44. The percentage earning ability E, as dependent upon the eyes is given by Magnus as

E = FVNK,

the

where F = functional ability, V = necessary knowledge, K = ability to compete (demand for him), x has one of the values 5, 7, or 10. Compute E for F

=

0.78792, V

=

1, x = 10, K

= 0.39396.

46. The following table indicates the amount of milk delivered at a creamery by different patrons (numbered) during one month:

From these data the amount due each patron is to be computed by multiplying each hundredweight of milk delivered, by the average test and then by the price. Find the amount due each patron.

73. The Slide Rule. The slide-rule is an instrument for carrying out mechanically the operations of multiplication and division. It is composed of two pieces, usually about the shape of an ordinary ruler; one of the pieces (called the slide,) fits in a groove in the other piece. Each piece is marked in divisions

(Fig. 22), such that the distance from one end (e. g., A) is equal to the logarithm of the number marked on it.

To multiply one number (e. g., 2.5) by another (e. g., 2) we

set the point marked 1 on scale B opposite the point marked 2.5 on scale A (see Fig. 23). Then the product appears on scale A

.

FIG. 23

opposite the point 2 on scale B. Thus 5 on scale A lies opposite 2 on scale B in Fig. 23. This follows from the fact that log 2.5 + log 2

=

log 5.

Likewise, if 1 on scale B is set opposite any number a on scale A, then we find opposite any number b on scale B the number ab on scale A.

Divisions can be performed by reversing this process. Thus if b on scale B be set opposite c on scale A, the 1 on scale B will be opposite c/b on scale A.

A little practice with such a slide-rule will make clear the actual procedure in any case.

Scales C and D are made just twice the size of scales A and B. It follows that any number on scale C, for example, is exactly opposite the square of that number on scale A. This facilitates the finding of squares and square roots, approximately.

Scales C and D may be used in place of scales A and B for multiplication and division. Indeed, after some practice, scales C and D will be preferred for this purpose.

More elaborate slide-rules, marked with several other scales,

are for sale by all supply stores. Descriptions of these and full directions for their use will be found in special catalogs issued by instrument makers.

One

A simple slide-rule can be bought at a moderate price. sufficient for temporary practice may be made by the student by cutting out the large figure printed on one of the fly-leaves of this book, and following the directions printed there.

The student should secure some form of slide-rule and he should use it principally in checking answers found by other processes.

As exercises the teacher may assign first very simple products and quotients. When the operation of the slide-rule has been mastered, the student may check the answers to the exercises on p. 86.

CHAPTER V

TRIGONOMETRY

74. Introduction. The sides and angles of a plane triangle are so related that any three given parts, provided at least one of them is a side, determine the shape and the size of the triangle. Geometry shows how, from three such parts, to construct the triangle.

Trigonometry shows how to compute the unknown parts of a triangle from the numerical values of the given parts.

Geometry shows in a general way that the sides and angles of a triangle are mutually dependent. Trigonometry begins by showing the exact nature of this dependence in the right triangle, and for this purpose employs the ratios of the sides.

75. Definitions of Trigonometric Functions. The three sides x, y, r of a right triangle ABC furnish six ratios:

If a second right triangle A1B1C1 be constructed with ang'e Ai equal to angle A, it will be similar to

Let us construct a third right triangle A2B2C2, making angle A2>A and the hypotenuse A2C2 = AC. From the construction

The ratios in this triangle are not equal to

the corresponding ratios in the first triangle.

X2

FIG. 25

B2

The foregoing considerations lead to the conclusion that these ratios depend for their values solely on the angle A, that is they change when A changes and they are constant when A is constant. This dependence is expressed mathematically by saying that the ratios are functions of the angle A. To distinguish them from other functions they are called trigonometric functions. The six trigonometric functions of any acute angle, as A, are denoted and defined as follows.

THE SIX TRIGONOMETRIC FUNCTIONS