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meet in one point, we thence conclude all is right, or so far as they do converge; but if we find a line of intersection to diverge or fly off from the rest, we may be sure that either a mistake has happened between the station the foregoing intersection was taken at and the station from whence the intersection line diverges, or there must be an error in the intersection; but to be assured in which of these the fault is, protract on to the next intersection, and having set it off, if it converges with the rest, though the foregoing one did not, we may conclude the fault was committed in taking the last intersection but one, and none in any station, and that so for is true as is protracted; but if this as well as the foregoing intersection diverge or fly from the point of concourse or converging point of the rest, the error must have its rise from some station or stations at or after that from whence the last converging intersection line was taken: so that by going to that station on the ground, and proceeding on to that where the next or from whence the following diverging intersection was taken, we can readily and with little trouble set all to rights. But in most tracts of land one object cannot be seen from every station, or from perhaps one-fourth of them; in this case we are under the necessity to move the pole after we begin to lose sight of it, to some other part of the land, where it may be seen from as many more stations as possible; which is easily done by viewing the boundary before it be surveyed: the pole then being fixed in an advantageous place, the first intersection to it is best to be made from the same station from whence the last one was taken, and then as often as may be thought convenient, as before; in like manner the whole may be done by the removal of the pole. When we here speak of stations, we do not mean such as are usually taken at every particular angle of the field: for it is to be apprehended, that every skilful surveyor, particularly such who use calculation, will take the longest distances possible, not only to lessen the number of stations, for the ease of either protraction or calculation, but with greater certainty to account for the land passed by, on the right-hand or on the left, which is taken by offsets: and surely it will be allowed that any measure taken on the ground, and the contents thence arithmetically computed, will be much more accurate than that which is obtained from any geometrical projection. From what has been said it is plain, that from this method any fault committed in a survey can be readily determined, and therefore must be much preferable to the present method of taking diagonals, or the bearings and lengths of lines across
land, to accomplish that end; which last method is too fre-
bearing or line to be adjusted where the fault was committed;
and till this be found, nothing can be certain.
To enlarge or diminish a map, or to reduce a map from one scale to another; also, the manner of uniting separate maps of lands which join each other into one map of any assigned size.
LAY the map you would enlarge over the paper on which you would enlarge it, and with a fine protracting pin prick through every angular point of your map; join these points on your paper (laying the map you copy before you) by pencilled or popped lines, and you have the copy of the map you are to enlarge: in this manner any protraction may be copied on paper, vellum, or parchment, for a fair map.
If you would enlarge a map to a scale which is double, or treble, or quadruple to that of the map to be enlarged, the paper you must provide for its enlargement must be two, or three, or four times as long and broad as the map; for which purpose in large things you will find it necessary to join several sheets of paper, and to cement them with white wafer or paste, but the former is best.
Then pitch upon any point in your copied map for a centre; from whence if distances be taken to its extreme points, and thenee if those distances be set in a right line with (but from) the centre, and these last points fall within your paper, the map may be increased on it to a scale as large again as its own; and if the like distances be again set outwards in right lines from the centre, and if these last points fall within your paper, it will contain a map increased to a scale three times as large as its own, &c.
PL. 12. fig. 2.
Let the pricked or popped lines represent the copy of a down or old survey, laid down by a scale of 80 perches to an inch, and let it be required to enlarge it to one laid down by 40 to an inch.
Pitch upon your centre as Q, from whence through a lay the fiducial edge of a thin ruler; with a fine-pointed pair of compasses take the distance from a to the centre @., and lay it by the ruler's edge from a to A ; in the like manner take the distance from the next station b to the centre @, and lay it over in a right line from b to B, and join the points A and B by the right line AB; in the like manner set over the distance from every station to the centre, from that station outwards, and you will have every point to enlarge to: the joining of these constantly as you go on by right lines will give you the enlarged map required.
In taking the distance from every station to the centre, set one foot of the compasses in the station, and the other very lightly over the centre-point, so lightly as scarcely to touch it, otherwise the centre-point will become so wide, that it may occasion several errors in the enlarged map : for if you err from the exact centre but a little, that error will become double, or treble, or quadruple, as you enlarge to a scale that is double, or treble, or quadruple of the given one; therefore great accuracy is required in enlarging a map.
When you have done with a station, give a dash with a pen or pencil to it, such as at the station a and b : by this means you cannot be disappointed in missing a station, or in laying your ruler over one station twice. From what has been said it is plain, that if a map is to be enlarged to one whose scale is double the given one, that the distances from the respective stations to the centre, being set over by the ruler's edge, will give the points for the enlarged one. And thus may a map be enlarged from a scale of 160 to one of 80, from one of 80 to one of 40, from one of 20 to one of 10 perches to an inch, &c. For to enlarge to a scale that is double, the mumber of perches to an inch for the enlarged map must be half of those to an inch for that to be enlarged: to enlarge to a scale that is treble the given one, the number of perches to an inch for the enlarged map will be one-third of those for the other; if to a scale that is quadruple the given one, the number of perches to an inch for the enlarged map will be one-fourth of those for the other, &c.: therefore, if you would enlarge a map which is laid down by a scale of 120 perches to an inch to one of 40 perches to an inch, the distance from the several stations to the centre, being set twice beyond the said stations, will mark out the several points required, for these points will be three times farther from the centre than the stationary points of the map are. In the same manner, if you would enlarge a map from a scale of 160 to one of 40 perches to an inch, the distance from the several stations to the centre, being set three times beyond said stations, will lay out the points for your enlarged map, for these points will be four times farther from the centre than are the stations of the map. When a map is enlarged to another, whose scale is double, or treble, or quadruple, &c. of the given one, every line, as well as the length and breadth of the enlarged map, will be double, or treble, or quadruple, &c. those of the given one, for it must be easy to conceive that those maps are like: but the area if the scale be double will be four times, if treble nine times, if quadruple sixteen times that of the given figure; that is, it will contain four, nine, or sixteen times as many square inches as the given one (for it has been shown that like polygons are in a duplicate proportion with the homologous sides). Yet these figures being cast up by their respective scales will produce the same contents. Thus much is sufficient for enlarging maps, and from hence, diminishing of them will be obvious; for one-fourth, one-third, or half the distances from the several stations to the centre
will mark out points which if joined will compose a map similar to the given one, whose scale will be four times, three times, or twice as small as the given one.
Thus, if we would reduce a map of from 40 to 80, from 20 to 40, from 10 to 20 perches to an inch, &c., half the distance of the stations from the centre will give the points requisite for drawing the map; if we would reduce from 40 to 120, from 20 to 60, from 10 to 30 perches to an inch, &c., one-third of the distances to the centre will give the points for the map; and if we would reduce from 40 to 160, from 20 to 80, from 10 to 40 perches to an inch, &c., one-fourth of the distances to the centre will give the points for the map.
By the methods here laid down I have reduced a map from a scale of 40 to one of 20 perches to an inch, which contained upwards of 1200 acres, and consisted of 224 separate divisions, without the least confusion from the lines; for none can arise if the methods here laid down be strictly observed.
I have also from the same methods reduced a large book of maps, each of which was an entire skin of parchment, and the whole contained upwards of 46,000 acres, to a pocket volume; and afterward connected all these maps into one map, which was contained in one skin of parchment: therefore, upon the whole I do recommend these methods for reducing maps to be much more accurate than any of the methods commonly used, such as squaring of paper, using a parallelogram, proportionable compasses, or any other method I ever met with, though the figures to be reduced were ever so numerous, irregular, or complicated.
To unite separate maps of lands which join each other into one map of any assigned size.
If there be several large maps contained in a book, each of which suppose to take up a skin of parchment, or a sheet of the largest paper, which maps of lands join each other, and it be required to reduce them to so small a scale that all of them when joined together may be contained in one skin, half a skin, or any assigned sized piece of parchment or paper:—
Having pricked off and copied the several maps on any kind of paper, unite them by cutting with scissors along the edge of one boundary which is adjoining the other, but not cutting by the edge of both," and throw aside the parts cut off: then lay these together on a large table, or on the floor, and where the boundaries agree they will fit in with each other as indentures do ; and after this manner they are easily connected: measure then the length and breadth of the entire connected maps, and