riable pendulum, as such an apparatus is called, are sensibly performed in equal times, or are isochronous, even when the arc gradually diminishes from retarding forces.

The length of a pendulum vibrating seconds in the latitude of New-York (40° 42′ 43′′ N.), in vacuo, at the level of the sea, has been accurately determined to be 39.102 inches.

The force of gravity, which, on account of the figure of the earth, varies slightly in intensity from the poles to the equator, may be determined for different places by the velocity of a standard pendulum, it being directly proportional to the squares of the number of vibrations in equal times: or as the lengths of pendulums are inversely proportionate to the square of the number of their vibrations in equal times, the force of gravity may be determined for any place by measuring the length of the pendulum which beats seconds at that place.

§ 29. Momentum may be accumulated to an enormous degree in very large masses thus suspended, by forces which, at their first application, appear to be totally inadequate to move them. By making repeated impulses coincide in time and direction with the first minute oscillations, the arc will gradually increase until a great degree of velocity is produced; and the force which is thus stored up is capable of producing a corresponding concentrated effect.

§ 30. According to the first law of gravity, the force with which the earth acts upon any body at its surface is in direct proportion to the quantity of matter which it contains; hence the measure of the force becomes the criterion of quantity, and the weight of a body is the exact amount of force, expressed with relation to some known standard, which is just sufficient to prevent that body falling to the ground. The commonest mode of ascertaining this is to oppose the known effect of the gravi

ty of certain pieces of metal which have been compared with some conventional standard with the greatest possible exactness, to the unknown gravity of the substance whose weight or quantity of matter has to be determined. A pound is a mass of matter which has been thus adjusted; and the common business of life has rendered most men familiar with the multiples and sub-multiples of this weight, and the denominations of tuns, hundred weights, ounces, grains, &c., which have been conferred on them. The standard liquid measure of the State of New-York is the gallon, and the weight of pure water which it contains is 8 lbs. The standard of dry measure is also called the gallon, and contains 10 lbs. of pure water; the bushel, in consequence, holds 80 lbs. The method of comparison, by means of the balance, is also well known.

31. The balance consists essentially of a uniform inflexible lever, delicately supported at its centre of gravity on a fine knife edge, and carrying scalepans freely suspended from points in the same horizontal line with the centre of gravity. If the weights or quantities of matter in each be equal, the one will counterbalance the other, and the beam will remain horizontal; if not, the heavier will preponderate. Lightness of construction and freedom of motion are secured by many ingenious contrivances, upon which it would be foreign to our present purpose to dwell. A good balance will indicate by its turn 1-5000th or 1-6000th of the weight which it is designed to carry, and will freely move with the difference of 1-1000th of a grain.

32. Equality in the length of the arms of the lever is, of course, the most important consideration in the construction of the balance; but when there may be any reason to doubt this essential point, it is well to know that any error may be avoided by the method of double weighing. This

consists in placing the object whose weight is to be ascertained in one scalepan, and exactly counterbalancing it in the other, not with the weights, but with sand or shot, or any other indifferent substance. The substance is then removed, and the weights applied in the same pan, till the counterpoise is balanced. By this contrivance the unknown quantity of matter is compared with the known, under exactly equal circumstances, and the result is independent of almost every source of error which can affect the comparison of one object with another.

33. The second law of gravity has little to do with the determination of weight, and for this purpose may safely be disregarded; for any variation of the distance from the centre of the earth, at which we may carry on our operations, is so small, with regard to the whole, as to be perfectly insignificant, although not inappreciable to the refinements of modern science. The mean radius of the earth, or the distance from the centre to the surface, is about 3941 miles; and supposing that we had to determine the weight of an object on the summit of a mountain one mile in height, the force of gravity would be decreased in the ratio 39422: 39412, which would make a difference of about one ounce in a tun weight. This difference would not, of course, be apparent in the usual manner of weighing by means of the balance; for the decrease of gravity would affect both the weight and the object to be weighed in the same degree; but it might be measured by the opposition of another force, as that of elasticity. A spring which would be bent to a certain degree by a tun weight at the surface of the earth, would require a tun weight and one ounce to bend it to the same amount on the summit of the mountain.

34. Every substance in nature, occupying a given space, is found to have, under the same circumstances, a weight specific or peculiar to itself;

or, in other words, the same volume of different kinds of matter contains different quantities of matter. The comparative weights of equal bulks of different bodies are called specific gravities. This is a very important distinctive property of matter, and one to which the chymist has perpetual occasion to refer. In comparing specific gravities it has been found convenient to refer them to a fixed standard, and water has been generally adopted for this purpose, as being easily procurable in most times and places.

35. The mode of taking the specific gravity of a liquid is very simple and easy of execution. A small stoppered flask is prepared to contain exactly 1000 grains of pure water; this is filled with the liquid, and placed upon the balance; in the opposite pan is placed the counterpoise of the bottle when' filled with water, and, when heavier than water, so much weight as will adjust the beam. If the liquid be lighter than water, the weight must be placed in the same pan with the bottle.

In the first case, the weight in grains added to 1000, and in the second case deducted from 1000, will give the specific gravity sought. Thus the same bottle which held 1000 grains of water was found to contain only 839 grains of spirits of wine; and, taking water as 1, the specific gravity of the spirit is said to be 0.839.

§ 36. It would be impossible thus directly to compare a given volume of any solid body with an equal volume of water; were there no other obstacle, it would require a nicety of measuration and workmanship which would be quite unattainable. But the same end may be obtained with perfect accuracy by means which are easily applied. The rule is to weigh the solid in air and afterward in water, and, having found the deficiency of the latter weight, to divide it by the former, and the quotient will be the required specific gravity. Now there is nothing

more injurious to the progress of a student in any science than the acquirement of such rules by rote, without a thorough comprehension of the principles upon which they are founded; they may thus be rendered available for mere practical purposes, but they are utterly useless as steps in his advancement. For instance, how many thousand persons are there who can (to make use of a common expression) work the rule of three, without in the least understanding the doctrine of proportion upon which it is founded? With less trouble than it takes them to learn and recover the rule, which they are in constant danger of forgetting, they might attain to a knowledge of its principles, which would not only be of general application, but a help to farther improvement. A little consideration will be sufficient to render the principles of the above process perfectly clear. When a solid is wholly immersed in water, it obviously displaces a bulk of that liquid exactly equal to its own, which bulk was supported in its place by a pressure from the surrounding particles equivalent to its own gravity: this may be easily proved by the familiar illustration of a bucket in a well. When the full bucket is wholly immersed in the water, it requires scarcely any effort to draw it to the surface; but the moment it rises above the water, the weight of the water is felt, being no longer supported by the surrounding liquid. In the process, therefore, of taking specific gravities, the solid immersed must also be supported by the surrounding water with a force exactly equal to the weight of the water which it has displaced, and thus the difference of its weight in water from that of its weight in air must be the weight of an equal bulk of water. The direct comparison of the substance with water having been thus effected, a number bearing the same ratio to unity is easily found by the rule of proportion, and this will be the specific gravity.