II. FUNDAMENTAL PROPOSITIONS IN MENSURATION.

26. The lateral area of a prism or a circular cylinder is equal to the product of a lateral edge or element, respectively, by the perimeter of a right section.

27. The volume of a prism (including any parallelopiped) or of a circular cylinder is equal to the product of its base by its altitude.

28. The lateral area of a regular pyramid or a right circular cone is equal to half the product of its slant height by the perimeter of its base.

29. The volume of a pyramid or a cone is equal to one-third the product of its base by its altitude.

30. The area of a sphere.

31. The area of a spherical polygon.

32. The volume of a sphere.

III. SUBSIDIARY THEOREMS.

33. If from an external point a perpendicular and obliques are drawn to a plane, (a) the perpendicular is shorter than any oblique; (b) obliques meeting the plane at equal distances from the foot of the perpendicular are equal; (c) of two obliques meeting the plane at unequal distances from the foot of the perpendicular, the more remote is the longer.

34. If two lines are cut by three parallel planes, their corresponding segments are proportional. 35. Between two lines not in the same plane there is one common perpendicular, and only one.

36. The bases of a cylinder are congruent.

37. If a plane intersects a sphere, the line of intersection is a circle.

38. The volume of two tetrahedrons that have a trihedral angle of one equal to a trihedral angle of the other are to each other as the products of the three edges of these trihedral angles.

39. In any polyhedron the number of edges increased by two is equal to the number of vertices increased by the number of faces.

40. Two similar polyhedrons can be separated into the same number of tetrahedrons similar each to each and similarly placed.

41. The volumes of two similar tetrahedrons are to each other as the cubes of any two corresponding edges.

42. The volumes of two similar polyhedrons are to each other as the cubes of any two corresponding edges.

43. If three face angles of one trihedral angle are equal, respectively, to the three face angles of another the trihedral angles are either congruent or symmetric.

44. Two spherical triangles on the same sphere are either congruent or symmetric if (a) two sides and the included angle of one are equal to the corresponding parts of the other; (b) two angles and the included side of one are equal to the corresponding parts of the other; (c) they are mutually equilateral; (d) they are mutually equiangular.

45. The sum of any two face angles of a trihedral angle is greater than the third face angle.

46. The sum of the face angles of any convex polyhedral angle is less than four right angles.

47. Each side of a spherical triangle is less than the sum of the other two sides. 48. The sum of the sides of a spherical polygon is less than 360°.

49. The sum of the angles of a spherical triangle is greater than 180° and less than 540°.

50. There can not be more than five regular polyhedrons.

51. The locus of points equidistant (a) from two given points; (b) from two given planes which intersect.

IV. SUBSIDIARY PROPOSITIONS IN MENSURATION.

52. The volume of a frustum of (a) a pyramid or (b) a cone.

53. The lateral area of a frustum of (a) pyramid or (b) a cone of revolution. 54. The volume of a prismoid (without formal proof).

Chapter VII.

THE FUNCTION CONCEPT IN SECONDARY-SCHOOL

MATHEMATICS.1

In Chapter II, and incidentally in later chapters, considerable emphasis has been placed on the function concept or, better, on the idea of relationship between variable quantities as one of the general ideas that should dominate instruction in elementary mathematics. Since this recommendation is peculiarly open to misunderstanding on the part of teachers, it seems desirable to devote a separate chapter to a rather detailed discussion of what the recommendation means and implies.

It will be seen in what follows that there is no disposition to advocate the teaching of any sort of function theory. A prime danger of misconception that should be removed at the very outset is that teachers may think it is the notation and the definitions. of such a theory that are to be taught. Nothing could be further from the intention of the committee. Indeed, it seems entirely safe to say that probably the word "function" had best not be used at all in the early courses.

What is desired is that the idea of relationship or dependence between variable quantities be imparted to the pupil by the examination of numerous concrete instances of such relationship. He must be shown the workings of relationships in a large number of cases before the abstract idea of relationship will have any meaning for him. Furthermore, the pupil should be led to form the habit of thinking about the connections that exist between related quantities, not merely because such a habit forms the best foundation for a real appreciation of the theory that may follow later, but chiefly because this habit will enable him to think more clearly about the quantities with which he will have to deal in real life, whether or not he takes any further work in mathematics.

Indeed, the reason for insisting so strongly upon attention to the idea of relationships between quantities is that such relationships do occur in real life in connection with practically all of the quantities with which we are called upon to deal in practice. Whereas there can be little doubt about the small value to the student who does

1 The first draft of this chapter was prepared for the national committee by E. R. Hedrick, of the University of Missouri. It was discussed at the meeting of the committee, Sept. 2-4, 1920; revised by the author, and again discussed Dec. 29-30, 1920, and is now issued as part of the committee's report.

not go on to higher studies of some of the manipulative processes criticized by the national committee, there can be no doubt at all of the value to all persons of any increase in their ability to see and to foresee the manner in which related quantities affect each other. To attain what has been suggested, the teacher should have in mind constantly not any definition to be recited by the pupil, not any automatic response to a given cue, not any memory exercise at all, but rather a determination not to pass any instance in which one quantity is related to another, or in which one quantity is determined by one or more others, without calling attention to the fact, and trying to have the student "see how it works.". These instances occur in literally thousands of cases in both algebra and geometry. It is the purpose of this chapter to outline in some detail a few typical instances of this character.

RELATIONSHIPS IN ALGEBRA.

The instance of the function idea which usually occurs to one first in algebra is in connection with the study graphs. While this is natural enough, and while it is true that the graph is fundamentally functional in character, the supposition that it furnishes the first opportunity for observing functional relations between quantities betrays a misconception that ought to be corrected.

1. Use of letters for numbers.-The very first illustrations given in algebra to show the use of letters in the place of numbers are essentially functional in character. Thus, such relations as I=prt and A = r2, as well as others that are frequently used, are statements of general relationships. These should be used to accustom the student not only to the use of letters in the place of numbers and to the solution of simple numerical problems, but also to the idea, for example, that changes in r affect the value of A. Such questions as the following should be considered: If r is doubled, what will happen to A? If p is doubled, what will happen to I? Appreciation of the meaning of such relationships will tend to clarify the entire subject under consideration. Without such an appreciation it may be doubted whether the student has any real grasp of the

matter.

2. Equations.-Every simple problem leading to an equation in the first part of algebra would be better understood for just such a discussion as that mentioned above. Thus, if two dozen eggs are weighed in a basket which weighs 2 pounds, and if the total weight is found to be 5 pounds, what is the average weight of an egg? If x is the weight in ounces of one egg, the total weight with the 2-pound basket would be 24x+32 ounces. If the student will first try the effect of an average weight of 1 ounce, of 1 ounces, 2 ounces, 24 ounces, the meaning of the problem will stand out clearly. In

every such problem preliminary trials really amount to a discussion of the properties of a linear function.

3. Formulas of pure science and of practical affairs.-The study of formulas as such, aside from their numerical evaluation, is becoming of more and more importance. The actual uses of algebra are not to be found solely nor even principally in the solution of numerical problems for numerical answers. In such formulas as those for falling bodies, levers, etc., the manner in which changes in one quantity cause (or correspond to) changes in another are of prime importance, and their discussion need cause no difficulty whatever. The formulas under discussion here include those formulas of pure science and of practical affairs which are being introduced more and more into our texts on algebra. Whenever such a formula is encountered the teacher should be sure that the students have some comprehension of the effects of changes in one of the quantities upon the other quantity or quantities in the formula.

As a specific instance of such scientific formulas consider, for example, the force F, in pounds, with which a weight W, in pounds, pulls outward on a string (centrifugal force) if the weight is revolved rapidly at a speed v, in feet per second, at the end of a string of Wv2

When

length r feet. This force is given by the formula F 32 r such a formula is used the teacher should not be contented with the mere insertion of numerical values for W, v, and r to obtain a numerical value for F.

The advantage obtained from the study of such a formula lies quite as much in the recognition of the behavior of the force when one of the other quantities varies. Thus the student should be able to answer intelligently such questions as the following: If the weight is assumed to be twice as heavy, what is the effect upon the force? If the speed is taken twice as great, what is the effect upon the force? If the radius becomes twice as large, what is the effect upon the force? If the speed is doubled, what change in the weight would result in the same force? Will an increase in the speed cause an increase or a decrease in the force? Will an increase in the radius r cause an increase or a decrease in the force?

As another instance (of a more advanced character) consider the formula for the amount of a sum of money P, at compound interest at r per cent, at the end of n years. This amount may be denoted by An. Then we shall have An=P (1+r)". Will doubling P result in doubling An? Will doubling n result in doubling An? Since the compound interest that has accumulated is equal to the difference between P and An, will the doubling of r double the interest? Compare the correct answers to these questions with the answers to the similar questions in the case of simple interest, in

which the formula reads An=P+Prn and in which the accumulated interest is simply Prn.

The difference between such a study of the effect produced upon one quantity by changes in another and the mere substitution of numerical-values will be apparent from these examples.

4. Formulas of pure algebra.—Formulas of pure algebra, such as that for (x+h)2, will be better understood and appreciated if accompanied by a discussion of the manner in which changes in h cause changes in the total result. This can be accomplished by discussing such concrete realities as the error made in computing the area of a square field or of a square room when an error has been made in measuring the side of the square. If x is the true length of the side, and if the student assumes various possible values for the error h made in measuring x, he will have a situation that involves some comprehension of the functional workings of the formula mentioned. The same formula relates to such problems as the change from one entry to the next entry in a table of squares.

A similar situation, and a very important one, occurs with the pure algebraic formula for (x+a) (y+b). This formula may be said to govern the question of the keeping of significant figures in finding the product xy. For if a and b represent the uncertainty in x and y, respectively, the uncertainty in the product is given by this formula. The student has much to learn on this score, for the retention of meaningless figures in a product is one of the commonest mistakes of both student and teacher in computational work.

Such formulas occur throughout algebra, and each of them will be illuminated by such a discussion. The formulas for arithmetic and geometric progression, for example, should be studied from a functional standpoint.

5. Tables.-The uses of the functional idea in connection with numerical computation have already been mentioned in connection with the formula for a product. Work which appears on the surface to be wholly numerical may be of à distinctly functional character. Thus any table, e. g., a table of squares, corresponds to or is constructed from a functional relation, e. g., for a table of squares, the relation y = x2. The differences in such a table are the differences caused by changes in the values of the independent variable. Thus, the differences in a table of squares are precisely the differences between x2 and (x+h)2 for various values of x.

6. Graphs. The functional character of graphical representations was mentioned at the beginning of this section. Every graph is obviously a representation of a functional relationship between two or more quantities. What is needed is only to draw attention to this fact and to study each graph from this standpoint. In addition to this, however, it is desirable to point out that functional