873 



QUADRATURE OF THE CIRCLE. 



QUADRATURES, METHOD OF. 



871 



instance, and promised to study more geometry before he proceeded 

 further; in a little while however he relapsed, and his work was 

 advertised, and, we believe, published. 



After the time of Newton, and the abundant means which were then 

 introduced to complete the quadrature, if such a thing were possible, 

 persons versed in mathematics seem to have dropped the attempt, and 

 the reign of the quadratora by instinct commences. It is true that a 

 serious diversion was made by the theory of gravitation, which drew 

 off against itself many of those who should have been quadrators ; but 

 enough remained to furnish a tolerable list. 



That of Montucla contains principally Frenchmen, though had the 

 history of mathematics been written by an Englishman, he could have 

 produced as great a number in this country. One Mathulon, in 1728, 

 promised in print 1000 crowns to any one who should convict his 

 solution of error, and was actually sentenced by the courts to pay the 

 sum to the Hotel Dieu at Lyon, to which charity Nicole, the exposer, 

 made over his claim. One Sullamar (as Montucla spells it), an English- 

 man, solved the problem by means of the number of the beast, 666, in 

 the year 1750 ; a M. de Causans, in 1753, found it by cutting a piece 

 of turf, and deduced from it the doctrines of original sin and of the 

 Trinity. He offered to bet 300,000 francs on the correctness of his 

 process, and deposited 10,000, which were claimed by several persons, 

 and among others by a young lady, who brought an action for them : 

 but the bet was declared void by the courts. Many more cases might 

 be added ; it is however enough to say that this problem is now" never 

 attempted (in print at least), except by those who are either altogether 

 ignorant of mathematics, or add a moat undue opinion of themselves 

 to an acquaintance with only the elements. Since 1755, the Academy 

 of Sciences has refused to examine any pretended solution ; and the 

 Royal Society in thia country came to the same resolution a few years 

 afterwards. In the announcement of their determination, the French 

 Academy, without pronouncing any dictum upon the problem, stated 

 that in a period of seventy years, those who had brought forward 

 quadratures had been persons who did not know the nature of the 

 difficulty of the problem. An experience of about half that time has 

 satisfied us that the same thing may be said of our own day : and with 

 this addition, that whereas the speculators who came before the 

 Academy seem for the most part to have had an idea that their chance, 

 if any, lay in the yeometrical quadrature, the squarers of our time, in 

 nine cases out of ten, announce the arithmetical quadrature, which has 

 been proved to be impossible. 



A few words may serve to prevent some one from making an attempt 

 upon thia enchanted castle, as it is supposed to be. When the diffi- 

 culty first began to be noticed, the circle stood alone among curves ; 

 and so remarkable a distinction between this, the only curve then 

 considered, and rectilinear figures, the only other figures then con- 

 sidered, could not but excite curiosity. Our position is now changed ; 

 not only does the now well recognised distinction of commensurable 

 and incommensurable prevent the circle frem presenting anything 

 peculiar to itself, but the curve is only one among an infinitely great 

 number, many of which have been investigated and their properties 

 examined. Consequently, with reference to the present state of 

 mathematics, the problem analogous to that of squaring the circle is, 

 " Given any curve whatsoever, to find its area." Now if the ingenuity 

 which is guided by the love of investigating hidden things, should 

 denire a field for its exertions, let it leave that of the circle, which baa 

 been cropped until it will yield no more, and, first acquiring sufficient 

 mathematical knowledge, let it spend its force upon some one of the 

 many real difficulties which abound, both in the pure and mixed 

 sciences : let it investigate the meaning of divergent series for example, 

 in all their varieties, or endeavour to extend the theory of discontinuous 

 expressions, or solve the equations of motion of the solar system by 

 some other method than that of series. For one point that should 

 strike the lover of the marvellous in the quadrature of the circle, there 

 are hundreds in the above-named subjects which surprise the mathe- 

 matician, however little he may possess that quality. Moreover, in 

 like manner as the quadrature of the circle was at one time, in the 

 hands of VVallia, Newton, Ac., a road to results which, though they did 

 not attain their end, yet answered many other purposes ; so the efforts 

 of the inquisitive on the actual difficulties of our own day may also 

 end in the promotion of science of every kind, if begun in knowledge 

 and directed by system. We owe the binomial theorem, now one of 

 the most important results of algebra, indirectly to the learned attempts 

 <it Wallis upon the quadrature of the circle, at a time when such 

 attempts were in season : and we might reasonably hope for collateral 

 successes something like those resulting from the labours of Wallis, if 

 those (not a few) whose minds compel them to inquiry into the curious, 

 wnuld but furnish themselves with a guide before they set out on their 

 travels. 



This problem still engages attention : and persona are found to 

 believe that they have attained even the arithmetical quadrature. 

 And it is a peculiarity of thia well-worn old problem, that it catches 

 the fancy of those who have not even attended to common geometry, 

 and frequently makes them think that the riddle is reserved for them 

 to read. And the reason is, that they treat it as a riddle : a thing to 

 be thought out by some sudden stroke of mother-wit. They do not 

 know that there are thousands of problems of the same kind in every- 

 thing but notoriety. Thirty years ago, when the ' Penny Magazine ' 



waa young, it contained an article on this subject which might have 

 been thought sufficient at least to establish the fact of a difficulty. A 

 man of high rank and great wealth, who had probably never opened a 

 book of geometry since his school days, if even then, happened to read 

 thia article, was caught, and in five minutes produced an idea which 

 he thought so likely to be the solution of the enigma, as he would 

 have it to be, that he forwarded it to the secretary of the Useful 

 Knowledge Society, for transmission to the author of the article. 



It has been stated in foreign newspapers within these few years that 

 the British government does offer, and always has offered, a large 

 reward for the solution of this problem. This, we need hardly say, is 

 a complete mistake : the government never at any time offered one 

 farthing for the quadrature of the circle. Some years ago, a labouring 

 man from the country came to town with a quadrature, and addressed 

 a letter to the Lord Chancellor, stating his claim, and desiring his lord- 

 ship to hand over forthwith 100,000?., being the sum due from govern- 

 ment for the discovery, as per proclamation : this letter we saw. The 

 only problem for the solution of which any reward was at any time 

 offered, was the practical mode of finding longitude : and even this 

 offer is now retracted ; rewards having been, in fact, received by several 

 persons. Should this article fall into the hands of any, at home or 

 abroad, who are working at any mathematical or astronomical difficulty 

 under the impression that our government is pledged to remunerate 

 them in case of success, they may rest assured that their information 

 is incorrect, and that they will find it to be so on inquiry. The only 

 person we know of who states definitely what the reward is, and by 

 whom offered, is Nicolas Ericius, in a tract on the quadrature " Funda- 

 mentalis figura geometrica . . . ." Copenhagen, 1755. The author states, 

 as known to all, that on the 2nd of June, 1747, the Royal Society of 

 London publicly advertised a reward of 30,000?. for the quadrature of 

 the circle and the discovery of the true nature of the magnet, over and 

 above some previously offered sum, the amount of which is not stated. 

 It is needless to say that the archives of the Royal Society contain no 

 account of any offer of the kind. 



Persons who are not acquainted with the subject are puzzled by one 

 consideration, which we admit is calculated to have weight. The 

 quadrators appeal to great discoveries which have at last been recog- 

 nised, though their promulgators were at first treated with contempt. 

 We pass over the last part of the assertion with the remark that 

 Copernicus, Galileo, Tycho Brahe", Harvey, &c,, were never treated 

 with contempt : some were opposed, some were persecuted, and all 

 were ridiculed by a section of their opponents ; for ridicule is a con- 

 stituent of all opposition whatsoever, though not from every opponent. 

 But those with whom Copernicus, &c., came in contact, whether in 

 person or in book, showed amply that they knew what manner of men 

 they had to deal with. The more important point is the following. 

 Invention in mechanics may have been sometimes attained not very 

 often by men who were ignorant of the doings of their predecessors. 

 But discovery in matters of speculation that is, conquest over old and 

 tried difficulties has never, that we have read of, been reached by any 

 one whose mind had not been trained by study of previous speculation. 

 All the men we have mentioned, and all of their stamp that we know 

 of, were learned men, acquainted with the history of the subject they 

 were destined to advance, and practised in its methods : they had all 

 shown their power over what was known, before they presented them- 

 selves as the promulgators of what was not. The squarer of the circle, 

 therefore, may be asked to ahow proof of his acquaintance with the 

 previous history of the subject : any one who really does this would, 

 even now, meet with some attention. But the general run of circle- 

 squarers are as ignorant of the past as the future will be of them. 



Again, we have to meet the very plausible objection, that many 

 difficulties have been overcome at last : why not then the quadrature 

 of the circle ? The answer is easy. The difficulties which have been 

 overcome at last have been overcome by the discovery of new means, 

 and the introduction of new powers ; but our problem is to square the 

 circle with the old allowance of means : Euclid's postulates and nothing 

 more. We cannot remember an instance in which a question to be 

 solved by a definite method was tried for centuries by the best heads, 

 and answered at last, by that method, after thousands of complete 

 failures. 



QUADRATURES, METHOD OF. The method of quadratures 

 derives its name from its earliest application, that of finding the areas 

 of curves, which was always called their quadrature, as being the 

 arithmetical process by which, when exact, squares equal to them 

 might be found. And since the AKEA of a curve can always be found 



vrbenfydx can be found, this term has also been applied to the deter- 

 mination of the definite values of integrals by approximation. We 

 shall first complete the reference in OFFSET. An area being bounded 

 by a curve, a part of the axis, and two extreme ordinates, if the base 

 on the axis be divided into any number of equal parts by ordinates 

 (offsets), the chords which join the ends of successive offsets, if 

 substituted for the arcs of the curve, will give very nearly the same 

 area. And this area is found by adding half the first and last offset 

 to the sum of all the intermediates, and multiplying by the common 

 base. This is the old surveyor's rule. Thomas Simpson gave one 

 of more exactness, by drawing a parabola, of axis perpendicular to the 

 base, through the first, second, and third points, another through 



