MECHANICS. 



would be deterred from encountering. 

 To avoid this, it is usual, in the first in- 

 stance, to omit the consideration of these 

 obstructions to the action of the power, 

 and to consider a machine as free 

 from them. Surfaces are considered 

 as perfectly smooth, cords as perfectly 

 flexible, bars and levers as perlectly in- 

 flexible, and so on. Although these 

 suppositions are absolutely ialse, yet 

 they are found to be, in the end, the 

 shortest road to truth. For having de- 

 termined what would be the relation 

 be; ween the power and weight in any 

 machine, were there no friction or rigi- 

 dity, it will be easy to correct the result 

 when the effects of these forces are sub- 

 sequently ascertained, and the process 

 is found to be, on the whole, not only 

 more simple arid intelligible to the stu- 

 dent, but more expeditious in actual 

 practice by taking this course, than if 

 the real state of the machine were taken 

 under consideration in the first instance. 

 These observations apply more or 

 less to every part of physical science. 

 The results which we obtain are rather 

 to be considered as constant approxima- 

 tions to truth, than truth itself. In our 

 first essays, false suppositions are ever 

 mixed up with true ones, and our 

 first conclusions are more or less 

 tainted with the errors of the source 

 from which they flow. Being, however, 

 aware of the deviations from the truth 

 in our primitive hypotheses, we are 

 enabled to perceive the consequences 

 which they produce and the errors 

 which they entail upon our results, and 

 we gradually remove these errors as 

 they are detected, and our conclusions 

 thus constantly come nearer to that 

 truth which is the great end of all our 

 researches. Thus the progress of the 

 mind in the acquisition of the know- 

 ledge of physical science resembles that 

 of an artist in the production of a pic- 

 ture or statue; the first rude attempt 

 bears but a remote and uncouth resem- 

 blance to the original, while every suc- 

 cessive stroke of the pencil or the chisel 

 removes some deviation from perfect 

 similitude, and the work gradually 

 approximates to a faithful copy of na- 

 ture. 



(5.) Viewing a machine, then, divested 

 ot those considerations to which we have 

 alluded, the problem which first presents 

 itself is the determination of the power, 

 which by its means would be capable of 

 supporting a given weight. Now it 

 happens, that notwithstanding the great 



variety of machines which have actually 

 been constructed, and the infinitely 

 greater variety which it is possible for 

 human invention to produce, there is 

 one great principle, simple in itself and 

 easily intelligible, which applies indiffe- 

 rently to all, and by which the power, 

 which is capable of supporting a given 

 weight, may be determined. The power 

 being connected with the weight so as to 

 act upon it by means of the machine, if 

 any motion be given to it, the weight will 

 receive a corresponding motion, and a 

 certain proportion will be found to sub- 

 sist between the velocity with which the 

 power descends in the vertical direction, 

 and that with which the weight ascends 

 in the vertical direction ; which propor- 

 tion depends entirely on the nature and 

 construction of the machine. But what- 

 ever proportion this may be, in order 

 that the power may be capable of sus- 

 taining the weight, it is only necessary 

 that it should have to the weight the 

 same ratio as the velocity of the weight 

 just mentioned has to the power ; or, to 

 express the same condition in other 



Words, THE POWER MULTIPLIED BY 

 THE SPACE THROUGH WHICH IT MOVES 

 J.Y THE VERTICAL DIRECTION MUST BE 

 EQUAL TO THE WEIGHT MULTIPLIED 

 BY THE SPACE THROUGH WHICH IT 

 MOVES IN THE VERTICAL DIRECTION.* 



This great principle, which is known 

 under the name of " the principle of 

 virtual velocities," may be considered as 

 the golden rule of mechanics. Indeed, 

 we may say that it implicitly contains 

 the whole science, statics and dyna- 

 mics ; and equally includes the resolu- 

 tion of all problems respecting bodies 

 and systems of bodies in equilibrium 

 and in motion: for it applies immedi- 

 ately and most evidently to all questions 

 respecting equilibrium or statical pro- 

 blems, and by means of another princi- 

 ple, known by the name of D'Alembert's 

 minciple, the whole region of dynamics 

 is brought under its dominion. We 

 cannot pretend, in a short popular trea- 

 tise like the present, to make the full 

 value of this principle apparent, nor 

 even to offer a general demonstration of 

 it ; not because it is incapable of rigor- 

 ous proof, nor because its results are 

 few or unimportant, but because its 

 general proof requires the aid of alge- 

 braic investigations of too difficult a 



* The principle of virtual velocities is much 

 more general tban that which is announced above ; 

 but we bhall not have Jp apply it ip its full gene- 

 rality. 



