RULE.*

Multiply the area of the base by 1.1416, and the product will be the superficies required.

* This rule may also be observed in elliptical groins, the error being too small to be regarded in practice.

In measuring works where there are many groins in a range, the cylindrical pieces between the groins, and on their sides, must be computed separately.

And to find the solidity of the brick or stone work, which forms the groin arches, observe the following

RULE. Multiply the area of the base by the height, including the work over the top of the groin, and this product lessened by the solid content, found as before, will give the solidity required.

The general rule for measuring all arches, is this:

From the content of the whole, considered as solid, from the springing of the arch to the outside of it, deduct the vacuity contained between the said springing and the under side of it, and the remainder will be the content of the solid part.

And because the upper sides of all arches, whether vaults or groins, are built up solid, above the haunches, to the same height with the crown, it is evident that the area of the base will be the whole content above mentioned, taking for its thickness the height from the springing to the top. And for the content of the vacuity to be deducted, take the area of its base, accounting its thickness to be two-thirds of the greatest inside height. But it may be noted that the area used in the vacuity, is not exactly the same with that used in the solid; for the diameter of the former is twice the thickness of the arch less than that of the latter.

And although I have mentioned the deduction of the vacuity as common to both the vault and the groin, it is reasonable to make it only in the former, on account of the waste of materials and trouble to the workman, in cutting and fitting them for the angles and intersections.

Whoever wishes to see this subject more fully handled, may consult La Théorie et la Pratique de la Geométrie, par M. l'Abbé Deidier, a work in which several parts of Mensuration and Practical Geometry are skilfully handled, the examples being mostly wrought out in an easy familiar manner, and illustrated with observations, and figures very neatly executed.

EXAMPLES.

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.

OF THE

CARPENTER'S RULE.

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 2s. 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.

PROBLEM I.

To find the product of two numbers, as 7 and 26.

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 product (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.

PROBLEM II.

To divide one number by another, as 510 by 12.

RULE.

Set the divisor (12) on A, to 1 on B; then against the dividend (510) on A, is the quotient (42) 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.

PROBLEM III.

To square any number, as 27.

RULE.

Set 1 upon D to 1 upon C; then against the number (27) upon D, will be found the square (729) upon C.

If you would square 270, reckon the 1 on D to be 100; and then the 1 on C will be 1000, and the product 72900.

PROBLEM IV.

To extract the square root of any number, as 4268.

RULE.

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.

PROBLEM V.

To find a mean proportional between any two numbers, as 27 and 450.

RULE.

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.

PROBLEM VI.

Three numbers being given, to find a fourth proportional; suppose 12, 28, and 57.

S