414 



SGIBNGE. 



[N. S. Vol. II. No. 39. 



that the course of intellectual events takes at 

 the present moment." (p. 7.) The work ex- 

 hibits this forcibly and repeatedly. Thus, by 

 an extension of the principles employed by 

 Stevinus in the study of hydrostatics, the au- 

 thor deduces a proposition which is now readily 

 recognizable as a special case of Green's Theo- 

 rem. "We may accordingly," says Professor 

 Mach, "see into the force-system of a fluid in 

 equilibrium, or, if you please, see out of it, sys- 

 tems of forces of greater or less complexity, 

 and thus reach by a short path propositions 

 a posteriori. It is a mere accident that Stevinus 

 did not light on these propositions. The 

 method here pursued corresponds exactly to 

 his." (p. 109.) 



The process from special cases to general 

 principles is of course one of economy, and we 

 might expect that any opportunity thus to econ- 

 omize would be at once seized upon. Says the 

 author, ' ' economy of commvmication and of ap- 

 prehension is of the very essence of science," 

 and this economy, serving at first to satisfy 

 mere bodily wants, becomes later a potent fac- 

 tor in the development of science in its more 

 advanced and specialized forms. At many 

 points in the book we are reminded of this the- 

 sis, but almost immediately after it is stated we 

 are brought face to face with a feature in the 

 history of science that seems in contradiction 

 to it, for after recounting the points which 

 Archimedes, in beginning his study of equilib- 

 rium, assumed as self-evident, and then pre- 

 senting that philosopher's mode of establishing 

 the law of the lever, we are introduced to a 

 succinct statement of the different methods by 

 which Galileo, Huygens, Lagrange and others 

 demonstrated the same law. We may believe 

 that, in part, various philosophers produced 

 new demonstrations because they saw or thought 

 they saw fallacies in the reasoning of their 

 predecessors, but this, we think, is not the prin- 

 cipal reason. The fact is rather an illustration 

 of the other fact that, in olden times, a problem 

 once stated, existed, in the estimation of many, 

 for the purpose of bringing out all the solutions 

 thai could be found. Hence the multiplicity of 

 solutions to various problems as, for example, 

 the many proofs of Euclid's Forty-seventh. 

 There does not seem to be much economy of 



time or labor in this. Professor Mach recog- 

 nizes and condemns this tendency, calling it a 

 ' mania for demonstration in science.' It is a 

 fact that variety in the solvitions of problems in 

 mechanicsi led to the development of principles 

 not before recognized, and thus resulted in an 

 expansion of the science. This is shown by 

 Professor Mach where the generalization of the 

 principle of the lever by Leonardo da Vinci 

 brings into prominence the principle of statical 

 moments ; and in like manner other advances 

 are introduced, but it was not for the sake of 

 these, nor yet in the interest of economy, that 

 the new demonstrations were produced. • 



It is shown that the celebrated investigation 

 of the inclined plane by Stevinus virtually in- 

 volves th« principle of the parallelogram of 

 forces, and the principle itself is then stated 

 and the fact commented on that Varignon as 

 well as Newton determined it. The importance 

 of the principle in both statics and kinetics is 

 very properly recognized, but surely it scarcely 

 needs pointing out that the statement and con- 

 ception of the principle in connection with the 

 parallelogram at this day is not miost econom- 

 ical in mental labor or in manual application. 

 It accords well with the cumbersome form in 

 which many statements were made early in the 

 development of science, and ia their time the 

 forms were excusable, but that a writer should 

 continue to employ this principle now in the 

 form in which it was enunciated by Newton is 

 not an indication of any economical tendency. 

 For the science has got beyond that. So soon 

 as the idea is accepted that the result of several 

 forces acting simultaneously upon a particle is 

 the same, whether they are considered inde- 

 pendently of one another or collectively, the 

 graphic composition of the forces by vectorial 

 addition becomes at once the simplest and most 

 rational method. This, for two forces and their 

 resultant, gives the triangle and dispenses with 

 the parallelogram and diagonal idea altogether, 

 besides serving equally well for three forces in 

 equilibrium. As good a treatise on mechanics 

 can be produced to-day without any reference 

 to the parallelogram of forces as with it, and 

 such is now the tendency. If the idea of 

 ; economy of communication and of apprehen- 

 sion ' is to prevail we must carry out this ten- 



