sions; it remains to point out the relation between some dissimilar solids.

As the circle, in surfaces, contains the greatest surface within the same bounds, so the sphere in solids contains the greatest solidity within the same surface; there being no corners in the one or the other. Both in solidity and surface, the sphere is its circumscribing cylinder, if we take into consideration the surface of the ends of the cylinder; the surface of the sphere being equal to the curve surface of the cylinder, without its ends.

If the diameter of the sphere be 1 inch, its solidity will be .52359, &c., of an inch, or as we generally say, .5236; and the solidity of the circumscribing cylinder will be .78539, &c., or as we may say for convenience .7854 of a cubic inch; of which .5236 is just .

By the "circumscribing cylinder" is meant a cylinder that may be just circumscribed around the sphere; or a cylinder, like a piece of stove-pipe, into which the sphere may be dropped, and it will just fit its cavity and be of the same height.

The surface of an inch globe is 3.1416 square inches; and the curve surface of an inch cylinder is 3.1416, to which .7854 being added for each end, we have the whole surface 4.7124, of which 3.1416, the surface of the globe, is just ; so that both in solidity and surface the globe is of its circumscribing cylinder. The surface of a sphere is equal to four times the area of a great circle of it.

But if the sphere bears to the circumscribing cylinder, the same ratio both in surface and solidity; how is it to contain greater solidity under the same surface than the cylinder does? The answer to this must be sought in the fact that solidity increases more rapidly than surface, and hence if you reduce the cylinder until it measures no more than the sphere enclosed in it, the surface will be made greater as compared with the solidity of it; but if you increase the sphere until it has 50 per cent. more matter in it, which will make it equal in solidity to the cylinder, the surface will not be equal to that of the cylinder ;-it will not be 4.7124, hence it has greater solidity under a given surface than the cylinder has. Surface increases as the square only, while solidity increases as the cube of the linear dimensions of the solid body: a globe 2 inches in diameter has four times the surface of one only an inch in diameter; and it has 8 times the solidity. If the diameter of the base of a cone be equal to the perpendicular height of the cone, then the solidity will be half that of a sphere whose

diameter is equal to the base of the cone, and of a cylinder of the same base and altitude as the cone.

From which it appears that the height and diameter being equal, the solidity of a cone, sphere and cylinder are as 1, 2, 3. If a cylinder of equal diameter and height be made to contain 3 pints of water, and a cone of equal diameter and height as the cylinder be introduced, one pint will run out; and if the cone be taken out and a sphere of the same diameter be introduced, another pint will run over. These proportions were amongst the discoveries of ARCHIMEDES, that prince of mathematicians.

It is frequently important for workmen to know how to determine the size that a given log will square, or on the other hand to know how large a log must be to make a square beam of a given size. To find what a tree will square, multiply the girt or circumference by .225, the result will be the side of the square that may be formed from it, near enough for all practical purposes. To ascertain the girt of a tree necessary to make a given sized square beam, multiply one side of the beam by 4.443.

But as these arbitrary multipliers are liable to be forgotten, a little investigation will enable the operator to make his calculations from principle.

Let the circle A C D B represent the end of a log, from which the largest practicable square beam is to be formed. The end of the beam will be represented by the square A B D C, of which C B is the hypotenuse, and is at the same time the diameter of the circle. Then if we know the circumference of a tree we can readily ascertain the diameter by dividing by 3.1416, and as the square of the hypotenuse is equal to the squares of the two legs, if the piece is to be square, the square root of half the square of the hypotenuse will be one side, A B or A C. If the piece is to be larger one way than the other, the diameter is still the hypotenuse; and its square must be divided accordingly.

If on the other hand we know the size of the timber, whether it be square or of unequal depth and thickness, we have but to ascertain its diagonal C B, and we know the diameter of the tree.

18*

GENERATION OF GEOMETRICAL FIGURES.

IMAGINING how bodies may be formed or generated by motion, or built up of parts, often assists in forming a clear idea of them.

The straight line may be considered as generated by the motion of a point constantly moving in the same direction, and leaving a track as it proceeds.

The circumference of a circle as formed by the motion of a movable point carried round a fixed point or centre, and kept constantly at the same distance from it.

A circle as formed by the motion of a straight line carried round a point; and if carried less than entirely round, a sector of a circle is produced.

A parallelogram as formed by the motion of a line carried at right angles to its length.

A triangle as formed by the movement of a point, while expanding into a line.

A cycloid is described by the motion of a point in the periphery of a wheel rolling forward.

A parallelopipedon is produced by the motion of a parallelogram, at right angles to its plane.

A cone by the revolution of a right angled triangle around its perpendicular; or the motion of a point expanding into a circle: if we can imagine the expansion of that which has no dimensions.

A cylinder by the motion of a circle at right angles to its plane, or the revolution of a parallelogram on one of its sides as an axis.

A prism by the motion of a polygon.

A sphere by the revolution of a semi-circle on its axis.

In this way we may imagine lines, surfaces and solids to be generated, and frequently our clearest ideas are thus formed. It is the basis of the doctrine of Fluxions, in which figures are not supposed to be made up of a collection of distinct parts, but as formed by the flowing of other figures; as a line by the flowing of a point, a surface by the flowing of a line, a solid by the flowing of a surface. It is from this the science takes its name; the word Fluxion, signifying a flowing. The notion of generating figures in this way was however thought of long before it was built up by LEIBNITZ and NEWTON into a beautiful and efficient science.

The doctrine of Differential and Integral calculus, which has in a great measure superseded the old form of Fluxions, supposes the quantities to be generated, not by a uniform increase or flowing motion, but by the successive additions of infinitely

small portions. The method which descends from quantities to their elements, is called the Differential Calculus; while that which ascends from the elements to the quantities is called the Integral Calculus.

After GULDIN and KEPLER had treated of figures as generated by a flowing motion, and supposed the extent of surface to be found by multiplying the line into the extent passed through in describing a surface, and the size of solids by multiplying the generating surface by its flowing space, ČAVALLERIUS advanced a new view of the subject. He supposed a line to be made up of an infinite number of points; a surface of an infinite number of lines; and a solid of an infinite number of surfaces. These were considered Indivisibles, or the elementary principles of the figures produced; and though we know that lines could never be produced by using points which have no dimension, any more than lines could form surfaces, or a pile of surfaces having no thickness could be ultimately built into a solid body, yet it was a step in the march of science.

We may conceive a prism to be made up of an infinite number of small prisms-a pyramid of small pyramids, and a sphere of a great number of small prisms uniting their apices at the centre as grains of corn upon the cob tend towards the centre. A large number of such small prisms having spherical bases, and properly proportioned sides, if laid so that their points would all meet at the centre, would form a perfect sphere. They must, however, be hexagonal, square, or equitriangular, in order to fit with each other, leaving no vacancies, unless the sides be made unequal.

In explaining the rule for finding the surface of figures, we may often make our explanations more clear by imagining the surface to be divided into many small squares; as if we are attempting to prove that an oblong 7 inches long and 4 wide, will contain 28 square inches, we may imagine the figure laid off as follows and really divided

into square inches.

So if we are seeking to find the cubic inches in a block 7 inches long, 4 inches wide, and

5 inches thick; we may imagine that the base is laid off into 28 square inches, and sawed into 28 square prisms, each containing 5 solid inches, and multiplying the number of little prisms (28) by 5, the solid inches in one, we have the solidity of the whole.

A tangent to a circle is a line drawn touching the circumference, and at right angles with a line drawn from the centre

to the point of contact. If a stone be whirled in a sling, and suddenly released, it will fly off at a tangent from the circle in which it had moved. The ancients used slings as a means of warfare, and the precision acquired in their use may be inferred from the fate of Goliah, as well as from what is said in the 20th chapter of Judges, "Among all this people there were seven hundred chosen men, left-handed, every one could sling stones at a hair breadth, and not miss." The fragments of a millstone, or grindstone, that bursts from its rapid circular motion, fly off in a tangent from the circle they described before the breaking. The expression "off at a tangent" is often used facetiously in common discourse to signify an abrupt departure.

LECTURE XIV.

THEORY OF WHEEL CARRIAGES.

ALTHOUGH the use of Wheel Carriages shows considerable advancement in science, the invention was made at a period too early for the light of history.

The savage, and even the brute beast, will drag his burden upon the ground when he is unable to carry it, and this seems to be the simplest form of transporting burdens by draught. But it is exceedingly objectionable in several respects. It would injure the body in many instances, and would produce a great amount of friction. Very soon the idea of a sled or slide would occur, as protecting the body from injury and diminishing friction. The hunter might drag home the fruit of the chase, as the beast of prey would drag his victim to his lair, but it would be neater and easier perhaps to place it upon a pole or other implement that would raise it from the ground; and soon he would learn to construct such a carriage permanently, especially in latitudes where snow favors its use. We accordingly find the Laplander and the rude inhabitants of