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fields on the plans, 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, though it be likely that some of the parts will be taken a small matter too little, and others too great, yet they will, on the whole, in all probability, very nearly balance one another, and give a sufficiently accurate result. 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 re-computed, till they nearly agree.
6. But the chief art 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 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 straightsided figure equal to the curved one; the content of which, being computed as before directed, will be the content of the crooked figure proposed.
Or, instead of the straight edge of the horn, a horse-hair, or fine thread, may be applied across the crooked sides in the same manner; and the easiest way of using the thread, is to string a small slender bow with it, either of wire, or cane, or whale-bone, or such-like slender 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.
Thus, let it be required to find the contents of the same figure as in Prob. ix, page 65, to a scale of 4 chains to an inch.
Draw the 4 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 4 sides, ABCD. Then draw the diagonal BD, which, by applying a proper scale to it, measures suppose 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 555152 square links, or 5 acres, 2 roods, 8 perches, the content of the trapezium, or of the irregular crooked piece.
As a general example of this practice, let the contents be computed of all the fields separately in the foregoing plan facing page 75, and, by adding the contents altogether, the whole sum or content of the estate will be found nearly equal to 103 acres. Then, to prove the work, divide the whole plan into two parts, by a pencil line drawn across it any way near the middle, as from the corner I on the right, to the corner near s on the left; then, by computing these two large parts separately, their sum must be nearly equal to the former sum, when the work is all right.
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 this blacked 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 inarks with common ink, or with Indian ink-Or, instead of blacking the rough plan, we 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, on 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.
Fifth Method.-But the neatest method of any, at least in copying from a fair plan, is this. Procure a copying 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.
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 twelfth 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. Vol. II.
'These four lines are logarithmic ones, and the three A, B, t, which are all equal to one another, are double lines, as they proceed twice over from 1 to 10. The other or lowest line, D, is a single one, proceeding from 4 to 40. It is also called the girt line, from its use in computing the contents of trees and timber; and on it are marked wg at 17:15, and AG at 18:95, the wine and ale gage points, to make this instrument serve the pupose of a gaging rule,
On the other part of this face, there is a table of the value of a load, or 50 cubic feet of timber, at all prices, from 6 pence to 2 shillings a foot.
When 1 at the beginning of any line is accounted 1, then the 1 in the middle will be 10, and the 10 at the end 100; but when I at the beginning is counted 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 And brickwork, either by the yard of 9 square feet, or by
the perch, or square rod or pole, containing 2724 square feet, or 30 square yards, being the square of the rod
or pole of 16 feet or 54 yards long. As this number 2724 is troublesome to divide by, the is often omitted in practice, and the content in feet divided only by the 272.
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.