SURVEYING

, the art, or act, of measuring land. This comprises the three following parts; viz, taking the dimensions of any tract or piece of ground; the delineating or laying the same down in a map or draught; and finding the superficial content or area of the same; beside the dividing and laying out of lands.

The first of these is what is properly called Surveying; the second is called plotting, or protracting, or mapping; and the third casting up, or computing the contents.

The first again consists of two parts, the making of observations for the angles, and the taking of lineal measures for the distances.

The former of these is performed by some of the following instruments; the theodolite, circumferentor, semicircle, plain-table, or compass, or even by the chain itself: the latter is performed by means either of the chain, or the perambulator. The description and manner of using each of these, see under its respective article or name.

It is useful in Surveying, to take the angles which the bounding lines form with the magnetic needle, in order to check the angles of the figure, and to plot them conveniently afterwards. But, as the difference between the true and magnetic meridian perpetually varies in all places, and at all times; it is impossible to compare two surveys of the same place, taken at distant times, by magnetic instruments, without making due allowance for this variation. See observations on this subject, by Mr. Molineux, Philos. Trans. number 230, p. 625, or Abr. vol. 1, p. 125.

The second branch of Surveying is performed by means of the protractor, and plotting scale. The description of which, see under their proper names.

If the lands in the survey are hilly, and not in any one plane, the measured lines cannot be truly laid down on paper, till they are reduced to one plane, which must be the horizontal one, because angles are taken in that plane. And in this case, when observing distant objects, for their elevation or depression, the following table shews the links or parts to be subtracted from each chain in the hypothenusal line, when the angle is the corresponding number of degrees.

A Table of the links to be subtracted out of every
chain in hypothenusal lines, of several degrees of
altitude or depression, for reducing them to hori-
zontal.
		links			links
4°	3′	1/4	19°	57′	6
5	44	1/2	21	34	7
7	1	3/4	23	4	8
8	7	1	24	30	9
11	29	2	24	50	10
14	4	3	27	8	11
16	16	4	28	22	12
18	12	5	29	32	13

|

For example, if a station line measure 1250 links, or 12 1/2 chains, on an ascent, or a descent, of 11°; here it is after the rate of almost two links per chain, and it will be exact enough to take only the 12 chains at that rate, which make 24 links in all, to be deducted from 1250, which leaves 1226 links, for the length to be laid down.

Practical surveyors say, it is best to make this deduction at the end of every chain-length while measuring, by drawing the chain forward every time as much as the deduction is; viz, in the present instance, drawing the chain on 2 links at each chain-length.

The third branch of Surveying, namely computing or casting-up, is performed by reducing the several inclosures and divisions into triangles, trapeziums, and parallelograms, but especially the two former; then finding the areas or contents of these several figures, and adding them all together.

1. Land is measured with a chain, called Gunter's chain, of 4 poles or 22 yards in length, which consists of 100 equal links, each link being 22/100 of a yard, or 66/100 of a foot, or 7.92 inches long, that is nearly 8 inches or 2/3 of a foot.

An acre of land is equal to 10 square chains, that is, 10 chains in length and 1 chain in breadth. Or it is 40 X 4 or 160 square poles. Or it is 220 X 22 or 4840 square yards. Or it is 1000 X 100 or 100000 square links. These being all the same quantity.

Also, an acre is divided into 4 parts called roods, and a rood into 40 parts called perches, which are square poles, or the square of a pole of 5 1/2 yards long, or the square of 1/4 of a chain, or of 25 links, which is 625 square links. So that the divisions of land measure will be thus:

625	sq. links	= 1 pole or perch
40	perches	= 1 rood
4	roods	= 1 acre

The length of lines, measured with a chain, are set down in links as integers, every chain in length being 100 links; and not in chains and decimals. Therefore, after the content is found, it will be in square links; then cut off five of the figures on the right-hand for decimals, and the rest will be acres. Those decimals are then multiplied by 4 for roods, and the decimals of these again by 40 for perches.

Ex. Suppose the length of a rectangular piece of ground be 792 links, and its breadth 385: to find the area in acres, roods, and perches.

792
385
3960
6336
2376
	ac. ro. p.
3.04920	Ans. 3   0  7
4
.19680
40
7.87200

2. Among the various instruments for surveying, the plain-table is the easiest and most generally useful, especially in crooked difficult places, as in a town among houses, &c. But although the plain-table be the most generally useful instrument, it is not always so; there being many cases in which sometimes one instrument is the properest, and sometimes another; nor is that surveyor master of his business who cannot in any case distinguish which is the fittest instrument or method, and use it accordingly: nay, sometimes no instrument at all, but barely the chain itself, is the best method, particularly in regular open fields lying together; and even when you are using the plain-table, it is often of advantage to measure such large open parts with the chain only, and from those measures lay them down upon the table.

The perambulator is used for measuring roads, and other great distances on level ground, and by the sides of rivers. It has a wheel of 8 1/4 feet, or half a pole, in circumference, upon which the machine turns; and the distance measured is pointed out by an index, which is moved round by clock work.

Levels, with telescopic or other sights, are used to find the level between place and place, or how much one place is higher or lower than another.

An offset-staff is a very useful and necessary instrument, for measuring the offsets and other short distances. It is 10 links in length, being divided and marked at each of the 10 links.

Ten small arrows, or rods of iron or wood, are used to mark the end of every chain length, in measuring lines. And sometimes pickets, or staves with flags, are set up as marks or objects of direction.

Various scales are also used in protracting and measuring on the plan or paper; such as plane scales, line of chords, protractor, compasses, reducing scales, parallel and perpendicular rulers, &c. Of plane scales, there should be several sizes, as a chain in 1 inch, a chain in 3/4 of an inch, a chain in 1/2 of an inch, &c. And of these, the best for use are those that are laid on the very edges of the ivory scale, to prick off distances by, without compasses.

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 sort of a fieldbook 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 that has formerly been much used. It is ruled into 3 columns: the middle, or principal column, is for the stations, angles, bearings, distances measured, &c; and those on the right and left are for the offsets on the right and lest, which are set against their corresponding distances in the middle column; as also for such remarks as may occur, and be proper to note for drawing the plan, &c.

Here Θ 1 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.

Draw a line under the work, at the end of every station line, to prevent confusion.

	Stations,
Offsets and Remarks	Bearings,	Offsets and Remarks
on the left.	and	on the right.
	Distances.
	Θ1
	105°25′
	00	25 corner
92	73
	248	Brown's hedge
cross a hedge 24	610	35
	954	00
	Θ2
	53°10′
	00	00
house corner 51	25	21
	120	29 a tree
34	734	40 a stile
	Θ3
	67°20′
	61	35
a brook 30	248
	639	16 a spring
foot path 16	810
cross hedge 18	973	20 a pond

But a few skilful surveyers now make use of a different method for the field book, namely, beginning at the bottom of the page and writing upwards; by which they sketch a neat boundary on either hand, as they pass it; an example of which will be given below, with the plan of the ground to accompany it.

In smaller surveys and measurements, a very 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 write the dimensions, as they are found, against the corresponding parts of the figure. And this method may be practised to a considerable extent, even in the larger surveys.

4. To measure a line on the ground with the chain: Having provided a chain, with 10 small arrows, or rods, to stick one into the ground, as a mark, at the end of every chain; two persons take hold of the chain, one at each end of it, and all the 10 arrows are taken by one of them who goes foremost, and is called the leader; the other being called the follower, for distinction's sake.

A picket, or station staff, being set up in the direction of the line to be measured, if there do not appear some marks naturally in that direction; the follower stands at the beginning of the line, holding the ring at the end of the chain in his hand, while the leader drags forward the chain by the other end of it, till it is stretched straight, and laid or held level, and the leader directed, by the follower waving his hand, to the right or left, till the follower see him exactly in a line with the mark or direction to be measured to: then both of them stretching the chain straight, and stooping and holding it level, the leader having the head of one of his arrows in the same hand by which he holds the end of the chain, he there sticks one of them down with it, while he holds the chain stretched. This done, he leaves the arrow in the ground, as a mark for the follower to come to, and advances another chain forward, being directed in his position by the follower standing at the arrow, as before; as also by himself now, and at every succeeding chain's length, by moving himself from side to side, till he brings the follower and the back mark into a line. Having then stretched the chain, and stuck down an arrow, as before, the follower takes up his arrow, and they advance again in the same manner another chain length. And thus they proceed till all the 10 arrows are employed, and are in the hands of the follower; and the leader, without an arrow, is arrived at the end of the 11th chain length. The follower then sends or brings the 10 arrows to the leader, who puts one of them down at the end of his chain, and advances with the chain as before. And thus the arrows are changed from the one to the other at every 10 chains' length, till the whole line is finished; the number of changes of the arrows shews the number of tens, to which the follower adds the arrows he holds in his hand, and the number of links of another chain over to the mark or end of the line. So if there have been 3 changes of the arrows, and the follower hold 6 arrows, and the end of the line cut off 45 links more, the whole length of the line is set down in links thus, 3645.

Let B and C be two ob- jects, or two pickets set up perpendicular; and let it be required to take their bearings, or the angle formed between them at any station A.

1st. With the Plain Table. The table being covered with a paper, and fixed on its stand; plant it at the station A, and fix a fine pin, or a point of the compasses, in a proper part of the paper, to represent the point A: Close by the side of this pin lay the fiducial edge of the index, and turn it about, still touching the pin, till one object B can be seen through the sights: then by the fiducial edge of the index draw a line. In the very same manner draw another line in the direction of the other object C. And it is done.

2d. With the Theodolite, &e. Direct the fixed sights along one of the lines, as AB, by turning the instrument about till you see the mark B through these sights; and there screw the instrument fast. Then | turn the moveable index about till, through its sights, you see the other mark C. Then the degrees cut by the index, upon the graduated limb or ring of the instrument, shew the quantity of the angle.

3d. With the Magnetic Needle and Compass. Turn the instrument, or compass, so, that the north end of the needle point to the flower-de-luce. Then direct the sights to one mark, as B, and note the degrees cut by the needle. Then direct the sights to the other mark C, and note again the degrees cut by the needle. Then their sum or difference, as the case is, will give the quantity of the angle BAC.

4th. By Measurement with the Chain, &c. Measure one chain length, or any other length, along both directions, as to b and c. Then measure the distance b c, and it is done.—This is easily transferred to paper, by making a triangle A b c with these three lengths, and then measuring the angle A as in Practical Geometry.

A h i k l m n being a crooked hedge, or river, &c. From A measure in a straight direction along the side of it to B. And in measuring along this line AB observe when you are directly opposite any bends or corners of the hedge, as at c d, e, &c; and from thence measure the perpendicular offsets, ch, di, &c, with the offset-staff, if they are not very large, otherwise with the chain itself; and the work is done. And the register, or field-book, may be as follows:

Offs. left.	Base line AB
	0	Θ	A
ch	62	45	Ac
di	84	220	Ad
ek	70	340	Ae
fl	98	510	Af
gm	57	634	Ag
Bn	91	785	AB

AP	794
AB	1321
PC	826

Having set up marks at the corners, which is to be done in all cases where there are not marks naturally; measure with the chain from A to P, where a perpendicular would fall from the angle C, and there measure from P to C; then complete the distance AB by measuring from P to B; setting down each of these measured distances. And thus, having the base and perpendicular, the area from them is easily found. Or having the place P of the perpendicular, the triangle is easily constructed.

Or, measure all the three sides with the chain, and note them down. From which the content is easily found, or the figure constructed.

Measure two sides AB, AC, and the angle A between them. Or measure one side AB, and the two adjacent angles A and B. From either of these ways the figure is easily planned: then by measuring the perpendicular CP on the plan, and multiplying it by half AB, you have the content.

AE	214	210	DE
AF	362	306	BF
AC	592

Measure along either of the diagonals, as AC; and either the two perpendiculars DE, BF, as in the last problem; or else the sides AB, BC, CD, DA. From either of which the figure may be planned and computed as before directed.

AP	110	352	PC
AQ	74	595	QD
AB	1110

Measure on the longest side, the distances AP, AQ, AB; and the perpendiculars PC, QD.

Measure the diagonal AC (see the first fig. above), and the angles CAB, CAD, ACB, ACD.—Or measure the four sides, and any one of the angles as BAD.

Thus	Or thus
AC	59<*>	AB	486
CAB	37°20′	BC	394
CAD	41 15	CD	410
ACB	72 25	DA	462
ACD	54 40	BAD	78°35′

Having set up marks at the corners, where necessary, of the proposed field ABCDEFG. Walk over the ground, and consider how it can best be divided into triangles and trapeziums; and measure them separately as in the last two problems. And in this way it will be proper to divide it into as few separate triangles, and as many trapeziums as may be, by drawing diago- | nals from corner to corner: and so, as that all the perpendiculars may fall within the figure. Thus, the following figure is divided into the two trapeziums ABCG, GDEF, and the triangle GCD. Then, in the first, beginning at A, measure the diagonal AC, and the two perpendiculars Gm, Bn. Then the base GC and the perpendicular Dq. Lastly the diagonal DF, and the two perpendiculars pE, oG. All which measures write against the corresponding parts of a rough figure drawn to resemble the figure to be surveyed, or set them down in any other form you choose.

Am	135	130	mG
An	410	180	nB
Ac	550
-------	-------
Cq	152	230	qD
CG	440
-------	-------
FO	206	120	oG
FP	288	80	pE
FD	520

Measure all the sides AB, BC, CD, DE, EF, FG, and GA; and the diagonals AC, CG, GD, DF.

Many pieces of land may be very well surveyed, by measuring any base line, either within or without them, together with the perpendiculars let fall upon it from every corner of them. For they are by these means divided into several triangles and trapezoids, all whose parallel sides are perpendicular to the base line; and the sum of these triangles and trapeziums will be equal to the figure proposed if the base line fall within it; if not, the sum of the parts which are without being taken from the sum of the whole which are both within and without, will leave the area of the figure proposed.

In pieces that are not very large, it will be sufficiently exact to find the points, in the base line, where the several perpendiculars will fall, by means of the cross, and from thence measuring to the corners for the lengths of the perpendiculars.—And it will be most convenient to draw the line so as that all the perpendiculars may fall within the figure.

Thus, in the following figure, beginning at A, and measuring along the line AG, the distances and perpendiculars, on the right and left, are as below.

Ab	315	350	bB
Ac	440	70	cC
Ad	585	320	dD
Ae	610	50	eE
Af	990	470	fF
AG	1020	0

Plant the table at any angle, as C, from whence all the other angles, or marks set up, can be seen; and turn the table about till the needle point to the flower-de-luce: and there screw it fast. Make a point for C on the paper on the table, and lay the edge of the index to C, turning it about there till through the sights you see the mark D; and by the edge of the index draw a dry or obscure line: then measure the distance CD, and lay that distance down on the line CD. Then turn the index about the point C, till the mark E be seen through the sights, by which draw a line, and measure the distance to E, laying it on the line from C to E. In like manner determine the positions of CA and CB, by turning the sights successively to A and B; and lay the lengths of those lines down. Then connect the points with the boundaries of the field, by drawing the black lines CD, DE, EA, AB, BC.

When all the other parts cannot be seen from one angle, choose some place O within; or even without, if more convenient, from whence the other parts can be seen. Plant the table at O, then fix it with the needle north, and mark the point O upon it. Apply the index successively to O, turning it round with the sights to each angle A, B, C, D, E, drawing dry lines to them by the edge of the index, then measuring the distances OA, OB, &c, and laying them down upon those lines. Lastly draw the boundaries AB, BC, CD, DE, EA.

When the figure is a wood or water, or from some other obstruction you cannot measure lines across it; begin at any point A, and measure round it, either within or without the figure, and draw the directions of all the sides thus: Plant the table at A, turn it with the needle to the north or flower-de-luce, fix it and mark the point A. Apply the index to A, turning it till you can see the point E, there draw a line; and then the point B, and there draw a line: then measure these lines, and lay them down from A to E and B. Next move the table to B, lay the index along the line AB, and turn the table about till you can see the mark A, and screw fast the table; in which position also the needle will again point to the flower-de-luce, as it will | do indeed at every station when the table is in the right position. Here turn the index about B till through the sights you see the mark C; there draw a line, measure BC, and lay the distance upon that line after you have set down the table at C. Turn it then again into its proper position, and in like manner find the next line CD. And so on quite round by E to A again. Then the proof of the work will be the joining at A: for if the work is all right, the last direction EA on the ground, will pass exactly through the point A on the paper; and the measured distance will also reach exactly to A. If these do not coincide, or nearly so, some error has been committed, and the work must be examined over again.

When all the angles can be seen from one point, as the angle C (last fig. but one), place the instrument at C, and turn it about till, through the fixed sights, you see the mark B, and there fix it. Then turn the moveable index about till the mark A is seen through the sights, and note the degrees cut on the instrument. Next turn the index successively to E and D, noting the degrees cut off at each; which gives all the angles BCA, BCE, BCD. Lastly, measure the lines CB, CA, CE, CD; and enter the measures in a field-book, or rather against the corresponding parts of a rough figure drawn by guess to resemble the field.

Plant the instrument at O, (last fig.) and turn it about till the fixed sights point to any object, as A; and there screw it fast. Then turn the moveable index round till the sights point successively to the other points E, D, C, B, noting the degrees cut off at each of them; which gives all the angles round the point O. Lastly, measure the distances OA, OB, OC, OD, OE, noting them down as before, and the work is done.

By measuring round, either within or without the field, proceed thus. Having set up marks at B, C, &c. near the corners as usual, plant the instrument at any point A, and turn it till the fixed index be in the direction AB, and there screw it fast: then turn the moveable index to the direction AF; and the degrees cut off will be the angle A. Measure the line AB, and plant the instrument at B, and there in the same manner observe the angle A. Then measure BC, and observe the angle C. Then measure the distance CD, and take the angle D. Then measure DE, and take the angle E. Then measure EF, and take the angle F. And lastly measure the distance FA.

To prove the work; add all the inward angles, A, B, C, &c, together, and when the work is right, their sum will be equal to twice as many right angles as the figure has sides, wanting 4 right angles. And when there is an angle, as F, that bends inwards, and you measure the external angle, which is less than two right angles, subtract it from 4 right angles, or 360 degrees, to give the internal angle greater than a semicircle or 180 degrees.

Otherwise. Instead of observing the internal angles, you may take the external angles, formed without the figure by producing the sides further out. And in this case, when the work is right, their sum altogether will be equal to 360 degrees. But when one of them, as F, runs inwards, subtract it from the sum of the rest, to leave 360 degrees.

With any of the instruments measure the lengths and positions of imaginary lines running as near the sides of the field as you can; and in going along them measure the offsets in the manner before taught; and you will have the plan on the paper in using the plain table, drawing the crooked hedges through the ends of the offsets; but in surveying with the theodolite, or other instrument, set down the measures properly in a field-book, or memorandum-book, and plan them after returning from the field, by laying down all the lines and angles.

So, in surveying the piece ABCDE, set up mark: a, b, c, d, dividing it into as few sides as may be. Then begin at any station a, and measure the lines ab, bc, cd, da, and take their positions, or the angles a, b, c, d; and in going along the lines measure all the offsets, as at m, n, o, p, &c, along every station line.

And this is done either within the field, or without, as may be most convenient. When there are obstructions within, as wood, water, hills, &c; then measure without, as in the figure here below. |

This is performed by choosing two stations, from whence all the marks and objects can be seen, then measuring the distance between the stations, and at each station taking the angles formed by every object, from the station line or distance.

The two stations may be taken either within the bounds, or in one of the sides, or in the direction of two of the objects, or quite at a distance, and without the bounds of the objects, or part to be surveyed.

In this manner, not only grounds may be surveyed, without even entering them, but a map may be taken of the principal parts of a country, or the chief places of a town, or any part of a river or coast surveyed, or any other inaccessible objects; by taking two stations, on two towers, or two hills, or such like.

When the plain table is used; plant it at one station m, draw a line m n on it, along which lay the edge of the index, and turn the table about till the sights point directly to the other station; and there screw it fast. Then turn the sights round m successively to all the objects ABC, &c, drawing a dry line by the edge of the index at each, as mA, mB, mC, &c. Then measure the distance to the other station, there plant the table, and lay that distance down on the station line from m to n. Next lay the index by the line nm, and turn the table about till the sights point to the other station m, and there screw it fast. Then direct the sights successively to all the objects A, B, C, &c, as before, drawing lines each time, as nA, nB, nC, &c: and their intersection with the former lines will give the places of all the objects, or corners, A, B, C, &c.

When the theodolite, or any other instrument for taking angles, is used; proceed in the same way, measuring the station distance mn, planting the instrument first at one station, and then at another; then placing the fixed sights in the direction mn, and directing the moveable sights to every object, noting the degrees cut off at each time. Then, these observations being planned, the intersections of the lines will give the objects as before.

When all the objects, to be surveyed, cannot be seen from two stations; then three stations may be used, or four, or as many as is necessary; measuring always the distance from one station to another; placing the instrument in the same position at every station, by means described before; and from each station observing or setting every object that can be seen from it, by taking its direction or angular position, till every object be determined by the intersection of two or more lines of direction, the more the better. And thus may very extensive surveys be taken, as of large commons, rivers, coasts, countries, hilly grounds, and such like.

If the estate be very large, and contain a great number of fields, it cannot well be done by surveying all the fields singly, and then putting them together; nor can it be done by taking all the angles and boundaries that inclose it. For in these cases, any small errors will be so multiplied, as to render it very much distorted.

1st. Walk over the estate two or three times, in order to get a perfect idea of it, and till you can carry the map of it tolerably in your head. And to help your memory, draw an eye draught of it on paper, or at least, of the principal parts of it, to guide you.

2d. Choose two or more eminent places in the estate, for your stations, from whence you can see all the principal parts of it: and let these stations be as far distant from one another as possible; as the fewer stations you have to command the whole, the more exact your work will be: and they will be sitter for your purpose, if these station lines be in or near the boundaries of the ground, and especially if two or more lines proceed from one station.

3d. Take angles, between the stations, such as you think necessary, and measure the distances from station to station, always in a right line: these things must be done, till you get as many angles and lines as are sufficient for determining all your points of station. And in measuring any of these station distances, mark accurately where these lines meet with any hedges, ditches, roads, lanes, paths, rivulets, &c, and where any remarkable object is placed, by measuring its distance from the station line, and where a perpendicular from it cuts that line; and always mind, in any of these observations, that you be in a right line, which you will know by taking back sight and foresight, along your station line. And thus as you go along any main station line, take offsets to the ends of all hedges, and to any pond, house, mill, bridge, &c, omitting nothing that is remarkable. And all these things must be noted down; for these are your data, by which the places of such objects are to be determined upon your plan. And be sure to set marks up at the intersections of all hedges with the station line, that you may know where to measure from, when you come to survey these particular fields, which must immediately be done, as soon as you have measured that station line, whilst they are fresh in memory. In this way all your station lines are to be measured, and the situation of all places adjoining to them determined, which is the first grand point to be obtained. It will be proper for you to lay down your work upon paper every night, when you go home, that you may see how you go on.

4th. As to the inner parts of the estate, they must be | determined in like manner, by new station lines: for, after the main stations are determined, and every thing adjoining to them, then the estate must be subdivided into two or three parts by new station lines; taking inner stations at proper places, where you can have the best view. Measure these station lines as you did the first, and all their intersections with hedges, and all offsets to such objects as appear. Then you may proceed to survey the adjoining fields, by taking the angles that the sides make with the station line, at the intersections, and measuring the distances to each corner, from the intersections. For every station line will be a basis to all the future operations; the situation of all parts being entirely dependant upon them; and therefore they should be taken of as great a length as possible; and it is best for them to run along some of the hedges or boundaries of one or more fields, or to pass through some of their angles. All things being determined for these stations, you must take more inner ones, and so continue to divide and subdivide, till at last you come to single fields; repeating the same work for the inner stations, as for the outer ones, till all be done: and close the work as often as you can, and in as few lines as possible. And that you may choose stations the most conveniently, so as to cause the least labour, let the station lines run as far as you can along some hedges, and through as many corners of the fields, and other remarkable points, as you can. And take notice how one field lies by another; that you may not misplace them in the draught.

5th. An estate may be so situated, that the whole cannot be surveyed together; because one part of the estate cannot be seen from another. In this case, you may divide it into three or four parts, and survey the parts separately, as if they were lands belonging to different persons; and at last join them together.

6th. As it is necessary to protract or lay down your work as you proceed in it, you must have a scale of a due length to do it by. To get such a scale, you must measure the whole length of the estate in chains; then you must consider how many inches in length the map is to be; and from these you will know how many chains you must have in an inch; then make your scale, or choose one already made, accordingly.

7th. The trees in every hedge row must be placed in their proper situation, which is soon done by the plain table; but may be done by the eye without an instrument; and being thus taken by guess, in a rough draught, they will be exact enough, being only to look at; except it be such as are at any remarkable places, as at the ends of hedges, at stiles, gates, &c, and these must be measured. But all this need not be done till the draught is finished. And observe in all the hedges, what side the gutter or ditch is on, and consequently to whom the fences belong.

8th. When you have long stations, you ought to have a good instrument to take angles with; and the plain table may very properly be made use of, to take the several small internal parts, and such as cannot be taken from the main stations, as it is a very quick and ready instrument.

15. Instead of the foregoing method, an ingenious friend (Mr. Abraham Crocker), after mentioning the new and improved method of keeping the field book by writing from bottom to top of the pages, observes that “In the former method of measuring a large estate, the accuracy of it depends on the correctness of the instruments used in taking the angles. To avoid the errors incident to such a multitude of angles, other methods have of late years been used by some few skilful surveyors; the most practical, expeditious, and correct, seems to be the following.

“As was advised in the foregoing method, so in this, choose two or more eminences, as grand stations, and measure a principal base line from one station to the other, noting every hedge, brook, or other remarkable object as you pass by it; measuring also such short perpendicular lines to such bends of hedges as may be near at hand. From the extremities of this base line, or from any convenient parts of the same, go off with other lines to some remarkable object situated towards the sides of the estate, without regarding the angles they make with the base line or with one another; still remembering to note every hedge, brook or other object that you pass by. These lines, when laid down by intersections, will with the base line form a grand triangle upon the estate; several of which, if need be, being thus laid down, you may proceed to form other smaller triangles and trapezoids on the sides of the former: and so on, until you finish with the enclosures individually.

“To illustrate this excellent method, let us take AB (in the plan of an estate, fig. 1, pl. 28) for the principal base line. From B go off to the tree at C; noting down, in the field-book, every cross hedge, as you measure on; and from C measure back to the first station at A, noting down every thing as before directed.

“This grand triangle being completed, and laid down on the rough-plan paper, the parts, exterior as well as interior, are to be completed by smaller triangles and trapezoids.

“When the whole plan is laid down on paper, the contents of each field might be calculated by the methods laid down below, at article 20.

“In countries where the lands are enclosed with high hedges, and where many lanes pass through an estate, a theodolite may be used to advantage, in measuring the angles of such lands; by which means, a kind of skeleton of the estate may be obtained, and the lane-lines serve as the bases of such triangles and trapezoids as are necessary to fill up the interior parts.”

The method of measuring the other cross lines, offsets and interior parts and enclosures, appears in the plan, fig. 1, last referred to.

16. Another ingenious correspondent (Mr. John Rodham of Richmond, Yorkshire) has also communicated the following example of the new method of surveying, accompanied by the field-book, and its corresponding plan. His account of the method is as follows.

The field-book is ruled into three columns. In the middle one are set down the distances on the chain line at which any mark, offset, or other observation is made; and in the right and left hand columns are entered, the offsets and observations made on the right and left hand respectively of the chain line.

It is of great advantage, both for brevity and per- | spicuity, to begin at the bottom of the leaf and write upwards; denoting the crossing of fences, by lines drawn across the middle column, or only a part of such a line on the right and left opposite the figures, to avoid confusion, and the corners of fields, and other remarkable turns in the fences where offsets are taken to, by lines joining in the manner the fences do, as will be best seen by comparing the book with the plan annexed, fig. 2, pl. 28.

The marks called, a, b, c, &c, are best made in the fields, by making a small hole with a spade, and a chip or small bit of wood, with the particular letter upon it, may be put in, to prevent one mark being taken for another, on any return to it. But in general, the name of a mark is very easily had by referring in the book to the line it was made in. After the small alphabet is gone through, the capitals may be next, the print letters afterwards, and so on, which answer the purpose of so many different letters; or the marks may be numbered.

The letter in the left hand corner at the beginning of every line, is the mark or place measured from; and, that at the right hand corner at the end, is the mark measured to: But when it is not convenient to go exactly from a mark, the place measured from, is described such a distance from one mark towards another; and where a mark is not measured to, the exact place is ascertained by saying, turn to the right or left hand, such a distance to such a mark, it being always understood that those distances are taken in the chain line.

The characters used, are for turn to the right band, for turn to the left hand, and placed over an offset, to shew that it is not taken at right angles with the chain line, but in the line with some straight fence; being chiefly used when crossing their directions, and is a better way of obtaining their true places than by offsets at right angles.

When a line is measured whose position is determined, either by former work (as in the case of producing a given line or measuring from one known place or mark to another) or by itself (as in the third side of a triangle) it is called a fast line, and a double line across the book is drawn at the conclusion of it; but if its position is not determined (as in the second side of a triangle) it is called a loose line, and a single line is drawn across the book. When a line becomes determined in position, and is afterwards continued, a double line half through the book is drawn.

When a loose line is measured, it becomes absolutely necessary to measure some line that will determine its position. Thus, the first line ab, being the base of a triangle, is always determined; but the position of the second side hj, does not become determined, till the third side jb is measured; then the triangle may be constructed, and the position of both is determined.

At the beginning of a line, to fix a loose line to the mark or place measured from, the sign of turning to the right or left hand must be added (as at j in the third line); otherwise a stranger, when laying down the work may as easily construct the triangle hjb on the wrong side of the line ah, as on the right one: but this error cannot be fallen into, if the sign above named be carefully observed.

In choosing a line to fix a loose one, care must be taken that it does not make a very acute or obtuse angle; as in the triangle pBr, by the angle at B being very obtuse, a small deviation from truth, even the breadth of a point at p or r, would make the error at B when constructed very considerable; but by constructing the triangle pBq, such a deviation is of no consequence.

Where the words leave off are written in the fieldbook, it is to signify that the taking of offsets is from thence discontinued; and of course something is wanting between that and the next offset.

The field-book above referred to, is engraved on plate 29, in parts, representing so many pages, each of which is supposed to begin at the bottom, and end at top. And the map or plan belonging to it, in fig. 2, pl. 28.

1st. Choose two, three, or four eminent places for stations; such as the tops of high hills or mountains, towers, or church steeples, which may be seen from one another; and from which most of the towns, and other places of note, may also be seen. And let them be as far distant from one another as possible. Upon these place raise beacons, or long poles, with flags of different colours flying at them; so as to be visible from all the other stations.

2d. At all the places, which you would set down in the map, plant long poles with flags at them of several colours, to distinguish the places from one another; fixing them upon the tops of church steeples, or the tops of houses, or in the centres of lesser towns.

But you need not have these marks at many places at once, as suppose half a score at a time. For when the angles have been taken, at the two stations, to all these places, the marks may be moved to new ones; and so successively to all the places you want. These marks then being set up at a convenient number of places, and such as may be seen from both stations; go to one of these stations, and with an instrument to take angles, standing at that station, take all the angles between the other station, and each of these marks, observing which is blue, which red, &c, and which hand they lie on; and set all down with their colours. Then go to the other station, and take all the angles between the first station, and each of the former marks, and set them down with the others, each against his fellow with the same colour. You may, if you can, also take the angles at some third station, which may serve to prove the work, if the three lines intersect in that point, where any mark stands. The marks must stand till the observations are finished at both stations; and then they must be taken down, and set up at fresh places. And the same operations must be performed, at both stations, for these fresh places; and the like for others. Your instrument for taking angles must be an exceeding good one, made on purpose with telescopic sights; and of three, four, or five feet radius. A circumferentor is reckoned a good instrument for this purpose.

3d. And though it is not absolutely necessary to measure any distance, because a stationary line being laid down from any scale, all the other lines will be | proportional to it; yet it is better to measure some of the lines, to ascertain the distances of places in miles; and to know how many geometrical miles there are in any length; and from thence to make a scale to measure any distance in miles. In measuring any distance, it will not be exact enough to go along the high roads; by reason of their turnings and windings, and hardly ever lying in a right line between the stations, which would cause endless reductions, and create trouble to make it a right line; for which reason it can never be exact. But a better way is to measure in a right line with a chain, between station and station, over hills and dales or level fields, and all obstacles. Only in case of water, woods, towns, rocks, banks, &c, where one cannot pass, such parts of the line must be measured by the methods of inaccessible distances; and besides, allowing for ascents and descents, when we meet with them. And a good compass that shews the bearing of the two stations, will always direct you to go straight, when you do not see the two stations; and in your progress, if you can go straight, you may take offsets to any remarkable places, likewise noting the intersection of the stationary line with all roads, rivers, &c.

4th. And from all the stations, and in the whole progress, be very particular in observing sea coasts, river mouths, towns, castles, houses, churches, windmills, watermills, trees, rocks, sands, roads, bridges, fords, ferries, woods, hills, mountains, rills, brooks, parks, beacons, sluices, floodgates, locks, &c; and in general all things that are remarkable.

5th. After you have done with the first and main station lines, which command the whole county; you must then take inner stations, at some places already determined; which will divide the whole into several partitions: and from these stations you must determine the places of as many of the remaining towns as you can. And if any remain in that part, you must take more stations, at some places already determined; from which you may determine the rest. And thus proceed through all the parts of the country, taking station after station, till we have determined all we want. And in general the station distances must always pass through such remarkable points as have been determined before, by the former stations.

6th. Lastly, the position of the station line you measure, or the point of the compass it lies on, must be determined by astronomical observation. Hang up a thread and plummet in the sun, over some part of the station line, and observe when the shadow runs along that line, and at that moment take the sun's altitude; then having his declination, and the latitude, the azimuth will be found by spherical trigonometry. And the azimuth is the angle the station line makes with the meridian; and therefore a meridian may easily be drawn through the map: Or a meridian may be drawn through it by hanging up two threads in a line with the pole star, when he is just north, which may be known from astronomical tables. Or thus; observe the star Alioth, or that in the rump of the great bear, being that next the square; or else Cassiopeia's hip; I say, observe by a line and plummet when either of these stars and the pole star come into a perpendicular; and at that time they are due north. There- fore two perpendicular lines being fixed at that moment, towards these two stars, will give the position of the meridian.

This may be done with any of the instruments for taking angles, but best of all with the plain table, where every minute part is drawn while in sight. It is proper also to have a chain of 50 feet long, divided into 50 links, and an offset-staff of 10 feet long.

Legin at the meeting of two or more of the principal streets, through which you can have the longest prospects, to get the longest station lines. There having fixed the instrument, draw lines of direction along those streets, using two men as marks, or poles set in wooden pedestals, or perhaps some remarkable places in the houses at the farther ends, as windows, doors, corners, &c. Measure these lines with the chain, taking offsets with the staff, at all corners of streets, bendings, or windings, and to all remarkable things, as churches, markets, halls, colleges, eminent houses, &c. Then remove the instrument to another station along one of these lines; and there repeat the same process as before. And so on till the whole is finished.

Thus, fix the instrument at A, and draw lines in the direction of all the streets meeting there; and measure AB, noting the street on the left at m. At the second station B, draw the directions of the streets meeting there; measure from B to C, noting the places of the streets at n and o as you pass by them. At the 3d station C, take the direction of all the streets meeting there, and measure CD. At D do the same, and measure DE, noting the place of the cross streets at p. And in this manner go through all the principal streets. This done, proceed to the smaller and intermediate streets; and lastly to the lanes, alleys, courts, yards, and every part that it may be thought proper to represent.

If the survey was taken with a plain table, you have a rough plan of it already on the paper which covered | the table. But if the survey was with any other instrument, a plan of it is to be drawn from the measures that were taken in the survey, and first of all a rough plan upon paper.

To do this, you must have a set of proper instruments, for laying down both lines and angles, &c; as scales of various sizes, the more of them, and the more accurate, the better; scales of chords, protractors, perpendicular and parallel rulers, &c. Diagonal scales are best for the lines, because they extend to three figures, or chains and links, which are hundredth parts of chains. And in using the diagonal scale, a pair of compasses must be employed to take off the lengths of the principal lines very accurately. But a scale with a thin edge divided, is much readier for laying down the perpendicular offsets to crooked hedges, and for marking the places of those offsets upon the station line; which is done at only one application of the edge of the scale to that line, and then pricking off all at once the distances along it. Angles are to be laid down either with a good scale of chords, which is perhaps the most accurate way; or with a large protractor, which is much readier when many angles are to be laid down at one point, as they are pricked off all at once round the edge of the protractor.

Very particular directions for laying down all sorts of figures cannot be necessary in this place, to any person who has learned practical geometry, or the construction of figures, and the use of his instruments. It may therefore be sufficient to observe, that all lines and angles must be laid down on the plan in the same order in which they were measured in the field, and in which they are written in the field-book; laying down first the angles for the position of lines, then the lengths of the lines, with the places of the offsets, and then the lengths of the offsets themselves, all with dry or obscure lines; then a black line drawn through the extremities of all the offsets, will be the hedge or bounding line of the field, &c. After the principal bounds and lines are laid down, and made to fit or close properly, proceed next to the smaller objects, till you have entered every thing that ought to appear in the plan, as houses, brooks, trees, hills, gates, stiles, roads, lanes, mills, bridges, woodlands, &c, &c.

The north side of a map or plan is commonly placed uppermost, and a meridian somewhere drawn, with the compass or flower-de-luce pointing north. Also, in a vacant place, a scale of equal parts or chains is drawn, and the title of the map in conspicuous characters, and embellished with a compartment. All hills must be shadowed, to distinguish them in the map. Colour the hedges with different colours; represent hilly grounds by broken hills and valleys; draw single dotted lines for foot-paths, and double ones for horse or carriage roads. Write the name of each field and remarkable place within it, and, if you choose, its content in acres, roods, and perches.

In a very large estate, or a county, draw vertical and horizontal lines through the map, denoting the spaces between them by letters, placed at the top, and bottom, and sides, for readily finding any field or other object, mentioned in a table.

In mapping counties, and estates that have uneven grounds of hills and valleys, reduce all oblique lines, measured up hill and down hill, to horizontal straight lines, if that was not done during the survey, before they were entered in the field-book, by making a proper allowance to shorten them. For which purpose, there is commonly a small table engraven on some of the instruments for Surveying.

1st. Compute the contents of the figures, whether triangles, or trapeziums, &c, by the proper rules for the several figures laid down in measuring; multiplying the lengths by the breadths, both in links; the product is acres after you have 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 was given in the description of the chain, art. 1.

2d. In small and separate pieces, it is usual to cast up their contents from the measures of the lines taken in surveying them, without making a correct plan of them.

Thus, in the triangle in art. 7, where we had

AP = 794, and AB = 1321
PC =  826
7926
2642
10568
2 ) 10.91146
5.45573	ac r p
4	Ans. 32 1 33 nearly
1.82292
40
32.91680

Or the first example to art. 8, thus:

AE 214  210	ED
AF 362  306	FB
AC 592
516	sum of perp.
592	AC
1032
4644
2580
3.05472		ac r p
4		Ans. 3 0 8
.21888
40
8.75520

|

Or the 2d example to the same article, thus:

AP 110	352	PC
AQ 745	595	QD
AB 1110
PC 352	PC 352	QD 595
AP 110	QD 595	QB 365
-------	----	----
2 APC 38720	sum 947	2975
-------	PQ 635	3570
	----	1785
	4735	-----
	2841	217175	= 2QDB
	5682	601345	= 2PCDQ
	-----	38720	= 2APC
2PCDQ	601345	-----
	-------	2) 8.57240	= dou. the whole
		4.2862
		4
ac r p		-----
Ans. 4 1 5		1.1448
		40
		-----
		5.7920
		-----

3d. 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 trapezoids. Thus, for the example to art. 6, where

Ac  45	62 ch
Ad 220	84 di
Ae 340	70 ek
Af 510	98 fl
Ag 634	57 gm
AB 785	91 Bn

Ac 45	ch 62	di 84	ek 70	fl 98	gm 57
ch  62	di 84	ek 70	fl 98	gm 57	Bn 91
90	146	154	168	145	148
270	cd 175	de 120	ef 170	fg 124	gB 151
2790	730	18480	11760	580	148
	1022		168	290	740
	146			145	148
			28560
	25550			17980	22348
		2790
		25550
		18480
		28560
		17980
		22348
	2 )	1.15708		ac r p
		.57854	Content 0 2 12
		4
		2.31416
		40
		12.56640

4th. Sometimes such pieces as that above, are computed by finding a mean breadth, by dividing the sum of the offsets by the number of them, accounting that for one of them where the boundary meets the station line, as at A; then multiply the length AB by that mean breadth.

Thus:

00	785	AB
62	66	mean breadth
84
70	4710	ac r p
98	4710	Content 0   2   2 by this method,
57		which is 10 perches too little.
91	.51810
	4
7) 462
66	2.07240
	40
	2.89600

But this method is always erroneous, except when the offsets stand at equal distances from one another.

5th. 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 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 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, it is likely that some of the parts will be taken a small matter 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 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.

A specimen of dividing into one triangle, or one trapezium, which will do for most single fields, may be seen in the examples to the last article; and a specimen of dividing a large tract into several such trapeziums and triangles, in article 9, where a piece is so divided, and its dimensions taken and set down; and again in articles 15, 16.

6th. But the chief secret in casting up, 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 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 trape- | zium. 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 lanthorn-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 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 contents 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 horsehair 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, or cane, or whale-bone, or such like slender springy 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.

Ex. Thus, let it be required to find the contents of the same figure as in art. 12, 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 as near 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, measures 1256. Also the perpendicular, or nearest distance, from A to this diagonal, measures 456; and the distance of C from it, is 428. Then

456	2 ) 11.10304
428	5.55152
	4
884
1256	2.20608
	40
5024
10048	8.24320
10048
1110304

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.

After the rough plan is completed, and a fair one is wanted; this may be done, either on paper or vellum, by any of the following Methods.

First Method.—Lay the rough plan upon the clean paper, and keep them always pressed flat and close together, by weights laid upon 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 the said black part upon the clean paper, upon 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, &c.—Or, instead of blacking the rough plan, you may keep constantly a blacked paper to lay between the plans.

Third Method.—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 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.

Fourth Method.—A fourth way is by the instrument called a pentagraph, which also copies the plan in any size required.

Fifth Method.—But the neatest method of any is this. Procure a copying frame or glass, made in this manner; namely, a 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 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 | upon the face of the plan to be copied. Lay them, with the back of the old plan, upon 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. And then another part. And so on till the whole be copied.

Then, take them asunder, and trace all the pencillines over with a sine pen and Indian ink, or with common ink.

And thus you may copy the finest plan, without injuring it in the least.

When the lines, &c, are copied upon the clean paper or vellum, the next business is to write such names, remarks, or explanations as may be judged necessary; laying down the scale for taking the lengths of any parts, a flower-de-luce to point out the direction, and the proper title ornamented with a compartment; and illustrating or colouring every part in such manner as shall seem most natural, such as shading rivers or brooks with crooked lines, drawing the representations of trees, bushes, hills, woods, hedges, houses, gates, roads, &c, in their proper places; running a single dotted line for a foot path, and a double one for a carriage road; and either representing the bases or the elevations of buildings, &c.

In the division of commons, after the whole is surveyed and cast up, and the proper quantities to be allowed for roads, &c, deducted, divide the net quantity remaining among the several proprietors, by the rule of Fellowship, in proportion to the real value of their estates, and you will thereby obtain their proportional quantities of the land. But as this division supposes the land, which is to be divided, to be all of an equal goodness, you must observe that if the part in which any one's share is to be marked off, be better or worse than the general mean quality of the land, then you must diminish or augment the quantity of his share in the same proportion.

Or, which comes to the same thing, divide the ground among the claimants in the direct ratio of the value of their claims, and the inverse ratio of the quality of the ground allotted to each; that is, in proportion to the quotients arising from the division of the value of each person's estate, by the number which expresses the quality of the ground in his share.

But these regular methods cannot always be put in practice; so that, in the division of commons, the usual way is, to measure separately all the land that is of different values, and add into two sums the contents and the values; then, the value of every claimant's share is found, by dividing the whole value among them in proportion to their estates; and, lastly, by the 24th article, a quantity is laid out for each person, that shall be of the value of his share before found.

23. It is required to divide any given Quantity of Ground, or its Value, into any given Number of Parts, and in Proportion as any given Numbers.

Divide the given piece, or its value, as in the rule of Fellowship, by dividing the whole content or value by the sum of the numbers expressing the proportions of the several shares, and multiplying the quotient severally by the said proportional numbers for the respective shares required, when the land is all of the same quality. But if the shares be of different qualities, then divide the numbers expressing the proportions or values of the shares, by the numbers which express the qualities of the land in each share; and use the quotients instead of the former proportional numbers.

Ex. 1. If the total value of a common be 2500 pounds, it is required to determine the values of the shares of the three claimants A, B, C, whose estates are of these values, 10000, and 15000, and 25000 pounds.

The estates being in proportion as the numbers 2, 3, 5, whose sum is 10, we shall have 2500 ÷ 10 = 250; which being severally multiplied by 2, 3, 5, the products 500, 750, 1250, are the values of the shares required.

Ex. 2. It is required to divide 300 acres of land among A, B, C, D, E, F, G, and H, whose claims upon it are respectively in proportion as the numbers 1, 2, 3, 5, 8, 10, 15, 20.

The sum of these proportional numbers is 64, by which dividing 300, the quotient is 4 ac. 2 r. 30 p. which being multiplied by each of the numbers, 1, 2, 3, 5, &c, we obtain for the several shares as below:

		Ac.	R.	P.
A	=	4	2	30
B	=	9	1	20
C	=	14	0	10
D	=	23	1	30
E	=	37	2	00
F	=	46	3	20
G	=	70	1	10
H	=	93	3	00
Sum	=	300	0	00

Ex. 3. It is required to divide 780 acres among A, B, and C, whose estates are 1000, 3000, and 4000 pounds a year; the ground in their shares being worth 5, 8, and 10 shillings the acre respectively.

Here their claims are as 1, 3, 4; and the qualities of their land are as 5, 8, 10; therefore their quantities must be as 1/5, 3/8, 2/5, or, by reduction, as 8, 15, 16. Now the sum of these numbers is 39; by which dividing the 780 acres, the quotient is 20; which being multiplied severally by the three numbers 8, 15, 16, the three products are 160, 300, 320, for the shares of A, B, C, respectively. |

24. To Cut off from a Plan a Given Number of Acres, &c, by a Line drawn from any Point in the Side of it.

Let A be the given point in the annexed plan, from which a line is to be drawn cutting off suppose 5 ac. 2 r. 14 p.

Draw AB cutting off the part ABC as near as can be judged equal to the quantity proposed; and let the true quantity of ABC, when calculated, be only 4 ac. 3 r. 20 p. which is less than 5 ac. 2 r. 14 p. the true quantity, by 0 ac. 2 r. 34 p. or 71250 square links. Then measure AB, which suppose = 1234 links, and divide 71250, by 617 the half of it, and the quotient 115 links will be the altitude of the triangle to be added, and whose base is AB. Therefore if upon the centre B, with the radius 115, an arc be described; and a line be drawn parallel to AB, touching the arc, and cutting BD in D; and if AD be drawn, it will be the line cutting off the required quantity ADCA.

Note. If the first piece had been too much, then D must have been set below B.

In this manner the several shares of commons, to be divided, may be laid down upon the plan, and transferred from thence to the ground itself.

Also for the greater ease and perfection in this business, the following problems may be added.

25. From an Angle in a Given Triangle, to draw Lines to the opposite Side, dividing the Triangle into any Number of Parts, which shall be in any assigned Proportion to each other.

Divide the base into the same number of parts, and in the same proportion, by article 22; then from the several points of division draw lines to the proposed angle, and they will divide the triangle as required.— For, the several parts are triangles of the same altitude, and which therefore are as their bases, which bases are taken in the assigned proportion.

Ex. Let the triangle ABC, of 20 acres, be divided into five parts, which shall be in proportion to the numbers 1, 2, 3, 5, 9; the lines of division to be drawn from A to CB, whose length is 1600 links.

Here 1 + 2 + 3 + 5 + 9 = 20, and 1600 ÷ 20 = 80; which being multiplied by each of the proportional numbers, we have 80, 160, 240, 400, and 720. Therefore make Ca = 80, ab = 160, bc = 240, cd = 400, and dB = 720; then by drawing the lines Aa, Ab, Ac, Ad, the triangle is divided as required.

26. From any Point in one side of a Given Triangle, to draw Lines to the other two Sides, dividing the Triangle into any Number of Parts which shall be in any assigned Ratio.

From the given point D, draw DB to the angle opposite the side AC in which the point is taken; then divide the same side AC into as many parts AE, EF, FG, GC, and in the same proportion with the required parts of the triangle, like as was done in the last problem; and from the points of division draw lines EK, FI, GH, parallel to the line BD, and meeting the other sides of the triangle in K, I, H; lastly, draw KD, ID, HD, so shall ADK, KDI, IDHB, HDC be the parts required.—The example to this will be done exactly as the last.

For, the triangles ADK, KDI, IDB, being of the same height, are as their bases AK, KI, IB; which, by means of the parallels EK, FI, DB, are as AE, EF, FD; in like manner, the triangles CDH, HDB, are to each other as CG, GD: but the two triangles JDB, BDH, having the same base BD, are to each other as the distances of I and H from BD, or as FD to DG; consequently the parts DAK, DKI, DIBH, DHC, are to each other as AE, EF, FG, GC.

The method of Surveying harbours, and of forming maps of them, as also of the adjacent coasts, sands, &c, depends on the same principles, and is chiefly conducted like that of common Surveying. The operation is indeed more complicated and laborious; as it is necessary to erect a number of signals, and to mark a variety of objects along the coast, with different bearings from one another, and the several parts of the harbour; and likewise to measure a great number of angles | at different stations, whether on the land or the water. For this purpose, the best instrument is Hadley's quadrant, as all these operations may be performed by it, not only with greater ease, but also with much more precision, than can be hoped for by any other means, as it is the only instrument in use, in which neither the exactness of the observations, nor the ease with which they may be made, are sensibly affected by the motion of a vessel: and hence a single observer, in a boat, may generally determine the situation of any place at pleasure, with a sufficient degree of exactness, by taking the angles subtended by several pairs of objects properly chosen upon shores round about him; but it will be still better to have two observers, or the same observer at different stations, to take the like angles to the several objects, and also to the stations. By this means, two angles and one side are given, in every triangle, from whence the situation of every part of them will be known. By such observations, when carefully made with good instruments, the situation of places may be easily determined to 20 or 30 feet, or less, upon every 3 or 4 miles. See Philos. Trans. vol. 55, pa. 70; also Mackenzie's Maritime Surveying.

Surveying Cross. See Cross.

Surveying Quadrant. See Quadrant.

Surveying Scale, the same with Reducing Scale.

Surveying Wheel. See Perambulator.