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OF COMPUTING AND DIVIDING.
PROBLEM XVI. To Compute the Contents of Fields. 1. COMPUTE the contents of the figures as divided into triangles, or trapeziums, by the proper rules for these figures laid down in measuring ; multiplying the perpendiculars by the diagonalş or bases, both in links, and divide by %; the quotient is acres, after haying cut off five figures on ihe right for decimals. Then bring these decimals to roods and perches, by multiplying first by 4. and then by 40. An example of which is given in the description of the chain, pag. 429.
2 In small and separate pieces, it is usual to compute 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 adding all the offsets together, and dividing the sum by the number of them, ac. counting that for one of them where the boundary meets the station-line, (which increases the number of them by 1, for the divisor, though it does not increase the sum or quantity to be divided); then multiply the length by that mean breadth.
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
filds on the plan, so as to divide them into trapeziums and triangles, the bases and perpendiculars of wiich are mea. sured 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 litile, and oiers too great, yet they will, on the whole, in all probabilny, very nearly balance one another, and give a sufficienily 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 suns 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 luule 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. 1x, page 441, to a scale of 4 chains to an incú.
C Draw the 4 dotted straight lines AB, BC, CD, DA, cutting off equal quantities on both sides of theni, which they du cis pear as the eye can judge: so is the crooked figure reduced to an equivalent right-lined one of 4 sides, ABCD Tuen draw the diagonal BD, which, by applying a proper scals 10 it, measures suppose 1256. Also the perpendicuiar, 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 roots, 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 in page 452, and by adding the contents altogether, the whole sum or content of the estate will be found nearly equat to 1034 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 l on the right, to the corner near 8 on the left ; then, by computing these iwo 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 p.per 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, 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 conve. nient 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 lage 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.
OF ARTIFICERS' WORKS,
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 muluplied 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 pre being above the numbers, and the other below. Vol I.