to a scale of a double size will be convenient, such a scale being also found on most instruments. In doing this, begin at the commencement of the field-book, or bottom of the first page, and draw the first line ah in any direction at pleasure, and then the next two sides of the first triangle bhj by sweeping intersected arcs; and so all the triangles in the same manner, after each other in their order; and afterwards setting the perpendicular and other offsets at their proper places, and through the ends of them drawing the bounding fences. Note. That the field-book begins at the bottom of the first page, and reads up to the top; hence it goes to the bottom of the next page, and to the top; and thence it passes from the bottom of the third page to the top, which is the end of the field-book. The several marks measured to or from, are here denoted by the letters of the alphabet, first the small ones, a, b, c, d,... and after them the capitals, A, B, C, D, ... But, instead of these letters, some surveyors use the numbers in order, 1, 2, 3, 4, &c.* * In surveying with the plain table, a field-book is not used, as every thing is drawn on the table immediately when it is measured. But in surveying with the theodolite, or any other instrument, some kind of a field-book must be used, to write down in it a register or account of all that is done and occurs relative to the survey in hand. This book every one contrives and rules as he thinks fittest for himself. The following is a specimen of a form which has been much used by country surveyors. It is ruled in three columns, as below. Here is the first station, where the angle or bearing is 105° 25′. On the left, at 73 links in the distance or principal line, is an offset of 92; and at 610 an offset of 24 to a cross hedge. On the right, at 0, or the beginning, an offset 25 to the corner of the field; at 248 Brown's boundary hedge commences; at 610 an offset 35; and at 954, the end of the first line, the 0 denotes its terminating in the hedge. And so on for the other stations. A line is drawn under the work, at the end of every station-line, to prevent confusion. 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 diagonals or bases, both in links, and divide by 2; the quotient is acres, after having 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 is given in the description of the chain, p. 462. 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 trapezoïds. 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, accounting 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 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 book, namely, beginning at the bottom of the page and writing upwards; sketching also a neat boundary on either hand, resembling the parts near the measured lines as they pass along; an example of which was given in the new method of surveying, in the preceding pages. In smaller surveys and measurements, a good way of setting down the work is to draw by the eye, on a piece of paper, a figure resembling that which is to be measured: and so writing the dimensions, as they are found, against the corresponding parts of the figure. This method may, also, be practised to a considerable extent, even in the larger surveys. 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 recomputed, 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 enclose 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 easy, and accurately performed in this manner :-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, 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 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 whalebone, 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. EXAMPLE. Thus, let it be required to find the contents of the same figure as in prob. 9. p. 499, to a scale of 4 chains to an inch. 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 506, 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. The content of irregular fields, farms, &c. when planned, may be readily and correctly found by the process of weighing *. If the plan be not upon paper, or fine drawing pasteboard of uniform texture, let it be transferred upon such. Then cut the figure separately close upon its boundaries, and cut out from the same paper or pasteboard a square of known dimensions, according to the scale employed in drawing the plan. Weigh the two separately in an accurate balance, and the ratio of the weight will be the same as that of the superficial contents. If great accuracy be required, cut the plan into four portions, called 1, 2, 3, 4. First, weigh 1 and 2 together, 3 and 4 together, and take their sum. Then weigh 1 and 3 together, 2 and 4 together, and take their sum. Lastly, weigh 1 and 4 together, 2 and 3 together, and take their sum. The mean of the three aggregate weights thus obtained, compared with the weight of the standard square, will give the ratio of their surfaces very nearly. PROBLEM XVII. To transfer a plan to another paper. 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 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. This 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. See p. 399. Fourth method. By the instrument called a pentagraph, which also copies the plan in any size required; for this purpose, also, Professor Wallace's eidograph may be advantageously employed. Fifth method. A very neat process, at least for copying from a fair plan, is this: procure a copying frame of glass, made in this manner; namely, a large * By a method like this, Dr. Long found the quantities of land and water on our globe to be very nearly as 2 to 5. He cut up the gores of a globe for the purpose. square of the best plate-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. You may thus copy the finest plan, without injuring it. ARTIFICERS' WORK AND TIMBER MEASURE. ARTIFICERS Compute their work in different ways: the chief distinctions of which are the following : 1. Glazing and masonry by the foot square *. 2. Painting, plaistering, paving, and paperhanging by the yard square. 3. Flooring, partitioning, roofing, and tiling by the square of 100 feet, or a square whose side is 10 feet. 4. The removal of earth, as in forming roads and railways, the purchase of stone, and other works on which volume is concerned, the measures are either the cubic foot or the cubic yard. All works, whether of superficial or solid measure, are computed by the rules proper to the figure of the magnitude concerned, and therefore come under one or other of the methods already explained for the mensuration of surfaces and solids. The only peculiarity of the operations as distinct from those already laid down, is the computation of the value of the work done or the materials supplied. The particular customary allowances to be made are detailed in the notes to the several kinds of work in which they occur. I. BRICKLAYERS' WORK. BRICKWORK is estimated at the rate of a brick and a half thick: but 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 †. * This is only the common mode of expressing the magnitude which has been previously designated as the " square foot." + The dimensions of a building may be 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, which, multiplied by the height, gives the content of the materials. Chimneys are commonly measured as if they were solid, on account of the trouble, deducting |