The art of accurately dividing circular arcs into the minute equal parts which serve as the units of astronomical measurement, remained, during the whole of the eighteenth century, almost exclusively in English hands. It was brought to a high degree of perfection by Graham, Bird and Ramsden, all of whom, however, gave the preference to the old-fashioned mural quadrant and zenith-sector over the entire circle, which Römer had already found the advantage of employing. The five-foot vertical circle, which Piazzi with some difficulty induced Ramsden to complete for him in 1789, was the first divided instrument constructed in what may be called the modern style. It was provided with magnifiers for reading off the divisions (one of the neglected improvements of Römer), and was set up above a smaller horizontal circle, forming an "altitude and azimuth" combination (again Römer's invention), by which both the elevation of a celestial object above the horizon and its position as referred to the horizon could be measured. In the same year, Borda invented the "repeating circle" (the principle of which had been suggested by Tobias Mayer in 1756[341]), a device for exterminating, so far as possible, errors of graduation by _repeating_ an observation with different parts of the limb. This was perhaps the earliest systematic effort to correct the imperfections of instruments by the manner of their use. The manufacture of astronomical circles was brought to a very refined state of excellence early in the nineteenth century by Reichenbach at Munich, and after 1818 by Repsold at Hamburg. Bessel states[342] that the "reading-off" on an instrument of the kind by the latter artist was accurate to about 1/80th of a human hair. Meanwhile the traditional reputation of the English school was fully sustained; and Sir George Airy did not hesitate to express his opinion that the new method of graduating circles, published by Troughton in 1809,[343] was the "greatest improvement ever made in the art of instrument-making."[344] But a more secure road to improvement than that of mere mechanical exactness was pointed out by Bessel. His introduction of a regular theory of instrumental errors might almost be said to have created a new art of observation. Every instrument, he declared in memorable words,[345] must be twice made--once by the artist, and again by the observer. Knowledge is power. Defects that are ascertained and can be allowed for are as good as non-existent. Thus the truism that the best instrument is worthless in the hands of a careless or clumsy observer, became supplemented by the converse maxim, that defective appliances may, through skilful use, be made to yield valuable results. The Königsberg observations--of which the first instalment was published in 1815--set the example of regular "reduction" for instrumental errors. Since then, it has become an elementary part of an astronomer's duty to