Sept. 21, 18761 



NA TURE 



461 



requires us to be consistent in our logic ; and thus, if we find 

 anything else in the physical world whose quantity we cannot 

 alter, we are bound to admit it to have objective reality as truly 

 as matter has, however strongly our senses may predispose us 

 against the concession. Heat therefore, as well as light, sound, 

 electricity, &c., though not forms of matter, must be looked 

 upon as being as real as matter, simply because they have been 

 found to be forms of energy — which in all its constant mutations 

 satisfies the test which we adopt as conclusive of the reality of 

 matter. We sTiall find that this test fails when applied to force. 



But you must again be most carefully warned to distinguish 

 between heat and the mere sensation of warmth ; just as you 

 distinguish between the motion of a cudgel and the pain pro- 

 duced by the blow. The one is the thinq to be measured, the 

 other is only the more or less imperfect reading or indication 

 given by the instrument with which we attempt to measure it in 

 terms of some one of its effects. So that when your muscular 

 sense impresses on you the notion that you are exerting force as 

 in pushing or pulling, you ought to be very cautious in forming 

 a judgment as to what is really going on ; and you ought to 

 demand much farther evidence before admitting the objective 

 reality of force. 



Until all physical science is reduced to the deduction of the 

 innumerable mathematical consequences of a few known and 

 simple laws, it will be impossible altogether to avoid some con- 

 fusion and repetition, whatever be the arrangement of its various 

 parts which we adopt in bringing them before a beginner. But 

 when we confine ourselves to one definite branch of the subject, 

 all of whose fundamental laws can be distinctly formulated, there 

 need be no such confusion. Here in fact the mathematician has 

 it all in his own hands. He is the skilled artificer with his plan 

 and his trowel, and the hodmen have handed up to him all the 

 requisite bricks and mortar. 



[Prof. Tait then gives a quotation in support of this view.] 



Whether there is such a thing as force or not I shall consider 

 presently. But in the meanwhile there can be no doubt that it 

 is a convenient term, provided it be employed in one definite 

 sense, and one only. Let us then first see how it is to be cor- 

 rectly used. Here we cannot but consult Newton. The sense 

 in which he uses the word "force," and therefore the sense in 

 which we must continue to use it if we desire to avoid intellec- 

 tual confusion, will appear clearly from a brief consideration of 

 his simple statement of the laws of motion. 



The first of these laws is : Every body continues in its state of 

 rest or of uniform motion in a straight line, except in so far as it 

 is compelled by ifnpressed forces to change that state. 



In other word.«, any change, whether in the direction or in the 

 rate of motion of a body is attributed to force. Thus a stone 

 let fall moves quicker and quicker, and we say that a force (viz. , 

 the weight of the stone, or the earth's attraction for it) is con- 

 tinually acting so as to increase the rate of the motion. If the 

 stone be thrown upwards, the rate of its motion continually di- 

 minishes, and we say that the same force (the stone's weight) is 

 continually acting so as to produce this diminution of speed. So 

 far, none of you probably feels the least difficulty. But we have 

 got only half of the information on this point which Newton's 

 first law affords. You see the moon revolving about the earth, 

 and the earth and other planets revolving about the sun — ap- 

 proximately, at least, in circles. Why is this ? Their directions 

 of motion are constantly changing ; in fact, a curved line is 

 merely a line whose direction changes from point to point, while 

 a straight line is one whose direction does not change ; but to 

 produce this change of direction force is required just as much as 

 to produce change of speed. That is supplied by the gravitation 

 attraction of the central body of the system. The old notion 

 was that a centripetal force was required to balance the so-called 

 centrifugal force, it being imagined that a body moving in a 

 circle had a tendency to fly outwards from the centre ! Newton's 

 simple law exposes fully the absurdity of this. If a body is to 

 be made to move in a curved line instead of its natural straight 

 path, you must apply force to compel it to do so ; certainly not 

 to prevent it from flying outwards from the centre, about which 

 it is for the moment revolving. In fact, inertia means, not revo- 

 lutionary activity, but dogged perseverance, ;and just as you 

 must apply force in the direction of motion to change the rate of 

 mt)tion, so must you apply force perpendicular to the direction 

 of motion to change that direction. 



Newton's second law is now required : Change of motion is 

 proportiottal to the impressed force, and takes place in the direction 

 of the straight line in which the force acts. 



Mark here most carefully that this one simple law holds for all 



kinds of force alike. There is no special law for gravitation- 

 force and others for electric and magnetic forces. All are defined 

 alike, without reference to their origin. 



Motion, as Newton has previously defined it, is here used as a 

 technical scientific term for what we now call momentum. It is 

 the product of the mass moving into the velocity with which it 

 moves. '* Change of motion, " therefore, is change of momentum, 

 or the product of the mass of the moving body into its change of 

 velocity. Now a change of velocity is itself a velocity, as we 

 see by the science of mere motion — kinematics — the purely 

 mathematical science of mixed space and time. 



Newton's words, however, imply more than this. Of 

 course, the longer a given force acts, the greater will be 

 the change of momentum which it produces ; so that to 

 compare forces, which is the essence of the process of mea- 

 suring them, we must give them equal times to act — or, ia 

 scientific language, we must measure a force by the rate at which 

 it produces change of momentum. Rate of change of velocity 

 is called in kinematics acceleration. Thus the measure of a 

 force is the product of the mass of the body moved into the 

 acceleration which the force produces in it. This is the so- 

 called Vis matrix, or "moving force" of the Cambridge text- 

 books : — the so-called Vis acceleratrix, or "accelerating force," 

 being really no force at all, but another name for the kinematical 

 quantity acceleration which I have just defined. 



Unit force is thus that force which, whatever be its source, pro- 

 duces unit momentum in unit of time. If we employ British 

 units — unit of force is that which, in one second, gives to one 

 pound of matter a velocity of one foot per second. Here you 

 must carefully notice that a /tfMwa' of matter is a certain wajj or 

 quantity of matter. When you buy a pound of tea, you buy a 

 quantity of the matter called tea, equal in mass to the standard 

 pound of platinum. The idea of weight does not enter primarily 

 into the process. In fact, the use of an ordinary balance depends 

 upon one clause of Newton's law of gravitation — which tells 

 us that in any locality whatever, the weights of bodies are equal 

 if their masses are equal. The weight of a pound of matter 

 varies from place to place on the earth's surface — it depends on 

 the attracting as well as the attracted body. The mass of a body is 

 its own property. The earth's attraction for a body, or the weight 

 of the body, is a force which produces in it in one second, a 

 velocity which (in this latitude, and at the sea-level) is about 

 32*2 feet per second. So that, in Glasgow the weight of a pound 

 — which we take as our standard of mass — is rather more than 

 thirty-two units of force, or, what comes to the same thing, the 

 British unit of force is about the former weight of a penny letter 

 — half an ounce. 



Some people are in the habit of confounding force with 

 momentum. No one having [sound ideas of even elementary 

 mathematics could be guilty of this or any similar monstro- 

 sity. He would as soon, as Hopkins used to say, measure 

 heights in acres, or arable land in cubic miles. But to show to 

 a non-mathematician that it is really monstrous to confound 

 force and momentum, it suffices to change the system of units 

 employed in measuring them, when it will be found that, if 

 numerically equal for any one system of units, they are neces- 

 sarily rendered unequal by a mere change of the unit employed 

 for time. Now two things which are really equal to one another 

 must necessarily be expressed by the same numerical quantity 

 whatever system of units be adopted. Let us try then unit of 

 force and unit of momentum, as defined by pound, foot, second, 

 units : and see what alterations a common change of these fun- 

 damental units will make in their numerical expression. 



Unit momentum is that of one pound of matter moving with 

 a velocity of one foot per second. Unit force is that force 

 which, acting for one second, produces in unit of mass a velocity 

 of one foot per second. In each of these statements you may 

 put an ounce or a ton, instead of a pound, and an inch or a mile 

 in place of a foot, and their relative value will not be altered. 

 But suppose we take a minute instead of a second as the unit of 

 time. One foot per second is sixty feet per minute — so this 

 change of the time unit increases sixty-fold the nominal value of 

 the momentum considered. But in the case of the force our 

 statement would stand thus : — What we formerly called unit of 

 force is that which, acting for one-sixtieth only of our new unit 

 of time produces in a mass of one pound, sixty-fold the new unit 

 of velocity. In other words the number expressing the momen- 

 tum is increased sixty-fold, while that representing the force is 

 increased three thousand six hundred fold. 



In fact, whatever be the system of units you employ — it you 

 increase in any proportion the unit of time, the measure of a 



