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INSTRUMENTAL ARITHMETIC.

OR UTILITY OF THE SLIDE RULE.

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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 A B C D ; 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.

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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 в C.

NUMERATION.

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 в opposite to the multiplier on a ; and against the number to be multiplied on в 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 7 on B is 28 on A.

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TO DIVIDE NUMBERS UPON THE RULE.

Set the divisor on в to 1 on A; and against the number to be divided on в is the quotient on A,

Divide 63 by 3.

Set 3 on в to 1 on a; and against 63 on в is 21 on A.

PROPORTION, OR RULE OF THREE DIRECT.

Rule.-Set the first term on в to the second on a; and against the third upon в is the fourth upon a. 1. If four yards of cloth cost 38 shillings, what will 30 yards cost at the same rate?

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 three cwt of iron, at what rate is that per ton? 1 ton=20 cwt.

Set 3 upon в to 31.5 upon a; and against 20 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 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 valve; what weight must there be placed on the end of the lever to equipoise a force or pressure of 40 Ms. 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, 1 yard 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 875 upon c is 15 yards

upon A.

SQUARE AND CUBE ROOTS OF NUMBERS.

On the engineer's rule, when the lines C and D are equal at both ends, c is a table of squares, and D a table of roots, as

Squares 1 4 9 16 25 36
Roots 1 2 3 4 5 6

49 64 81 on c.

7 8

9 on D.

To find the geometrical mean proportion between two numbers.

Set one of the numbers upon c to the same number upon D; and against the other number upon c is the mean number or side of an equal square upon D.

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.

To cube any number, set the number upon c to 1 or 10 upon D; and against the same number upon D is the cube number upon c.

Required the cube of 4.

Set 4 upon c to 1 or 10 upon D; and against 4 upon D is 64 upon c.

To extract the cube root of any number, invert the slide, and set the number upon в to 1 or 10 upon D; and where two numbers of equal value coincide on the lines B D, is the root of the given number.

Required the cube root of 64.

Set 64 upon в to 1 or 10 upon D; and against 4 upon в is 4 upon D, or root of the given number.

On the common rule, when 1 in the middle of the line c is set opposite to 10 on D, then c is a table of squares, and D a table of roots.

To cube any number by this rule, set the number upon c to 10 upon D; and against the same number upon D is the cube upon c.

F

MENSURATION OF SURFACE.

1. Squares, Rectangles, &c.

Rule. When the length is given in feet and the breadth in inches, set the breadth on в to 12 on a; and against the length on A is the content in square feet on B.

If the dimensions are all inches, set the breadth on в to 144 upon A; and against the length upon a is the number of square feet on B.

Required the content of a board 15 inches broad and 14 feet long.

Set 15 upon в to 12 upon A; and against 14 upon a is 175 square feet on B.

2. Circles, Polygons, &c.

Rule.-Set 7854 upon c to 1 or 10 upon D; then will the lines C and D be a table of areas and diameters. Areas 3.14 7.06 12.56 19.63 28.27 38-48 50.26 63.61 upon c. Diam. 2 4 5 6 7 89

upon D.

3 In the common rule, set ·7854 on c to 10 on D; then c is a line or table of areas, and Dof diameters, as before.

Set 7 upon в to 22 upon A; then в and A form or become a table of diameters and circumferences of circles. Cir. 3.14 6.28 9.42 12.56 15.7 18.85 22 25-13 28.27 upon a. Dia. 1 3 4 5 6 7 8 9 upon B. Polygons from 3 to 12 sides.-Set the gauge-point upon c to 1 or 10 upon D; and against the length of one side upon D is the area upon c.

Sides

2.

3 5 6 7 8 9 10 11 12 Gauge-points 433 1.7 2.6 3.63 4.82 6.18 7.69 9.37 11.17 Required the area of an equilateral triangle, each side 12 inches in length.

upon D; and against 12 upon D are

Set 433 upon c to 62.5 square inches upon c.

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