A fourth way is by the inftrument called a pentagraph, which alfo copies the plan in any fize required.

FIFTH METHOD.

But the neatest method of any is this. Procure a copying frame or glafs, made in this manner; namely, a large fquare of the beft window glafs, fet in a broad frame of wood, which can be raised up to any angle, when the lower fide of it refts 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 feveral pins quite around, to keep them together, the clean paper being laid uppermoft, and upon the face of the plain to be copied. Lay them, with the back of the old plan, upon the glafs, namely, that part which you intend to begin at to copy firit; and, by means of the light fhining through the papers, you will very diftinctly perceive every line of the plan through the clean paper. In this ftate then trace all the lines on the paper with a pencil. Having drawn that part which covers the glafs, flide another part over the glass, and copy it in the fame manner. And then another part. And fo on till the whole be copied.

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

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

When the lines, &c, are copied upon the clean paper or vellum, the next bufinefs is to write fuch names, remarks, or explanations as may be judged neceffary; laying down the fcale for taking the lengths of any parts, a flower-de-luce to point out the direc

tion, and the proper title ornamented with a compartment; and illuftrating or colouring every part in fuch manner as fhall feem moft natural, fuch as fhading rivers or brooks with crooked lines, drawing the reprefentations of trees, bufhes, hills, woods, hedges, houses, gates, roads, &c, in their proper places; running a fingle dotted line for a foot path, and a double one for a carriage road; and either reprefenting the bafes or the elevations of buildings, &c.

PROBLEM IV.

To change a Figure from One Scale to Another.

From one angle A draw lines AC, AD, AE, &c, to all the other angles of the figure given ; then augment or diminish one fide AB till Ab be to AB

in the given propor

tion of the fcales; and by means of a parallel ruler, draw be parallel to BC and meeting ac in c, and in the fame manner cd parallel to CD, de parallel to DE, ef parallel to EF; fo fhall Abcdefa be the figure required.

CHAPTER V.

Of the Divifion of Lands.

In the divifion of commons, after the whole is furveyed and caft up, and the proper quantities to be allowed for roads, &c, deducted, divide the net. quantity remaining among the feveral proprictors, by the rule of Fellowship, in proportion to the real

value of their eftates, and you will thereby obtain their proportional quantities of the land. But as this divifion supposes the land, which is to be divided, to be all of an equal goodnefs, you must observe that if the part in which any one's fhare 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 fhare in the fame proportion*.

PROBLEM I.

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 fum of the numbers expreffing the proportions of the feveral shares, and multiplying the quotient feverally by the faid proportional numbers for the refpective fhares required, when the land is all of the fame quality. But if the fhares be of different qualities,

*Or, which comes to the fame 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 arifing from the divifion of the value of each perfon's eftate, by the number which expresses the quality of the ground in his fhare.

But these regular methods cannot always be put in practice; fo that, in the divifion of commons, the ufual way is, to measure feparately all the land that is of different values, and add into two fums the contents and the values; then, by the first part of the following problem 1, the value of every claimant's fhare is computed, by dividing the whole value among them in proportion to their eftates; and, laftly, by the 2d problem, a quantity is laid out for each perfon, that fhall be of the value of his fhare before found.

then divide the numbers expreffing the proportions or values of the fhares, by the numbers which exprefs the qualities of the land in each share; and use the quotients instead of the former proportional num

bers.

If the total value of a common be 2500 pounds, it is required to determine the values of the fhares of the three claimants A, B, C, whofe eftates are of these values 10000, and 15000, and 25000 pounds.

The eftates being in proportion as the numbers 2, 3, 5, whofe fum is 10, we fhall have 2500 ÷ 10 =250; which being severally multiplied by 2, 3, 5, the products 500, 750, 1250, are the values of the fhares required.

EXAMPLE II.

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

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

1

EXAMPLE III.

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 fhares being worth 5, 8, and 10 fhillings the acre refpectively.

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,,, or, by reduction, as 8, 15, 16. Now the fum of thefe numbers is 39; by which. dividing the 780 acres, the quotient is 20; which being multiplied feverally by the three numbers 8, 15, 16, the three products are 160, 300, 320, for the fhares of A, B, C, refpectively.

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 fuppofe 5ac.

2r. 14p.

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 4ac. 3r. 20p. which is less than 5ac. 2r. 14p. the true quantity, by o ac. 2r. 34p. B or 71250 fquare links. Then meafure AB, which fuppofe 1234 links, and divide 71250, by 617

D

A

the half of it, and the quotient 115 links will be the altitude of the triangle to be added, and whofe bafe