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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 separate them, and trace all the pencil lines over with a fine
pen and Indian ink, or with common ink. And thus you may copy the finest plan, without injur. ing it in the least.
When the lines, &c. are copied on the clean paper or vellum, the next business is to write such names, remarks, or explanations as may be judged necessary ; drawing the scale for taking the lengths of any parts, a flower-de-luce to point out the direction, and the proper title, ornamento, ed with a compartment ; and illustrating or colouring every part in such manner, as shall seem most natural ; 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 representing either the bases or the elevations of buildings, &c.
To change a figure from one scale to another.
of a parallel rule, draw bc parallel to BC, and meeting AC in c, and in the same manner cd parallel to CD, de parallel to DE, ef parallel to EF ; so shall Abcdefa be the figure required
DIVISION OF LAND.
In the division of commons, after the whole is surveyed and computed, 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 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 be always 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, by the first part
It is required to divide any given quantity of ground, or its
value, into any given number of parts, and in proportion to 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 sequired, 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.
1. If the total value of a common be 2500l. it is required to determine the values of the shares of the three claimants A, B, C, whose estates are respectively of these values, 10000, 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.
the following Problem I. the value of every claimant's share is computed by dividing the whole value among them in proportion to their estates ; and, lastly, by Problem II. a quantity is laid out for each person, that shall be of the value of his share before found.
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 spectively in proportion as the numbers 1, 2, 3, 5, 8, 10,
The sum of these proportional numbers is 64, by which 300 being divided, the quotient is 4ac. ar. 3op. which being multiplied by each of the numbers 1, 2, 3, 5, &c. we have for the several shares as follow :
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, or, by reduction, as 8, 15, 16. Now the sum of these numbers is 39 ; by which the 780 acres being divided, 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.
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, sac. 2r. 14p.
Draw AB cutting off the part ABC as nearly as can be judged equal to the quantity proposed ; and suppose the true quantity of ABC, when calculated, be only 4ac. 3r. 20p. which is less than gac. 2r. 14p. the true quantity, by
D 2r.34p. or 71 250 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 its base is AB. Therefore, if on the centre B, with the radius 11); an arc le 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 on the plan, and transferred thence to the ground itself.
Also for the greater ease and perfection in this business, the following Problems may be added.