International projectionist (Oct 1931-Sept 1933)

MATHEMATICS FOR THE PROJECTIONIST Siegfried S. Meyers INSTRUCTOR IN PHYSICS, STUYVESANT HIGH SCHOOL, NEW YORK IT is the purpose of this article to excite the interest of the projectionist in an elementary study of mathematics. Explanations of certain equipments, prints, graphs, schematics and the like are often presented to a projectionist who has never had an opportunity of familiarizing himself with the elements of mathematics, the lack of knowledge of which often renders valueless much material which otherwise would offer much of interest. This series of articles will attempt to make up this deficiency. This and succeeding articles will be prepared on the assumption that the reader knows nothing about mathematics. Those who are familiar with some of the material presented herein are asked to bear with their fellow craftsmen and the author until the series is more advanced ; but in any case, even this elementary material will serve this class of readers as a review. In the average textbooks on algebra and geometry is found a mass of data which is of little practical use to the projectionist. For this reason the reproduction herein of a major part of such textbooks would be unjustified. A special effort has been made to deal only with the practical aspects of mathematics, and it is believed that this course will best serve the purpose of these articles. Mathematical Language It is necessary that the reader have an adequate knowledge of the language of mathematics — and by this is meant an understanding of the terms and symbols used, just as in music there is a language comprised of terms and symbols. Let us first consider that which is commonly referred to as a "formula." What is a "formula"? A formula is simply a mathematical equation, which we may consider as a hopper, into which one pours certain information, then turns the crank — and out comes the answer. The mechanical analogy of a formula to a hopper is indeed illustrative. Working out a formula is a process from which the unknown is derived by supplying the known factors. Beyond the use of formulae lie terms like "perpindicular," "bisection," "vertex,' and the like. Let us first familiarize ourselves with these terms. By definition, a "straight line" is a continuous succession of points. "Points" are the boundaries of these lines. A "curve" is spoken of as a series of infinitely small straight lines. An "angle" is the space bounded by two insersecting straight lines. By "bisecting an angle" is meant the cutting of an angle into two smaller equal angles, the sum of which is equal to the original angle. In "trisecting an angle" we do the same thing, but in three equal parts. This latter process is very difficult. The "vertex" of an angle is the point of intersection of the two bounding intersecting lines. "Perpendicular" is a line which is set at right angles to to another straight line. In any circle there are 360 degrees. If two diameters intersect each other within a circle, and do so perpendicularly, we have four right angles included by these diameters (Fig. 1). Ratio and Proportion Ratio and proportion is a common algebraic process knowledge of which is extremely valuable to the projectionist. For example, this process is used to determine the proportionality of two factors, when two other factors vary in some definite ratio. By this is meant that a certain unknown number can be found when three others are known. Example : If a current of one ampere can flow through 10 ohms resistance, how many amperes will flow through 5 ohms resistance? Discussion. 1. The variable terms are the resistances. 2. As the resistance becomes one-half as great, the current becomes twice as great. Solution : 10 ohms is to 5 ohms as X amperes is to 1 ampere, or 10:5 = X:l (No. 1) This may also be written as: 10 X = (No. 2) 5 1 In solving for X, it is necessary that we make X equal to all other terms. In equation No. 1 the colon ( : ) represents the ratio, and is read as follows: "10 is to 5." This means that 10 ohms is compared with 5 ohms. The "equals" sign ( = ) represents the proportion or comparison and is read: "As." Now, reading equation No. 1 in its entirety we say: "10 is to 5 as X is to 1." Equation No. 2 is read the same way, the division sign meaning the same as the colon, namely: "is to." In solving equation No. 1 there are two parts — the "means" and the "extremes." The "means" has reference to the two numbers located on either side of the equals ( = ) sign. The "extremes" are the two numbers at the ends of equation No. 1. We multiply the means by each other, and let them equal the product of the extremes. This gives us: 10 X 1 "Extremes" Or: 5X = 10 10 X = = 2 amperes, 5 Answer. It will be seen that this proportion does not increase directly as the ratio increases. The ratio of the resistances Fisure 1 LINE CURVE VERTEX BISECTING AN ANGLE A CIRCLE HAS 360 DECREES [24]