To find the Solidity of a Sphere or Globe.

RULE I. Multiply the surface by the diameter, and take of the product for the content *. Or, which is the 410 same thing, multiply the square of the diameter by the cir cumference, and take of the product.

RULE II. Take the cube of the diameter, and multiply it by the decimal 5236, for the content.

RULE III. Cube the circumference, and multiply by 01688.

Ex. 1. To find the content of a sphere whose axis is 12.

Ans. 904 7808. Ex 2. To find the solid content of the globe of the earth, supposing its circumference to be 25000 miles.

Ans. 263858149120 miles.

PROBLEM IX.

To find the Solid Content of a Spherical Segment. RULE I. From 3 times the diameter of the sphere také

* For, put d the diameter, c = the circumference, and s = the surface of the sphere, or of its circumscribing cylinder; also, a the number 3.1416.

Then, s is the base of the cylinder, or one great circle of the sphere; and d is the height of the cylinder; therefore ds is the content of the cylinder. But 4 of the cylinder is the sphere, by th. 117, Geom. that is, of ds, or ds is the sphere; which is the first rule.

Again, because the surfaces is ad2; therefore ds = {ad3 5236d3, is the content, as in the 2d rule. Also, d being c÷a, therefore ad3 = ‡c2 ÷a2 = •01638, the 3d rule for the content. † By corol. 3, of theor. 117, Geom. it appears that the spheric segment PFN, is equal to the difference between the cylinder ABLO, and the conic frustum ABMQ.

But, putting d AB or FH the diameter of the sphere or cylinder, h FK the height of the segment, rPK the radius of its 3'1416; then the content of the cone ABI is ad× FI= 24ad3; and by the similar cones

base, and a

A

B

E

IG

D

ABI, QMI, as

take double the height of the segment; then multiply the remainder by the square of the height, and the product by the decimal 5236, for the content.

RULE II. To 3 times the square of the radius of the segment's base, add the square of its height; then multiply the sum by the height, and the product by 5236, for the

content.

Ex. 1. To find the content of a spherical segment, of 2 feet in height, cut from a sphere of 8 feet diameter.

Ans. 41.888.

Ex. 2. What is the solidity of the segment of a sphere, its height being 9, and the diameter of its base 20? Ans. 1795 4244.

Note. The general rules for measuring all sorts of figures. having been now delivered, we may next proceed to apply them to the several practical uses in life, as follows.

F1a : KI3 :: Tad3 : 14ad3 (d) the cone QMI;

therefore the cone ABI

zd

the cone QMI = {ad3 — ad3 X

2)3 = £ad2h — ±adh2 + Zah3 is — the conic frustum ABMQ.

And ad h is the cylinder ABLO.

Then the difference of these two is Zadh — Zah3 — Zah2 × (3d2h), for the spheric segment PFN; which is the first rule.

Again, because PK2 = FK × KH (cor. to theor. 87, Geom.) or r2 = h (d – h), therefore d

h

+h, and 3d

Note. By subtracting a segment from a half sphere, or from another segment, the content of any frustum or zone may be

Grund.

LAND

[53]

LAND SURVEYING.

SECTION I.

DESCRIPTION AND USE OF THE INSTRUMENTS.

1. OF THE CHAIN.

LAND is measured with a chain, called Gunter's Chain, from its inventor, the length of which is 4 poles, or 22 yards, or 66 feet. It consists of 100 equal links; and the length of each link is therefore of a yard, or of a foot, or 7.92 inches.

22

66

100

Land is estimated in acres, roods, and perches. An acre is equal to 10 square chains, or as much as 10 chains in length and 1 chain in breadth. Or, in yards, it is 220 × 22 = 4840 square yards. Or, in poles, it is 40 × 4 = 160 square poles. Or, in links, it is 1000 x 100 = 100000 square links: these being all the same quantity.

3

Also, an acre is divided into 4 parts called roods, and rood into 40 parts called perches, which are square poles, or the square of a pole of 5 yards long, or the square of 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 best 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. These decimals are then multiplied by 4 for roods, and the decimals of these again by 40 for perches.

EXAM. 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.

2. OF THE PLAIN TABLE.

THIS instrument consists of a plain rectangular board, of any convenient size: the centre of which, when used, is fixed by means of screws to a three-legged stand, having a ball and socket, or other joint, at the top, by means of which, when the legs are fixed on the ground, the table is inclined in any direction.

To the table belong various parts, as follow.

1. A frame of wood, made to fit round its edges, and to be taken off, for the convenience of putting a sheet of paper on the table. One side of this frame is usually divided into equal parts, for drawing lines across the table, parallel or perpendicular to the sides; and the other side of the frame is divided into 360 degrees, to a centre in the middle of the table; by means of which the table may be used as a theodolite, &c.

2. A magnetic needle and compass, either screwed into the side of the table, or fixed beneath its centre, to point out the directions, and to be a check on the sights.

3. An index, which is a brass two-foot scale, with either a small telescope, or open sights set perpendicularly on the ends. These sights and one edge of the index are in the same plane, and that is called the fiducial edge of the index.

To use this instrument, take a sheet of paper which will cover it, and wet it to make it expand; then spread it flat on the table, pressing down the frame on the edges, to stretch it and keep it fixed there; and when the paper is become dry, it will, by contracting again, stretch itself smooth and flat from any cramps and unevenness. On this paper is to be drawn the plan or form of the thing measured.

Thus, begin at any proper part of the ground, and make a point on a convenient part of the paper or table, to represent that place on the ground; then fix in that point one. leg of the compasses, or a fine steel pin, and apply to it the fiducial edge of the index, moving it round till through the sights you perceive some remarkable object, as the corner of a field, &c; and from the station-point draw a line with the point of the compasses along the fiducial edge of the index, which is called setting or taking the object: then set another object or corner, and draw its line; do the same by another; and so on, till as many objects are taken as may be thought fit. Then measure from the station towards as many of the objects as may be necessary, but not more, taking the requisite offsets to corners or crooks in the hedges, laying the measures down on their respective lines on the table,

Then,

Then at any convenient place measured to, fix the table in the same position, and set the objects which appear from that: place; and so on, as before. And thus continue till the work is finished, measuring such lines only as we necessary, and determining as many as may be by intersecing lines of direction drawn from different stations.

Of shifting the Paper on the Plain Table.

When one paper is full, and there is occasion for more; draw a line in any manner through the farthest point of the last station line, to which the work can be conveniently laid down; then take the sheet off the table, and fix another on, drawing a line over it, in a part the most convenient for the rest of the work; then fold or cut the old sheet by the line drawn on it, applying the edge to the line on the new sheet, and, as they lie in that position, continue the last station line on the new paper, placing on it the rest of the measure, be ginning at where the old sheet left off. And so on from sheet to sheet.

When the work is done, and you would fasten all the sheets together into one piece, or rough plan, the aforesaid lines are to be accurately joined together, in the same manner as when the lines were transferred from the old sheets to the new ones. But it is to be noted, that if the said joining lines, on the old and new sheets, have not the same inclination to the side of the table, the needle will not point to the original degree when the table is rectified; and if the needle be required to respect still the same degree of the compass, the easiest way of drawing the lines in the same po sition, is to draw them both parallel to the same sides of the table, by means of the equal divisions marked on the other two sides.

3. OF THE THEODOLITE.

THE theodolite is a brazen circular ring, divided into 360 degrees, &c, and having an index with sights, or a telescope, placed on the centre, about which the index is moveable; also a compass fixed to the centre, to point out courses and check the sights; the whole being fixed by the centre on a stand of a convenient height for use.

In using this instrument, an exact account, or field-book, of all measures and things necessary to be remarked in the plan, must be kept, from which to make out the plan on returning home from the ground.

Begin at such part of the ground, and measure in such directions as are judged most convenient; taking angles or directions to objects, and measuring such distances as appear

necessary,