Extracts from

The Mechanic's Calculator
by William Grier
(seventh edition 1839)

THE SLIDING RULE

(pages 24 to 28)

    35. We are indebted for the invention of this useful instrument to Edmund Gunter. It is a kind of logarithmic table, whose great use is to obtain the solution of arithmetical questions by inspection, in the multiplication, division, and extraction of the roots of numbers. It consists of two equal pieces of boxwood, each 12 inches long, joined together by a brass folding joint. In one of those pieces there is a brass slider. On the face of this instrument, there are engraven four lines, marked by the letters A, B, C, and D; at the beginning of each line, the lines A and D being marked on the wood part of the rule, and B and C on the brass slider.

    36. Before the use of the sliding rule can be explained, it is necessary that a correct idea should be formed of the method of estimating the values of the several divisions on these lines. Let it be observed, then, that whatever value is given to the first 1 from the left, the numbers following, viz. 2, 3, 4, 5, &c. will represent twice, thrice, four times, &c., that value. If 1 is reckoned 1 or unity, then 2, 3, 4, &c., will just signify two, three, four, &c.; but if 1 is reckoned 10, then 2, 3, 4, &c., will represent 20, 30, 40, &c. If the first 1 is reckoned 100, then 2, 3, 4, &c., will represent 200, 300, 400, &c. The value of the 1 in the middle of the line is always ten times that of the first 1; the value of the second two is ten times that of the first two: so that if the value of the first 1 be 10, that of the second 1 will be 100; the first 2 will be 20, and the second 2 will be 200, &c. The value of these divisions being understood, we may now attend to the minute divisions between these. Now, on the lines A, B, and C, there are 50 small divisions betwixt 1 and 2, 2 and 3, 3 and 4, &c.; and it follows, from the nature of the larger divisions, that if the first 1 be reckoned 1, or unity, each of these small divisions between 1 and 2, 2 and 3, &c., will be 1/50 or .02; and supposing still the first 1 to be unity, then the small divisions from the second 1 to 2, 2 to 3, &c., will each be ten times greater than a 1/50, or .02, that is, each of them will be 10/50, or 1/5, or .2. In the same way, if the first 1 represents 100, the first 2 will be 200; if the second 1 be 1000, the second 2 will be 2000, &c.; and on the same principle as above the small divisions or 50th parts will represent each 1/50 of 100, or 2, in the first half, or from the first 1 to 2, 2 to 3, &c., and 1/50 of 1000, or 20, in the second half; or from the second 1 to 2, 2 to 3, &c.

    37. These divisions being understood, we may proceed to shew the method of using this rule in the solution of arithmetical questions.

    38. To find the product of two numbers:
    Move the slider, so that 1 on B stands against one of the factors on A; then the product will be found on the line A, against the other factor on the line B.
    Thus, to find the product of 3 by 8:
    Set 1 on B to 3 on A: then against 8 on B will be found the product 24 on A.
    For the product of 34 by 16:
    Set 1 on B against 16 on A, then look on B for 34, and against it on the line A will be found the product 544.

    39. To find the quotient of two numbers:
    This may be done in two ways, - either set 1 on the slider B against the divisor on A, then against the dividend on A the quotient will be found on B. Or, set the divisor on B against 1 on A, then the quotient will be found on A against the dividend on B; therefore, in general, it is to be remembered, that the quotient must always be found on the same line on which 1 was taken, and the divisor and dividend on the other line.
    Thus to find the quotient of 96 divided by 6:
    Move the slider till 1 on B stands against 6 on A; then the quotient 16 will be found on B against the dividend 96 on A.
    In like manner, to find the quotient of 108 divided by 12, we may take the latter form of the rule, thus:
    Set 12 on B against 1 on A; then on the line A will be found the quotient 9 against 108 on B.

    40. To solve questions in the rule of three or simple proportion:
    Set the first term on the slider B to the second on A; then on the line A will be found the fourth term, standing against the third term on B.
    If 4 lbs. of brass cost 36 pence, what will 12 lbs. cost?
    Move the slider so, that 4 on B will stand against 12 on A; then against 36 on B will be found the fourth term 108 on A.

    41. To extract the square root:
    Move the slider so, that the middle division on C, which is marked 1 stands against 10 on the line D, then against the given number on C the square root will be found on D.
    It is to be observed before applying this rule, that if the given number consists of an even number of places of figures, as two, four, six, &c., it is to be found on the left hand part of the line C; but if it consists of any odd number of places, as three, five, seven, &c., it is to be found on the right hand side of C, 1 being the middle point of the line.
    To find the square root of 81:
    Here the number of places are even, being two; therefore, the number 81 is sought for on the left hand side of the line C.
    Set 1 on C against 10 on D; then against 81 on C will be found 9, the square on D.
    For the square root of 144:
    Set 1 on C to 10 on D; then against 144 on C will be found the square root 12 on D.

    42. To find the area of a board or plank:
    The rule is, to multiply the length by the breadth, the product will be the area; therefore, by the sliding rule,
    Set 12 on B against the breadth in inches on A; then on the line A will be found the surface in square feet, against the length in feet on the line B.
    To find the area of a plank 18 inches broad and 10 feet 3 inches long:
    Move the slider so, that 12 on B stands against 18 on A; then will 10 1/4 on B stand against 15 3/4 on A, which 15 3/4 is square feet.
    This may be proved by cross multiplication.

    43. For the solid content of timber.
    The rule is to multiply length, breadth, and thickness all together
    Set the length in feet on C to 12 on D; then on C will be found the content in feet against the square root of the product of the depth and breadth in inches on D.
    What is the content of a square log of timber, the length of which is ten feet, and the side of its square base is 15 inches.
    Set 10 on C against 12 on D; then will 15 on D stand against the content 15 5/8 on C.

    44. Other particulars on the measurement of timber will be given hereafter, when we come to Mensuration.

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TIMBER MEASURE

(pages 108 to 110)

    EXAMPLES of timber measuring have already been given in the department allotted to arithmetic, but it is necessary to be here somewhat more particular. The surface of a plank is found:-
    1st. By multiplying the length by the breadth. When the board tapers gradually, add the breadth at both ends together, and take half of this sum for the mean breadth.
    2nd. By the sliding rule.-Set the length in inches on B to 12 on A, and against the length in feet on B will be the area in square feet and decimals on A.
    Ex.-A board is 12 feet 6 inches long and 1 foot 3 inches broad; hence,

    1st. For the content of squared timber, length × mean breadth × mean thickness = content.
    2nd. By the sliding rule.-Find the mean proportional between the breadth and thickness, then set the length on C to 12 on D, and against the mean proportional on D the solid content on C. If the mean proportional be in feet, reduce to inches.
    Ex.-A log is 24 feet long, the mean depth and breadth being each 13 inches.

    For round timber.-1st. Take one fourth of the mean girth and square it, this multiplied by the length will give the content.
    2nd. By the sliding rule.-Set the length in feet on C to 12 on D, then against the quarter girth in inches on D, will be the content on C.
    This gives no allowance for bark, but there is usually a deduction made of about an inch to the foot of quarter girth. The rule given above gives the customary, but not the true content; the following gives the true content.
    One-fifth of the girth squared and multiplied by twice the length = content.
    Ex.-The mean girth of a tree being 5 feet 8 inches, and its length 18 feet, the two rules will apply as below:-

    Trees very seldom have an equal girth throughout, one end being generally much smaller than the other: the girth taken above is the mean girth; that is to say, the girths of both ends added together, and their sum halved for the mean girth. It is to be observed, however, that, if the difference of the girths is great, it will be best to find the content of the tree as if it were a conic frustum.-The method of using the sliding-rule in the measurement of timber has been given before.

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This page was last updated on 30 July 2000.