THE GRAPHICAL SOLUTION OF PROBLEMS. BY PAUL H. GEIGER, Ann Arbor, Mich. In the November, 1917, number of SCHOOL SCIENCE AND MATHEMATICS appears a paper, entitled, "Two Uses for Graphs." The first part of this article is concerned with the equation used in solving elementary problems about lenses and mirrors, and the author after presenting a graph of that equation shows how to obtain the approximate numerical answers to typical problems. The graphical method thus set forth has certain pedagogical advantages, in that the subject material is clearly and simply displayed. But the numerical results may be obtained, without calculation, by other methods which are sufficiently general to warrant their value and interest. They were brought to the author's attention by Mr. Herbert Bell. The first is an extension of the idea of the slide rule. The second is a very simple example of the straight line nomogram. In an ordinary slide rule the addition of logarithms is facilitated by two parallel and similar scales which, when they are properly displaced, give a product, just as a sum may be obtained by two ordinary yardsticks. The essential feature of the instrument is that while the ordinary numbers appear along the scales, the distance out to them is proportional, not to the number, but to its logarithm. Such a rule does not help us much with the equation, the origin to any number was proportional to the reciprocal of that number, then two such scales, side by side, would solve a problem of our particular class, just as the ordinary rule gives us a product. Such a scale appears in Figure 1. With a pair of dividers, problems may be solved with this scale alone. For, supposing p is 6 and q is +2, we can with the dividers add the reciprocal of 6 to that of 2, giving the reciprocal of 3/2, which is f. Two such scales, side by side, would be operated just as the oldinary slide rule. The scale may be constructed by actual Naturat Scale computation and measurement, but more simply by projection as is shown in Figure 2. Here the scale of reciprocals appears along AB, natural numbers being set down along CD, and O being the center from which lines are drawn. Evidently, if Dr. Sleator had laid off scales of reciprocals along his axis and then plotted his curve, it would have been a straight line. Evidently, also, the method is not limited in its application to this one type of problem, but suitable scales may be constructed for any equation expressing an additive relation between simple functions. In case the equation to be solved contains a product or a quotient, one obtains by passing to logarithms a new equation which contains an algebraic sum, thereby permitting suitable scales to be constructed. The second method is that of the straight line nomogram. No attempt will be made to give the theory of the nomogram, only a single application which, though originally obtained by using the general method of the nomogram, is so simple that it can be understood without any knowledge of the theory. The formula, B 1 2 B' On two parallel straight lines EB and E'B' (Figure 3) are laid .off scales of reciprocals as explained under the first method, with the exception that the distances are laid off in both directions from the origins O and O' instead of in one direction only. Let AC be a line midway between EB and E'B' and parallel to them. Now if p and q are known we may connect point P (Figure 3) on EB (where the distance OP represents the reciprocal of p) with point Q on line E'B' (where the distance O'Q corresponds to the value of P the reciprocal of q) by a straight line PQ. The distance FD between the intersection of this line with AC and the origin then represents the average length of AC, OP and O'Q, which in mathematical terms is 1/p+1/q 2 This by equation B is equal to 1/2f, so if we lay off on line AC scales similar to those on EB and E'B' but with the unit only one-half as great, we may read directly the solution. Letting 3 C 3455 10 3 EB be the p axis, AC the fƒ axis, and E'B' the q axis, we can find either p, q, or ƒ when the other two are given. laying a straight edge across the two known This is done by points on their corresponding axes and reading the unknown value on the remaining axis. When values are negative, the distances are to be measured below (or to the left of, in case the lines are horizontal) the line OO'. It is evident that the use of these methods would be extremely limited in case only problems having values which are shown on the scale, can be solved. However, in the calculations we may disregard the decimal point and therefore our scales will apply to any values. The location of the decimal point in the answer is usually obvious; if not it may be determined by easy approximations, as it is determined when the ordinary form of slide rule is used. The accuracy of these methods is the same as that of an ordinary slide rule. With scales a foot long, results accurate to three significant figures may be obtained; with longer scales the accuracy can be made as great as we may desire. DAYLIGHT SAVING IN THE UNITED STATES. Teachers College, Cleveland, Ohio. The geography class will find much material for discussion in the provisions of the Daylight Saving Act which has recently been passed by Congress and signed by the President. The student will quickly perceive that this bill is a national affair, and also that a legal status is given the various standard time belts of the United States and Alaska. Daylight saving, in a nutshell, is to advance the standard time one hour during a period of five months every year, beginning at 2 o'clock a. m. of the last Sunday in March, and retarding one hour beginning at 2 o'clock a. m. of the last Sunday in October. There is no suggestion that hours of work be lengthened or shortened. The shortening of the old-time work day of "sun-up to sunset" to the present day of ten hours or eight hours, has resulted in a waste of the morning sunlight hours in the summer months. On a June day the sun rises at 4 a. m. and we start work at 8 a. m. Thus we sleep through hours of sunlight in the morning and have our period of recreation at the close of the day in artificial light. We waste the daylight and use the artificial light and heat which could easily be conserved if this measure became a law. The practical value in the conservation of daylight has been fully established where this plan is in operation in Austria, Denmark, France, England, Italy, Holland, Norway, Portugal, and Sweden. If the student will plat the time of sunrise and sunset for his locality for the entire year, he will readily see how the daylight is thrown away and the artificial light substituted. The Nautical Almanac will give all the necessary data for such a graph. Let the student figure the saving in the electric light bill in his family and city. Careful estimates give $40,000,000 as the amount to be saved in this country in a single year by this plan. France is believed to have saved $10,000,000 each year by this plan, and England about $12,000,000. There is not much but the inertia of tradition to be urged against daylight saving. The entire provisions of the Daylight Saving Act are to be found in the following paragraphs. From a reading of this act and the graph of daylight and night, have the pupil explain the reason for the change of time in March and October. THE DAYLIGHT SAVING ACT. To Save Daylight and to Provide Standard Time for the United States. Section 1. Be it enacted by the Senate and House of Representatives of the United States of America in Congress assembled, That, for the purpose of establishing the standard time of the United States, the territory of continental United States shall be divided into five zones in the manner hereinafter provided. The standard time of the first zone shall be based on the mean astronomical time of the seventy-fifth degree of longitude west from Greenwich; that of the second zone on the ninetieth degree; that of the third zone on the one hundred and fifth degree; that of the fourth zone on the one hundred and twentieth degree; and that of the fifth zone, which shall include only Alaska, on the one hundred and fiftieth degree. That the limits of each zone shall be defined by an order of the Interstate Commerce Commission, having regard for the convenience of commerce and the existing junction points and division points of common carriers engaged in commerce between the several states and with foreign nations, and such order may be modified from time to time. Section 2. That within the respective zones created under the authority hereof the standard time of the zone shall govern the movement of all common carriers engaged in commerce between the several states or between a state and any of the |