As yet we have mentioned only mathematicians, or men of eminence in something : but we have also the Spaniard Falcon (15S9), who dialogues with the circle in verse: Gephirmider and Alphonse de I Molina, who attribute their discovery to inspiration; the Latter over turns Euclid, and found another who with willing to admit his dis coveries and translate them into Latin. A merchant in Rochelle, De la Len, not only found out the problem by inspiration, bit showed that the conversion of Jews, Turks, and Pagans depended upon it. Montucla gives some account of several other visionaries, the chief of whom is one Cluver, who found that the problem depended upon another, namely, " Construere mundum divine menti analoguin," which, if it be translateable, is, " to make a world analogous to the divine mind." He also mentions Richard White, an English Jesuit, who stands out among the solvers of the problem as the only one who ever was convinced of his error. The writer of this article once pointed out the example of White to another Roman Catholic clergy man, who had come from South America to England, to publish a quadrature of the circle. The party addressed seemed struck by the instance, and promised to study more geometry before he proceeded further; in a little while however he relapsed, and his work was advertised, and, we believe, published.
After the time of Newton, and the abundant means which were then introduced to complete the quadrature, if such a thing were possible, persons versed in mathematics seem to have dropped the attempt, and the reign of the quadrators by instinct commences. It is true that a serious diversion was made by the theory of gravitation, which drew off against itself many of those who should have been quadrators; but enough remained to furnish a tolerable list.
That of Montucla contains principally Frenchmen, though had the history of mathematics been written by an Englishman, he could have produced as great a number in this country. One Mathulon, in 1728, promised in print 1000 crowns to any one who should convict his solution of error, and was actually sentenced by the courts to pay the sum to the Hotel Dieu at Lyon, to which charity Nicole, the exposer, made over his claim. One Sullamar (as Montucla spells it),an English man, solved the problem by means of the number of the beast, 666, in the year 1750 ; a M. de Causans, in 1753, found it by cutting a piece of turf, and deduced from it the doctrines of original sin and of the Trinity. He offered to bet 300,000 francs on the correctness of his process, and deposited 10,000, which were claimed by several persons, and among others by a young lady, who brought an action for them : but the bet was declared void by the courts. Many more cases might be added ; it is however enough to say that this problem is non' never attempted (in print at least), except by those who are either altogether ignorant of mathematics, or add a most undue opinion of themselves to an acquaintance with only the elements. Since 1755, the Academy of Sciences has refused to examine any pretended solution; and the Royal Society in this country came to the same resolution a few years afterwards. In the announcement of their determination, the French Academy, without pronouncing any dictum upon the problem, stated that in a period of seventy years, those who had brought forward quadratures bad been persons who did not know the nature of the difficulty of the problem. An experience of about half that time has
satisfied us that the same thing may be said of our own day : and with this addition, that whereas the speculators who came before the Academy seem for the most part to have had an idea that their chance, if any, lay in the geometrical quadrature, the squarers of our time, in nine cases out of ten, announce the arithmetical quadrature, which has been proved to be impossible.
A few words may serve to prevent some one from making an attempt upon this enchanted castle; as it is supposed to be. When the diffi culty first began to be noticed, the circle stood alone among curves ; and so remarkable a distinction between this, the only curve then considered, and rectilinear figures, the only other figures then con sidered, could not but excite curiosity. Our position is now changed ; not only does the now well recognised distinction of commensurable and incommensurable prevent the circle from presenting anything peculiar to itself, but the curve is only one among an infinitely great number, many of which have been investigated and their properties examined. Consequently, with reference to the present state of mathematics, the problem analogous to that of squaring the circle is, " Given any curve whatsoever, to find its area." Now if the ingenuity which is guided by the love of investigating hidden things, should desire a field for its exertions, let it leave that of the circle, which has been cropped until it will yield no more, and, first acquiring sufficient mathematical knowledge, let it spend its force upon some one of the many real difficulties which abound, both in the pure and mixed sciences : let it investigate the meaning of divergent series for example, in all their varieties, or endeavour to extend the theory of discontinuous expressions, or solve the equations of motion of the solar system by some other method than that of series. For one point that should strike the lover of the marvellous in the quadrature of the circle, there are hundreds in the above-named subjects which surprise the mathe matician, however little he may possess that quality. Moreover, in like manner as the quadrature of the circle was at one time, in the lauds of Wallis, Newton, &c., a road to results which, though they did not attain their end, yet answered many other purposes ; so the efforts of the inquisitive on the actual difficulties of our own day may also end in the promotion of science of every kind, if begun in knowledge and directed by system. We owe the binomial theorem, now one of the most important results of algebra, indirectly to the learned attempts of 1Valli/3 upon the quadrature of the circle, at a time when such attempts were in season : and we might reasonably hope for collateral successes something like those resulting from the labours of Wallis, if those (not a few) whose minds compel them to inquiry into the curious, would but furnish themselves with a guide before they set out on their travels.