QUADRATURE OF THE fli:i l.K 



QUADRATURK (>K TIIK CHBOUL 



RTJ 



Them excesses of calculation are useful in showing the way, ami in 

 destroying that brlk-f iu the iui|>nctical>ility of XMndfctaf what has 

 bean done which has retarded the progress of many subjects. The 

 607 decimal places give no guftkient notion i>f tli<-ir iiniotint of accu- 

 racy; or rather, give no Milli.-ient notion of the impossibility of pLtcing 

 the amount of accuracy before the imagination. But the following 

 illustration has a tendency to do what is wanted. 



The blood-globules of sonic nninmlcules are a millionth of an inch 

 in diameter. Let there be an inhabited globe no large that our great 

 globe U but fit to be a blood-globule in the body of one of its animal- 

 cules; and call this the first globe abort ut. Let the tecond globe 

 bore us be so large that the tint globe is but as one of the blood- 

 globules in it ; and so on to the tireittittk globe above us. Next, let 

 one of the blood-globules on our globe be an inhabited globe, with 

 . ver\ thing in proportion ; and let this be the first globe Mow ut. Let 

 the tecond globe below us be but a blood-globule in an animalcule of 

 the >*; and so on to the twentieth globe below us. Then if the 

 inhabitants of the twentieth globe above \ts were to calculate the 

 circumference of their globe from its diameter by help of the 607 

 decimals, the inhabitants of the twentieth globe below us could not 

 detect the error with their best microscopes, unless their Tulleys and 

 Rosses were much greater masters of their art than ours. 



The newest attempt at quadrature of our century is that of Mr. R. 

 Ambrose Smith, of Aberdeen. It was undertaken as a test of the 

 theory of probabilities. If a thin rod, not so long as the breadth of a 

 plank, be thrown at hazard upon a planked floor, the chance which the 

 rod has of intersecting a seam between two planks is the fraction 

 which the length of the rod is of the quadrant having the breadth of 

 a plank for its radius. In 8204 tosses with a rod three-fifths of the 

 breadth of plank, the experimenter found 11 mere contacts, and 1213 

 decided intersections. He counted the contacts as intersections, and, 

 from the principle [PROBABILITY] that the result of a large number of 

 trials is nearly that of the long mn, he presumed that 3 is to the quad- 

 rant of radius 5 nearly as 1224 to 3204. This gives 3'1412 for the ratio 

 of the circumference to the diameter ; which comes nearer than Archi- 

 medes. If, which perhaps ought to have been done, the contacts had 

 been equally divided between intersections and non-intersections, the 

 result would have been 3'1553 : and even this is a remarkable approach, 

 the nature of the method being considered. 



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 (' Mdm. Acad. 

 Berlin ' for 1 761 ), 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 on 

 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. 

 Buffon, and Panckouke, the editor of the ' Encyclopedic Mcthodique," 

 have attempted to give metaphysical reasons for this incommensura- 

 bility, apparently in order that the squarers 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 Montucla 

 Mem* to admit the proof at last,* though he only said that it was rrry 

 like demoiutratio* 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 Huygncns to admit that Gregory 

 h/l succeeded in proving the im|>ossibility 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. Bui 

 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 

 Thus 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 ratit 

 expressible by a finite number of algebraical terms. The consequence 

 of this, if established, must be drawn as follows : Since geometry 



The following wu the addition left by him, and printed by the editor o 

 the Hcond edition : " Aprca roir rcflechl encore plui uttcntircmcnt ur If 

 ratMnnemenU de Gregory, II me prlt avoir en rainon d'cn dtdulrc Hinpon- 

 pibilite de la quadrature memo deftnie du ccrclr." For ournclve*, we left of 

 unable to dlapute, but feeling the liuecuritjr which attache* to almost al 

 negative proofi. We thought It an convincing Abel'n proof of the im 

 poMlbillty of algebraic solution of the fifth degree, an that proof atood In Abel's 

 own memoir. 



circles of given or found radii and centres, the points of iuU-r- 

 section may bo determined by formula) derived from tin- root* of an 

 e> {nation not exceeding the second degree, the roots of which can be 

 expressed in a finite form : the two ends of the line equal to the circle 

 could therefore be assigned in a finite form, and hence the length 

 teelf. And the area of any polygon (whose arcs are obtained by 

 continual bisection) described about a given circle could also be ex- 

 pressed in a finite form, from which (the area of the circle being 



:pressible by means of it circumference) the ratio of the area to 



_*tof the einiimscribed polygon would also be expressible. But if 



11 be true, this area cannot be expressed in a finite 



orm ; "neither then can any construction allowable in geometry attain 



.he circumference of a circle. 



The indefinite quadrature was shown to bo impossible by Newton 

 .Principia, book i., lemma 28), in a manner which lies open t 

 objection ; we shall not therefore produce this proof. In fact, it 

 seems as difficult to induce geometers to agree in any proof of the 

 mposaibility of the indefinite quadrature, as others to leave off trying 

 ,hcir powers upon what geometers themselves have ceased to attempt.. 



Moutucla lias given a tolerable list of those who have signalised 

 themselves by attempting this problem without the requisite pre- 

 aminary of studying geometry ; if preliminary that may be called, 

 which would have made them give up the attempt. We shall piv- 

 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 1 will 

 set about it immediately," and such seems to have been the g. 

 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 proj>r- 

 tional between the inscribed and circumscribed squares, which h:i 

 to be the content of the inscribed octagon. Next we have Cardinal Cusa, 

 Orontius Fineus.and Simon a Quercu, or Duchesnc, already mentioned. 

 At the time when the problem really was of practical importance, 

 every ijuadrator 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 Kegiomontanus, Orontius by Buteo and Nonius, and 

 Duchesne 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 Duchesne. But inde- 

 pendently of what we have already said about Metius, it is not true 

 that Ihieliesne's quadrature required any new accuracy of limits to 

 expose it. He produced 3'14t>60.. . : and Archimedes had already 

 shown that 3'1 4285 . . . . 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 his hand on this and 

 other problems in 1592, and was met by Vieta, Adrianus Romauus, 

 and Clavius. Longomontonus, 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 fake quadrature. The works of the last against 

 geometry and geometers were the consequence of the mortification 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 connection 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, Koberval, 

 Huyghens, and Luotaud, who soon despatched him. 



As yet we have mentioned only mathematicians, or men of eminence 

 in something : but we have also the Spaniard Falcon (1589), who 

 dialogues with the circle in verse : Gephirauder and Alphonso de 

 Molina, who attribute their discovery to inspiration ; the latter over- 

 turns Euclid, and found another who was willing to admit his dis- 

 coveries and translate them into Latin. A merchant in Hochelle, De 

 la Leu, not only found out the problem by inspiration, but showed 

 that the conversion of Jews, Turks, and Pagans depended upon it. 

 Montucla gives some account of several other visionaries, the clu'ef of 

 whom is one Cluver, who found that the problem depended upon 

 another, namely, " Construere munduui divimr menti analogum," 

 which, if it be translateable, is, " to make a world analogous to the 

 divine niiniL" Ho also mentions Richard White, an English Jesuit, 

 who stands out amoug the solvers of the problem as the only one 

 who ever was convinced of his error. The writer of this article once 

 pointed out the example of VVTiite to another Roman Catholic clergy- 

 man, who had come from South America to England, to publish a 

 quadrature of the circle. The party addressed seemed struck by the 



