himself by a trial, with respect to the areas of all the similar figures: and by analogy, the contents of all similar solid figures are to one another in the proportion of the cubes of their corresponding or homologous sides or lines, Thus the solid contents of a sphere of 10 inches diameter will be 261,773 nearly, and the solidity of another sphere of 20 inches diameter will be 2094,1829 nearly, then the cube of the diameter 10 being 1000, and that of the dia meter 20 being 8000, the solid contents 261,773 will be to the solid contents 2094,1829 as the cube 1000 is to the cube 8000.

If it were required to form a solid body similar to a given solid, and whose contents were to be in a given proportion to the contents of the given solid, we must discover the line or side which, when cubed, would bear to the homologous, line or side of the given solid the same proportion with that of their proposed solidities: thus if, for example, it were required to determine the diameter of a sphere whose solidity would be 8 times greater than that of a sphere whose diameter is 10 inches, we would state this proportion; as the solidity of the given sphere which may be expressed by 1 to the solidity of the sphere required, which is 8 times greater, or 8, so is the cube of the given diameter 10 1000 to a fourth proportional 8000, which will be the cube of the diameter of the required sphere; and extracting the cube root 20, (Arith. page 266,) this will be the diameter of a sphere whose solidity will be 8 times greater than that of one whose diameter is 10 inches.

=

As in bodies consisting of the same substance, the weights are in proportion to the quantities of matter in each body, so by knowing the weight of any regular figure, as a ball or sphere of a determinate diameter, if we wish to learn the weight of another ball or sphere of the same sub

வ.

stance,

stance, but of a different diameter, we state this proportion; as the cube of the diameter of the first sphere whose weight is known to the cube of the diameter of the second, so is the weight of the first sphere to the weight of the second.

SURVEYING OR LAND-MEASURING.

By Surveying is generally understood the art of measur ing the contents of a field, an estate, a parish, a county, &c. and it comprehends three several operations, viz. 1st. making the survey, or measuring the dimensions of the ground; 2dly, delineating or laying down upon paper a correct representation of the ground surveyed, every part being exhibited in its due situation and proportion with respect to the others; and, 3dly, calculating the area or superficial contents of the several portions and of the whole. Of these operations the first is what is properly called surveying, the second is called plotting, protracting, or mapping, and the third is termed casting up, or computing the contents.

1st. The first operation, or surveying, consists of two parts, the making of observations for the angles, and the taking of lineal measures for the sides: the former operation being performed with one or other of the following instruments, viz. the theodolite, the circumferentor, the semicircle, the graphometer, the plain-table, the compass, or by the measur ing chain alone: the latter operation is performed by the chain or the perambulator.

The theodolite is made in various ways, according to the taste and skill of the artist, in order to render it the more simple and manageable, or the more accurate and convenient in use; but in general it consists of a brass circle of 6, 9, or 12 inches diameter, strengthened by four cross bars meet3 R 2 ing

ing in the centre. The circular part, or limb, as it is termed, is divided into 360 degrees, each of which is again subdivided into minutes. To the limb are adapted two pillars supporting an axis to bear a telescope for viewing objects at a distance and on the centre of the instrument moves a circular plate or index, having in the middle a mariner's compass, the meridian line of which corresponds to the fiducial edge of one side of the index; and two other pillars on the index support another telescope, of which the line of collimation corresponds to the fiducial edge on the other side of the index. At each end of these telescopes are also adapted plain sights for viewing objects near the observer; and the ends of the index are cut circularly to suit the limb of the instrument, so that when the limb is divided by means of concentric circles and diagonals, the fiducial line, on the end of the index, points out the degrees and minutes on the Timb. The theodolite is also fitted with cross spirit levels to direct the observer in placing the circle truly horizontal, and with a vertical circle for taking angles of elevation and depression. The whole instrument is mounted on a fulcrum, or three-legged staff, with a ball and socket, or other contrivance, for rendering it moveable in all directions without deranging the position of the fulcrum.

The circumferentor consists of a brass circle and index all of one piece: the diameter of the circle is about seven inches, the length of the index about fourteen, with a breadth of an inch and a half. On the circle is a chart whose meridian line answers to the middle of the breadth of the index, and the circumference is divided into 360 degrees. On the centre of the instrument is adapted a magnetic needle; and there are two sights to screw on and slide up and down the index, with a contrivance to render the instrument moveable on the head of its fulcrum.

Angles are measured by the circumferentor in this way: the instrument being placed with its centre precisely over

the

the angular point, with the flower-de-luce of the compass towards the observer, he directs the sights along one of the lines forming the angle to be measured, and observes what degrees are cut by the south end of the needle: then turning the instrument round until the sights bear along the other line forming the angle, he again observes what degrees are cut by the needle; when subtracting the less from the greater of these two quantities, the difference gives the quantity of the angle in question.

The graphometer, or semicircle, is, as its name imports, a figure bounded by the diameter and half the circumferenee of a circle: the limb or arch is divided into 180 degrees, which are again by diagonals, or otherwise, divided into minutes, &c. On the centre of the instrument are usually fixed a compass-box and needle, together with a moveable index equal to the diameter of the semicircle, on whose extremities, as well as on those of the fixed diameter, are fitted sights for viewing objects at a distance; the whole instrument being mounted with a proper apparatus on a fulcrum.

In taking an angle with the graphometer, the instrument is to be placed, by means of a plummet, directly over the angular point, with the fixed diameter in the direction of one of the lines forming the angle, then moving the index round until through the sights some distant objects is observed in the direction of the other side of the angle, the number of degrees and minutes on the limb indicated by the extremity of the index will be the measure of the given angle.

These directions are also applicable to the theodolite, for the graphometer is only one half of that instrument, and the index and compass on either of them may be employed instead of the circumferentor.

The plain-table is an instrument by which the draught

or

or plot of a piece of ground is taken on the spot, as the survey or measurement goes on, without the necessity of any future protracting or planning. This instrument consists commonly of a plain rectangular board, of any convenient size, fixed by means of screws and other machinery in the centre to the fulcrum, so as to be moveable in all directions vertically and horizontally. To the table are applied, 1st, a wooden frame made to fit close round its edges, for the purpose of fixing on a sheet of paper. One side of the frame is divided into a scale of equal parts, by which to draw lines across the table, parallel or perpendicular to the sides, and the other side is divided into degrees from a centre in the middle of the table, so that the instrument can be used as a theodolite, &c.-2dly, a magnetic needle and compass screwed into the side of the table, to point out the bearings of objects, and serve as a check upon the sights.-3dly, an index, which is a brass two-feet scale, having either a small telescope fitted to it, or sights erected perpendicularly on the ends, the middle of which must be parallel to the fiducial edge of the index.

In using the plain-table a sheet of paper is spread flat on the surface, having been previously moistened to make it expand, and the edges are fastened down by the frame before mentioned: when the paper becomes dry it will shrink a little and become perfectly flat and smooth, so as to be fit to receive the draught of the ground to be surveyed. The surveyor begins his work by setting up the instrument in some convenient spot, making a point on the paper to represent his position; he then fixes in that point the foot of a pair of compasses, or a steel pin, and applying to it the fiducial edge of the index, he moves it round close to the pin until through the sights he observes some object such as the corner of a wall, a pole, or picket set up on purpose, &c. ; then from the first station point he draws a faint line along the edge of the index, and turning it round to some other

object,