46

40

212

3

30

40

30

20

20

10

20

30

40

off

60

70

801

8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46

20

30

Vernier to arched limb

London. John Weale 1848.

4U111

Marquois's Scale

Vernier to straight limb

૮

Plate.2.

point B, in the third square below g h on the top line; and a Îine drawn from A in the copy, through these several points to B, will be a correct reduced copy of the original line. Proceed in like manner with every other line on the plan, and its various details, and you will have the plot or drawing, laid down to a small scale, yet bearing all the proportions in itself exactly as the original.

It may appear almost superfluous to remark, that the process of enlarging drawings by means of squares is a similar operation to the above, except that the points are to be determined in the smaller squares of the original, and transferred to the larger squares of the copy. The process of enlarging, under any circumstances, does not, however, admit of the same accuracy as that of reducing.

Having now completed the description of those instruments, applicable to the purposes of geometrical drawing, to the consideration of which we propose for the present to limit ourselves, in accordance with the plan of our little work, we now propose to add thereto a description of Coggeshall's Sliding Rule, and then to conclude this part of our subject with some practical hints, which we think may prove acceptable to the commencing student.

COGGESHALL'S SLIDING RULE. (Plate II. Fig. 4.)

Coggeshall's, or the Carpenter's Sliding Rule, is the instrument most commonly used for taking the dimensions and finding the contents of timber. It consists of a rule one foot long, having on its face a groove throughout its entire length, in which a second rule of the same length slides smoothly. On the face of the rule are four logarithmic lines marked at one end A, B, C, and D. The three lines A, B, C, are called double lines, because the figures from 1 to 10 are contained twice in the length of the rule, and are, in fact, repetitions of the loga rithmic line of numbers already described (p. 25). The fourth line, D, is a single line numbered from 4 to 40, and is called the Girt Line, because the girt dimensions are estimated upon it in computing the contents of trees and timber. The lengths upon this line denote the logarithms of the squares of the numbers, from 4 to 40, placed against the several divisions; and enable us, as will be seen, to obtain approximately the contents of a solid by a single operation.

* Extracted from a treatise on drawing instruments, by F. W. Simms, Civil Engineer and Surveyor.

The line c is used with the girt line D, and the two lines A and B, enable us to perform more readily all such operations as have been already described as being performed by the logarithmic line of numbers with the aid of the compasses, the second line B, upon the slider, supplying the place of the compasses.

On the girt line is a mark at the point 18.79, lettered G (gallons), which is the imperial gauge point*, enabling us to compute contents in imperial gallons.

The back of the rule has a decimal scale of one foot divided into one hundred equal parts, by which dimensions are taken in decimals of a foot; and also a scale of inches, numbered from 1 to 12, which scale is continued on the slider and numbered from 12 to 24, so that, when the slider is pulled out, a two feet rule is formed, divided into inches.-The vacant spaces on the rule are filled up with various other scales and tables, which may be selected to suit the purposes of the various purchasers.

The method of notation on the rule, and the manner of estimating any number upon it, are the same as have already been fully explained, when treating of the line of logarithmic numbers (p. 28).

Problem 1. To multiply two Numbers together.-Set 1 on B to the multiplier on A, and against the multiplicand on B will be found the required product on A. Example.-To multiply 33 by 23. Set 1 on в to 2.3 on A, and against 3.3 on B will be found 7.59 on A, and 759 is therefore the product required +.

Problem 2. To divide one Number by another.-Set 1 on B to the dívisor on A, and against the dividend on a will be found the required quotient on B. Example.-To divide 759 by 23. Set 1 on B to 2·3 on A, and against 75·9 on a will be found 33 on B, which is the quotient required.

Problem 3. To find a Fourth Proportional to three given Numbers.-Set the first term on B to the second term on A, and against the third term on B will be found the required fourth term on A. Or, against the first term on A, set the second term on B, and against the third term on a will be found the required fourth term on B. Example. To find a fourth

18-79 is the diameter of a cylindrical vessel to contain one gallon for each inch of depth. The gauge point for the old wine gallon was at 17.15, lettered W. G., and for the old ale gallon at 18.95, lettered A. G. These marks are consequently found upon rules constructed prior to January, 1826.

The tens must be supplied mentally, as explained at page 28.

proportional to the three numbers 34, 11, and 14.

Set 3,

or 3.5, on B to 11 on A, and against 14 on в will be found on A, 44, the fourth proportional required.

Problem 4. To find a Third Proportional to two given Numbers. This is the same problem as the preceding, repeating the second number for the third term of the proportion. Example.-To find a third proportional to the two numbers 3 and 11. This is to find a fourth proportional to the three numbers 3, 11, and 11. Set therefore 34, or 3.5, on в to 11 on A, and against 11 on B will be found on a 34-6, the third proportional required.

Problem 5. To square a given Number. First Method, by means of the Lines A and B.-Set 1 on B to the given number on A, and against the given number on в will be found its square upon A. Example.-Required the square of 23. Set 1 on в to 23 on A, and against 23 on в will be found its square 529 on A. Second Method, by means of the Lines c and D.—If the number to be squared lie between 1 and 4, or 10 and 40, or 100 and 400, &c., so that its first significant digit is less than 4; set the 1 on c to 10 on D, and against the digits on D, expressing the given number, will be found on c the digits expressing the required square. Then, the square of 1 being 1, of 10, 100, of 100, 10,000, &c., and of 1 being 01, of 01 being 0001, &c., the digits upon c must be estimated at the actual values represented by them as numbered upon the scale, viz., 1, 2, &c., to 16, or at 100 times their values, from 100 up to 1600, or at 10,000 times their values from 10,000 up to 160,000, &c., or, again, at the Tooth part of these values from 01 up to 16, or at Toth part of these values from 0001, up to 0016, &c., according as the highest denomination in the number to be squared is units, or tens, or hundreds, &c., or, again, tenths, hundredths, &c. Example.-Required the square of 23. The 10 on D being set against the 1 on c, against 23 on D will be found 5.29 on c, and, the highest denominations in 23 being tens, the square required is 529. Also the squares of 23, 230, 2300, 23, 023, would be 5.29, 52,900, 5,290,000, 0529, 000529, respectively, the highest denominations in the proposed numbers being respectively units, hundreds, thousands, tenths, and hundredths. Third Method, by means of the Lines c and D.—If the number to be squared lie between 4 and 10, or 40 and 100, or 400 and 1000, &c., so that its first significant digit is not less than 4; set the 100 on c against the 10 on D, and against the digits on D, expressing

D 3

the given number, will be found on c the digits expressing the required square. Example.-Required the square of 15. The 100 on c being set against the 10 on D, against 5.1 on D will be found 26 on c, and, the highest denomination in 51 being tens, the square required is 2600 *.

on D,

Problem 6. To extract the Square Root of a given Number. -This problem being the converse of the preceding, set the rule in the same manner, with the 1 on c against the 10 if the given square be between 1 and 16, or 100 and 1600, or 10,000 and 160,000, &c., or again between 01 and ∙16, or 0001 and 0016, &c., and with the 100 on c against the 10 on D, if the given square be between 16 and 100, or 1600 and 10,000 &c., or again between 16 and 1, or 0016 and 01, &c. ; and then against the given number on c will stand its square root on D. Example 1.-Required the square root of 529. The given number being between 100 and 1600, set the 1 on c against the 10 on D, and against 5.29 on c will be found 23 on D, the square root required. Example 2.-Required the square root 2601. The given number being between 1600 and 10,000, set the 100 on c against the 10 on D, and against 26 on c will be found 5.1 on D, and 51 is therefore the root sought.

Problem 7. To find a mean Proportional between two given Numbers.-Set one of the numbers upon c to the same number on D, and against the other number on c will be found upon D the mean proportional required t. Example.-Required a mean proportional between 4 and 49. Set 4 on c to 4 on D, and against 49 on c will be found on D 14, the mean proportional required.

If one number exceed the other so much that they cannot both be taken off from the line c, the 7th part of the larger may be taken, and the mean proportional then found, multiplied by 10, will give the mean proportional required. Also if the second number on c be situated beyond the scale D, the Tth part of such second number may be substituted for it, and the result multiplied by 10; or 100 times such number may

* The accurate square is 2601, but the fourth figure cannot be estimated upon a foot rule, and the third figure only approximately. The solution, in fact, may be considered as obtained to within the 200th part of the whole, but, if greater accuracy is required, arithmetical methods must be resorted to.

† Ifax: x: b, then a : b:: a2: x2; and, therefore,

log. b

log. α = log. x2- log. a2; whence the rule given in the text.