give in to the plainest reasoning. He will sooner deny the most elementary propositions of geometry, like Moliensis Cana, who found no less than twenty-seven false propositions in the first book of Euclid. Scaliger replied with bitterness to the geometricians who had censured his quadrature; he treated them with contempt, especially Clavius who had already wounded him by an answer to his attack on the Gregorian Calender. Unfortunately for the honor of Scaliger abuse is not reasoning, and the established fact remains that Scaliger, an eagle in literature, was nothing in geometry.

Scarcely had Scaliger disappeared when one Thomas Gephirander came to take his place. But he had not Scaliger's pride; he acknowledged, even in the title of his book, that his discovery was simply the result of divine Grace. We shall see many others gifted with this same spirit of humility. The paralogism of Gephirander nevertheless was palpable; for it consisted in the pretension that if between two magnitudes there is any geometrical relation whatever, the same relation will exist if the same quantity is taken from each. Thus, according to this illuminate, the same relation exists between 2 and 5 as between 3 and 6, since only the same quantity, viz., unity is subtracted from each of these numbers. But scarcely any of the follies with which false reasoning, a false mind, and the conceit of never recanting one's errors inspire these visionaries have equaled those of Alph. Cano, of Molina, in a book entitled: Nuevos descubrimientos Geometricos. He remodels the whole of Euclid, and scarcely one of his propositions is spared by him. Yet who would believe it! he found another fool, named Janson or Jansen, who translated him into Latin under the title of Nova reperta Geometrica, etc. Moreover Cano admitted that he had not the least idea of geometry until the Deity, whose delight it is to humble the proud and enlighten the ignorant, had inspired him. A similar wiseacre presented at the same time in France his paralogisms upon the quadrature of the circle and the duplication of the cube. It was a merchant of Rochelle, called De Laleu. This one also pretended to have received the solution of these problems by divine revelation, and announced that the union of the Jews, Turks, and Pagans to the Christian religion depended upon the manifestation of this truth. In fact, according to him, the quadrature of the circle was the quadrature of the heavenly Temple, and the duplication of the cube that of the elementary, terrestrial, and aquatic altar, whence was to flow the conversion of the Jews, idolaters, etc. Accordingly some religion

ists, overexcited by meditation, dabbled with the matter, and the superior of the Jesuits even invited some able geometricians of that time, like Mydorge, Hardy, etc., to confer with Laleu. The result can easily be foreseen-it is impossible to reason with persons who have no principles in common with us. Hardy showed the incorrectness of the solutions in a manner satisfactory to all geometricians; for, as this Laleu gave several solutions of the same problem, Hardy made it appear that these did not even agree with each other. But Laleu, backed by Perjos, his book-keeper, and by a Scotchman named J. de Dunbar, did not give up the contest until after his death. We pass lightly on some other quadratures of the circle proposed by an anonymous writer of his time, and by one Benoit Scotto, whom St. Clair, royal professor, and Hardy refuted, to come to Longomontanus, who defiled, so to speak, the last years of his life, by his pretensions on the quadrature of the circle. That astronomer, formerly a disciple of Tycho Brahé, and known by a good work on astronomy, imagined, in 1622, that he had discovered the solution of this celebrated problem, which he published under the title of Cyclometria lunulis reciproci demonstrata, etc. He claimed to have found that the diameter is to the circumference as 1. is to 3.14185. In vain did Snellius, Henry Briggs, Guldin, warn him with moderation of his mistake, by showing him that the diameter being 1, the circumference is more than 3.14159, and less than 3.14160. But Longomontanus was not at all inclined to give in. He heaped a thousand bad reasons against the calculations of Vieta, Adrianus, Romanus, Ludolph, Vanceulen, Snellius, who were unanimously opposed to him. Soon he saw the quadrature of the circle in the mysterious. properties of the numbers 7, 8, 9, and of the proportion sesqui tertiam, or of 3 to 4; he spent the last years of his life in publishing new vagaries his different quadratures do not even agree together. The geometrician Pell attempted, about 1644, to set him right; he made. him see, by a calculation, without the extraction of any root, that his relation would make the circumference greater than the circumscribed polygon of 236: the stubborn and irritable old man died in 1647, persuaded that he alone was right against all.

About the same time a new pretender to the honor of squaring the circle appeared in France, in the person of one S. Oudart, of Ogen, author of a work entitled, Supplementum Supplementi Continens. He gave a geometrical construction quite ingenious, and which would, in fact, give a line equal to the circumference, if three points which he

supposes in a straight line were so in fact; besides, he did not deduce from it any numerical relation. His reasoning was about the same as that of Mallemant, of Messange, who, among many insignificant works, gave, in 1685, a quadrature of the circle, with a pretty rational history of the problem. He supposed that the three points were in a circular line, for which he had no warrant. They were not deemed worthy of a refutation; both might have been undeceived by making his construction only 2 feet in diameter.

The famous Hobbes appeared shortly after, about 1650, with his claims, not only to the quadrature of the circle, but also the rectification of the parabola, etc. His pretended solution having been refuted by Wallis, he took occasion to write against geometricians and geometry itself. Almost every year he wrote something new on this subject, and went always from paralogism to paralogism. One of his writings is entitled: Rosetum Geometricum, or the Geometrical Boquet. He abused geometricians a great deal, and Wallis in particular, and showed in several ways that his pretended discoveries were ridiculous.

Bertrand la Costa published in 1666, and again in 1677, a work entitled: Demonstration of the Quadrature of the Circle. But it was of no value and was treated with contempt by the French Academy of Sciences.

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Three other semi-mystical visionaries presented to the public some vagaries on the quadrature of the circle. One John Bachon, of Lyons, announced in 1657 his discovery by a work entitled: Demonstratio divini Theorematis Quadraturae Circuli, theologica, philisophica, Geometrica et Mecanicacum ratione quantitatum incommensurabilium. The geometrical quadrature would have been enough; and people may judge of the author from his mixing together all these different methods.

An anonymous writer proclaimed in 1671 that the reign of the greatest king of the universe was to be rendered illustrious by this most brilliant discovery, and undertook to prove it by a pamphlet in 4to, entitled: Demonstration of the Divine Theorem of the Quadrature of the Circle, and the relation of this theorem with the Visions of Ezechiel and the Revelation of St. John. He does not fail, after the example of his colaborers, to ascribe his discovery to a special favor of the Divinity, according to this passage of the Scriptures: Revelasti ea parvulis. In fact, there is found at the end of this work a large mysterious board, presenting on a common center four decreasing pyramids of circles and angles, which represent the Angelic Hierarchy.

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The third fool was named Dethlef Cluver, grandson or nephew of the celebrated geographer of that name. By ransacking the science of the infinite, on which he promised a great treatise, he finitely discovered that this problem, to find the quadrature of the circle, reduces itself to this one: to construct a world analogous to the divine intelligence: Construere Mundum divinae Menti Analogum; he promised to give geometrically and vigorously the solution of the first. Meanwhile, he (unsquared) dequarrail the parabola, and claimed that all that the geometricians had found on the curve figures was incorrect. (See Acta lipsiensia Julii, 1686, October, 1687). Leibnitz propounded doubtless to amuse himself some doubts on these vagaries. He wanted to pitch this Cluver against Nieuventit, who at the same time accumulated pitiable objections against the new calculations of the infinite that would have amused the geometricians. This trick did not succeed.

But among all the discoveries of the quadrature of the circle one is a kind of phenomenon, the only one who as yet admitted his mistake. It is Richard Albinus (White), an English Jesuit, author of a work entitled: Chrysespis sen Quadratura Circuli, in which he gives a false solution of the problem. But some friends opened his eyes, and he also acknowledged his error on the rectification of the spiral.

A better knowledge of geometry did not keep the 18th century from similar follies; there is not even a doubt that succeeding ages will resemble in this respect the past. In 1713, a Mr. G. A. Roerberg undertook to show that the circle is equal to the square of the side of an inscribed equilateral triangle; he did not perceive that it follows that the circumference would be exactly three times the diameter, or equal to the inscribed hexagon.

The solution of the three problems which have so long puzzled narrow intellects, the quadrature of the circle, perpetual motion, and the trisection of the angle was also announced in 1714 with much emphasis. The first discovery was that of one S. Daniel Wayvel, a Dutchman, and it was a palpable paralogism. From it followed that the diameter being 1. the circumference was, 3.142 exactly, which is altogether too much.

Usually the (quadrateurs) come off with seeing their discoveries neglected or scoffed at by their contemporaries; but in 1728, Mathulon, of Lyons, was more unfortunate. He announced to the learned world his signal discoveries of the quadrature of the circle and perpetual motion. He was so confident of his success that he appropriated 1,000

crowns for whoever would demonstrate to him that he was mistaken on either of these points. But Nicoli, then very young, demonstrated his error, and Mathulon admitted it; but he objected to the payment of this sum which Nicoli had abandoned to the Hôtel Dieu of Lyons. The matter was brought before the courts of that city, and the 1,000 crowns adjudicated to the poor.

In spite of this ill-success a new aspirant to the honor of squaring the circle was soon after seen. It was Basselin, professor of the University; his calculations were so complicated and so long that no one would have been willing to follow and verify them. But there is in such a case a means of detecting the mistake. Barben du Bourg, who later devoted himself to medicine, employed it by showing that the results of Basselin fell outside the known limits; for the rest, he was such a novice in geometry that he did not know that Archimedes had squared the parabola. Yet I have seen a beautiful Latin poem which celebrated the glory of Basselin and that of the college which his discovery had made illustrious.

The abbot Falconet, brother of the celebrated academician of that name, also published, about 1740, a little work in which he claimed to have discovered the quadrature of the circle. His process was less awkward than many others, but la Land, who was his friend, vainly endeavored, a few years later, to undeceive him.

Leistner, an officer in the service of the emperor, made more noise, and succeeded in having an imperial commission appointed to judge of the truth, real or assumed, of his discovery. He was, like many others, persuaded that in the series of numbers there are two which express the ratio of the diameter to the circumference, and that if the quadrature of the circle was not found it is because no one has been fortunate enough to put his finger on these numbers. In the first place that is a very false idea, since it has been demonstrated that there is no two numbers which express exactly the ratio of the side of a square to the diagonal; and it is also demonstrated that there is none which express the ratio of the diameter to the circumference, but Leistner thought he had found these highly favored numbers in the following: 1225 and 3844, for these numbers are two squares, and even prime to each other. They are derived from 35 and 31, which, according to Leistner, express the relation of the square to the circumscribed circle, and squaring them both and quadrupling the last they must express the relation of the diameter to the circumference. But Marinoni, reporter of the Com