and submitted to the commissioners and their clerk; if admitted, it is entered into a book for that purpose made by the clerk, and a copy is delivered to the surveyor, who has to make out a calculation sheet, or general reference, ruled in columns under every head of the different claims, and every particular of value and quantity.

A scheme is then submitted to the proprietors, showing as near as possible the situation they are to be placed to receive their allotment or allotments, in lieu of their original holdings and rights, after proportionate deductions have been made for manorial rights, tithe, gravel-pits, drains, roads, public fencing, recreation-ground, &c.

The most difficult part of an inclosure is in giving general satisfaction to the proprietors in the situation their allotments are placed, according to the judgment of the commissioners, that being at variance with the proprietors as to value and location.

When the scheme is pretty well matured, a division book is made, according to the lines drawn on the plan, the pieces and parts of pieces of land are numbered and scaled, and the proportionate value adjusted, until each proprietor has his exact share. This is a very delicate operation to the surveyor, as the whole of the allotments must balance with the original quantity and value.

All roads and drains are set out prior to any of the allot

ments.

After the allotments have been staked out, and the plan to correspond, the clerk then divides them in their just proportion of copyhold and freehold, &c., which is accurately laid down on the plan by the surveyor.

The surveyor has to make out a rate according to the expense of inclosing the parish, submit it to the commissioners, on the approval of which it is then given to the clerk to collect.

Example. Open field properties and old inclosures charged with a rate to defray expenses.

The rate to be raised on the above property amounts to

shill. dec.

23971. 2s. 74d., or 47942.625; and it is supposed 23s. in the pound on the open field, and 12s. 6d. in the pound on the old inclosures, will raise the above sum. But, in case it should not, the said sums are to be increased or diminished in exact proportion, so as to raise the exact sum-viz. 23971. 2s. 74d.

First reduce the required sum into shillings and decimals. Then find factors for the open field land and the old inclosures, thus:

Then multiply the annual value of the open fields and the old inclosures by their relative factors, the amount of which will give the total sum according.

As 39060.303225: 35.5 :: 47942.625 43.5727 the required sum to be raised

Therefore upon the same principle can any rate for a parish or county be calculated.

ON PLANE TRIGONOMETRY.

Plane trigonometry is the art of measuring and computing the sides of plane triangles, or of such whose sides are right lines.

As this work is not intended to teach the elements of mathematics, it will be sufficient to point out a few of the principles, and give the rules of plane trigonometry for those cases that occur in surveying. In most of these cases it is required to find lines or angles, whose actual admeasurement is difficult or impracticable; they are discovered by the relation they bear to other given lines or angles, a calculation being instituted for that purpose; and as the comparison of one right line with another right line is more convenient and easy than the comparison of a right line to a curve, it has been found advantageous to measure the quantities of angles, not by the arc itself, which is described on the angular point, but by certain lines described about that arc.

The circumference of every circle is supposed to be divided into 360 equal parts, called degrees; each degree into 60 minutes; each minute into 60 seconds; and so on.

The sine of three angles of every triangle, or two right angles, are equal to 180 degrees.

The sum of two angles in any triangle, taken from 180 degrees, leaves the third angle.

In a right angled plane triangle, if either acute angle be

taken from 90 degrees, the remainder will give the other acute angle.

When the sine of an obtuse angle is required, subtract such obtuse angle from 180 degrees, and take the sine of the remainder, or supplement.

If two sides of a triangle are equal, a line bisecting the contained angle will be perpendicular to the remaining side, and divide it equally.

Before the required side of a triangle can be found by calculation, its opposite angle must first be given, or found.

The required part of a triangle must be the last term of four proportionals, written in order under one another, whereof the three first are given or known.

In four proportional quantities, either of them may be made the last term; thus, let A B C D be proportional quantities: As first to second, so is third to fourth, A: B :: C: D As second to first, so is fourth to third, B: A :: D : C As third to fourth, so is first to second, C: D: A: B As fourth is to third, so is second to first, D: C:: B: A Against the three first terms of every proposition or stating must be written their respective values taken from the proper tables.

If the value of the first term be taken from the sum of the second and third, the remainder will be the value of the fourth term or thing required, because the addition and subtraction of logarithms correspond with the multiplication and division of natural numbers.

If to the complement of the first value be added the second and third values, the sum rejecting the borrowed index will be the tabular number expressing the thing required. This method is generally used when radius is not one of the proportionals.

The complement of any logarithm, sine, or tangent, in the common table, is its difference from the radius 10.000.000, or, its double, 20.000.000.

The complement of an arc is what it wants of 90 degrees.

The supplement of an arc is what it wants of 180 degrees.

A sine or right sine of an arc is a line drawn from one extremity of the arc perpendicular to the diameter, as B F, or the supplement to the arc B D E. (Fig. 1, Plate 28.)

The versed sine of an arc is that part of the diameter intercepted between the arc and its sine; as A F is the versed sine of the arc A B.

The tangent of an arc is a line perpendicular to the diameter touching the circle, as A H.

A secant is a line drawn from the centre C through any point of the circumference until it intersects the tangent as at H; then CH is the secant of the arc A B; also EI is the tangent and C I the secant of the supplemental arc BDE; and this latter tangent and secant are equal to the former, but are accounted negative, as being drawn in an opposite direction to the former.

The co-sine, co-tangent, co-secant of an arc, are the sine, tangent, and secant of the complement of that arc, the co. being only a contraction of the word complement. Thus, the arcs A B and B D being the complements of each other, the sine, tangent, or secant of one of these is the co-sine, co-tangent, or co-secant of the others; so B F the sine of AB, is the co-sine of B D; and B K the sine of B D, is the co-sine of A B. In like manner, A H the tangent of A B, is the co-tangent of BD; and DL the tangent of D B, is the co-tangent of A B. Also CH the secant of A B, is the co-secant of B D; and C L the secant of B D, is the co-secant of A B.

Corollary. Hence several remarkable properties easily follow from their definitions; as

1st. That an arc and its supplement have the same sine, tangent, and secant; but the two latter, the tangent and secant, are counted negative when the arc is greater than a quadrant, or 90 degrees.

2nd. When the arc is zero, or nothing, the sine and tangent are nothing, but the secant is the radius C A.

3rd. Of any arc A B, the versed sine A F and co-sine B K