Where one diagonal is taken, it may perhaps close or meet with one part of the survey and not with the other; in this case, if the surveyor would discover his error, he must survey that part of the land which did not close, and this may be half or more, of the whole. And should the diagonal close with neither part, but be too long, or toc short, or should it fall on either side of the assigned point it was to close with, he ought to go over the whole, and make a new survey of it in order to discover his error.

A number of diagonals are frequently taken, the sum of the lengths of which very often exceeds the circuit of the ground, and after all they are but approximations, and the content remains uncertain as before; therefore he who returns a map, made up by the assistance of diagonals, where there remains a misclosure in any one part, runs the risque of being detected in an error, and must suffer uneasiness in his mind, as he cannot be certain of the return he makes.

The frequent misclosures which are botched up by diagonals, occasion the many and frequen scandalous broils and animosities between sureyors, which tend to the loss of character of the one or the other, and indeed often to the disrepute o both, as well as to that of the science they profess

But these may be easily remedied by intersections, and the bearing or line to be adjusted where the fault was committed, and till this be found, nothing can be certain.

SECTION VI.

To ENLARGE OR DIMINISH MAPS.

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 he 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 thence 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 @, from whence thro' a lay the fiducial edge of a thin ruler, with a fine pointed pair of compasses, take the distance from a to the centre O, and lay it by the ruler's edge from a to A: in the like manner take the distance from the next station 6 to the centre, and lay it over in a right line from 6 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 centrepoint, 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 number 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 further from the centre than the stationary points of the map are.

Mm

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 further 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 shewn 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 content.

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 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