and precise tests for equality are needed. The writer feels that the difficulty of making accurate measurements, their approximate character, and the feeling of repulsion which the abler students manifest for too much measuring, is not sufficiently

appreciated by teachers. In an informal, conversational way students can be led to see weakness in their old methods of reasoning and to feel a need for better reasons for their beliefs; under a little skilful guiding they soon find delight in seeking reasons for their beliefs which will stand up against all attacks upon either their truth or relevance. Only after the students have attained some appreciation of the purpose of proofs should they be led to put their final results into the formal, precise, and elegant deductive form.

Having considered general principles, we shall now give them concrete application. The students should come to the first class provided with paper, pencil, ruler, compass, and protractor. Through oral directions teach the following constructions: (1) To bisect a given line segment.

(2) To bisect a given arc of a circle. (3) To bisect an angle.

(4) To erect a perpendicular to a line at a point in the line.. (5) To drop a perpendicular to a line from a point without the line.

(6) To trisect a right angle.

(7) To inscribe a circle in a triangle.

(8) To pass a circle through three points not in a straight

line.

(9) To draw a line parallel to a given line through an exterior point.

Make no attempt here to give the usual proofs of these constructions; if students question their truth give them approximate verification by measurement. Next have the students write a description of each construction in a notebook. Thus for (3) they would write something like the following:

To bisect any triangle BAC.

With A as a center and any radius r, describe an arc intersecting the angle sides in M and L. With M as a center describe an arc of a circle; with L as a center, and with the same radius describe another arc intersecting the latter arc in a point K. Join K to A and the line AK is the required bisector.

While learning these constructions and writing up their descriptions the students are introduced to the symbolism of geometry; they learn how to read lines and angles; use the simpler single letter notation wherever possible. While this work is being done in class periods under the direction of the teacher, assign for home work related construction problems, such as,

"Construct a square," "Inscribe a square in a circle," "Construct an inscribed hexagon by bisecting the sides of an inscribed triangle," etc. Geometric designs involving the fundamental constructions may be used for home work. (Hedrick's Constructive Geometry will suggest home work to the teacher.)

During all this work the teacher should have before him a list of geometric terms which he expects to fix in the students' minds; such as point (having position only), straight line (length only, no breadth or thickness), plane, solid, rectilinear figure, parallel lines, curves, circle, radius, diameter, chord, angle, right angle, acute, obtuse, complement, supplement, interior, exterior, vertical angles, bisect, trisect, etc. Develop the meaning of each of these terms by means of questions and illustrations; ask the students to point out examples of these terms on their figures; ask them to find examples in objects about them; translate old expressions, like "corners fit," "square corner," etc., into geometric language; compare their definitions with dictionary and text definitions; make these terms meaningful by continued and careful use. The originality and questioning skill of the teacher will direct this work of fixing fundamental notions and correlating them with their daily experiences.

If we are to avoid attempting to prove that which seems obvious to the students, there must be a broader foundation of assumptions than is common in beginning geometry. So the teacher should also have before him a carefully prepared list of obvious geometric facts which he expects to use as assumptions in later proofs, such as the following: (1) All straight angles are equal; (2) All right angles are equal; (3) The shortest path between two points is the line segment joining the points; (4) Two distinct points determine a straight line; (5) Any side of a triangle is less than the sum of the other two sides; (6) A diameter bisects a circle; (7) A straight line intersects a circle at most in two points; (8) Complements (or supplements) of equal angles are equal; (9) Two lines parallel to the same line are parallel to each other, etc. (The introductory chapter in Young and Schwartz's, or Wentworth-Smith's Geometry, or the Report of the National Committee of Fifteen on Geometry, p. 20, will assist a teacher in making such a list of fundamental geometric assumptions.) In most cases the emphatic statement of these fundamental facts is sufficient to bring conviction of their truth: informal discussion will bring acceptance of the truth of all; draw figures illustrating these fundamental facts; seek illustra

tions in everyday experiences. After oral discussion has fixed. these facts they should be carefully listed in the students' notebooks for future reference.

In order to make progress toward proving theorems, next develop with the class proofs for the simple theorems: (1) Vertical angles are equal; and (2) The bisectors of vertical angles form one and the same straight line.

Next take up the construction of triangles, from given data, as from two sides and their included angle, considering the usual four cases. Consider also their application to right triangles with necessary modifications in statement.

From this consideration of triangle construction make the transition to the formal proofs of congruent triangles as given in most texts, and begin from this point on to make use of the textbooks. By certain judicious omissions it is believed that this introductory oral work can be here connected up with any one of the textbooks now in common use.

Within the space here available, it is impossible to give all the details of the plan for introductory work here presented, but it is hoped that enough suggestions have been given to enable the teacher to complete all the details and adapt the scheme to his particular class and text. The length of time given to such an introduction must be determined by the preparation and the particular needs of each class. It may be quite short if students have had preliminary courses in constructive or concrete geomtry. The prolonging of such informal work when not needed. will tend to disgust the abler students.

The writer knows of no text on demonstrative geometry which gives a sufficiently informal introduction, and doubts whether such work can be put into a text with as effective results as when presented in class by the teacher. Some texts (Wentworth-Smith, Wells-Hart, Young-Schwartz, Long-Brenke, for illustration) however have greatly improved the introductory work.

This is one way of introducing demonstrative geometry. I hope the suggestions will assist the skilful teacher in making the study of geometry real, interesting, and valuable.

THE RELATION OF HIGH SCHOOL CHEMISTRY TO GENERAL CHEMISTRY IN COLLEGES.1

By C. L. FLEECE,

Princeton University, Princeton, N. J.

In teaching general chemistry in colleges we cannot ignore the fact that a considerable portion of our students have studied chemistry in high school or preparatory school and that this elementary training will have an influence on their advancement. There is no doubt that we can teach these students more chemistry than beginners. But it is a problem to determine the exact advantage they may have and what credit may be given to their previous study of the subject.

This problem affects the colleges more seriously than the high schools, since a very small percentage of the high school graduates enter college and less than half of these have offered chemistry as an entrance subject. In an indirect way, however, it does have an important bearing on high school chemistry. If we cannot take high school graduates and use their knowledge of chemistry as a basis of a more advanced course, there is something radically wrong with their elementary training. And when I say this I would not have you think that high school chemistry should be taught as a preparatory course to future chemistry in college. In fact our elementary science courses in colleges, as well as in secondary schools, are frequently to be criticized because we look upon our pupils as future scientists and may teach them little that is of general educational value. Those who teach elementary science in college are naturally inclined to consider that the ideal methods of teaching high school science are those that are miniatures of their own pet schemes. But we should expect high school students to come to us with a training meant to fit them for after life. It is the duty of the high schools to supply this and, if we put them in an advanced course in general chemistry, it should be one based on this idea. It is the only logical basis for such a course and probably the best that the high schools could possibly give us.

Very few colleges give two distinct courses in general chemistry, one for beginners and one for men who have already had an elementary training in the subject. Columbia has such an arrangement. At Harvard I believe the laboratory work is different. At Princeton we have different quiz groups, the lab

Part of an address before the Chemistry Section of the New Jersey Science Teachers' Association at Newark, December 9, 1916.