xxi.] THEORY OF APPROXIMATION. 461 



utterly inappreciable in a practical point of view, if the 

 "bar be a good stout one ; but in a theoretical point of 

 view they entirely prevent our saying that we have solved 

 a natural problem. The faculties of the human mind, 

 even when aided by the wonderful powers of abbreviation 

 conferred by analytical methods, are utterly unable to cope 

 with the complications of any real problem. And had 

 we exhausted all the known phenomena of a mechanical 

 problem, how can we tell that hidden phenomena, as yet 

 undetected, do not intervene in the commonest actions ? 

 It is plain that no phenomenon comes within the sphere of 

 our senses unless it possesses a momentum capable of 

 irritating the appropriate nerves. There may then be 

 worlds of phenomena too slight to rise within the scope of 

 our consciousness. 



All the instruments with which we perform our measure- 

 ments are faulty. We assume that a plumb-line gives a 

 vertical line ; but this is never true in an absolute sense, 

 owing to the attraction of mountains and other inequalities 

 in the surface of the earth. In an accurate trigonometrical 

 survey, the divergencies of the plumb-line must be ap- 

 proximately determined and allowed for. We assume a 

 surface of mercury to be a perfect plane, but even in the 

 breadth of 5 inches there is a calculable divergence from a 

 true plane of about one ten-millionth part of an inch ; and 

 this surface further diverges from true horizontality as the 

 plumb-line does from true verticality. That most perfect 

 instrument, the pendulum, is not theoretically perfect, 

 except for infinitely small arcs of vibration, and the 

 delicate experiments performed with the torsion balance 

 proceed on the assumption that the force of torsion of a 

 wire is proportional to the angle of torsion, which again is 

 only true for infinitely small angles. 



Such is the purely approximate character of all our 

 operations that it is not uncommon to find the theoretically 

 worse method giving truer results than the theoretically 

 perfect method. The common pendulum which is not 

 isochronous is better for practical purposes than the 

 cycloidal pendulum, which is isochronous in theory but 

 subject to mechanical difficulties. The spherical form is 

 not the correct form for a speculum or lense, but it differs 

 so slightly from the true form, and is so much more easily 



