International projectionist (July-Dec 1934)

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INTERNATIONAL PROJECTIONIST VOLUME VII NUMBER 1 JULY 1934 MATHEMATICS FOR THE PROJECTIONIST Gordon S. Mitchell Lesson IX. General Review IN CONCLUDING this series of mathematics for the projectionist, we will in this lesson review the subjects of logarithms, the slide rule, and the fundamentals of algebra, — the subjects of addition, subtraction, multiplication and division requiring no further review. In the next and final lesson, we will review the general subject of equations, the solution of simultaneous linear equations, the use of graphs and curves, and the solution of quadratic equations. Logarithms take their importance mainly from the fact that the fundamentals of the design and operation of the slide rule, are based upon logarithmic principles. A logarithm is subdivided into two parts — the characteristic, which is that part of the logarithm to the left of the decimal point, and the mantissa, which is that part of the logarithm to the right of the decimal. While the mantissa is based wholly upon the combination of figures making up the number under consideration, and is obtained from the log tables, the characteristic is entirely independent of the actual figure under consideration, and is based upon the number of whole numbers there are in that figure. For example, the figure 4,345 has four whole numbers, and the characteristic of the log of 4,345, (determined as one less than the number of whole numbers in the figure) would be 3; the characteristic of the log of 547,909 would be 5, while the characteristic of the log of 18 would be 1. To briefly recapitulate the fundamental logarithmic operations: (1) To multiply two numbers together, we add their logarithms, and the anti-log of the sum is the product of our multiplication. (When we have a number which we know to be a logarithm, and wish to determine the figure of which our number is the logarithm, we determine the figure from the log tables by exactly the reverse operation of finding the logarithm of a number, — the figure so obtained being known as an "anti-log".) (2) To divide two numbers, we subtract their logarithms, and the antilog of the difference is the answer to our division problem. (3) To obtain any desired root of a number, we divide the logarithm of that number by the root index, and the antilog of the quotient is the desired root. (4) To raise any number to any desired power, we multiply the logarithm of that number by the exponent, and the anti-log of the product of that multiplication is the correct result. In working out short-cuts for mathematical operations, mathematicians have devised multiplication and division scales, log tables, trigonometric tables, and various types of slide rules. Of the many types of slide rules, the project tionist should have at least a working acquaintance with the log slide rule (based upon logarithmic tables) the type rule most commonly used by engineers. The slide rule consists essentially of two scales, placed so that one will slide upon the other. Although the ordinary inch scale with which we are all familiar is marked off in such a way that consecutive integers are placed equidistant from each other along the length of the scale, on the slide rule the scale is so marked that the distance between each consecutive integer decreases towards the right — the distance between each being based upon a logarithmic scale-. Contrasted to the ordinary measuring scale, which begins at zero, the scale on the slide rule begins at 1. The slide rule is a specialized and [5]