SUMMARY

331. This chapter has taught the meaning of the following words and phrases: formula, solving a formula, evaluating a formula, applying a formula, centigrade, Fahrenheit.

332. A formula is a conveniently abbreviated form of some practical rule of procedure.

333. A clear understanding of a formula implies:

1. An analysis of some arithmetical situation so as to arrive at some rule of procedure.

2. Translating the rule into a formula.

3. The ability to solve for any letter in terms of the other letters in the formula.

4. The ability to apply the formula to a particular problem and to evaluate the formula.

334. The preceding steps were illustrated in detail by applications to interest problems, to problems involving motion, to work problems, to thermometer problems, and to geometric problems.

335. The graphical interpretations suggested economical methods of manipulating a formula. For example:

1. Simple-interest problems were solved by the formulas I= Prt and A= P+Prt.

2. A problem involving uniform motion in a straight line was solved by the formula d = rt.

3. The relation between centigrade and Fahrenheit readings was expressed by the formula C=§(F-32).

336. While the important thing in this chapter is the power of manipulating and evaluating a formula, the student was given the 'meaning of most of the formulas in order to have him realize from the very outset that both the formulas and their manipulation refer to actual situations.

HISTORICAL NOTE. The development of the formula belongs to a very late stage in the development of mathematics. It requires a much higher form of thinking to see that the area of any triangle can be expressed by A than to find the area of a particular

= ab

lot whose base is two hundred feet and whose altitude is fifty feet. Hence, it was very late in the race's development that letters were used in expressing rules.

The early mathematicians represented the unknown by some word like res (meaning "the thing "). Later, calculators used a single letter for the unknown, but the problems still dealt with particular cases. Diophantus, representing Greek mathematics, stated some problems in general terms, but usually solved the problems by taking special cases.. Vieta used capital letters (consonants and vowels) to represent known and unknown numbers respectively. Newton is said to be the first to let a letter stand for negative as well as positive numbers, which greatly decreases the number of formulas necessary.

While the race has had a difficult time discovering and understanding formulas, it takes comparatively little intelligence to use a formula. Many men in the industrial world do their work efficiently by the means of a formula whose derivation and meaning they do not understand. It is said that even among college-trained engineers only a few out of every hundred do more than follow formulas or other directions blindly. Thus, it appears that for the great majority only the immediately practical is valuable. However, we can be reasonably sure that no one can rise to be a leader in any field by his own ability without understanding the theoretical as well as the practical.

The formula is very important in the present complex industrial age. A considerable portion of the necessary calculation is done by following the directions of some formula. Therefore to meet this need the study of the formula should be emphasized. In discussing the kind of mathematics that should be required Professor A. R. Crathorne (School and Society, July 7, 1917, p. 14) says: "Great emphasis would be placed on the formula, and all sorts of formulas could be brought in. The popular science magazines, the trade journals and catalogues, are mines of information about which the modern boy or girl understands. The pupil should think of the formula as an algebraic declarative sentence that can be translated

into English. The evaluation leads up to the tabular presentation of the formula. Mechanical ability in the manipulation of symbols should be encouraged through inversion of the formula, or what the Englishman calls 'changing the subject of the formula.' We have here also the beginning of the equation when our declarative sentence is changed to the interrogative."

Archimedes (287-212 B.C.), a great mathematician who studied in the university at Alexandria and lived in Sicily, loved science so much that he held it undesirable to apply his information to practical use. But so great was his mechanical ability that when a difficulty had to be overcome the government often called on him. He introduced many inventions into the everyday lives of the people.

His life is exceedingly interesting. Read the stories of his detection of the dishonest goldsmith; of the use of burning-glasses to destroy the ships of the attacking Roman squadron; of his clever use of a lever device for helping out Hiero, who had built a ship so large that he could not launch it off the slips; of his screw for pumping water out of ships and for irrigating the Nile valley. He devised the catapults which held the Roman attack for three years. These were so constructed that the range was either long or short and so that they could be discharged through a small loophole without exposing the men to the fire of the enemy.

When the Romans finally captured the city Archimedes was killed, though contrary to the orders of Marcellus, the general in charge of the siege. It is said that soldiers entered Archimedes' study while he was studying a geometrical figure which he had drawn in sand on the floor. Archimedes told a soldier to get off the diagram and not to spoil it. The soldier, being insulted at having orders given to him and not knowing the old man, killed him.

The Romans erected a splendid tomb with the figure of a sphere engraved on it. Archimedes had requested this to commemorate his discovery of the two formulas: the volume of a sphere equals twothirds that of the circumscribing right cylinder, and the surface of a sphere equals four times the area of a great circle. You may also read an interesting account by Cicero of his successful efforts to find Archimedes' tomb. It will be profitable if the student will read Ball's "A Short History of Mathematics," pp. 65-77.

CHAPTER XII

FUNCTION; LINEAR FUNCTIONS; THE RELATED IDEAS OF FUNCTION, EQUATION, AND FORMULA INTERPRETED GRAPHICALLY; VARIATION

337. Function the dependence of one quantity upon another. One of the most common notions in our lives is the notion of the dependence of one thing upon another. We shall here study the mathematics of such dependence by considering several concrete examples.

EXERCISES

1. Upon what does the cost of 10 yd. of cloth depend? 2. If Resta drives his car at an average rate of 98.3 mi. per hour, upon what does the length (distance) of the race depend?

3. A boy rides a motor cycle for two hours. Upon what does the length of his trip depend?

4. How much interest would you expect to collect in a year on $200?

5. Upon what does the length of a circular running track depend?

6. A man wishes to buy wire fencing to inclose a square lot. How much fencing must he buy?

7. State upon what quantities each of the following depends: (a) The amount of sirloin steak that can be bought for a dollar. (b) The number of theater tickets that can be bought for a dollar.

(c) The height of a maple tree that averages a growth of 4 ft. per year.