Sir Charles Wheatstone devised, many years ago, a small and very efficient apparatus for the compounding of rectangular vibrations. A drawing, Fig. 186, and a description of this beautiful little instrument, for both of which I am indebted to its eminent inventor, may find a place here: _a_ is a steel rod polished at its upper end so as to reflect a point of light; this rod moves in a ball-and-socket joint at _b_, so that it may assume any position. Its lower end is connected with two arms _c_ and _d_, placed at right angles to each other, the other ends of which are respectively attached to the circumferences of the two circular disks _e_ and _f_. The axis of the disk _e_ carries at its opposite end another large disk _g_, which gives motion to the small disk _h_, placed on the axis which carries the disk _f_; and, according as this small disk _h_ is placed nearer to or further from the centre of the disk _g_, it communicates a different relative motion to the disk _f_. The nut and screw _i_ enable the disk _h_ to be placed in any position between the centre, and circumference of the larger disk _g_; and by means of the fork _j_ the disk _f_ is caused to revolve, whatever may be the position of the disk _h_. By this arrangement, while the wheel _k_ is turned regularly, the rod _a_ is moved backward and forward by the disk _e_ in one direction, and by the disk _f_, with any relative oscillatory motion, in the rectangular direction. The end of the rod is thus made to describe and to exhibit optically all the beautiful acoustical figures produced by the composition of vibrations of different periods in directions rectangular to each other. A lever _l_, bearing against the nut _i_, indicates, on a scale _m_, the numerical ratio of the two vibrations.[81] I close these remarks on the combination of rectangular vibrations with a brief reference to an apparatus constructed by Mr. A. E. Donkin, of Exeter College, Oxford, and described in the “Proceedings of the Royal Society,” vol. xxii., p. 196. In its construction great mechanical knowledge is associated with consummate skill. I saw the apparatus as a wooden model, before it quitted the hands of its inventor, and was charmed with its performance. It is now constructed by Messrs. Tisley and Spiller. SUMMARY OF CHAPTER IX By the division of a string Pythagoras determined the consonant intervals in music, proving that, the simpler the ratio of the two parts into which the string was divided, the more perfect is the harmony of the sounds emitted by the two parts of the string. Subsequent investigators showed that the strings act thus because of the relation of their lengths to their rates of vibration.