anticipates the objection made by some of those who reject his account, for the reason that he could not precisely extract the square roots of the several numbers used in his calculation. But I have known some of these individuals, and I have never found a single one who knew Archimedes otherwise than by name. We still further know by the testimony of Simplicius, that Nicomedes and Appolonius had tried to square the circle; the first by means of the curve, which he calls quadrans or the quadratrix, the discovery of which, however, is usually ascribed to Dinostratos; and the second, by means of a certain line which he called the sister of the tortuous line, or the spiral, and which was nothing else but the quadratrix of Dinostratos. This quadratrix, invented in truth at first to divide the angle in any way whatever, would give the quadrature if its extreme limit on the radius could be found. Perhaps Appolonius or Nicomedes discovered this property; be this so or not, Eutocious tells us that Appolonius had carried further than Archimedes the close relation of the diameter to the circumference, and that another geometrician named Philo, of Gadares or Gades, had gone still further, so that the error did not exceed the 100000th. The moderns have carried this accuracy much beyond this point. Finally, among the ancients there were many of those persons unworthy the name of geometrician, who pretended to have found in different ways the quadrature of the circle. Jamblicus, cited by Simplicius, says so expressly. But their false reasoning has not reached us, and no doubt did not deserve it. The Arabs, who followed the Greeks in the Culture of the Sciences, must also have had their quadratures; but all we know about it is that some of them supposed they had discovered that the diameter being 1, the circumference is the square root of 10; a very grave error; for it exceeds 3.162, and the circumference, according to the account of Archimedes, is not quite 3.142857. For the rest we see in the catalogues of Arabian writings several works entitled de quadratura Circuli; like several others on the trisection of the angle, the duplication of the cube, etc. We pass rapidly over the centuries of ignorance which produced a few treatises on the quadrature of the circle, manuscripts left remaining in the dust of libraries, until we reach the period of the revival of let. ters among ourselves. About this time the famous CARDINAL DE CUSA distinguished himself by his unfortunate attempts at the solution of this problem. Nevertheless he tried an ingenious method; he rolled a circle on a plane or line, and supposing that its circumference was applied to it wholly until the point which had first touched it touched it again; he therefore justly inferred that this line would be equal to the circumference. He even conceived the outline of the curve, which the point that first touched the straight line was to describe which formed the curve, since called the Cycloid. But he supposed, with Charles de Bovelle, in the following century, that this curve was itself an arc of a circle, and from this he claimed to determine it by a geometrical construction which was entirely arbitrary, resting on no real property of this movement. He also tried another method, according to which he gave the following solution of the problem: a circle being given, add to its radius the side of the inscribed square, and with this line as diameter describe a circle, in which is inscribed an equilateral triangle, the perinuter of this triangle will, says Cardinal de Cusa, be equal to the circumference of the first circle. It was not difficult for Regiomontanus to prove that Cusa was mistaken; this relation of the circumference to the diameter fell outside the limits demonstrated by Archimedes; that is according to this relation the diameter would be to the circumference as one to a number greater than 34 already too large. Besides, the Cardinal learned for his age, though very much addicted to astrology, he presents in the collection of his works several geometrical tracts which are full of paralogisms. We have just spoken of Charles de Bovelle or Carolus Bovillus, distinguished at the time by the title of noble philosopher. He signalized himself by the strangest ideas. He gave in 1507 a work entitled : Introductionum Geometricum, translated into French and republished in 1552 under the auspices of Oronce Finée, under the title of Geometrie Pratique, Composée par le noble Philosopha, Maitre Charles de Bovelle, etc. He claims to give there the quadrature of the circle according to the idea of the Cardinal de Cusa, which, he says, came to him by seeing a wheel moving on the pavement. But the construction by which he pretends to give the length of the line to which is applied the circumference of the rolling circle is absolutely arbitrary, and it would follow that the diameter is to the circumference as 1 is to the square root of 10, or 3.1618, which is far from the limits of Archimedes. What is also singular, is that in this same book, and in an appendix added to the first volume of the preceding works, he speaks of the quadrature of the circle made by a poor peasant, according to which the circle having 8 for diameter is equal to the square having 10 for diagonal, that is to 50, which is false; for the circle is in this case less than 50%, and more than 5011, and the quadrature of Bovelle does not agree with that of the peasant, which he considers as true; for the latter gives the relation of the diameter to the circumference exactly as 1.0000 to 3.1250; the noble philosopher even wanders further from the truth than the peasant does below; and he might have been told that when one is mistaken he ought not at least to contradict himself. Bovelle says, falsely, that these relations coincide. Either he had not performed the calculation himself, or he did not know enough arithmetic to extract by approximation a square root; these works of Bovelle are pitiable; his manner of cubing the sphere is preeminently absurd. We are sorry to find in the same class a royal professor of the 16th century, named Oronce Finée, who, by his numerous works, acquired a kind of fame. He gave in his Protomathesis a quadrature of the circle, a little more ingenious, in truth, than that of Bovelle; but which is, nevertheless, a paralogism. On the point of dying, in 1555, he urgently advised his friend, Mizault de Monthuçon, to publish his discoveries, not only upon this subject, but also on the most famous problems of geometry, such as the trisection of the angle, and the duplication of the cube, and the inscription in the circle of all regular polygons. Mizault kept his word, and in 1556 published this assemblage of paralagisms under the title of De rebus Mathematicus hactenus desideratis, libri IV. Most of these problems are solved in various ways by him; it happens that his different solutions of the same problem do not agree with one another, nor with those of Bovelle, and of his rural geometrician which he had approved by publishing them, it was the height of false reasoning in geometry; consequently he was easily refuted by the geometrician Buteon, who had been his disciple at the College Royal, by Momus or Nunez, a Portuguese geometrician, and several others; but still he died satisfied, fully persuaded that his name would be placed on a level with those of Archimedes and Apollonius. This scandal was renewed among the royal professors in 1600, when Monantheuil, one of their number, published a quadrature of the circle. One Simon a Quercu (doubtless Duchêne or Van Eck) appeared on the arena a few years later, in 1585, and proposed a quadrature of the circle. His pretended discovery was much less wide from the truth than those of his predecessors and fell within the limits of Archime of this problem. Nevertheless he tried an ingenious to which he gave the following solution of the problem : It was not difficult for Regiomontanus to prove that C taken; this relation of the circumference to the diamete the limits demonstrated by Archimedes; that is accordin lation the diameter would be to the circumference as one greater than 34 already too large. Besides, the Cardinal his age, though very much addicted to astrology, he pre collection of his works several geometrical tracts which paralogisms. →s obliged to We have just spoken of Charles de Bovelle or Carolus Bo tinguished at the time by the title of noble philosopher. He himself by the strangest ideas. He gave in 1507 a work Introductionum Geometricum, translated into French and re in 1552 under the auspices of Oronce Finée, under the title c trie Pratique, Composée par le noble Philosopha, Maitre C. Bovelle, etc. He claims to give there the quadrature of the cording to the idea of the Cardinal de Cusa, which, he says, him by seeing a wheel moving on the pavement. But the cons by which he pretends to give the length of the line to whic plied the circumference of the rolling circle is absolutely arbitra it would follow that the diameter is to the circumference as 1 is square root of 10, or 3.1618, which is far from the limits of Archi What is also singular, is that in this same book, and in an api. added to the first volume of the preceding works, he speaks 11 sooner deny the most eleliensis Cana, who found no the first book of Euclid. tricians who had censured pt, especially Clavius who attack on the Gregorian aliger abuse is not reaScaliger, an eagle in Thomas Gephirander inder the tted that ght it is him. |