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demands great care and system, particularly when the plan is large. A very ordinary method that is adopted is, to lay the original on to the blank paper, and then with a fine needle-point to prick through every corner and bend. These points being thus determined on the copy, the pencil is applied to complete the work, which, after being compared with the original, is completed with the drawing-pen and Indian-ink. This method, however, by no means improves the original plan, which may be of great value, even in respect of finished drawing only, and we consider it in every way inferior to the following mode. The original plan should be laid down on a flat table, of dimensions corresponding to the size of the drawing to be copied ; over it should be laid clear tracing-paper, as large as can be obtained, if the size of the original requires it; the whole should then be loaded with any long flat rules that are to be come at, and leaden weights, so that the whole shall lay perfectly flat; the plan is then traced off
, in one or half-a-dozen sheets, as may be required. This need not be in ink, a sharp, clean-pointed pencil will do very well; but take care that in tracing sheet after sheet sufficient work is repeated along the length and width of each sheet of tracing paper to determine accurately their relative positions if they were all joined together. The correctness of a plan as a whole in a great measure depends on this particular feature, and great care should be taken when first laying down the tracing-paper, not only that it lays flat, but also without any twist in its length or breadth. The blank paper which is to receive the copy is now laid down on the board, and over it thin sheets of tissue-paper that have had the under-side well rubbed with black-lead; two or three of these may be laid down one alongside of the other, if size requires it; over the whole is laid one of the tracings, loaded as before, so as to lay flat and even, and with a tracing. point the plan is thus transferred to the blank sheet of paper underneath, by means of the black-lead paper; this tracer we manufacture ourselves, by driving the point of a fine needle into a pencil, and making use of the head as a tracer.
The first sheet being now accurately transferred, the next tracing is laid down so that the abovementioned repeated work along the length and breadth of the tracing shall perfectly coincide with the pencil-work which has just been transferred to the blank paper ; it is then fixed in this position, and the black-leaded paper is gently slipped underneath; the whole is then loaded as before and this second sheet thus transferred, and by repe. tition the whole of the plan to be copied. The whole may then be inked in.
Another very good method, though, as we think, much more tedious than the above, is to triangulate the original, as in surveying; trans. fer, or rather lay down the triangles on the blank paper, and by a system of offsets copy the whole.
The reduction of a plan is often effected by a system of proportional squares, as one-quarter, or one-fourth, as may be required; these must be all perfectly rectangular, and the perpendiculars should all be carefully drawn in, by means of a large and perfectly correct set square,
of the square.
taking care, both on the original and the copy, to use the same side
The reduction is, however, much more rapidly effected by means of the two following instruments.
The pentagraph (Fig. 143) consists of four flat rulers, made either of wood or brass ; the two outside ones are generally from 15 to 24 inches long, and the others about half that length. The longer ones, A and B, are united together by a pivot, about which they turn like the legs of a pair of compasses, and the two smaller rulers are similarly attached to each other at D, and to the longer rulers. A sliding box is placed on each of the arms, A and B, which may be fixed by a clamp-screw at any part of the ruler ; these slides carry a tube, to contain either a blunt tracing-point, a pencil, or pen, or the fulcrum G, which is a heavy weight of lead, having a point on the under side to pierce the drawingboard and remain immovable in its proper position, it being the centre upon which the whole instrument turns. Several ivory castors support the surface of the machine parallel to the paper, as well as facilitate its motions.
The arms, D and B, are graduated and marked with the ratios, }, }, &c., so that when a copy of a plan is required to be made in any of these proportions, it is only requisite to fix, at the required ratio, the slides carrying the fulcrum, Ĝ, and the tube at D, with a pencil or pen, and the instrument will be ready for operation. Thus, suppose it were required to make a copy of a plan exactly one-half the size of the original, the slide carrying the pencil, and that working on the fulcrum are each fixed by their respective clamp-screws at the divisions marked t, the pentagraph being first spread out so as to give room for the tracing-point to be passed over every line on the plan, whilst the pencil is making corresponding marks on the copy, which it is evident will be equal to one-half the size of the original. A fine string is attached to the pencil-holder, and passed round to the tracing-point, the pulling at which raises the pencil a small quantity above the paper, to prevent false or improper marks upon
copy It should also be remarked, that the cup represented on the top of the pencil-holder is intended to receive a weight, to keep the pencil down upon the paper, or when a stronger mark is required.
, When the instrument is set for work, the tracing-point, the pencil, and the fulcrum must in all cases be in a straight line, which may be proved by stretching a fine string over them.
When it is required to make an enlarged copy of a plan, the setting of the instrument is precisely the same as above-stated, only the tracing-point and pencil must change, places. But when a copy is to be made of the same size as the original, the fulcrum must be placed in the middle.
With regard to an enlarged copy of a plan, by whichever of the above methods it may be done, it can never be considered a very correct representation of the lands surveyed; and when an enlarged plan is required, a fresh plot should be made from the field-book, according to the scale required.
The eidograph is an improved instrument for copying and reducing plans, and illustrated at Fig. 144. After what has been said about the former instrument, a little experience will fully explain the use of the latter and its advantages.
The Planimeter and Computation of Areas. Those who have had occasion to perform. much scaling and computations of areas are aware of the tedious nature of the task. With irregular figures, such as are constantly met with, it is in the first place often troublesome to give and take correctly, and indeed it is in practice at the best of times a mere guess, however near it may be to accuracy. The figure has then to be divided into triangles, of which one side and one perpendicular, have to be scaled, another constant source of error, however small, or the polygon has to be carefully reduced to one triangle, which requires a great deal of care, and the perpendicular and base again scaled ; the computations have then to be made by multiplication, and all of these have to be carefully checked before the work can be depended on. There will therefore be nothing at all surprising when
. we observe that many inventions have been suggested from time to time either to shorten the modus operandi, or to facilitate the calculations. None of them, however, appear to have met with success, and the old fashioned mode has had to be adhered to. The planimeter, however, bids fair to make the laborious task much more easy, inasmuch as with it we have only to trace the periphery of the figure with a tracer, and then merely read off the contents. This instrument, illustrated at Fig. 145, and lately introduced by Messrs. Elliott, has already been received with a great deal of favour. E and F are two points at the ends of the moveable arms, A and B; when in use the instrument rests on these two points, and on a third, being that part of the graduated wheel D which is tangent to the paper. G is a graduated disc connected with an endless screw, and H is a vernier to the wheel D. When the instrument is applied to the operation of computing, the point E is firmly fixed at any convenient point on the paper. A very little thought and experience will show that the object sought in fixing this point will be to place the point of the tracer F sO that it may be made to run over the greatest possible portion of the periphery of the figure, the contents of the area of which are required. Let it be considered that in the drawing the instrument is so placed, and that the point of the tracer is at a convenient point of the periphery of the irregular figure to be traced. Read off the instrument, trace the contour of the figure, and read off a second time. Subtract the first reading from the second; multiply the remainder by 10, which will give the area required in square inches, with two places of decimals. Let the first reading of the disc G be 2, that of D equal to 80, but with the vernier reading instead of 80 let it be 805; then the reading of the instrument at starting is to be set down as 2.805; let the second reading be 7.553; this minus the first is equal to 4.748, which multiplied by 10 gives 47 48 as the contents in square inches which was required. In carrying the tracer round, care must be taken to observe whether the disc G has performed an entire revolution, for
in that case 10 must be added to the unit in the last reading. For instance, in the above example, had the second reading been 7-553, plus an entire circuit, then we should have had 7.553 +10= 17:553, from which subtracting 2.805, and multiplying the remainder by 10, we should get 147.48. If the figure is too large for the instrument to contour at once,
be divided into two or three. The contents of each being taken separately, they are all to be added together.
The instrument is constructed to give areas in other denominations than inches, but we prefer the square inch because it is a convenient constant to work from, whatever the scale of the plan. Suppose, for instance, that in the above example we have been calculating an area from the Ordnance map, having a scale of one inch to the mile; one square inch is one square mile, we should therefore have very nearly 472 square miles; again, let us suppose that we have been measuring an area from a plan plotted to a scale of five chains to one inch ; one inch or five chains square, equal to 250,000 square links; from this we obtain the constant by which to multiply the square inches found by the planimeter to reduce the area to square links for a five chain scale ; thus, 47 48 x 2500 = 11,870,000. Then for a 1 chain scale we shall have
3600 by which to multiply the square inches found by the planimetric operations, to reduce them to square links, which will be the more easily remembered from the multipliers being merely the square of the number of chains to one inch, with two zeros added.
As before explained, if from square links we strike off five figures to the right, we have the areas on the left, and the little tablet at the end of the volume will give the roods and perches due to the square Jinks first cut off. We are aware that old hands generally object to have their practice of routine broken in upon by any novelty. It is natural to all of us; but if they will give this a trial, they will find the labour of scaling and computing very greatly reduced, and, as we think, more accurately performed.
EXPLANATIONS FOR THE USE OF THE TABLES.
TABLE I.-Reference to what has been said as to chaining over inclined ground sufficiently explains the use of this table.
TABLE II. is for facilitating the process of computing areas ; one example will explain its use. The area of an enclosure has been found equal to 25.69981; point off five figures to the right, as shown by the decimal point, and in one of the columns marked 1 R, 2 R, 3 R, look for the first three figures of those pointed off. The heading to the column R will give the roods, and in the column headed Pls. and on a line with the figures found, will be the number of perches; the nearest figures to 69981 will be found to be 700 in column 2 R, and on the same line will be found 32, which gives 2 roods, 32 perches, and the total contents are found to be 25A. 2R. 32P.
TABLE III.-The Lengths of Circular Arcs. The application of this table has been explained under the head of length of curve, in the chapter on setting out curves.
TABLE IV.—The use of this table has already been very amply explained. When offsets are required for setting out curves, refer to the particnlar radius required. In the first horizontal column are the lengths of the arcs in chains; in the second, the lengths of A c in chains, from which the right-angled offset is to be set off
, and in the third, the lengths of the offsets in feet. Fig. 146 illustrates the whole.
TABLE V.-Corrections for curvature and refraction, when required in levelling operations.
TABLE VI.--Natural sines and tangents to every ten minutes ; example illustrating their application :-Required the sine of 9° 47'; from the tables take out the sines of 9° 40' and 9o 50"; subtract the lesser from the greater ; divide the difference by 10, and multiply the quotient by the number of odd minutes, here equal to 7.
Sine of 9° 50', sine of 9° 40'=0028669 the difference which divided by 10, and multiplied by 7 is equal to .0020068 and sine of 9° -40'= 1679159
•1699227=Natural of 9° 47' as required. Let it now be required to find the cosine of 9° 47'. Repeat the above operations on the cosines in tables; but instead of adding the quantity found to the cosine of 9° 40', it is to be subtracted, and the cosine of go 47' will thus be found equal to .9854565, near enough for all the purposes we require in practice.