1 64 HISTORY OF SCIENCE. 



obviate, namely, the resistance of the medium/' He goes on to state 

 that ihe resistance of the medium (the atmosphere) continually in- 

 creasing would at length prevent further acceleration of the velocity 

 of the moving body, which would continue its motion at a uniform 

 velocity. 



In a treatise, published in 1592, Galileo sets in the clearest light the 

 real advantages of machines, " which," he says, " I have thought it 

 necessary to do, because, if I mistake not, I see almost all mechanics 

 deceiving themselves in the belief that, by the help of a machine, they 

 can raise a greater weight than they are able to lift by the exertion of 

 the same effort without it. Now, if we take any determinate weight, 

 and any force, and any distance whatever, it is beyond doubt that we 

 can move the weight to that distance by means of that force, because, 

 even although the force may be exceedingly small, if we divide the 

 weight into a number of fragments, each of which is not too much for 

 our force, and carry these pieces one by one, at length we shall have 

 removed the whole weight ; nor can we reasonably say at the end of 

 our work that this great weight has been removed and carried away 

 by a force less than itself, unless we add that the force has passed 

 several times over the space through which the whole weight has gone 

 but once. From which it appears that the velocity of the force (under- 

 standing by velocity the space gone through in a given time) has been 

 as many times greater than that of the weight as the weight is greater 

 than the force ; nor can we, on that account, say that a great force is 

 overcome by a small one, contrary to nature : then only might we say 

 that nature is overcome when a small force moves a great weight as 

 swiftly as itself, which we assert to be absolutely impossible Avith any 

 machine either already or hereafter to be contrived. But since it may 

 occasionally happen that we have but a small force, and want to move 

 a great weight without dividing it into pieces, then we must have re- 

 course to a machine, by means of which we shall remove the given 

 weight with the given force through the required space. But, neverthe- 

 less, the force, as before, will have to travel over that very same space 

 as many times repeated as the weight surpasses its power, so that at 

 the end of our work we shall find that we have derived no other benefit 

 from our machine than that we have carried away the same weight 

 altogether which, if divided into pieces, we could have carried without 

 the machine, by the same force, through the same space, in the same 

 time. This is one of the advantages of a machine, because it often 

 happens that we have a lack of force but abundance of time, and that 

 we wish to move great weights all at once." This compensation of 

 force and velocity has been fancifully expressed by saying that Nature 

 cannot be cheated. The same principle is treated of in systematic 

 treatises on mechanics, under the name of " Virtual Velocities." 



Galileo explained the concords and dissonances of musical sounds 

 by the concurrence or opposition of vibrations in the air striking upon 



