no calculation is to be carried out, the problems of real life frequently involve the ability to think correctly about the nature of the relationships which exist between related quantities. Specific mention has been made already of this type of problem in connection with interest on money. In everyday affairs, such as the filling out of formulas. for fertilizers or for feeds or for spraying mixtures on the farm, the similar filling out of recipes for cooking (on different scales from that of the book of recipes), or the proper balancing of the ration in the preparation of food, many persons are at a loss on account of their lack of training in thinking about the relations between quantities. Another such instance of very common occurrence in real life is in insurance. Very few men or women attempt intelligently to understand the meaning and the fairness of premiums on life insurance and on other forms of insurance, chiefly because they cannot readily grasp the relations of interest and of chance that are involved. These relations are not particularly complicated, and they do not involve any great amount of calculation for the comprehension of the meaning and of the fairness of the rates. Mechanics, farmers, merchants, housewives, as well as scientists and engineers have to do constantly with quantities of things, and the quantities with which they deal are related to other quantities in ways that require clear thinking for maximum efficiency.

One element that should not be neglected is the occurrence of such problems in public questions which must be decided by the votes of the whole people. The tariff, rates of postage and express, freight rates, regulation of insurance rates, income taxes, inheritance taxes, and many other public questions involve relationships between quantities-for example, between the rate of income taxation and the amount of the income-that require habits of functional thinking for intelligent decisions. The training in such habits of thinking is therefore a vital element toward the creation of good citizenship.

It is believed that transfer of training does operate between such topics as those suggested in the body of this paper and those just mentioned, because of the existence of such identical or common elements, whereas the transfer of the training given by courses in mathematics that do not emphasize functional relationships might be questionable.

While this account of the functional character of certain topics in geometry and in algebra makes no claim to being exhaustive, the topics mentioned will suggest others of like character to the thoughtful teacher.

It is hoped that sufficient variety has been mentioned to demonstrate the existence of functional ideas throughout elementary algebra and geometry. The committee feels that if this is recognized, algebra and geometry can be given new meaning to many children, and indeed to many educators, and that all students will be better able to control the actual relations which they meet in their own lives.

CHAPTER VIII

Terms and Symbols in Elementary Mathematics'

A. Limitations imposed by the committee upon its work.— The committee feels that in dealing with this subject it should explicitly recognize certain general limitations, as follows:

1. No attempt should be made to impose the phraseology of any definition, although the committee should state clearly its general views as to the meaning of disputed terms.

2. No effort should be made to change any well-defined current usage unless there is a strong reason for doing so, which reason is supported by the best authority, and, other things being substantially equal, the terms used should be international. This principle excludes the use of all individual efforts at coining new terms except under circumstances of great urgency. The individual opinions of the members, as indeed of any teacher or body of teachers, should have little weight in comparison with general usage if this usage is definite. If an idea has to be expressed so often in elementary mathematics that it becomes necessary to invent a single term or symbol for the purpose, this invention is necessarily the work of an individual; but it is highly desirable, even in this case, that it should receive the sanction of wide use before it is adopted in any system of examinations.

3. On account of the large number of terms and symbols now in use, the recommendations to be made will necessarily be typical rather than exhaustive.

B.

I. Geometry

Undefined terms.-The committee recommends that no attempt be made to define, with any approach to precision, terms whose definitions are not needed as parts of a proof.

1 The first draft of this chapter was prepared by a subcommittee consisting of David Eugene Smith (chairman), W. W. Hart, H. E. Hawkes, E. R. Hedrick, and H. E. Slaught.

Especially is it recommended that no attempt be made to define precisely such terms as space, magnitude, point, straight line, surface, plane, direction, distance, and solid, although the significance of such terms should be made clear by informal explanations and discussions. C. Definite usage recommended. It is the opinion of the committee that the following general usage is desirable:

1. Circle should be considered as the curve; but where no ambiguity arises, the word "circle" may be used to refer either to the curve or to the part of the plane inclosed by it.

2. Polygon (including triangle, square, parallelogram, and the like) should be considered, by analogy to a circle, as a closed broken line; but where no ambiguity arises, the word polygon may be used to refer either to the broken line or to the part of the plane inclosed by it. Similarly, segment of a circle should be defined as the figure formed by a chord and either of its arcs.

3. Area of a circle should be defined as the area (numerical measure) of the portion of the plane inclosed by the circle. Area of a polygon should be treated in the same way.

4. Solids. The usage above recommended with respect to plane figures is also recommended with respect to solids. For example, sphere should be regarded as a surface, its volume should be defined in a manner similar to the area of a circle, and the double use of the word should be allowed where no ambiguity arises. A similar usage should obtain with respect to such terms as polyhedron, cone, and cylinder.

5. Circumference should be considered as the length (numerical measure) of the circle (line). Similarly, perimeter should be defined as the length of the broken line which forms a polygon; that is, as the sum of the lengths of the sides.

6. Obtuse angle should be defined as an angle greater than a right angle and less than a straight angle, and should therefore not be defined merely as an angle greater than a right angle.

7. The term right triangle should be preferred to "right-angled triangle," this usage being now so standardized in this country that it may properly be continued in spite of the fact that it is not international. Similarly for acute triangle, obtuse triangle, and oblique triangle.

8. Such English plurals as formulas and polyhedrons should be used in place of the Latin and Greek plurals. Such unnecessary Latin abbreviations as Q. E. D. and Q. E. F. should be dropped.

9. The definitions of ariom and postulate vary so much that the committee does not undertake to distinguish between them.

D. Terms made general.-It is the recommendation of the committee that the modern tendency of having terms made as general as possible should be followed. For example:

1. Isosceles triangle should be defined as a triangle having two equal sides. There should be no limitation to two and only two equal sides.

2. Rectangle should be considered as including a square as a special case.

3. Parallelogram should be considered as including a rectangle, and hence a square, as a special case.

4. Segment should be used to designate the part of a straight line included between two of its points as well as the figure formed by an arc of a circle and its chord, this being the usage generally recognized by modern writers.

E. Terms to be abandoned.-It is the opinion of the committee that the following terms are not of enough consequence in elementary mathematics at the present time to make their recognition desirable in examinations, and that they serve chiefly to increase the technical vocabulary to the point of being burdensome and unnecessary: 1. Antecedent and consequent.

2. Third proportional and fourth proportional.

3. Equivalent. An unnecessary substitute for the more precise expressions "equal in area" and "equal in volume," or (where no confusion is likely to arise) for the single word "equal."

4. Trapezium.

5. Scholium, lemma, oblong, scalene triangle, sect, perigon, rhomboid (the term "oblique parallelogram" being sufficient), and reflex angle (in elementary geometry).

6. Terms like flat angle, whole angle, and conjugate angle are not of enough value in an elementary course to make it desirable to recommend them.