Ex. 3.-Required the area of the whole surface of the earth, its diameter beIng 7957 miles, and its circumference 25000 miles.

Ans. 198943750 sq. miles. Ex. 4.-The axis of a sphere being 42 inches, what is the convex superficies of the segment, whose height is 9 inches? Ans. 1187-5248 inches.

Ex. 5.-Required the convex surface of a spherical zone, whose breadth or neight is 2 feet, and cut from a sphere of 12 feet diameter. Ans. 78.54 feet

PROBLEM VIIL

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.

RULE II. Multiply the cube of the diameter by the decimal 5236, for the

content.

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. 263,857,437,760 miles.

PROBLEM IX.

To find the solid content of a spherical segment.

RULE 1. From three times the diameter of the sphere 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 three 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 in 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.

LAND SURVEYING.

SECTION I.

DESCRIPTION AND USE OF THE INSTRUMENTS.

1.-OF THE CHAIN.

6 6 100

I AND is measured with a chain, called Gunter's Chain, from its inventor, of 4 poles or 22 yards, or 66 feet in length. 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. Land is estimated in acres, roods, and perches. An acre is equal to 10 square chains, that is, 10 chains in length and one chain in breadth. Or it is 220 × 22 = 4840 square yards. Or it is 40 × 4 = 160 square poles. Or it is 1000 × 100 = 1,000,000 square links. These being all the same quantity. Also, an acre is divided into four parts called roods, and a 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:

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 These decimals are then multiplied by 4 for roods, and the decimals of these again by 40 for perches.

acres.

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.

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 upon the table. The 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 from a centre which is in the middle of the table; by means of which the table is to be used as a theodolite, &c.

2. A needle and compass screwed into the side of the table, to point out the directions, and to be a check upon the sights.

3. An index, which is a brass two-foot scale, with either a small telescope, or open sights erected perpendicularly upon the ends. These sights and one edge of the index, are in the same plane, and that edge 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 upon the table, pressing down the frame upon 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,

Then, begin at any part of the ground the most proper, and make a point on a convenient part of the paper or table, to represent that point of 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; then set another object or corner, and draw its line; do the same by another, and so on, till as many objects are set as may be thought fit. Then measure from the station, towards as many of the objects as may be necessary, and no more, taking the requisite offsets to corners or crooks in the hedges, laying the measures down on their respective lines on the table. Then, at any convenient place, measured to, fix the table in the same position, and set the objects which appear from thence, &c. as before; and thus continue till the work is finished, measuring such lines as are necessary, and determining as many as may be, by intersecting lines of direction drawn from different stations.

OF SHIFTING THE PAPER ON THE PLAIN TABLE.

When one paper is full, and you have 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 upon 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, beginning 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 compass, the easiest way of drawing the lines in the same position, 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, 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 you judge most convenient; taking angles or directions to objects, and measuring such distances as appear necessary, under the same restrictions as in the use of the plain table. And it is safest to fix the theodolite in the original position at every station, by means of fore and back objects, and the compass, exactly as in using the plain table; registering the number of degrees cut off by the index when directed to each object; and, at any station, placing the index at the same degree as when the direction towards that station was taken from the last preceding one, to fix the theodolite there in the original position.

The best method of laying down the aforesaid lines of direction, is to describe a pretty large circle; then quarter it, and lay on it the several numbers of degrees cut off by the index in each direction, and drawing lines from the centre to all these marked points in the circle. Then, by means of a parallel ruler, draw, from station to station, lines parallel to the aforesaid lines drawn from the centre to the respective points in the circumference.

4. OF THE CROSS.

The cross consists of two pair of sights set at right angles to each other, upon a staff having a sharp point at the bottom to stick in the ground.

The cross is very useful to measure small and crooked pieces of ground. The method is to measure a base or chief line, usually in the longest direction of the piece, from corner to corner; and while measuring it, finding the places where perpendiculars would fall on this line, from the several corners and bends in the boundary of the piece, with the cross, by fixing it, by trials, on such parts of the line, so that through one pair of the sights both ends of the line may appear, and through the other pair you can perceive the corresponding bends or corners: and then measuring the lengths of the said perpendiculars.

REMARKS.

Besides the fore-mentioned instruments, which are most commonly used, there are some others; as the circumferentor, which resembles the theodolite in shape and use; and the semicircle, for taking angles, &c.

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 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. And in measuring any sloping or oblique line, either ascending or descending, a small pocket level is useful for showing how many links for each chain are to be deducted, to reduce the line to the true horizontal length.

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 scale, parallel and perpendicular rules, &c. Of plane scales, there should be several sizes, as a chain in 1 inch, a chain in ₹ of an inch, a chain in § 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.

5. OF THE FIELD-BOOK.

In surveying with the plane 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 field-book 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 which has been formerly 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 left, which are set against their corresponding distances in the middle column; as also for such remarks as may occur, and may be proper to note in drawing the plan, &c.

On the

Here is the first station, where the angle or bearing is 105° 25′. left, at 73 links in the distances 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.

A line is drawn under the work, at the end of every station line, to prevent confusion.