much less, shall also be commensurable to it. It is a consequence of the theorem which gives the tangent of the sum and difference of two arcs whose tangents are given; for there being no extraction of square roots in this formula, if the arcs have their tangents rational, the tangent of the sum will be so also; and vice versa, an arc with a rational tangent can be divided into two arcs, whose tangents much smaller will be rational. Thus the arc 45° can be divided into two (incommensurable really to each other), the tangent of one of which will be, that of the other 3. We shall therefore find by the series of the arc by the tangent each of these arcs, and their sum (though irrational to each other and to the radius) will, nevertheless, be to the are of 45°, which, multiplied by 4, will give the ratio of the relation of the semi-circumference to the radius, or of the circumference to the 1 1 1 diameter; for the first of these series will be 2 3.23 1 1 1 1. 1 5.25—7.27+ 3.23 +5.25 24 160896460822528' 1 1 1 1 1 2 1 1225 1 5.357.37 etc. 15309177147 17537553 165526791' 17537553+165526791' Now, in both these series, the terms diminish rapidly enough to attain very nearly their real values; for, in the second, by taking only seven terms the error would already be less than 186,000,000th. Euler shows that it is possible to obtain this result even more rapidly; for he remarks that the arc whose tangent is can be divided into two whose tangents will be and, which gives the arc of 45° equal to twice 1 1 1 the second of the above series, plus this one 7 1 1 1129+89035. can be Finally, we shall observe that the arc whose tangent is divided into two whose tangents shall be and, which affords the means of obtaining two series still more converging than the one which gives the arc answering to the tangent. It would be easier to find by these 3 or 4 series the circumference of a circle to 200 decimals than it was for Viete or Romanus to calculate by their methods to 10 or 15 decimals. We pass in silence over the other artifices of calculations presented by Euler in the same treatise, and in another, Vol. XI, for 1739, by which it would require only 80 hours of work to find 128 figures of Lagny; there are also some in Stirling, Summatione Serierum, in Simpson; The Doctrine of Fluxions, 1750. Kraft, in the 13th volume of Petersbourg for 1741, page 121, gave some mechanical constructions very simple and very near. All these methods which, undoubtedly, mutually confirm each other, leave no doubt as to the numerical expression of the approximate ratio of the diameter to its circumference; this ratio is the true touchstone to try the pretended quadratures of the circle without entering into the maze of the pitiable and often obscure reasoning of those who claim to be the happy explainers of this enigma. Nothing better can be done than by leaving them to their pleasing allusion; for experience has shown attempts to open their eyes would be fruitless; that is what induced the Academy of Sciences to give notice that it would examine no more quadratures of the circle any more than trisections of the angle, or duplications of the cube, or perpetual motions. Histoire de l'Academie, 1775, page 64. This is the way in which the Secretary of the Academy, Condorset, who was himself a very great geometrician expresses himself. This solution may be considered in two lights. In fact, we may look for the quadrature of the entire circle or the quadrature of any of its sectors whose chord is assumed as known. The last of these problems is considered as having no solution. Gregory and Newton, whose authority is so great, even in a science where authority goes for so little, have given different demonstrations of the impossibility of the indefinite equation. John Bernoulli has proved that the required sector was expressed by a real logarithmetical function, but which in form contains imaginary quantities. It follows that no real functions, either algebraical or logarithmetical and real in form, can represent the value of the sector of an indefinite circle; that the equation between the sector and the chord can not be constructed by the intersection of the lines of real or curve surfaces, and we may infer from this consideration the absolute impossibility of the indefinite quadrature. Geometricians are not as well agreed as to the impossibility of the first problem, for it often happens that for particular values quantities are found whose expression is in general impossible; but an experience of more than 70 years taught the Academy that none of those who used to send in solutions of these problems understood either their nature or their difficulties, that none of the methods which they used would have led them to the solution. This long experience sufficed to satisfy the Academy of the little utility that would accrue to the sciences by examining all these pretended solutions. Other considerations have also decided the Academy. There is a popular report that governments have promised considerable rewards to the person who might succeed in solving the problem of the quadrature of the circle, and that this problem is the subject of the investigations of the most celebrated geometricians. On the strength of these reasons a crowd of men, much more considerable than is supposed, have given up useful occupations to devote themselves to the discovery of this problem, often without understanding it, and always without the information necessary to try its solution with success. Nothing was better calculated to undeceive them than the declaration the Academy thought proper to make. Many had the misfortune to believe they had succeeded, and would not yield to the reasoning with which geometricians attacked their solutions. Often they could not comprehend them, and ended by accusing them of envy or bad faith. Sometimes their obstinacy degenerated into madness ; but it is not considered as such if the opinion which forms this folly does not clash with the received ideas of men, if it has not influence upon the conduct of life, and if it does not disturb the order of society. The madness of the quadrateurs would therefore have no other inconvenience for them but loss of time often useful to their families ; but unfortunately madness is seldom confined to a single object, and the habit of reasoning falsely is acquired and extends like that of reasoning accurately. That happened more than once to the quadrateurs. Besides, unable not to realize how singular it would be if they should discover, without study, truths which the most celebrated men have sought in vain, they almost all become persuaded that it is by a special protection of Providence that they have succeeded, and there is only one step from this idea and believing all the strange combinations of ideas which occur to them are so many inspirations. Humanity, therefore, required that the Academy, persuaded of the uselessness of the examination it might have made of these solutions of the quadrature of the circle, should seek to dispel, by a public declaration, popular opinions which have been fatal to many families. Considerations so wise could not excite the animadversion of a writer like Linguet in his Political Annals. He had also found that it is not true that the pictures of exterior objects are taken inverted (upside down) in the retina, and the tide of the Amazon does not ascend up to Pauxis, where Condamine noticed it. Nothing surprises me more than to see persons of understanding have so little good sense as to persist in things they do not understand, with as much assurance and warmth as if they had occupied their whole lives in studying them, and had acquired a real superiority in the same; but mankind is subject to these inconsist encies. 3 PRACTICAL REMARKS AND EXAMPLES ILLUSTRATING THE QUADRATURE OF THE CIRCLE. THE Author, who proposes for investigation a new theory upon any subject in the present advanced state of science, when nearly every truth concerning the same is believed to have been discovered, fully demonstrated and applied, must be happy if he succeeds in establishing the truth of what he advances; for, if he does not have to disprove all former theories he of necessity calls them in question and creates a doubt of their correctness before he can hope to succeed in establishing his own. This is especially so with regard to the science of mathematics; for, when a fact has been once established by actual mathematical demonstration which is universally admitted to be true, it is adhered to as firmly as one's own faith, and it is next to impossible to doubt its correctness. Any discovery, therefore, made after such fact has been so demonstrated and received, which goes to prove even a slight error in the result obtained is received with great caution; but if it should seriously call in question the truth of former theories or directly contradict a result already established, it must be brilliant indeed to secure the attention and merit the consideration of the learned. It would also have to be profitable, for in this age the money tree is pruned before all others; even imperfect theories are often preferred after they have been proved erroneous, because they are better understood, and there are so few who can bear the mental exertion necessary to examine newer and truer ones. There are two classes of propositions in mathematics which deserve to be considered with attention, the first of which needs only to be referred to here, as the truths upon which they are based have been put beyond any question of doubt; such for example as THEOREM 1. In any right-angled triangle the square described on the side subtending the right angle is equal to the sum of the squares described on the sides which contain the right angle. |