HARVARD COPYRIGHT 1912 BY AMERICAN SCHOOL OF CORRESPONDENCE Entered at Stationers' Hall, London All Rights Reserved PRACTICAL MATHEMATICS PART I INTRODUCTION No one who is at all acquainted with the demands of the Engineering profession will deny the need of a good foundation in elementary mathematics any more than he will deny the need of a solid underpinning on which to rest the walls of a big business block. The simplest problems of the contractor and workman, such as the number of feet of lumber required for a house, the number of cubic yards of excavation for a ditch or cellar, the proper understanding of plans and specifications, and the laying off of measurements according to these plans, all require a knowledge of this important subject. The size of a concrete retaining wall, the dimensions of a girder for a steel structure, the amount of iron in the field of a dynamo, or the capacity of the cylinders of an engine, is certainly not left to the arbitrary judgment of a foreman but is carefully worked out by mathematics and by a knowledge of the properties of the materials used. - Mathematics might be likened to a kit of tools which the workman carries; the master workman carries more than the apprentice and the more tools each man has in his kit and knows how to use, the more things he can do and the greater is his earning power. Each mathematical process is a tool to be used as the occasion demands. Some of them are used in every problem which comes up, others less frequently, but the more advanced the work the greater the number of tools required. It is with this keen demand in mind, therefore, that we are requiring of each student at the outset of his course this work or its equivalent in Practical Mathematics. We want him to fill his kit with enough tools to meet the steady demands of the work ahead of him, and we feel sure that, once provided with this equipment, his progress will be assured. Copyright, 1911, by American School of Correspondence. In the preparation of this work the authors have intentionally lost sight of the material usually found in the school books on this subject, and have kept in mind only the particular parts which are of special importance to the engineering student. Not only the topics discussed but all of the problems have been made exceptionally practical, and the aim has been at all times to give the student the satisfaction of knowing that whatever he is learning will be of use in his work and will also count for his advancement. DEFINITIONS AND MATHEMATICAL SIGNS 1. Definitions. Mathematics is the science which treats of quantity, and its fundamental branches are Arithmetic, Algebra, and Geometry. Quantity is anything which can be increased, diminished, or measured; for example: numbers, lines, space, motion, time, volume, and weight. A unit is a single thing, or one. A number is a unit or a collection of units and is either concrete or abstract. A concrete number is one whose units refer to particular things, as, for example 5 rivets, 7 bolts. An abstract number does not refer to any particular thing. For example, 5, 23, etc., used without designating any particular objects, are abstract numbers. 2. Mathematical Signs. For the sake of brevity, signs are used in mathematics to indicate processes. Those signs most used in Arithmetic are +, , = () and, The sign + is read "plus" and is the sign of addition. It shows that the quantities between which it is placed are to be added together. If 2 and 2 are to be added it is expressed, thus: 2 + 2 are four. The sign is read “minus" and is the sign of subtraction. It means that the quantity which follows this sign is to be subtracted or taken away from the quantity which precedes it, thus: 64 are 2. The sign is read "times" and is the sign of multiplication. It means that the quantity which precedes this sign is to be multiplied by the quantity which follows it, thus: 2 × 5 are 10. The sign is read "divided by" and is the sign of division. It means that the quantity which precedes this sign is to be divided by the quantity which follows it, thus: 42 are 2. = = The sign is read "equals" or "is equal to" and is the sign of equality. It means that the expressions between which it is placed are identical in value, thus: 4+3 103. This sign is very often misused. Great care should be taken at all times to make sure that the quantities connected by it are equal. For example, it would be absurd to say that 59 14 ÷ 2 = 7, because 5+ 9 does not equal 7. = The parenthesis ( ) and vinculum are used to show that two or more quantities are to be treated as one; or in other words, that the operations indicated within the parenthesis or under the vinculum are to be carried out first, thus: (205) +32 +3 = (15) +3 (5) = 13. NOTATION 3. Notation is the art of writing numbers in words, in figures, and in letters. There are two methods of notation in common use; the Roman and the Arabic. 4. Roman Notation. In the Roman notation, 7 capital letters are used, as follows: All other numbers are expressed by repetitions or by combinations of these seven letters according to the following rules: By repeating a letter the value denoted by the letter is doubled; thus: XX means twenty; CC means two hundred. By placing a letter denoting a less value before a letter denoting a greater, their difference of value is represented; thus: IV denotes four or one less than five; XL denotes forty or ten less than fifty. |