SQUARING, or Quadrature of the Circle, signifies the finding a square exactly equal to the area of a given circle. This problem, however, has not been, and probably cannot be, strictly resolved by the commonly admitted principles of geometry; mathematicians having hitherto been unable to do more than to find a square that shall differ from the area of any proposed circle by as small a quantity as they please. The problem is of the same degree of difficulty, and indeed may be regarded as identical with another geo metrical problem, namely, the rectification of the circle, or the finding a straight line equal to its circumference; for the area of a circle is equal to that of a rectangle contained by the radius and a straight line equal to half the circumference; therefore, if a straight line exactly equal to the circumference could be found, a rectlineal space precisely equal to the area might also be found, and the contrary. But although no perfectly accurate resolution of the problem has been obtained under either form, we can always find approximate values of the area and circumference; and it is now customary to apply the terms quadrature and rectification of the circle also to these.

The problem of the quadrature of the circle appears to have engaged the attention of geometers at a very early period; for we are told that Anaxagoras, who lived about live hundred years before Christ, attempted its solution while confined in prison on account of his philosophical opinions. We are ignorant of the result of his researches; but al though we cannot suppose they were attended with any success, we may reasonably conclude that we are indebted to them for the discovery of some of the properties of the figure, which are now known as elementary propositions in geometry.

Hippocrates of Chios was likewise engaged in trying to resolve the same problem, and it was no doubt in the course of his inquiries into this subject, that he discovered the quadrature of the curvilineal space, which is now known by the name of the Lune of Hippocrates. The nature of this discovery may be briefly explained as follows. Let ABCD be a circle, H its centre, AC its diameter, ADC a triangle inscribed in the semicircle, having its sides, AD, DC equal to one another. On

D as a centre, with DA or DC as a ra dius, let the quadrantal arch AEC be described, then shall the curvilineal space bounded by the semicircle ABC

and the quadrantal arch AEC (which is the Lune of Hippocrates), be equal to the rectilineal triangle ADC.

Although Hippocrates’s discovery has led to no important conclusion either relating to the quadrature of the circle or that of any other curve, yet at the time it was made, it might be regarded as of some consequence, chiefly because it shewed the possibility of exhibiting a rectilineal figure equal to a space bounded by curve lines, a thing which we have reason to suppose was then done for the first time, and might have been fairly doubted, considering the insuperable difficulty that was found to attend the quadrature of the circle or its rectification.

Aristotle speaks of two persons, namely, Bryson and An tiphon, who, about his time, or a little earlier, were occu pied with the quadrature of the circle. The former appears, according to the testimony of Alexander Aphrodiseus, to have erred most egregiously; he having concluded that the circumference was exactly 3¾ times the diameter. And the latter seems to have proceeded in the same manner as Archimedes afterwards did in squaring the parabola, that is, by first inscribing a square in the circle, then an isoceles tri angle in each of the segments of the curve, having for its base a side of the square; and next again a series of triangles in the segments, having for their bases the sides of the former series, and so on. This mode of procedure, how ever, was not attended with success, as these spaces do not, as in the case of the parabola, admit of being absolutely summed.

It may be supposed that Archimedes exerted his utmost efforts to resolve this problem; and probably it was only[7:20:553] after long meditation on the subject that he lost all hopes of success, and contented himself with that approximation to the ratio of the diameter to the circumference which is contained in his treatise De Circuli Dimensione. He found his approximation to the ratio, by supposing a regular polygon of ninety-six sides, to be described about the circle, and another of the same number, to be inscribed in it, and by shewing that the perimeter of the circumscribing polygon was less than 3 10 / 70 , or 3 1 / 7 times the diameter, but that the perimeter of the inscribed figure was greater than 3 10 / 71 times the diameter; now, the circumference of the circle being less than the perimeter of the one polygon, but greater than that of the other; it follows that the circumference must be less than 3 1 / 7 times the diameter, but greater than 3 10 / 71 times; so that, taking the first of these limits, as being expressed by the smallest numbers, the circumference will be to the diameter as 3 1 / 7 to 1, or as 22 to 7 nearly.

Although the ratio found by Archimedes be the oldest known in the western world, yet one more accurate was known at a much earlier period in India. This we learn from the Institutes of Akbar (Ayeen Akberry), where it is said that the Hindoos suppose the diameter of a circle to be to its circumference as 1250 to 3927. Now, this ratio must have required the inscription of a polygon of 768 sides in the circle, and must have been attended with nine ex tractions of the square root, each carried as far as ten places of figures.

We learn from Simplicius that Nicoedes and Apollonius both attempted to square the circle, the former by means of a curve, which he called the quadratrix; the invention of which, however, is ascribed to Dinostratus, and the latter, by the help of a curve, denominated the sister to the tortuous line, or spiral, and which was probably the quadratrix of Dinostratus; the nature of which, and the manner of its application to the subject in question, we shall briefly explain.

Let AFB be a quadrant of a circle,

(fig. 2), and C its centre; and conceive the radius CF to revolve uniformly about C, from the position CA, until at last it coincide with CB; while at the same time a line DG is carried with an uniform motion from A towards CB; the former line continuing always parallel to the latter, until at last they coincide; both motions being supposed to begin and end at the same instant; the point E, in which the revolving radius CF, and the moveable line DG intersect one another, will generate a certain curve line AEH, which is the quadratrix of Dinostratus.

Draw EK, FL both perpendicular to CB; then because the radius AC and the quadrantal arch AFB, are uniformly generated in the same time by the points D and F, the contemporaneous spaces described will have to one another the same ratio as the whole spaces; that is, AD: AF:: AC: AB; hence we have AC: AB:: DC, or EK: FB. Now, as the moveable point F approaches to B, the ratio of the straight line EK to the arch FB will approach to, and will manifestly be ultimately the same as the ratio of the straight line EK to the straight line FL, which again is equal to the ratio of CE to CF; therefore the ratio of the radius AC to the quadrantal arch AFB is the limit of the ratio of CEtoCF,and consequently equal to the ratio of CH to CB, H being the point in which the quadratrix meets CB. Since therefore CH: CB:: CA, or CB: quad, arch AFB, if by any means we could determine the point H, we might then find a straight line equal to the quadrantal arch, (by finding a third proportional to CH and CB), and consequently a straight line equal to the circumference. The point H, however, can not be determined by a geometrical construction, and there fore all the ingenuity evinced by the person who first thought of this method of rectifying the circle, (which certainly is considerable), has been unavailing.

The Arabs, who succeeded the Greeks in the cultivation of the sciences, would no doubt have their pretended squavers of the circle. We however know nothing more than that one of them believed he had discovered that the diameter being unity, the circumfernnce was the square root of 10; a very gross mistake; for the square root of 10 exceeds 3162; but Archimedes had demonstrated that the circumference was less than 3·143.

It appears that, during the dark ages, some attempts were made at the resolution of this famous problem, which, how ever, have always remained in manuscripts, buried in the dust of old libraries. But upon the revival of learning, the problem was again agitated by different writers, and particularly by the celebrated Cardinal De Cusa, who distinguished himself by his unfortunate attempt to resolve it. His mode of investigation, which had no solid foundation in geometry, led him to conclude, that if a line equal to the sum of the radius of a circle, and the side of its inscribed square were made the diameter of another circle, and an equilateral triangle were inscribed in this last, the perime ter of this triangle would be equal to the circumference of the other circle. This pretended quadrature was refuted by Regiomontanus.

It would be trespassing too much upon the patience of our readers, were we to mention all the absurd and erroneous attempts which have been made during the last three centuries to square the circle. In a supplement to Montucla’s Histoire des Mathematiques, we find upwards of forty pretenders to the honour of this discovery enumerated. They were almost all very ignorant of geometry; and many of them were wild visionaries, pretending to discover inexplicable relations between the plain truths of mathematics and the most mysterious doctrines of religion. From such persons as have generally pursued this inquiry, no improvement whatever of the science was to be expected; although, in some instances, it has derived advantage from the labours of euch as have undertaken to expose the ab surdity of their conclusions; as in the case of Metius, who in refuting the quadrature of one Simon à Quercu, found a much nearer approximation to the ratio of the diameter to the circumference than had been previously known in Eu rope, namely, that of 113 to 355.

Among the most remarkable of those who have under taken to resolve this problem, we cannot fail to enumerate Joseph Scaliger, a man of stupendous erudition. Full of confidence in his own powers, he believed that, entering upon the study of geometry, he could not fail to surmount by the force of his genius, those obstacles which had completely stopped the progress of all preceding inquirers. He gave the result of his meditations to the world in 1592, under the title Nova Cyclometria; but he was refuted by Clavius, by Vieta, and others, who shewed that the magnitude which he had assigned to the circumference was a little less than the peri meter of the inscribed polygon of 192 sides, which proved beyond a doubt that he was wrong. Scaliger, however, was not to be convinced of the absurdity of his conclusion; and indeed, in almost every instance, pretenders to this dis covery have not been more remarkable for their egregious errors, than for their obstinacy in maintaining that they were in the right, and all who held a contrary opinion in the wrong.

The famous Hobbes came also upon the field about the year 1650, with pretensions not only to the quadrature of the circle, but also to the trifection of an angle, the reclification of the parabola, &c.; but his pretended solutions were refuted by Dr. Wallis. And this circumstance aftbrd ed him occasion to write not only against geometers, but even against the science of geometry itself.[7:20:554] We find it recorded by Montucla, as a sort of phenomenon, that one Richard White, an English Jesuit, having happened upon what he conceived to be a quadrature of the circle, which he published under the title, Chrysaespis, seu Quadratura Circuli, suffered himself at last to be convinced by some of his friends that he was wrong. But a solution of the same problem, found out by one Mathulen of Lyon, did not end in so much advantage to its author. This man in 1728 announced to the world that he had discovered both the quadrature of the circle, and a perpetual motion; and he was so certain of the truth of these discoveries, that he he consigned 1000 ecus (about L.125), to be paid to any one who should demonstrate that he was deceived in either. The task was not difficult. Nicoe of the Academy of Sciences demonstrated that he was wrong, and he himself allowed it; but he hesitated to pay the money, which Nicoe had relinquished in favour of the Hotel Dieu of Lyon. The affur went before a court of justice, which adjudged the money to be paid, as Nicoe had destined it, to the poor. At a later period, namely, in 1753, the Chevalier de Causons, a French officer, and a man who was never sup posed to be a mathematician, suddenly found a quadrature of the circle in procuring a circular piece of turf to be cut; and rising from one truth to another, he explained by his quadrature the doctrine of original sin, and the Trinity. He engaged himself, by a public writing, to deposit with a notary the sum of 300,000 francs, to be wagered against such as should oppose him, and he actually lodged 10,000, which were to devolve to him who should demonstrate his error. This was easily done, as it resulted from his disco very that a circle was equal to its circumscribing square, that is, a part to the whole. Some persons came forward to answer his challenge, and in particular a young lady sued him at one of the courts of law; but the French king judg ed that the chevalier’s fortune ought not to suffer on ac count of his whim; for, setting aside this piece of folly, in every other respect he was a worthy man. The procedure was therefore stopped, and the wager declared void.

We shall not enter further into the history of these vain and absurd attempts to resolve this important problem, but proceed to state what has actually been done by men of sound minds and real mathematical acquirements towards its solution. And in the first place, it may be observed that the problem admits of being proposed under two diffe rent forms; for it may be required to find either the area of the whole circle, or, which is the same thing, the length of the whole circumference; or else to find the area of any proposed sector or segment, or, which is equivalent, the length of the arch of the sector or segment. The former is termed the definite, and the latter the indefinite quadrature of the circle. The latter evidently is more general than the former, and includes it as a particular case. Now, if we could find by any means a finite algebraic equation that should express the relation between any proposed arch of a circle, and some known straight line, or lines, the magnitude of one or more of which depended on that arch, then we would have an absolute rectification of the arch, and consequently a rectification or quadrature of the whole circle. We here speak of an analytical solution; the ancients, who were almost entirely ignorant of this branch of mathematical science, must have endeavoured to treat it entirely upon geometrical principles. It is now well known, how ever, that all geometrical problems may be subjected to analysis; and that it is only by such a mode of proceeding they have in many cases been resolved.

With respect to the definite quadrature of the circle, no unexceptionable demonstration of its impossibility has hitherto been published. It is true, that James Gregory, in his Vera Circuli et Hyperbola Quadratura, has given what he considered as such a demonstration; but it was objected to by Huygens, one of the best geometers of his time. It is certain that the ratio of the diameter to the circumference, also, that the ratio of the square of the diameter to the square of a straight line equal to the circumference, cannot be expressed by rational numbers, for this has been strictly demonstrated by Lambert in the Berlin Memoirs for 1761. A demonstration is also given in Legendre’s Geometrie. As to the indefinite quadrature, if Newton’s demonstration of the twenty-eighth lemma of the first book of his Principia, be correct, the thing ought to be absolutely impossible. For the object of that proposition is to prove, that in no oval figure whatever, that returns into itself, can the area cut off by straight lines at pleasure, be universally found by an equation of a finite dimension, and composed of a finite number of terms. If this be true, then it will be impossible to express any sector of a circle taken at pleasure in finite terms. It is however to be remarked, that the accuracy of the reasoning by which Newton has attempted to establish the truth of the general proposition, has been questioned by no less a geometer than D’Alembert; and in deed we know one oval curve, which returns into itself, and which according to Newton’s proposition ought there fore not to admit of an indefinite quadrature; yet this is by no means the case, for it does really admit of such a quad rature. The curve we mean is the lemniscata, whose figure is nearly that of the numeral character 8. Upon the whole, then, we may infer that an unexceptionable demonstration of the impossibility of expressing either the whole circle, or any proposed sector of it, by a finite equation, is still among the desiderata of mathematics.

Of the different methods which have been found for approximating to the area or to the circumference, that of Archimedes is the only one found by the ancients in the western world that has descended to modern times, and it appears to have been the most accurate known, until about the year 1585, when Metius, in refuting a pretended quadrature, found the more accurate ratio of 113 to 355, as we have already noticed. About the same time Vieta and Adrianus Romanus published their ratios, the former carrying the approximation to ten decimals instead of six, (which was that of Metius’s ratio), and the latter extending it to 17 figures. Vieta also gave a kind of series, which being continued to infinity, gave the value of the circle.

These approximations, however, were far exceeded by that of Ludolph Van Ceulen, who, in a work published in Dutch in 1610, carried it as far as 36 figures, showing that if the diameter were unity, the circumference would be greater than 3.14159,26535,89793,23846,26433,83279,50288, but less than the same number with the last figure increased by an unit. In finding this approximation, Van Ceulen followed the method of Archimedes, doubling continually the number of sides of the inscribed and circumscribed poly gons, until at length he found two which differed only by an unit in the 36th place of decimals in the numbers ex pressing their perimeters. This was rather a work of patience than of genius; indeed the labour must have been prodigious. He seems to have valued highly this singular effort; for in imitation of Archimedes, whose tomb was adorned with a sphere and cylinder, he directed that the ratio he had found might be inscribed on his tomb.

Snellius abridged greatly the labour of calculation; and although he did not go beyond Van Ceulen, yet he verified his result. Des Cartes also found a geometrical construction, which being repeated continually, gave the circumference, and from which he might easily have deduced an expression in the form of a series. Gregory of St. Vincent distinguished himself also on this subject; he however committed a great error in supposing he had dis covered the quadrature of both the circle and hyperbola. Gregory’s mistake was the cause of a sharp controversy carried on between his disciples on the one side, and by Huygens, Mersenne, and Lestaud, on the other; and it[7:20:555] was this that gave Huygens occasion to consider particularly the quadrature of the circle, and to investigate various new and curious theorems relating to it, which are contained in his Theoremata de Quadratura Hyperboles, Ellipsis, et Circuli, 1651; and in his work De Circuli Magnitudine inventa, 1654. In particular he showed, that if c denote the chord of an arch, and s its sine, then the arch itself will be greater than c+ 1 / 3 (c—s), but less titan c+ 4c + s

——— ×½(c—i): he also showed that the arch is less than 2e+3jt a the sum of 2 / 3 of its sine and 1 / 3 of its tangent.

James Gregory, in his Vera Circuli et Hyperbolae Quad ratura, gave several curious theorems upon the relation of the circle to its inscribed and circumscribed polygons, and their ratios to one another; and by means of these he found with infinitely less trouble than by the ordinary methods, and even by those of Snellius, the measure of the circle as far as 20 places of figures. He gave also, after the example of Huygens, constructions for finding straight lines nearly equal to arches of a circle, and of which the degree of accuracy was greater. For example, he found that if A be put for the chord of an arch of a circle, and B for twice the chord of half the arch, and C be taken such that A+B: B:: B: C, then the arch itself is nearly equal to

—∆, but a little less, the error in the case of a 15

complete semicircle being less than its, j> δβ part; and when the arch does not exceed 120°, it is less than its 1 / 400000 part; and finally, for a quadrant the error is not greater than its 1 / 300000 part. And farther, that if D be such that A: B::

„ v v . , , 12C + 4B—D ,

B: D, then the arch is nearly equal to ⅛, , but

J 5

a little greater, the error in the semicircle being less than its 1 / 1000 part, and in a quadrant less than its 1 / 60000 part.

Dr. Wallis gave, in his Arithmetica Infinitorum, a singular expression for the ratio of the circle to the square of its diameter. He found that the former was to the latter as 1 to the product 3×3×5×5×7×7×9×9 × 1 1 ×I1, & c. 2×4×4×6×6×8×8×10×10×12

the fractions 5 / 2 , 5 / 4 , 5 / 4 , 5 / 6 , &c., being supposed infinite in number. The products being supposed continued to infinity, we have the ratio exactly; but if we stop at any finite number of terms, as must necessarily be the case in its application, the result will be alternately too great and too small, according as we take an odd or an even number of terms of the numerator and denominator.

An expression of another kind for the ratio of the circle to the square of the diameter was found by Lord Brounker. He showed that the circle being unity, the square of the diameter is expressed by the continued fraction 1

H 9

2+ 2+∞

2+l?_

2+,&c which is supposed to go on to infinity, the numerators 1, 9, 25, 49, Sic., being the squares of the odd numbers 1, 3, 5, 7, &c. By taking two, three, four, &c., terms of this fraction, we shall have a series of approximate values, which are alternately greater and less than its accurate value.

Such were the chief discoveries relating to the quadrature of the circle made before the time of Newton; many others, however, were quickly added by that truly great man, as well as by his contemporaries. In particular, James Gregory found, that t being put for the tangent, the arch is expressed by the very simple series t 3 t i f . t i t —3^^^ 5—7" + 9— & c

By supposing that t =1, in which case the arch is one eighth of the circumference, we have the corresponding arch expressed by the series 4 <⅛M+⅛+* c ∙> which was also given by Leibnitz as a quadrature of the circle in the Acta Eruditorum in the year 1682, but was dis covered by him 1673. Gregory had however found the series under its general form several years before. This series is altogether inapplicable in its present form, on ac count of the slowness of its convergency; for Newton has observed, that to exhibit its value exact to twenty places of figures, there would be occasion for no less than five thou sand millions of its terms, to compute which would take up above a thousand years.

The slowness of the convergency has arisen from our supposing t =1. If we had supposed t greater than 1, then the series would not have converged at all, but on the contrary diverged. But by giving to t a value less than 1, then the rate of convergency will be increased, and that so much the more, as t is smaller.

If we suppose the arch of which t is the tangent to be 30°, then t will be √ 1 / 3 = 1 / 3 √3, and therefore half the circumference to radius unity, or the circumference to the diameter unity, will be

*∕ 12 (1 X3 + 53 3 ~73 s + 9∑∙ &C ')·

Mr. Machin, enticed by the easiness of the process, was induced, about the beginning of the last century, to continue the approximation as far as 100 places of figures, thus finding the diameter to be to the circumference as 1 to 3.14159,26535,89793,23846,26433,83279,50288,41971,69 399,37510,58209,74944,59230,78164,06286,20899,86280, 34825,34211,70680. After him, De Lagny continued it as far as 128 figures. But he has also been outdone; for in Radcliffe’s library at Oxford, there is a manuscript in which it is carried as far as 150 figures!

Although this last series, which was first proposed by Dr. Halley, gives the ratio of the diameter to the circumference with wonderful facility when compared with the operose method employed by Van Ceulen, yet others have been since found which accomplish it with still greater ease.

We have given such a method in our treatise on Algebra, (see article 272.) In the same treatise, (article 273), we have given a very elementary method, which may be understood by the mere elements of geometry. For the analytical methods, see Fluxions.