Professor Willis seems to be the first to have pointed out the proper basis of this interchangeability in cycloidal gearing. With the clearness which characterized all his work he states: “If for a set of wheels of the same pitch a constant describing circle be taken and employed to trace those portions of the teeth which project beyond each pitch line by rolling on the exterior circumference, and those which lie within it by rolling on its interior circumference, then any two wheels of this set will work correctly together.... The diameter of the describing circle must not be made _greater_ than the radius of the pitch-circle of any of the wheels.... On the contrary, when the describing circle is _less_ in diameter than the radius of the pitch-circle, the root of the tooth spreads, and it acquires a very strong form.... The best rule appears to be that the diameter of the constant describing circle in a given set of wheels shall be made equal to the least radius of the set.”[65] This practice is standard for cycloidal gearing to this day. In his “Principles of Mechanism,” Willis did the work on involute gearing which Hawkins set before himself; and also describes “a different mode of sizing the teeth” which had “been adopted in Manchester,” for which he suggests the name “diametral pitch.”[66] [65] Willis: “Principles of Mechanism,” Articles 114-116. London, 1841. See also “Transactions of the Institution of Civil Engineers,” Vol. II, p. 91. [66] Diametral pitch, which is credited to John George Bodmer, was long known as “Manchester pitch.” CHAPTER VII FAIRBAIRN AND BODMER With the improvement in machinery came improvement in millwork and power transmission. We quote in the next chapter Nasmyth’s description of the millwork of his boyhood.[67] Two of the mechanics most influential in the change from these conditions were Sir William Fairbairn and his younger brother, Sir Peter Fairbairn. They were born in Scotland but spent their boyhood in poverty in the neighborhood of Newcastle, in the same village with George Stephenson. [67] See page 85. Sir William Fairbairn went to London in 1811 and obtained work with the Rennies. The shop, however, was filled with union men who set their shoulders against all outsiders. After struggling for a foothold for six weeks, he was set adrift, almost penniless, and turned his face northward. He obtained odd jobs in Hertfordshire as a millwright, and returned again to London in a few weeks, where he finally found work and remained for two years, most of the time at Mr. Penn’s engine shop in Greenwich. In the spring of 1813 he worked his way through southern England and Wales to Dublin, where he spent the summer constructing nail-making machinery for a Mr. Robinson, who had determined to introduce the industry into Ireland. The machinery, however, was never set at work owing to the opposition of the workmen, and the trade left Ireland permanently.