straight lines, if that was not done during the survey, before they were entered in the field-book, by making a proper allowance to shorten them. For which purpose there is commonly a small table engraven on some of the instruments for surveying.

PROBLEM II.

To compute the contents of fields.

1. Compute the contents of the figures, whether triangles or trapeziums, &c., by the proper rules for the several figures laid down in measuring; multiply the lengths by the breadths, both in links, and divide by 2; the quotient is acres, after you have cut off five figures on the right for decimals. Then bring these decimals to roods and perches, by multiplying first by 4, and then by 40. An example of which has been already given in the description of the chain,

2. In small and separate pieces, it is usual to cast up their contents from the measures of the lines taken in surveying them, without making a correct plan of them.

3. In pieces bounded by very crooked and winding hedges, measured by offsets, all the parts between the offsets are most accurately measured separately as small trapezoids.

4. Sometimes such pieces as that last mentioned, are computed by finding a mean breadth, by dividing the sum of the offsets by the number of them, accounting that for one of them where the boundary meets the station line; then multiply the length by that mean breadth. - But this method is commonly in some degree erroneous.

5. But in larger pieces, and whole estates, consisting of many fields, it is the common practice to make a rough plan of the whole, and from it compute the contents quite independent of the measures of the lines and angles that were taken in surveying. For, then, new lines are drawn in the fields in the plan, so as to divide them into trapeziums and triangles, the bases and perpendiculars of which are measured on the plan by means of the scale from which it was drawn, and so multiplied together for the contents. In this way, the work is very expeditiously done, and sufficiently correct; for such dimensions are taken as afford the most easy method of calculation; and, among a number of parts, thus taken and applied to a scale, it is likely that some of the parts will be taken a small matter too little, and others too great; so that they will, upon the whole, in all probability, very nearly balance one another. After all the fields and particular parts are thus computed separately, and added all together into one sum, calculate the whole estate independent of the fields, by dividing it into large and arbitrary triangles and trapeziums, and add these also together. Then if this sum be equal to the former, or nearly so, the work is right; but if the sums have any considerable difference, it is wrong, and they must be examined and recomputed, till they nearly agree.

6. But the chief secret in computing consists in finding the contents of pieces bounded by curved or very irregular lines, or in reducing such crooked sides of fields or boundaries to straight lines, that shall inclose the same or equal area with those crooked sides, and so obtain the area of the curved figure by means of the right-lined one, which will commonly be a trapezium. Now, this reducing the crooked sides to straight ones, is very easily and accurately performed in this manner: Apply the straight edge of a thin, clear piece of lanthorn-horn to the crooked line which is to be reduced, in such a manner, that the small parts

M M

cut off from the crooked figure by it, may be equal to those which are taken in: which equality of the parts included and excluded you will presently be able to judge of very nicely by a little practice; then with a pencil or point of a tracer, draw a line by the straight edge of the horn. Do the same by the other sides of the field or figure. So shall you have a straight-sided figure equal to the curved one; the content of which, being computed as before directed, will be the content of the curved figure proposed.

Or, instead of the straight edge of the horn, a horse-hair may be applied across the crooked sides in the same manner; and the easiest way of using the hair, is to string a small slender bow with it, either of wire, or cane, or whalebone, or such like slender or elastic matter; for, the bow keeping it always stretched, it can be easily and neatly applied with one hand, while the other is at liberty to make two marks by the side of it, to draw the straight line by.

EXAMPLE.

Thus, let it be required to find the contents of the same figure as in problem IX. of the last section, to a scale of 4 chains to an inch.

B

Draw the four dotted straight lines AB, BC, CD, DA, cutting off equal quantities on both sides of them, which they do as near as the eye can judge: so is the crooked figure reduced to an equivalent right-lined one of four sides ABCD. Then draw the diagonal BD, which, by applying a proper scale to it, measures 1256. Also the perpendicular, or nearest distance, from A to this diagonal, measures 456; and the distance of C from it, is 428.

Then, half the sum of 456 and 428, multiplied by the diagonal 1256, gives 555,152 square links, or 5 acres, 2 roods, 8 perches, the content of the trapezium, or of the irregular crooked piece.

PROBLEM III.

To transfer a plan to another paper, &c.

After the rough plan is completed, and a fair one is wanted, this may be done by any of the following methods:

First Method.-Lay the rough plan on the clean paper, keeping them always pressed flat and close together, by weights laid on them. Then, with the point of a fine pin or pricker, prick through all the corners of the plan to be copied. Take them asunder, and connect the pricked points, on the clean paper, with lines; and it is done. This method is only to be practised in plans of such figures as are small and tolerably regular, or bounded by right lines.

Second Method. -Rub the back of the rough plan over with black-lead powder; and lay the said black part on the clean paper on which the plan is to

be copied, and in the proper position. Then with the blunt point of some hard substance, as brass, or such like, trace over the lines of the whole plan; pressing the tracer so much as that the black-lead under the lines may be transferred to the clean paper: after which, take off the rough plan, and trace over the leaden marks with common ink, or with Indian ink. Or, instead of blacking the rough plan, you may keep constantly a blacked paper to lay between the plans. Third Method. Another method of copying plans, is by means of squares. This is performed by dividing both ends and sides of the plan which is to be copied, into any convenient number of equal parts, and connecting the corresponding points of division with lines; which will divide the plan into a number of small squares. Then divide the paper, upon which the plan is to be copied, into the same number of squares, each equal to the former when the plan is to be copied of the same size, but greater or less than the others, in the proportion in which the plan is to be increased or diminished, when of a different size. Lastly, copy into the clean squares the parts contained in the corresponding squares of the old plan; and you will have the copy, either of the same size, or greater or less in any proportion.

-

Fourth Method. A fourth method is by the instrument called a pentagraph, which also copies the plan in any size required.

Procure a copying

Fifth Method. But the neatest method of any is this. frame or glass, made in this manner: namely, a large square of the best window glass, set in a broad frame of wood, which can be raised up to any angle, when the lower side of it rests on a table. Set this frame up to any angle before you, facing a strong light; fix the old plan and clean paper together with several pins quite around, to keep them together, the clean paper being laid uppermost, and over the face of the plan to be copied. Lay them, with the back of the old plan, on the glass, namely, that part which you intend to begin at to copy first; and, by means of the light shining through the papers, you will very distinctly perceive every line of the plan through the clean paper. In this state then trace all the lines on the paper with a pencil. Having drawn that part which covers the glass, slide another part over the glass, and copy it in the same manner. Then another part. And so on, till the whole is copied.

Then take them asunder, and trace all the pencil lines over with a fine pen and Indian ink, or with common ink.

And thus you may copy the finest plan, without injuring it in the least. When the lines are copied on the clean paper, the next business is to write such names, remarks, or explanations as may be judged necessary; laying down the scale for taking the lengths of any parts, a flower-de-luce to point out the direction, and the proper title ornamented with a compartment; illustrating or colouring every part in the manner that shall seem most natural, such as shading rivers or brooks with crooked lines; drawing the representations of trees, bushes, hills, woods, hedges, houses, gates, roads, &c., in their proper places; running a single dotted line for a footpath, and a double one for a carriage road; and either representing the bases or the elevations of buildings, &c.

OF ARTIFICERS' WORKS

AND

TIMBER MEASURING.

I. OF THE CARPENTER'S OR SLIDING RULE.

THE Carpenter's or Sliding Rule, is an instrument much used in measuring of timber and artificers' works, both for taking the dimensions, and computing the contents.

The instrument consists of two equal pieces, each a foot in length, which are connected together by a folding joint.

One side or face of the rule is divided into inches, and eighths, or half quarters. On the same face also are several plane scales, divided into 12th parts by diagonal lines; which are used in planning dimensions that are taken in feet and inches. The edge of the rule is commonly divided decimally, or into tenths; namely, each foot into ten equal parts, and each of these into ten parts again; so that by means of this last scale, dimensions are taken in feet, tenths and hundredths, and multiplied as common decimal numbers, which is the best way.

On the one part of the other face are four lines, marked A, B, C, D; the two middle ones, B and C, being on a slider, which runs in a groove made in the stock. The same numbers serve for both these two middle lines,—the one being above the numbers, and the other below.

These four lines are logarithmic ones, and the three A, B, C, which are all equal to one another, are double lines, as they proceed twice over from one to ten. The other or lowest line D, is a single one, proceeding from four to forty. It is also called the girt line, from its use in computing the contents of trees and timber; and upon it are marked WG at 17·15, and AG at 18.95, the wine and ale gauge points, to make this instrument serve the purpose of a gauging rule. On the other part of this face, there is a table of the value of a load, 50 cubic feet of timber, at all prices, from sixpence to two shillings a foot.

When 1 at the beginning of any line is accounted 1, then the I in the middle will be 10, and the 10 at the end 100; but when 1 at the beginning is accounted 10, then the 1 in the middle is 100, and the 10 at the end 1000; and so on. And all the smaller divisions are altered proportionally.

II. ARTIFICERS' WORK.

ARTIFICERS Compute the contents of their works by several different measures;

as,

Glazing and masonry by the foot;

Painting, plastering, paving, &c., by the yard, of 9 square feet;

Flooring, partitioning, roofing, tiling, &c., by the square of 100 square feet.

And brickwork, either by the yard of 9 square feet, or by the perch, or square rod or pole, containing 2724 square feet, or 304 square yards, being the square of the rod or pole of 161⁄2 feet of 51⁄2 yards long.

is often omitted in But when the exact

As this number 2724 is troublesome to divide by, the practice, and the content in feet divided only by the 272. divisor 272 is to be used, it will be easier to multiply the feet by 4, and then divide successively by 9, 11, and 11. Also to divide square yards by 304, first multiply them by 4, and then divide twice by 11.

All works, whether superficial or solid, are computed by the rules proper to the figure of them, whether it be a triangle or rectangle, a parallelopiped or any other figure.

III-BRICKLAYERS' WORK.

BRICKWORK is estimated at the rate of a brick and a half thick. So that, if a wall be more or less than this standard thickness, it must be reduced to it, as follows:

Multiply the superficial content of the wall by the number of half bricks in the thickness, and divide the product by 3.

The dimensions of a building are usually taken by measuring half round on the outside and half round on the inside; the sum of these two gives the compass of the wall,—to be multiplied by the height, for the content of the materials. Chimneys are by some measured as if they were solid, deducting only the vacuity from the hearth to the mantle, on account of the trouble of them.

And by others they are girt or measured round for their breadth, and the height of the story is their height, taking the depth of the jambs for their thickness. And in this case, no deduction is made for the vacuity from the floor to the mantle-tree, because of the gathering of the breast and wings, to make room for the hearth in the next story.

To measure the chimney shafts, which appear above the building, girt them about with a line for the breadth, to multiply by their height. And account their thickness half a brick more than it really is, in consideration of the plastering and scaffolding,

All windows, doors, &c., are to be deducted out of the contents of the walls in which they are placed. But this deduction is made only with regard to materials; for the whole measure is taken for workmanship, and that all outside measure too, namely, measuring quite round the outside of the building, being in consideration of the trouble of the returns or angles. There are also some other allowances, such as double measure for feathered gable ends, &c.

EXAMPLES.

Ex. 1.-How many yards and rods of standard brick-work are in a wall whose length or compass is 57 feet 3 inches, and height 24 feet 6 inches; the walls being 2 bricks or 5 half bricks thick? Ans. 8 rods, 17 yards.

Ex. II.

-Required the content of a wall 62 feet 6 inches long, and 14 feet 8
Ans. 169-753 yards.

inches high, and 21⁄2 bricks thick ?

Ex. 1.-A triangular gable is raised 171⁄2 feet high, on an end wall whose length is 24 feet 9 inches, the thickness being two bricks; required the reduced content? Ans. 32-08 yards.