Peirce was at the same time calculating the planet's perturbations. The approximate results of each furthered the computations of the other, so that within eighteen months from the discovery of the planet these two Americans had attained a remarkably accurate statement of its theory.
In conjunction with Prof. A. D. Bache, Superintendent of the Coast Survey, Walker developed the method of determining differences of longitude by telegraph. "What was the separate share of each of these two men in this work will probably never be known, for each ascribed the chief merit to the other. One feature introduced by Walker was the application of the method of coincidence of beats to the comparison of timekeepers one indicating mean, the other sidereal time at the two ends of a telegraphic line. These beats were signalized from one station to the other by taps of an observer upon the telegraph key. Such signals are, of course, subject to the errors that always attend the action of human nerves and muscles, so the next problem was to make the clock give its own signals. Two methods had been proposed, but there were fears groundless they have since been proved that either of these would injuriously affect the running of the clock. Mr. Walker sought diligently for some apparatus that would not arouse any such fears. He propounded the problem to several astronomers, and two or three contrivances were devised for the purpose.
This mode of observation and the apparatus invented to meet its requirements proved valuable not alone for determinations of longitude, but also for all other astronomical observations requiring minute precision in the determination of time. The mental effort required of the observer being reduced to a minimum, many more transits could be observed at a single meridian passage. Walker immediately modified the transit instrument to suit the new requirements, and, instead of five, seven, or at most nine threads, he provided it with several tallies of five threads each. There remained but one requisite to complete the American method of observation. This was some mechanical contrivance for securing a uniform rotary motion of the record sheet. It had not been attained when Walker died, although some progress toward the solution of the problem had been made.
It is proper for the biographer to point out the share which Walker personally had in this series of inventions, although he was far from making any such claims for himself. With a fine comradeship he was jealous only for the credit of the organization of which he was a member the United States Coast Survey. Speaking to the American Association for the Advancement of Science, Walker said: "With the single exception of the experiment between Baltimore and Washington, in 1844, I know of