face in the shape of a crescent, bounded by two arcs and exactly equal to a given square. He also found two unequal lines which were together equal to a rectilineal figure, so that if their relation could have been found the solution of the problem would have been obtained. But this no one has yet been able to do, nor is it likely ever to be done. We are also indebted to Simplicius for the history of a disciple of Pythagoras, named Sextus, who claimed to have solved the problem, but his reasoning has not been transmitted to us. Finally, this inquiry at that early day became so famous that Aristophanes in ridiculing Meton makes him appear on the stage, in his "Comedy of the Clouds," as promising to square the circle about 430 years before Christ. This was all the more amusing from the fact that people generally suppose that to attempt to square the circle is the same thing as trying to make a circle square, which implies a manifest contradiction. Yet this was that Meton, so renowned for his discovery of the cycle of nineteen years, of whom, together with Socrates, the comedian made a public laughing stock.

Aristotle mentions two of his contemporaries, Bryson and Antiphon who worked at the quadrature of the circle. Nothing could have been more grossly inaccurate than Bryson's pretended quadrature; for he made the circumference of a circle equal to 32 times the diameter. But Antiphon stated that having inscribed a square in a circle, if an isoceles triangle having the chord for the base be inscribed in each of the remaining segments, and similar triangles in the remaining eight segments, and so on, the sum of all these rectilineal figures would be equal to the circle; nothing could be more true than this, and Aristotle was undoubtedly wrong in calling Antiphon a paralogist, for one of Archimedes' two quadratures of the parabola depends upon the same operation; but this method has not yet succeeded with the circle.

It might be supposed that Archimedes applied himself to the solution of this problem, and that he gave his approximate measure of the circumference of a circle only for want of the long-sought-for vigorously exact measure. His discoveries on the spiral, if they have preceded his book on the dimensions of the circle, might well inspire him with the hope of finding the length of the circumference. However, that may be, Archimedes showed, about the year 250 before Christ, that if the diameter of a circle is 1, its circumference is less than 310 or 34, and more than 310; by taking 34 the error is less than the 47 of the diameter. The calculation of Archimedes is singularly skillful, and

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 letters 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

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des. So Peter Metius, who undertook to refute him, was obliged to seek for a closer relation of the diameters to the circumference, and found that the one was to the other as 113 to 355. The pretended quadrature of Duchêne could not stand this test, and must be named only because it led to the curious and elegant discovery of Metius; for this relation of 113 to 355, reduced to decimals, is the same as 10000000 to 31415929; which is at the most but 10,000,000 of the diameter in excess. The diameter of the earth being only 6542816 (toises) or 13936912 yards, the error made by this relation of the circumference of a circle of that size would scarcely be 2 (toises) or 4 yards. If those who connect in their minds the problem of the quadrature of the circle with that of longitudes, knew what we have just said they would soon see their mistake; for, if these problems were connected with each other, what would be an error of 4 yards on a track around the earth? The Spanyard, Sir Jaime Falcon, of the order of Our Lady (Notre Dame), of Montesa, published in 1587, at Antwerp, his paralogism on the quadrature of the circle. His book is rendered amusing by a dialogue in verse between himself and the circle, which thanks him very affectionately for squaring it; but the good and model knight ascribes all the honor thereof to the holy patroness of his order. The paralogism was apparently so gross that no one took the trouble to refute it.

But a man much more famous than the foregoing challenged the attention of learned Europe by his pretensions on the quadrature of the circle; it was the celebrated Joseph Scaliger. Full of self conceit, he supposed that he had only to present himself on the field of geometry and that nothing that had baffled geometricians until then could resist a man of letters with his powers. He therefore undertook to find the quadrature of the circle, and put forward, with much braggadocia, his discoveries on this subject in a book which appeared in 1592: Nova Cyclometria; but he had no cause to congratulate himself on having thus wished to place himself among geometricians. For he was refuted by Clavius, Viete, Adrianus Romanus, Christman, etc., who showed each in his own way that the size which he assigned to the circumference of the circle was only a little less than the inscribed polygon of 192 sides; which being absurd, demonstrated the incorrectness of Scaliger's reasoning; but he did not surrender; and never did a man who thought he had discovered the quadrature of the circle, the trisection of the angle, the duplication of the cube, or perpetual motion,