mission, made it appear that this ratio of 1225 to 3844 is not even as accurate as that of 1 to 34, and that it gives a circumference which falls below the least limits of 34 and 314, the last of which is less than the polygon of 196 sides.

There is a quantity of other couples of numbers enjoying the properties deemed so wonderful by Leistner, and which gives a value nearer the circumference, as was shown by Lambert in his Beytrage or Memoires de Mathematiques, Vol. II, 1770, page 156. What Leistner thought he had found by a special favor of God can be found in a thousand ways by an analytical process. The Commission, on the report of Marinoni, rejected the discovery of Leistner, who, like his compeers, appealed from the judgment in a work entitled Nodus Gordius. It afforded Marinoni the occasion of publishing a work, where the process of determining the ratio of the diameter to the circumference is developed and expounded in a manner to convince any one but a man who thinks he has found the quadrature of the circle.

In 1751 a pastor or preacher of Kattembourg announced to the public this beautiful discovery, and soon after an inhabitant of Rostock enrolled himself as a volunteer for the same object. One of the two claimed that having entertained a Frenchman the latter had seen some of his papers and taken unfair advantage of the circumstance. Yet, about 1750, Henry Sullamar announced in England the quadrature of the circle, and found it in the number 666 of the Apocalypse; he published periodically every two or three years some pamphlet in which he endeavored to prop his discovery. Paris soon enjoyed a similar spectacle. In 1753 an officer in the guards, Sir de Causans, who until then had never had any suspicion of geometry, suddenly found the quadrature of the circle in having a circular piece of sod cut, and then, rising from truth to truth, explain by his quadrature original sin and the Trinity. He pledged himself by a public writing to deposit at a notary to the amount of 300,000 francs to wager with those who might wish to appear against him, and actually deposited 10,000 francs for the benefit of any who might show him his mistake. That was certainly not difficult; for it followed from his discovery that the square circumscribed about the circle was equal to it and, consequently, the whole to its part. Some persons undertook to win the 10,000 francs, among others a young lady sued him; some others, in answer to his challenge, deposited different sums of money with the notaries. But the king adjudged that the fortune of a man, who was really innocent, ought not

to suffer from such eccentricities of mind;

for, on every other subject, Sir de Causans was a very estimable man. The suit was quashed and the bets declared void. Yet the knight found the means of obtaining the judgment of the academy which considerately refused to speak, but which was finally compelled to give its opinion.

We shall pass more rapidly over the names of other discoverers of the quadrature of the circle. Fondee, of Nangis, found it, not by measuring curves by straight lines, but straight lines by curves. Liger filled les Mercurés with similar follies on the quadrature of the circle. He demonstrated it by the Méccanisme en plein des figures, which gave him, independently of the quadrature of the circle, the commensurability of the side of the square and its diagonal, by making out that 288 are equal to 289. Sir de Culant fell on the same discovery some years ago, and would also no doubt have discovered the quadrature of the circle had not death taken him away. La Frenaye, footman to the Duke of Orleans, spent twenty years in wandering from paralogism to paralogism, and in sifting the numbers 7, 8, and 9, which, according to him, contained the whole mystery of the quadrature. Clerget saw some contradiction in the relation more or less near of the diameter to the circumference given decimally, and had besides found the size of the point of contact of a sphere with a plain. Some years ago Maure used to weary all those that would listen to him by the recital of the injustice of geometricians and the Academie of Sciences. He intended to cross to England where he was sure to find more equitable judges in the Royal Society.

We must not forget one of the modern Quadrateurs who outdid many others in sanguineness and absurdity. It is Rohberger de Vausenville. The challenges he had made to the geometricians of all nations, even Turk and Arabian, as well as to all academies, the suit brought forward against the Academy of Sciences to secure for himself the capital of the prize established by Count de Meslay, his indecent attacks upon all geometricians who tried to enlighten or teach him, have made him famous among those who have followed this path.

His final theorem is that the square of the diameter is to that of the circumference, as 22 times the radius multiplied by the square root of 3 is to 432 times the radius. A more experienced geometrician would have said as eleven times the square root of three is to 216. That made the circumference of the circle whose radius is one equal to 3.36,

which differs from the known proportion even in the 2d figure. It would follow from the pretended discovery of Vausenville that the circumference of a circle would exceed the circumscribed polygon of 12 sides.

Even at the present time, citizen Tardi, an old engineer, applies to the institute, the Corps Legislatif and all the world, to show his quadrature. He is having pamphlets printed, but is waiting for the proceeds of subscription. We have also just received a print with the title: Final Solution of the diameter of the Circle to its Circumference, or the discovery of the Quadrature of the Circle, by Christian Lowenstein, Architect, Cologne, 1801. His method consists in applying to a great quarter of a circle a strip of iron and he finds the circumference to be 3.1426.

These publications come to us more especially in the Spring of the year, when fits of folly are more frequent, and cit. de la Land, who spent a year at Berlin, says it was the season when the Academy of Berlin received most writings of that kind.

We were, perhaps, wrong in dwelling so long upon these follies; we now pass to a more important article about this subject.

The impossibility of finding the quadrature of the circle was maintained by James Gregory, a Scotch geometrician, in a treatise entitled : Vera Circuli et hyperbola quadratura, Patav, 1664, in 4to, for he understood by the quadrature that which he obtained by approximation. One is disposed to think this quadrature impossible to the human intellect when the useless efforts of geometricians of all times are considered; I do not speak of the pitiable efforts of those we have just been discussing, but of the efforts of such modern geometricians as St. Vincent, Wallis, Newton, Leibnitz, Bernoulli Euler, etc., who have found new methods of determining the area of curves, and who, by their reasoning, have found that of a quantity of other curves less complicated in appearance than the circle, whereas the latter has always eluded their efforts.

Besides, a distinction must be made in this respect: there are two quadratures of the circle, one definite, the other indefinite. The definite quadrature is the one that would give the precise measure of the entire circle or of a given sector or segment, without giving indefinitely that of any sector or segment whatever. The indefinite quadrature, which would be the most perfect by giving the quadrature of any

part whatever, would evidently include the other. Scarcely any but the first is sought by quadrateurs in general.

The conviction is general that there is no demonstration absolutely convincing that the definite quadrature is impossible. Yet James Gregory claimed that he gave an irrefragable demonstration.

It rested upon the progressive course represented by the increase and decrease of the inscribed and circumscribed polygons whose limit is the circle itself. But this demonstration did not appear conclusive to Huygens, and it was the cause of a contest between these two geometricians which occupied the newspapers of the time. It must be admitted that though the reasoning is worthy of a head like that of Gregory, one of the forerunners of Newton, yet as the last limit of which he speaks is placed, so as to speak, in the mists of the infinite, the mind is not struck by an irresistible conviction. Still I would not put in the same category the assumed demonstration of this impossibility by Hanow. It is only a pitiable reasoning. An anonymous writer, some years ago, gave a little tract entitled: Demonstration of the incommensurability, etc. He claimed to have proved the impossibility of the quadrature of the circle. His calculations are exact, although more complicated than necessary; but it proves neither the incommensurability of the circumference and diameter, nor the impossibility of measuring the former; for a complication of incommensurable quantities does not prove demonstratively the incommensurability of the product or quotient. Two irrational quantities may, when multiplied together, give a rational quantity. The same is true of a larger number. A quantity may be composed of an infinite number of irrational quantities. and represent only a rational quantity. But citizen Legendre, at the end of his Geometry, edition of 1800, page 320, demonstrates that the ratio of the circumference to the diameter and its square are irrational numbers, and that had been already demonstrated by Lambert, Mein de Berlin, 1761.

An irrational quantity is susceptible of a geometrical construction. Thus, supposing the circumference to be irrational or incommensurable with the diameter, it could, nevertheless, be determined geometrically, and this would undoubtedly be to find the quadrature of the circle.

As for the indefinite quadrature, Newton seems to have demonstrated that no enclosed (fermée) curve continually receding upon itself, as the circle, is capable of it. (Princ. phil. nat. math. lib. I.; Lem. XXVIII, p. 106.) This demonstration is connected with the theory of angular

sections and of equations. I undertook, in 1754, to make it more plain and develop it more fully in my Histoire des Recherches, etc. I will have to refer to it and deem it convincing. Besides, although geometry presents numberless examples of squared curves, I know of none among the enclosed curves or curve continually receding (retournant) upon itself, that can be. Still D.'Alembert, in the fourth volume of his Opuscules, 1768, says, that he can scarcely assent to Newton's reasoning to prove the impossibility of the quadrature of the circle. I see, says he, that a similar course of reasoning, applied to the rectification of the cycloid, would lead to a false conclusion, the only difference, it seems to me, is that the circle is a receding curve and the cycloid is not. But I see nothing in Newton's reasoning which can be changed by disparity, more particularly, since the cycloid, if it is not a receding curve like the circle, is a continued curve whose sides (branches) are not separated; in a word, the reasoning of Newton rests solely on this supposition that in the circle an infinite number of areas corresponds to the same abscissa, whence he infers that the equation between the arc and the abscissa must be of an infinite degree, and consequently is not algebraically rectifiable; now, by applying his reasoning to the cycloid, I would infer that the equation between the abscissa and the corresponding are must also be of an infinite degree, and therefore the arc is not rectifiable algebraically, which is false. D.'Alembert made the calculation and concluded by saying, it seems to me that these reflections might deserve the attention of the geometricians and induce them to look for a more vigorous demonstration of the impossibility of the quadrature and of the indefinite rectification of oval curves.

We shall now give a brief account of the principal discoveries on the quadrature of the circle, as most of them are included among the geometrical discoveries already discussed in the former volumes, I will only give them here without going into details. Archimedes first discovered that the circumference is less than 348 or 34, and more than 310 times the diameter. Some of the ancients, as Appolonius and Philo, found nearer relations, but it is not known what they were. About 1585 Peter Metius, in impugning the false quadrature of Duchêne, gave his near ratio of 113 to 355. It was shown above how near he was right. About the same time Viete and Adrianus Romanus also published relations expressed decimally which came much nearer to the truth. Viete carried the approximation to 10 decimal places instead of 6, and taught besides several somewhat simple constructions which