If the Density of water be called 1, that is, if water be taken as a standard, to measure Density, then the Density of lead will be properly called 11, or 114.

In the same manner as it has been explained how the Density of lead is estimated with respect to the Density of water, the Densities of any other substances, whether solid or fluid, may be determined with respect to that of water.

9. Definition of " MEASURE OF FORCE".

In Statics a FORCE is measured by the weight which it would support. In other words, the amount of a Statical Force is expressed by stating the number of pounds it would support, if the Force were made to act directly opposite to the Force of Gravity.

Thus, if the weight of a body were P pounds, and it were prevented from moving towards the earth's surface by a hand placed beneath it, the resistance offered by the hand to the communication of motion (that is, the force exerted by the hand), would be P pounds; and if this same pressure were produced by the hand in any other direction, it would be described in the same manner, by saying that it was "equal to P pounds", or that it was P pounds". If, therefore, a Force be represented by P, it is meant that P is the number of pounds which the Force would support, on the supposition that the Force is made to act directly opposite to the Force of Gravity. In other words, P is the number of pounds, which the Force is just able to lift.

FORCES, in STATICS, also called PRESSURES.

10. In whatever direction a Force tends to produce motion, its magnitude, as has already been stated, is measured by the weight of the body which would exert the same effect to produce motion downwards, as the Force under consideration exerts in the line in which it endeavours to produce motion. And that such a method of measuring Forces is allowable appears from this consideration, viz, that the effect produced by the weight of a heavy body* may be made to take place in any direction whatever; horizontally, as in the case of a string being attached to an object lying on a table and kept stretched by a heavy body (W), which hangs over the edge o

By a heavy body', in MECHANICS, is simply meant a body acted on by the Force of Gravity.

the table, as in fig. (1); or vertically upwards, by passing the string

line BA, which is inclined at any angle to that (AW) in which it acts itself.

11. Definitions with respect to the action of FORCES.

(1) The point at which a Force acts upon a body is called the "point of application" of the Force.

(2) The line in which a Force, acting alone, produces, or tends to produce, motion, is called "the line of the Force's action"; and any line which is parallel to the line of a Force's action is said to be "in the direction of the Force's action", or "in the direction of the Force".

When the direction of a Force's action, (or, as it is generally called, "the direction of the Force",) is indicated by a line, either the very line is given in which the Force acts, or some line which is parallel to it. "The line of a Force's action", and "the direction of the Force", must by no means be confounded together. If the former be known, the latter is necessarily known also; but if only the latter be given, the precise line in which the Force produces, or tends to produce, motion, is uncertain; and all that can be said respecting it is, that the line of action of the Force is either that given line, or some other line which is parallel to it.

(3) If two or more Forces be applied to a body, or at some point, and no motion is produced, they are said to "counteract", or to "balance", one another, or to be “in equilibrium”.

12. Forces properly represented by geometrical straight lines.

Since lines may be drawn of any length, and in any direction, from a point, the lines in which Forces act, and the ratios which the Forces bear to one another, may be represented by drawing lines, which coincide with the lines in which the Forces act, and whose lengths bear to one another the same ratios that the Forces themselves bear to one another.

Among other advantages which attend this method of expressing the magnitudes and directions of Forces, the addition and subtraction of such Forces as act at a point in the same straight line are easily effected. Thus, if a certain Force act at A in the line AH, and AB be taken to represent it, and another Force, half as great as the former, act at A in the same direction, and also tend to move the body from A towards H, then, by taking BC equal to the half of AB, the line AC will represent the whole pressure at A, both with respect to the magnitude of that pressure, and to the line in which it acts.

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And, in like manner, if a Force equal to half the original Force AB act at A in the line AH, but tends to move the body at A from A towards K, half the pressure of the former Force will be counteracted by this new Force. Cutting off from the line AB, therefore, a part BD equal to the half of AB, the effective pressure still remaining will be properly represented by AD, with respect both to its magnitude, and to its line of action.

13. N.B. It will be gathered from the above, that a Force AB applied to A has not the same effect as a force BA applied at that point; for a Force AB would tend to move a body at A in the line KH towards H, but a Force BA would tend to move a body at A in the line KH from A towards K. It is not, therefore, indifferent whether the words "a Force AB”, or “a Force BA”, be used; since, though the two Forces represented by AB, and BA, are the same in magnitude, and also act in the same straight line, yet they tend to produce motions directly opposite to one another, the Force AB tending to move the body at A towards H, and the force BA tending to move the body at A towards K.

14. The effect produced at a point by any Force is the same at whatever point in its line of action the Force is applied, provided the latter point be supposed rigidly connected with the former.

Thus, if a body P be suspended by a string CP, the Force necessary to prevent P falling to the earth is found to be the same whether that Force be applied at A, or B, or C; the weight of the string being either neglected, or the weight of that portion of it which is supported along with the heavy body P, being counterbalanced. And although, in this case, the points A, B, C, are not, in fact, rigidly connected with one another, and with P, the result

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is the same as if they were, but in certain other cases the rigidity of the system is necessary.

15. DEF. If a string, fastened at one end, be pulled by a Force applied at the other end, the resistance to motion made by the string at any point in it is called the TENSION of the string at that point.

If the string be supposed to be without weight, it will follow, from Art. 14, that the tension at every point of it is the same; namely, the Force by which the string is pulled.

16. To recapitulate the substance of this Chapter.

(1) The precise amount of the Force, or pressure, with which any particular body endeavours to move towards the earth, is called the WEIGHT of that body. Art. 3.

This WEIGHT of a body is measured by comparing the tendency of the body to move towards the earth with that of some other given body (of a certain size and formed of a certain material) which is taken for a standard, and to which the name of a grain, an ounce, a pound, or a ton is given, as the case may be. Art. 4.

(2) MATTER is the substance of which all bodies are composed. It is found, by universal experience, to have a tendency to move towards the earth. Art. 6.

(3) THE QUANTITIES OF MATTER contained in different bodies (whether the bodies be great or small, rare like gas, or dense like lead) are proportional, (not equal,) to the weights of the bodies. Art. 6.

(4) The DENSITIES of different substances are proportional, (not equal,) to the weights of equal bulks of the substances. Art. 8.

(5) FORCE. I. Whatever be the cause which moves, or tends to move, matter, existing under any form whatsoever, it is called a

FORCE.

FORCES which prevent motion taking place,—that is, STATICAL FORCES,—are measured by the number of pounds they would support if they acted vertically upwards. Arts. 2, 9.

II. THE LINE OF A FORCE'S ACTION is the actual line in which the force tends to produce, or to prevent, motion. Art. 11. And if a Force be said to be P, it is meant that it is equal to P pounds.

III. THE DIRECTION OF A FORCE is indicated either by the line of its action, or by any line which is parallel to the line of its action.

Art. 11.

IV. The magnitudes of Forces, and the lines (or the directions) in which they act, may be represented by means of straight lines. Art. 12.

v. If a straight line AB represent a Force acting on a material point placed at A, and BA represent another Force acting on the same point, the two Forces AB and BA are equal, and tend to move the point in the same straight line, but in opposite ways from A. Art. 13.

VI. A Statical Force produces the same effect at whatever point in its line of action it is applied. Art. 14.

VII. To investigate the effects produced by a Force, there must be given :

1st. The magnitude of the Force;-which is known, if the number of pounds be known which the Force would support.

2nd. The point at which the Force is applied.

3rd. The line in which the Force acts,-or the direction; for knowing the direction of the Force, and the point it is applied at, the line in which the Force acts may be determined by drawing a line through the given point parallel to the given direction.

QUESTIONS ON CHAP. I.

(1) What are the two great divisions of Mechanics called, and how are they separately distinguished?

(2) What is Force? Give an example.

(3) Define Weight, and explain how it is reduced to numbers, so as to become the subject of calculation.

(4) Is the Weight of the same body at the same place invariable? Is the Weight of a body changed by changing its figure?

(5) How do you define Matter? and Quantity of Matter?

(6) How is the Quantity of Matter in a body ascertained and measured?

(7) Which has the greater Quantity of Matter, a feather-bed weighing 50 lbs., or a child of the same weight?

(8) What is Density? How is it reduced to numbers, and measured?

(9) If the weights of equal bulks of two substances are in the ratio 3 1, what is the ratio of their Densities?