50 



NATURE 



[November 15. 1S94 



physician the weak points of a demonstration ; it is a 



pity then that he seems unacquainted with Duchayla's 

 proof of the Parallelogram of Forces, which is unfor- 

 tunately so popular with writers on mechanics in this 

 country, as he would have revelled in pointing out the 

 weakness of a logic which prides itself above all things 

 on its rigour. 



The Principle of Virtual Velocities, employed as funda- 

 mental by Lagrange in his " M<5canique analytique " in 

 preference to the principle of the Parallelogram of 

 Forces, was enunciated very clearly by Stevinus in its 

 application to systems of pulleys ; and here we are com- 

 pelled to call attention to a flaw in Fig. 39, r, the only 

 one that we have met in the course of the work ; the 

 system cannot possibly be in equilibrium with the central 

 portion of the thread askew, as drawn in the diagram. 



The Principle of Virtual \'elocities is important in the 

 historical development of Mechanics as the first sketch 

 and shadowing forth of the modern Principle of the Con- 

 servation of Energy ; but it is unfortunate that the name 

 should still sur\-ive, as it is confusing and meaningless. 

 Prof Mittag Leffler was eloquent at the meeting of the 

 British .■Association at Oxford in his denunciation of the 

 habit of attaching to theorems certain names of indi- 

 viduals, real or quasi discoverers ; and he might have 

 qjoted the Principle of Virtual \'elocities as an instance 

 of the disadvantage of inventing a descriptive title of too 

 great generality to a newly discovered theorem. 



Lagrange has attempted an experimental verification 

 of the Principle of \'irtual X'elocities, and it is a tradition 

 that an apparatus was constructed on these indications by 

 a former professor of mathematics at Cambridge. It is 

 probable that the description of this demonstration and 

 of the apparatus to be found in Todhunter's Analytical 

 Sialics was purposely ironical ; and that, in popular 

 language, Todhunter wrote this with his tongue in his 

 cheek, knowing the story of the sceptical student who 

 had tried the experiment himself 



The demonstration amounts to proving that a certain 

 weight is no more likely to rise than to fall, and therefore 

 (here Todhunter says he follows Lagrange's words very 

 closely) the weight should remain stationary. The 

 student, however, found that the weight did not remain 

 stationary, and wanted to know why ; the professor told 

 him in confidence that it was prudent to make use of 

 an invisible pin to keep the weight in order. 



This is not the only case in which it is desirable for 

 the professor to keep a card up his sleeve, as the saying 

 is ; in the Foucault experiment of the pendulum which 

 shows the rotation of the Earth, the slightest current of 

 air will destroy and reverse the desired motion ; so that 

 it is advisable in showing the experiment to have an 

 elastic ball concealed in the palm of the hand, which can 

 send a slight current of air on the bob of the pendulum, 

 and thus accelerate the initial precession of the plane of 

 the vibration so as to gratify the eyes of the audience and 

 diminish their impatience at the slowness of the motion ; 

 .ifterwards the motion can be checked so that the total 

 advance is made to agree with the theoretical result. 

 Very undignified and dishonest, some will say ; but the 

 experiment is otherwise bound to fail from its delicacy 

 >¥hen shown to a large audience, except under the most 

 favourable conditions. 



NO. 1307. VOL. 51] 



While Statics, both as a Science and an Art, can be 

 traced back through Archimedes and the existing monu- 

 ments of the Eg>'ptians, Greeks, Romans, and of medieval 

 architects, the Principles of Dynamics, discussed in 

 chapter ii., were first laid down clearly by Galileo ; and 

 the great fallacy to be destroyed before any real advance 

 could be made was that of the Aristotelians, who main- 

 tained that heavy bodies fall faster than light ones, 

 because the upper parts weigh down on the under parts 

 and accelerate their descent. But in that case, retoried 

 Galileo, a small body tied to a larger body, must, if it 

 possesses in se the property of less rapid descent, retard 

 the larger ; ergo a larger body falls more slowly than a 

 smaller body. 



The entire fundamental assumption is wrong, as 

 Galileo says, because one portion of a falling body can- 

 not by its weight under any circumstances press another 

 portion; although, according to WohhviU (.Appendix I.), 

 even Galileo himself only very gradually abandoned the 

 Aristotelian conceptions. 



But discarding all metaphysical argument, the Aristo- 

 telian fallacy was demolished once for all by the 

 cxpcrimailum cruets carried out by Galileo, of letting 

 bodies of different weight fall from the Leaning Tower 

 of Pisa, when all were found to take the same time of 

 descent ; any slight discrepancies were afterwards 

 accounted for by the resistance of the air. 



A still more delicate experimental verification is to be 

 found in the pendulum, as pointed out by Galileo ; a 

 plummet at the end of a thread has the same period of 

 oscillation whatever be the weight or the material of the 

 plummet. 



The theory of the pendulum, when composed of a body 

 of finite size, as required in a clock, was completed by 

 Huygens in his Horfl/ot^iumoscillaloriiim, 1673, in which 

 the mutual controlling influence of a number of separate 

 plummets is investigated when the plummets are rigidly 

 attached together ; and thus for the first time the idea of 

 a moment of inertia and of a centre of oscillation was 

 introduced into Mechanics. In his further researches 

 into the theory of the clock, Huygens was led to the 

 discovery that isochronisin for all amplitudes can be 

 secured by making the phiiiimet oscillate in a cycloid ; 

 and to do this practically he found that the thread must 

 wrap and unwrap on an equal cycloid, and thereby he 

 made the first step in the doctrine of evolutes and the 

 theory of the circle of curvature. 



Having demolished the Aristotelian fallacies on falling 

 bodies, Galileo had still to determine the true laws ; his 

 first conjecture that the velocity grew uniformly with the 

 distance, or that v — gs, having proved untenable, 

 Galileo stumbled upon the true law that the velocity 

 grows at a constant rate, or that '' = gl. 



The next step in the theory, to prove that in conse- 

 quence s — hgl', was not an easy matter for Galileo, who 

 sought in general an experimental proof of his theorems 

 (as, for instance, his attempted quadrature of the cycloid 

 by weighing it made in sheet lead) ; once this was 

 established, however, it was comparatively an easy matter 

 to demonstrate that the path of an unresisted projectile 

 is a parabola, and to prove it experimentally by rolling a 

 ball obliquely on an inclined table. 



Galileo's difficulty was to measure lapse of time with 



