THE MECHANIC'S, MACHINISTS, AND ENGINEER'S PRACTICAL BOOK OF REFERENCE: CONTAINING TABLES AND FORMULÆ FOR USE IN SUPERFICIAL AND SOLID MENSURATION; STRENGTH AND WEIGHT OF MATERIALS; MECHANICS; MACHINERY: HYDRAULICS; HYDRODYNAMICS; MARINE ENGINES; CHEMISTRY: AND MISCELLANEOUS RECIPES. ADAPTED TO AND FOR THE USE OF ALL CLASSES OF PRACTICAL MECHANICS. EDITED BY CHARLES W. HACKLEY, Professor of Mathematics in Columbia College, N. . THE PRACTICAL BOOK OF REFERENCE. ARITHMETICAL SIGNS. THE following definitions of arithmetical signs which are employed in mechanical calculations, will be found of great value to those who do not yet understand them, and of some interest to those who are already familiar with their meanings. This is the sign of equality, and signifies equal to. ple: 12 inches = 1 foot (12 inches is equal to 1 foot). For exam This is the sign of addition, and signifies plus, or more. example: 5 + 3 = 8 (5 added to 3 is equal to 8). For This is the sign of subtraction, and signifies minus, or less. For example: 10 8 = 2 (10 minus 8 leaves or is equal to 2). x This is the sign of multiplication, and signifies multiplied by, or into. For example: 10 x 3 = 30 (10 multiplied by 3 is equal to 30). ÷ This is the sign of division, and signifies divided by. For example: 156 ÷ 6 =26 (156 divided by 6 is equal to 26); or, 24 ÷ 4 = 6 24 (24 divided by 4 is equal to 6); or = 6 (24 fourths are equal to 6 wholes). 4 :::: This is the sign of proportion, and signifies proportion. For example: 4 : 6 :: 8: 12 (as 4 is to 6, so is 8 to 12); or 3: 5 :: 9:15 (that is, as 3 is to 5, so is 9 to 15); 3 9 5 15 When it is placed before 25), it means that the square root of that 255, because 5 x 5 = 25; 64 = 8, because 8 × 8 = 64. When it is placed before a that number is required. === 16, and 4 × 16 = 64); × 36 = 216). This is the sign of the CUBE root. number, it means that the cube root of For example: 644 (that is, 4 × 4 or, 2166 (that is, 6 x 6 = 36, and 6 When this mark is added to a number (thus, 6), it means that that number is to be squared. For example: 52= 25 (that is, 5 x 5 = 25); or 62 == = 36 (that is, 6 × 6 = 36). = = • When this mark is added to a number, it means that that number is to be cubed. For example; 55 x 5 x 5 = 125 (that is, 5 × 525, and 5 × 25: 125; or, 7: 343 (that is, 7 x7 = 49, and 7 x 49: 243). The index or power (as the small figure annexed is called) shows how many times a number is to be multiplied by itself. This is called the bar. It signifies that all the numbers or quantities under it are to be taken together. For example: 3 + 5 × 432 (3 plus 5 are equal to 8, and that, multiplied by 4, is equal to 32); or, 7 − 3 + 8 = 12 (7 less 3 is equal to 4, and that, if added to 8, is equal to 12); or, 5 × 4 + 3 = 35 (that is, 4 and 3 are 7, which, if multiplied by 5, is equal to 35); or, 5 × 6 + 4 = = 50 (that is, 6 and 4 are 10, and ten times 5 are 50). The parenthesis () is sometimes used in place of the bar, thus: (6 + 4) × 5 = 50. .. The meaning of this sign is therefore. ... This sign signifies because. The meaning of this sign is perpendicular, ~This sign denotes difference, and is placed between two quantities (as x~ y) when it is not known which of them is the greater. > or The meaning of these signs is GREATER than. For example: AB> CD (that is, AB is greater than CD). <orThe meaning of these signs is LESS than. For example: ABCD (that is, AB is less than CD). a 125 This is a decimal point. When placed before a number (thus, .1), it means that that number has a unit (1) for its denominator. For example: .1 is the same as; .125 is the same as 100%; .01 is the same as To; .001 is the same as Too; .0001 is the same as Toooo; 42.85 is the same as 42,85; 57.217 is the same as 57. This is a degree mark. It is written and printed as follows: 25° (that is, 25 degrees). 10000; о 'This is a minute sign. "These two accents signify seconds. "These three accents signify thirds. They read thus: 57° 17' 43" 39" (that is, 57 degrees, 17 minutes, 43 seconds, and 39 thirds). ALGEBRAIC SYMBOLS. The advantage of these, in a work like the present, may be thus illustrated: Let / denote the length, b the breadth, and d the depth of an iron beam. If it be desired to express the product of the length and breadth, divided by the depth, it is done as follows: That is to say, multiplication is expressed by simply writing the letters which represent numbers one after the other; division, by drawing a line under the dividend, and writing the divisor below. The sum of the length and breadth, divided by the depth, would be expressed briefly thus: 1+ b The square of the length, multiplied by the cube of the breadth, thus: The square root of the length, divided by the fourth root of the breadth, thus: The square root of the difference of the length and breadth, divided by the depth, thus: d The square root of the quotient of the sum and difference of the length and breadth, thus: Any other letters-as a, b, c, &c.-may stand for the given dimensions. These explanations will serve to give the sense of the symbols which will be met with throughout the work. |