་ 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 num 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 eyeloid, 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 are 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 arc 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 31% 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 gave the value of the circle, or the circumference to within a few millionths. Adrianus Romanus carried the approximation to 17 figures. But all that is far below what was done by Ludolph Van Ceulen, and which he published in his book de Circulo et adscriptis, of which Snellius published a Latin translation, at Leyden, in 1619. Ceulen, assisted by Petrus Cornelius, found with inconceivable labor a ratio. of 32 decimals; see V. II, p. 6. Snellius found the means of shortening this calculation by some very ingenious theorems, and if he did not excel Van Ceulen he verified his result, which he put beyond attack. His discoveries on this subject are found in the book entitled Willebrordi Snelli Cyclometricus de Circuli dimensione, etc. Descartes also found a geometrical construction which, carried to infinity, would give the circular circumference, and from which he could easily deduce an expression in the form of a series. (See his Opera posthuma.) Gregoire de Saint-Vincent is one of those who are most distinguished in this field; true, he claimed incorrectly to have found the quadrature of the circle and of the hyperbola, but the failure in this respect was preceded by so great a number of beautiful geometrical discoveries, deduced with much elegance according to the method of the ancients, that it would have been unjust to have placed him among the paralogists we have mentioned. He announced, in 1647, his discoveries in a book entitled: Opus Geometricum quadraturae Circuli et Sectionem Coni libris, X, Comprehensum. All the beautiful things contained in this book are admired; only the conclusion is impugned. Gregoire de Saint-Vincent lost himself in the maze of his proofs which he calls proportionalities, and which he introduces in his speculations. It was the subject of quite a lively quarrel between his disciples on the one hand, and his adversaries on the other, Huygens, Mersenne, and Leotand, from 1652 to 1664. If that skillful geometrician had not been mistaken, it would only have followed from his investigations that the quadrature of the circle depends upon logarithms, and consequently on that of the hyperbola. That would still be a handsome discovery, but it did not even have that advantage. This furnished Huygens the occasion of divers investigations on this subject. He demonstrated several new and curious theorems on the quadrature of the circle: Theoremata de quadratura hyper, ellipsis et Circuli, 1651; De Circuli Magnitude inventa, 1654. He gave several methods of approaching his quadrature much shorter than the usual way. He demonstrated a theorem which Snellius had taken for granted. There are also many very simple geometrical constructions which give lines singularly near any given area. If, for example, the arc is 60° the error is scarcely oooth. James Gregory distinguished himself in this controversy, and whatever may be our opinion of his demonstration of the impossibility of the definite quadrature of the circle, he can not be denied the authorship of many curious theorems on the relation of the circle to the inscribed and circumscribed polygons, and their relation to each other. By means of these theorems he gives with infinitely less trouble than by the usual calculations, and even those of Snellius the measure of the circle and of the hyperbola (and consequently the construction of the logarithms) to more than twenty decimal places. Following the example of Huygens, he also gave constructions of straight lines equal to arcs of the circle, and whose error is still less For example, let the chord of the arc of a circle be a, the sum of the two equal inscribed chords equal to b, then make this proportion: A+B: B:: B: C; if you take the following quantity, it does not exceed the 30th for a semicircle, and for 120° it would be less than finally the error for a quarter of a circle will not be goobooth. 8c+8 B-A 15 booth; The discoveries of Wallace, found in his Arithmetica infinitum, published in 1655, lead him to a singular expression of the relation of the circle to the square of its diameter; it is a fraction in this 3X3X5X5X7X7×9×9×11x11 form, 2X+X+X6X6X8X8X10X10×129 etc. This fraction, carried to infinity, expresses exactly the above relation, Arithmet. infinit., prop. 191; but if we confine ourselves, as is necessary, to a finite number of terms we have alternately a relation greater or less than the true one according as we take an odd or an even number of terms of the numerator and denominator. Thus gives too great a relation, and 3X3 give too small a relation. The frac 5.5.7 2.4.4.6.6.8 2.4 2.4. tion 337 is too small, and 335577 too great. But to bring the one near to the other, Wallis directs to multiply this product by the square root of a fraction formed by the unity plus unity, divided by the last figure which ends the series. Then the product, although much nearer, will be too large if the figure is the last of the numer ator, and too small if the last of the denominator. The values of 8:9:5:5√ 1+3; 3:2:¦§§; √1+7; 3:4:§:8:7:31 ́ 1+7; 3:3:9:5:7:3 .3. 6.6.8 3.3.5. |