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5. But in large pieces, and whole estates, consisting of many fields, it is the common practice to make a rough plan of the whole, and from it to compute the contents, quite independently of the measures of the lines and an. gles, that were taken in the field. 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 then 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 soraewhat 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 together, calculate the whole estate independently 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.
A specimen of the division of a large tract into trapeziums and triangles may be seen in Prab. VI. of the Practice of Surveying.
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 an equal area with these crooked sides, and so obtaining 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, thus :- Apply the straight edge of a thin, clear piece of lantern 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,
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 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, cane, 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, by which the straight line may be drawn.
Let it be required to find the content of the same figure as in Prob. IX. of the Practice of Surveying, to a scale of 4 chains to an inch.
Draw the four dotted straight lines AB, BC, CD, DA, cutting off equal quantities on both sides of them, which they do ąs nearly 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, is found to be 1256. Also the perpendicular, or nearest distance from A to this diagonal, is 456 ; and the distance of C from it 428.
And thus the content of the trapezium, and consequently of the irregular figure, to which it is equal, is easily found to be 5 acres, 2 roods, 8 perches.
To transfer a plan to another paper, &
After the rough plan is completed, and a fair one is wanted; this may be obtained, either on paper or vellum, by any of the following methods.
Lay the rough plan on the clean paper, and keep 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. Separate them 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.
Rub the back of the rough plan over with black lead powder ; and lay the said black part upon 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, &c. trace over the lines of the whole plan ; pressing the tracer so much, 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, &c., Or, instead of blacking the rough plan, you may keep constantly a blacked paper to lay between the plans.
Another way 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 one of the former, when the plan is to be copied of the same size; but greater or less in the proportion, in which the plan is to be increased or dimina ished, wlien of a different size. . Lastly, copy into the clean squares the parts contained in the corresponding squares of the other plan ; and you will have the copy either of the same size, or greater or less in the required proportion.
A fourth way is by the instrument, called a Pantographs which also copies the plan in any size required.
But the nearest method of any is this. Procure a copying frame or glass, made in this manner ; namely, 2 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 at any angle before you, facing a strong light ; fix the plan and clean paper together with pins in the edges, to keep them together ; the clean paper being laid, uppermost, and on the face of the plan to be copied. Lay them with the back of the plan on the glass, namely, the part you intend to copy first, and by means of the light shining through