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1. What is the curve superficies of a circular groin arch, one side of its square being 12 feet?
Here 122 x 1.1416 = = 164.3904–superficies required.
2. What is the concave superficies of a circular groin arch, one side of its square being 9 feet? Ans. 92.4696.
This instrument is commonly called Cogeshall's sliding rule. It consists of two pieces, of a foot in length each, which are connected together by means of a folding joint.
On one side of the rule, the whole length is divided into inches and half quarters, for the purpose of taking dimensions. And on this face there are also several plane scales, divided by diagonal lines into twelve parts, which are designed for planning such dimensions as are taken in feet and inches.
On one part of the other face there is a slider, and four lines marked A, B, C, and D; the two middle ones B and C being upon the slider. Three of these lines, A, B, C, are double ones,
because they proceed from 1 to 10 twice over: and the fourth line D is a single, one, proceeding from 4 to 40, and is called the girth line.
The use of the double lines A, and B, is for working proportions, and finding the areas of plane figures. And the use of the girth line D, and the other double line C, is for measuring solids.
When 1 at the beginning of any line is counted 1, then the 1 in the middle will be 10, and the 10 at the end 100. And when 1 at the beginning is counted 10, then the 1 in the middle is 100, and the 10 at the end 1000, &c., and all the small divisions are altered in value accordingly.
Upon the other part of this face, there is a table of the value of a load of timber, at all prices, from 6d. to 28. a foot.
Some rules have likewise a line of inches, or a foot divided decimally into 10th parts; as well as tables of board measure, &c. but these will be best understood from a sight of the instrument.
THE USE OF THE SLIDING RULE.
Set 1 upon A, to one of the numbers (26) upon B; then against the other number (7) on A, will be found the produet (182) upon B.
Note.--If the third term runs beyond the end of the line, seek it on the other radius, or part of the line, and increase the product 10 times.
To divide one number by another, as 510 by 12.
Set the divisor (12) on A, to 1 on B; then against the dividend (510) on A, is the quotient (421) on B.
Note.-If the dividend runs beyond the end of the line, diminish it 10 or 100 times to make it fall on A, and increase the quotient accordingly.
RULE. Set 1 upon D to 1 upon C; then against the number (27) upon D, will be found the square (729) upon C.
would square 270, reckon the 1 on D to be 100; and then the 1 on C will bé 1000, and the product 72900.
To extract the square root of any number, as 4268.
Set 1 upon C, to 1 upon D; then against (4268) the number on C, is (65.3) the root on D.
To value this right, you must suppose the 1 on C to be some of these squares, 1, 100, 1000, &c., which is the nearest to the given number, and then the root corresponding will be the value of the 1 upon D.
To find a mean proportional between any two numbers, as
27 and 450.
Set one of the numbers (27) on C, to the same on D, then against the other number (450) on C, will be the mean (110.2) on D.
Note.--If one of the numbers overruns the line, take the 100th part of it, and augment the answer 10 times.
Three numbers being given, to find a fourth proportional;
suppose 12, 28, and 57.
Set the first number (12) upon A, to the second (28) upon B; then against the third number (57) on A, is the fourth (133) on B.
Note.- If one of the middle numbers runs off the line take the tenth part of it only, and augment the answer 10 times.
The finding a third proportional is exactly the same, the second number being twice repeated.
Thus, suppose a third proportional was required to 21 and 32.
Set the first 21 on B, to the second 32 on A; then against the second 32 on B, is 48.8 on A, which is the third proportional required.
* The use of the rule in board and timber measure will be shown in .what follows.
If the breadth of a board be given; to find how much in length will make a square foot.
RULE. If the board be narrow, it will be found in the table of board measure on the rule; but, if not, shut the rule, and seek the breadth in the line of board measure, running along the rule, from that table; then over against it, on the opposite side, is the length in inches required.
The side of the square of a piece of timber being given; to find how much in length will make a foot solid.
RULE. If the timber be small, it will be found in the table of timber measure on the rule; but, if not, look for the side of the square, in the line of timber measure, running along the rule, from that table, and against it in the line of inches is the length required.