The never-ending character of these numerals, so far as they were tried, led to an early suspicion that the ratio must be really in commensurable. This was actually proved by Lambert (' Mem. Acad. Berlin' for 1761), and the demonstration has been given in an abridged form by Legendre, in the notes to his work on geometry. This demonstration is perfectly complete, and leaves no manner of doubt ou the subject. Those who persist in asserting that they can assign two numbers which are in the ratio of the circumference to the diameter, should first learn geometry and algebra enough to refute this proof, for they may depend upon it that no mathematician will lend them a moment's attention until this preliminary step has been taken. Bunn, and Panckouke, the editor of the' Encyclopedic Itlethodique,' have attempted to give metaphysical reasons for this incommensura bility, apparently In order that the squarcrs of the circle might not have all the nonsense on their side of the question.
Proof of the impossibility of the geometrical quadrature, as above described, was attempted by James Gregory, in 1668; and hlontucla see= to admit the proof at Last,* though he only said that it was very like demonstration in the first edition of his work on the history of this problem. The objections made by Huyghens to this proof, and the controversy which ensued, obliged Huyglione to admit that Gregory had succeeded in proving the impossibility of what is called the in definite quadrature of the circle, by which is meant the finding of a method of squaring any given sector of the circle whatsoever. But since it is well known that there are curves, particular portions of which may be squared, this may happen in the case of the circle. Thun it might be possible to give a geometrical rule for squaring the whole circle, even though the rule would not apply to every given sector. The proposition which Gregory imagines himself to prove, is that no sector of a circle can have to the circumscribed polygon a ratio expressible by a finite number of algebraical terms. The consequence of this, if established, must be drawn as follows :—Since geometry allows the use of nothing hut definite circles and Straight lines, a straight lino equal to the circle in length (which being found, the whole difficulty is overcome) must be ascertained, if at all, by a construction in which points are successively determined by the lute:sections of straight lines or circles, or a straight line and a circle. The most complicated construction in Euclid may, if we begin from the first principles, be reduced to the determination of a succession of points in this manner. Now, in the most complicated case of the intersection of two circles of given or found radii and centres, the points of inter section may be determined by formuhe derived from the roots of an semation not exceeding the second degree, the roots of which am be expressed in a fiuite form : the two ends of the line equal to the circle could therefore be assigned in a finite form, and hence the length itself. And the area of nny polygon (whose arcs are obtained by continual bisection) described about a given circle could also be ex premed in a finite form, from which (the area of the circle being expressible by means of its circumference) the ratio of the area to that of the circumscribed polygon would also be expressible. But if Gregory's proposition be true, this area cannot be expressed in n finite form ; neither then can any construction allowable in geometry attain the circumference of a circle.
The indefinite qiiadrature was shown to be impossible by Newton (Priucipia, book i., lemma 2S), in a manner which lies open to eulid objection ; we shall not therefore produce this proof. In fact, it seems as difficult to induce geometers to agree in any proof of the impossibility of the indefinite quadrature, as others to leave off trying their powers upon what geometers themselves have ceased to attempt.
Montucla has given n tolerable list of those who have signalised themselves by attempting this problem without the requisite pre liminary of studying geometry ; if preliminary that may be called, which would have made them give up the attempt. We shall pre sently mention some of them ; but first we have a list of those who were, most of them mathematicians, or, if not, known in other things. " Only prove to me that it is impossible," said some one, "and I will set about it immediately," and such seems to have been the general feeling of the quadrators, as Montucla calls them. They existed in crowds in the time of Archimedes; and the race is not yet extinct. One Bryso, a Greek, heads the list ; he made the circle a mean propor tional between the inscribed and circumscribed squares, which happens to he the content of the inscribed octagon. Next we have Cardinal Cusa, Orontius Fillet's, and Simon h Quercu, or Niche-lino, already mentioned. At the time when the problem really was of practical importance, every quadrator raised up an opposing mathematician; and the quadrature was sometimes so ingenious, and so near the truth, that it could only be opposed by new approximations to the truth. Thus Cusa was met by Itegiomentanus, Orontius by Buteo and Nonius, and Ditches:le by Peter Metius, who (it is said) was compelled to discover the very close approximation we have given under his name by that of Archimedes being insufficient to expose Duchesue. But inde pendently of what we have already said about Metius, it is not true that Duchesne's quadrature required any new accuracy of limits to expose it. He produced and Archimedes had already !shown that was too great Quadrators whose results forced further investigation were of use, and if some of them would now arise, no one would object to fallacies so ingenious that new truths must be discovered to oppose them. The celebrated Joseph Scaliger, a mere tyro in geometry, tried hie hand on this and other problems in 1592, and was met by Vieta, Adrianus Emmaus, and Clavius. Isongoniontanus, the astronomer (refuted by Pell), J. B. Porta, and Hobbes (refuted by Wallis), are three names well known in other pursuits who must go down to posterity as having had dis tinguished success in false quadrature. The works of the last against geometry and geometers were the consequence of the mortifucatiou he felt at not having been admitted to have succeeded in his attempt. Before his time however, Gregory St.. Vincent, an acute mathematician (to whom is due the discovery of the connectiou between the hyper bolic area and logarithms), had made the most elaborate attempt which ever was published, in his work on the quadrature of the circle (Antwerp, 1647). Such a challenger raised up Des Cartes, ltoberval, linyghens, and Leotaud, who soon despatched him.