And it is related that in the year 1788, one of these deluded individuals, a M. de Vausenville, actually brought an action against the French Academy of Sciences to recover a reward to which he felt himself entitled. It ought to be needless to say that there never was a reward offered for the solution of this or any other of the problems which are discussed in this volume. Upon this point De Morgan has the following remarks: "Montucla says, speaking of France, that he finds three notions prevalent among the cyclometers [or circle-squarers]: 1. That there is a large reward offered for success; 2. That the longitude problem depends on that success; 3. That the solution is the great end and object of geometry. The same three notions are equally prevalent among the same class in England. No reward has ever been offered by the government of either country. The longitude problem in no way depends upon perfect solution; existing approximations are sufficient to a point of accuracy far beyond what can be wanted. And geometry, content with what exists, has long pressed on to other matters. Sometimes a cyclometer persuades a skipper, who has made land in the wrong place, that the astronomers are in fault for using a wrong measure of the circle; and the skipper thinks it a very comfortable solution! And this is the utmost that the problem ever has to do with longitude." In the year 1775 the Royal Academy of Sciences of Paris passed a resolution not to entertain communications which claimed to give solutions of any of the following problems: The duplication of the cube, the trisection of an angle, the quadrature of a circle, or any machine announced as showing perpetual motion. And we have heard that the Royal Society of London passed similar resolutions, but of course in the case of neither society did these resolutions exclude legitimate mathematical investigations--the famous computations of Mr. Shanks, to which we shall have occasion to refer hereafter, were submitted to the Royal Society of London and published in their Transactions. Attempts to "square the circle," when made intelligently, were not only commendable but have been productive of the most valuable results. At the same time there is no problem, with the possible exception of that of perpetual motion, that has caused more waste of time and effort on the part of those who have attempted its solution, and who have in almost all cases been ignorant both of the nature of the problem and of the results which have been already attained. From Archimedes down to the present time some of the ablest mathematicians have occupied themselves with the quadrature, or, as it is called in common language, "the squaring of the circle"; but these men are not to be placed in the same class with those to whom the term "circle-squarers" is generally applied.