INSTRUMENTAL ARITHMETIC; OR, UTILITY OF THE SLIDE RULE. The slide rule is an instrument by which the greater portion of operations in arithmetic and mensuration may be advantageously performed, provided the lines of division and gauge points be made properly correct, and their several values familiarly understood. The lines of division are distinguished by the letters ABCD, A B and C being each divided alike, and containing what is termed a double radius, or double series of logarithmic numbers, each series being supposed to be divided into 1000 equal parts, and distributed along the radius in the following manner: From 1 to 2 contains 301 of those parts, being the log. of 2. 66 3 66 477 3. 4. 5. 6. 7. 8. 9. 1000 being the whole number. The line D, on the improved rules, consists of only a single radius; and although of larger radius, the logarithmic series is the same, and disposed of along the line in a similar proportion, forming exactly a line of square roots to the numbers on the lines B C. Numeration teaches us to estimate or properly value the numbers and divisions on the rule in an arithmetical form. Their values are all entirely governed by the value set upon the first figure, and, being decimally reckoned, advance tenfold from the commencement to the termination of each radius: thus, suppose 1 at the joint be one, the 1 in the middle of the rule is ten, and 1 at the end one hundred. Again, suppose 1 at the joint ten, 1 in the middle is 100, and 1 or 10 at the end is 1000, &c., the intermediate divisions on which complete the whole system of its notation. To Multiply Numbers by the Rule. Set 1 on B opposite to the multiplier on A; and against the number to be multiplied on B is the product on A. Multiply 6 by 4. Set 1 on B to 4 on A: and against 6 on B is 24 on A. The slide thus set, against ין To divide Numbers upon the Rule. Set the divisor on B to 1 on A, and against the number to be divided on B is the quotient on A. Divide 63 by 3. Set 3 on B to 1 on A, and against 63 on B is 21 on A. Proportion, or Rule of Three Direct. Rule. Set the first term on B to the second on A, and against the third upon B is the fourth upon A.. 1. If 4 yards of cloth cost 38 shillings, what will 30 yards cost at the same rate? 1 Set 4 on B to 38 on A, and against 30 on B is 285 shillings on A. 2. Suppose I pay 31s. 6d. for 3 cwt. of iron, at what rate is that per ton! 1 ton 20 cut. Set 3 upon B to 315 upon A, and against 2 upon B is 210 upon A. Rule of Three Inverse. Rule. Invert the slide, and the operation is the same as direct proportion 1. I know that six men are capable of performing a certain given portion of work in eight days, but I want the same performed in three: how many men must there be employed? Set 6 upon C to 8 upon A, and against 3 upon C is 16 upon A. 2. The lever of a safety valve is 20 inches in length, and 5 inches between the fixed end and centre of the valves what weight must there be placed on the end of the lever to equipoise a force or pressure of 40 lbs. tending to raise the valve? Set 5 upon C to 40 upon A, and against 20 on C is 10 on A. 3. If 8 yards of cloth, 12 yards in width, be a sufficient quantity, how much will be required of that which is only ths in width, to effect the same purpose! Set 1'5 on C to 8 75 on A, and against 8'75 upon C is 15 yards upon A. Square and Cube Roots of Numbers. Squares, 1 4 9 16 25 36 49 64 81 on C. To find the geometrical mean proportion between two numbers. Required the mean proportion between 20 and 45. Set 20 upon C to 20 upon D, and against 45 upon C is 30 on D. Mensuration of Solidity and Capacity. General Rule. Set the length upon B to the gauge-point upon A; and against the side of the square, or diameter on D are the cubic contents, or weight in lbs. où C. 1. Required the cubic contents of a tree 80 feet in length, and 10 inches quarter girt. Set 20 upon B to 144 (the gauge-point) upon A; and against 10 upon D is 20'75 feet upon C. 2. In a cylinder 9 inches in length and 7 inches diameter, how many cubic inches? Set 9 upon B to 1273. (the gauge point) upon point) upon A; and against and against 7 on D is.346 inches on C. 3. What is the weight of a bar of cast iron 3 inches square, and 6 feet long? Set & upon B to 32 (the gauge-point) upon A; and against 3 upon D is 168 lbs. apon C. By the commm rule 4. Required the weight, of a cylinder of wrought iron 10 inches long, and 54 diameters Set 18 upon B to 233 (the gauge-point) upon A; and against 5% upon D is 66 65 lbs. on C. 5. What is the weight of a dry rope 25 yards long, and 4 inches circumference? Set 25 upon B to 47 (the gauge-point) upon A; and against 4 on Dis 536 lbs. en C. 6. What is the weight of a short linked chain 30 yards in length, andths of an inch in diameter? Set 30 upon B to 52 (the gauge-point) upon A; and against 6 on D is 129:5 lbs. on C: Land Surveying. ⠀ If the dimensions taken are in chains, the gauge point is 1 or 10; if in perches, 160; and if in yards, 4840 Rule. Set the length upon B to the gauge-point on A; and against the breadth upon A is the content in acres upon B. 1. Required the number of acres or contents of a field 20 chains 50 links in length, and 4 chains 40 links in breadth. Set 20'5 on B to 1 on A; and against 4'4 on A is 9 acres on B. 2. In a piece of ground 440 yards long, and 44 broad, how many acres? Set 440 upon B to 4840 on A; and against 44 on A is 4 acres on B Power of Steam-Engines. Condensing Engines -Rule. Set 3.5 on C to 10 on D; then D is a line of diameters for cylinders, and C the corresponding number of horse power; thus, A on C. 33 37% on D The same is effected on the common rule by setting 5 on C to 12 on D. Non-condensing Engines Rule. Set the pressure of steam in lbs. per square inch on B to 4 upon A; and against the cylinder's diameter on D is the number of horse power upon C. Required the power of an engine, when the cylinder is 20 inches diameter, and steam 30 lbs. per square inch. Set 30 on B to 4 on A; and against 20 on D is 30 horse power on C. How many superficial feet are contained in a boiler 23 feet in length and 5 in width ban hacer tray Set 1 upon B to 23 upon A; and against 5'5 upon B is 125'5 square feet upon A. If 5 square feet of boiler surface be sufficient for each horse power, how many horse power of engine is the boiler equal to ? Set 5 upon B to 1265 upon A, and against 1 upon B is 25'5 upon A. Or the mass of a body is equal to its weight divided by 32.19. to la bangus W = 35 × 32·19 — 112·66 lbs.%) 13 The gravity of a material body is its weight. Falling bodies fall through the same space in the same time, whatever may be their weight. A body one ton will fall to the ground no faster than a body one pound. The velocity of a body is the number of feet passed over in one second. Put the velocity of a falling body, at the end of t seconds, The quantity 32 19 is the velocity of a falling body at the end of dsnot-mcore le nowel one second. Rule, to find the Velocity of a Falling Body at the end of any Number of Seconds. i Multiply the number of seconds by 32-19, the product will be the velocity. EXAMPLE. Find the velocity of a body falling from a height in nine seconds: Velocity 32-19 × 9289-71. Puts for the number of feet a falling body falls through in t seconds: 32:19 t Rule to find the Space passed over by a by a Falling Body in any Number of Seconds. Square the number of seconds, and multiply the result by 1609, the product will be the distance passed over in feet. EXAMPLE. A stone fell from the top of a chimney to the bottom in four seconds; find the height of the chimney: Height of chimney v2 16:09 x 16 257:44 feet. = 3 64.39 where v is the velocity. Rule to find the Space passed over by a Falling Body when the Velocity is given. Square the velocity, and divide by 64.39; the quotient will be the number of feet passed over. The quantity 32-19 is frequently called the accelerating force of gravity, and is denoted by The following formulæ include all eases that can occur in falling bodies. ftz & space passed over=' 2 v = velocity at the end of (t) seconds = The above formulæ and rules are applicable only to the case when the body is acted upon by the force of gravity. Rules and Formule when a body is acted on by any force. Put Mmass acted on by a force of F pounds. avelocity at the end of a second, which is called acce lerating foree. 8 space passed over in (t) seconds, producing a velo city (v). |