epithet applied to a ship whose yards are very long. It is also used in contradistinction to all vessels whose sails are extended by stays or lateen-yards, or by booms and gaffs; the usual situation of which is nearly in the plane of the keel; and hence,

Square-Sail, is a sail extended to a yard which hangs parallel to the horizon, as distinguished from the other sails which are extended by booms and stays placed obliquely. This sail is only used in fair winds, or to scud under in a tempest. In the former case, it is furnished with a large additional part called the bonnet, which is then attached to its bottom, and removed when it is necessary to scud. See Scudding.

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 quadrature of the circle is a problem of the same degree of difficulty, and indeed may be regarded as identical with another geometrical 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 (Geometry, Sect. VI. Prop. 3): therefore, if a straight line exactly equal to the circumference could be found, a rectilineal 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 therefore 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 500 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 although 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 curvilinear 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 (Plate D. fig. 1.), 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 radius, let the quadrant arch AEC be described, then shall the curvilinear space bounded by the semicircle ABC and the quadrant arch AEC (which is the Lune of Hippocrates) be equal to the rectilineal triangle ADC. For because circles are to one another as the squares of the radii (Geometry, Sect. VI. Prop. 4); the circle having DA for its radius will be to the circle having HA for its radius as the square of DA to the square of HA, that is, as 2 to 1; hence the former of these circles will be double the latter, and consequently one fourth of the former will be equal to one half of the latter; that is, the quadrant AECD will be equal to the semicircle ABC; from these equals take away the common space bounded by the diameter AC and the arch AEC, and there will remain the triangle ADC equal to the lunular space AECBA.

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 showed 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, viz. Bryson and Antiphon, who about his time, or a little earlier, were occupied with the quadrature of the circle. The former appears, according to the testimony of Alexander Apollonius, to have erred most egregiously; he having concluded that the circumference was exactly 3\(\frac{1}{7}\) times the diameter. And the latter seems to have proceeded pretty much in the same manner as Archimedes afterwards did in squaring the parabola, that is, by first inscribing a square in the circle, then an isosceles triangle 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, however, could not be attended with any success, as it is well known that the spaces thus formed do not, as in the case of the parabola admit of being absolutely summed.

It may naturally be supposed that Archimedes exerted his utmost efforts to resolve this problem; and probably it was only 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, which has been preserved from the period in which he wrote, about 250 years before Christ, to the present times. He found his approximation to the ratio, by supposing a regular polygon of 96 sides to be described about the circle, and another of the same number of sides to be inscribed in it, and by shewing that the perimeter of the circumscribing polygon was less than \(3\frac{1}{7}\) or \(3\frac{1}{7}\) times the diameter, but that the perimeter of the inscribed figure was greater than \(3\frac{1}{7}\) times the SQUARING or Quadrature of the Circle.

PLATE D.

STARCH. Manufacture of.

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7 SQU

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 $3\frac{1}{7}$ times the diameter, but greater than $3\frac{4}{7}$ times; so that, taking the first of these limits as being expressed by the smallest numbers, the circumference will be to the diameter as $3\frac{1}{7}$ to $1$, or as $22$ to $7$ nearly.

Although the approximate ratio investigated by Archimedes be the oldest known to have been found in the western world, yet one more accurate seems to have been known at a much earlier period in India. This we learn from the Institutes of Akbar (Aycen Akberry) where it is said that the Hindoos suppose the diameter of a circle to be its circumference as $1250$ to $3927$. Now this ratio, which is the same as that of $1$ to $3.1416$, when found in the simplest and most elementary manner must have required the inscription of a polygon of $758$ sides in the circle, and must have been attended with nine extractions of the square root, each carried as far as ten places of figures.

We learn from Simplicius that Nicomedes 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 also by the help of a curve denominated the sister to the tortuous line or spiral, and which was probably no other than 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. Then 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 quadrant 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 quadrant arch AFB is the limit of the ratio of CE to CF, 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 quadrant arch, (by finding a third proportional to CH and CB) and consequently a straight line equal to the circumference. The point H, however, cannot be determined by a geometrical construction, and therefore 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 squares of the circle. We however know nothing more than that some one of them believed he had discovered that the diameter being unity, the circumference was the square root of $10$; a very gross mistake; for the square root of $10$ exceeds $3.162$; but Archimedes had demonstrated that the circumference was less than $3.143$.

It appears that, even during the dark ages, some attempts were made at the resolution of this famous problem, which however 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 perimeter of this triangle would be equal to the circumference of the other circle. This pretended quadrature of the cardinal's was refuted by Regiomontanus; and indeed the task was not difficult; for, according to his construction, the diameter being $1$, the circumference was greater than $3\frac{1}{7}$; a conclusion which must be absurd, seeing that Archimedes had demonstrated that it must be less than that number.

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 excellent work, Histoire des Mathematiques, we find upwards of forty pretenders to the honour of this discovery enumerated. These 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. If those who have sought the quadrature of the circle had been previously initiated in the doctrines of geometry, although they missed attaining the object they had in view, yet they could not have failed to have extended the boundaries of the science by the discovery of many new propositions. From such persons, however, as have generally pursued this inquiry, no improvement whatever of the science was to be expected; although, indeed, in some instances, it has derived advantage from the labours of such as have undertaken to expose the absurdity of their conclusions; as in the case of Metius, who in refusing the quadrature of one Simon à Quercy, found a much nearer approximation to the ratio of the diameter to the circumference than had been previously known, at least in Europe, viz. that of $\frac{113}{355}$, which reduced to decimals is the same as the ratio of $1$ to $3.1415929$, differing from the truth only in the seventh place of decimals.

Among the most remarkable of those who have recorded their own folly by publishing erroneous resolutions of the problem, we may reckon the celebrated Joseph Scaliger. Full of self-conceit, he believed that, entering entering upon the study of geometry, he could not fail to surmount by the force of his genius those obstacles which had completely stopt the progress of all preceding inquirers. He gave the result of his meditations to the world in 1592, under the title Nova Cyclometrica; but he was refuted by Clavins, by Vieta, and others, who shewed that the magnitude he had assigned to the circumference was a little less than the perimeter 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 discovery have not been more remarkable for their folly in committing absurd blunders, than for their obstinacy in maintaining that they were in the right, and all who held a contrary opinion in an error.

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 trisection of an angle, the rectification of the parabola, &c.; but his pretended solutions were refuted by Dr Wallis. And this circumstance afforded him occasion to write not only against geometers, but even against the science of geometry itself.

We find it recorded by Montuela, 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 Chrysespis seu Quadratura Circuli, suffered himself at last to be convinced by some of his friends that he was wrong both in his quadrature of the circle, and in his rectification of the spiral. But a solution of the same problem found out by one Mathulen of Lyons, did not produce in the end so much advantage to its author. This man in 1728 announced to the learned 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 consigned 1000 ecus (about 125l.) to be paid to any one who should demonstrate that he was deceived in either. The task was not difficult. Nicole of the Academy of Sciences demonstrated that he was wrong, and he himself allowed it; but he hesitated to pay the money, which Nicole had relinquished in favour of the Hotel Dieu of Lyons. The affair went before a court of justice, which adjudged the money to be paid, as Nicole had destined it, to the poor. At a later period, viz. in 1753, the Chevalier de Causans, a French officer, and a man who was never expected 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 discovery 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 judged that the Chevalier's fortune ought not to suffer on account of his whim; for, setting aside this piece of folly, in every other respect he was a worthy man. The procedure was therefore stopt, and the wager declared void.

We shall not enter farther 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 different 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 also of the whole circle. We here speak of an analytical solution of the problem; the ancients, however, 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, however, 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, it is commonly understood that no unexceptionable demonstration of its impossibility has hitherto been published. It is true that James Gregory, in his vera circular et Hyperbolic quadratura, has given what he considered as such a demonstration; but it has been objected to, particularly by Huygens, one of the best geometers of his time. We are, however, certain that the ratio of the diameter to the circumference, as 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 Geometric. As to the indefinite quadrature, if Newton's demonstration of the 28th 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 indeed we know one oval curve, which returns into itself, and which, according to Newton's proposition, ought therefore not to admit of an indefinite quadrature; yet this is by no means the case, for it does really admit of such a quadrature. The curve we mean is the lemniscata, the equation of which is \((x^2+y^2)^2=a(x^2-y^2)\), where \(x\) and \(y\) denote its coordinates. ordinates, and \(a\) is put for a given line. The figure of the curve 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.

We come now to speak of the different methods which have been found for approximating to the area or to the circumference. We have already noticed the approximation to the ratio of the diameter to the circumference found by Archimedes, and the earlier and more accurate approximation of the Indian mathematicians. Archimedes's ratio 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 \(\frac{113}{355}\), as we have already noticed. About the same time Vieta and Adrianus Romanus published their ratios expressed in decimals, the former carrying the approximation to ten decimals instead of six, (which was the number of accurate figures expressed by 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 \(3.141592653589793\ldots\), but less than the same number with the last figure increased by an unit. This work was translated into Latin by Saelius, and published under the title, *De Circulo et Descriptis*. In finding this approximation, Van Ceulen followed the method of Archimedes, doubling continually the number of sides of the inscribed and circumscribed polygons, until at length he found two which differed only by an unit in the 36th place of decimals in the numbers expressing their perimeters. This, however, must have been rather a work of patience than of genius; and 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, in commemoration of his discovery of the proportion which these solids bear to one another, he requested that the ratio he had found might be inscribed on his tomb; which was accordingly done.

Snellius found means to abridge greatly the labour of calculation by some very ingenious theorems; and although he did not go beyond Van Ceulen, yet he verified his result. His discoveries on this subject are contained in a work called *Willebrordi Snelli Cyclometricus de Circuli Dimensione*, &c., Lugd. Bat. 1621.

Descartes found also 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. It is true he committed a great error in supposing he had discovered the quadrature of both the circle and hyperbola; but he had previously made so many beautiful geometrical discoveries deduced with much elegance after the manner of the ancients, that Squaring it would be wrong to number him with those absurd pretenders which we have already noticed. Gregory's mistake was the cause of a sharp controversy carried on between his disciples on the one side, and by Huygens, Mercenius, and Lestaud, on the other; and it 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 Hyperbolae, 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 \(c + \frac{s}{2} (c-s)\), but less than \(c + \frac{4c+s}{2c+3s} \times \frac{3}{2} (c-s)\): he also showed that the arch is less than the sum of \(\frac{3}{2}\) of its sine and \(\frac{1}{4}\) of its tangent. These theorems greatly shorten the labour of approximating to the ratio of the diameter to the circumference, by means of inscribed and circumscribed figures, in somuch that by the inscribed polygons of 6 and 12 sides, we may obtain it to the same degree of accuracy as Archimedes did by the inscribed and circumscribed polygons of 66 sides.

James Gregory, in his *Vera Circuli et Hyperbolae Quadratura*, 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 \(A+B:B::2B:C\), then the arch itself is nearly equal to \(\frac{8C+8B-A}{15}\), but a little less, the error in the case of a complete semicircle being less than its \(\frac{1}{10000}\) part; and when the arch does not exceed \(120^\circ\), it is less than its \(\frac{1}{100000}\) part; and finally, for a quadrant the error is not greater than its \(\frac{1}{1000000}\) part. And farther, that if \(D\) be such that \(A:B::B:D\), then the arch is nearly equal to \(\frac{12C+4B-D}{15}\), but a little greater, the error in the semicircle being less than its \(\frac{1}{10000}\) part, and in a quadrant less than its \(\frac{1}{100000}\) part.

The discoveries of Dr Wallis, delivered in his *Arithmetica Infinitorum* published in 1655, led him to 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

\[ \frac{3 \times 3 \times 5 \times 5 \times 7 \times 7 \times 9 \times 9 \times 11 \times 11 \times \ldots}{2 \times 4 \times 4 \times 6 \times 6 \times 8 \times 8 \times 10 \times 10 \times 12} \]

the fractions \(\frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \ldots\) 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. Thus Thus the fraction \( \frac{1}{2} \) is too great; on the other hand,

\[ \frac{3 \times 3}{2 \times 4} = \frac{9}{8} \text{ is too small, and } \frac{3 \times 3 \times 5}{2 \times 4 \times 4} = \frac{45}{32} \text{ too great,} \]

and so on. But to approach as near as possible in each case, Wallis directs to multiply the product by the square root of a fraction formed by adding to unity the reciprocal of the last factor in either its numerator or denominator; then the result, although much nearer, will be too great if the number whose reciprocal is taken be the last in the numerator, but too small if it be the number in the denominator. Thus the following series of expressions will give approximate values of the infinite product

\[ \frac{3 \times 3 \times 5 \times 5 \times 7 \times 7 \times \cdots}{2 \times 4 \times 4 \times 6 \times 6 \times 8 \times 8 \times \cdots} \]

which are alternately too great and too small.

\[ \frac{3 \times 3 \times 5}{2 \times 4 \times 6} \sqrt{(1 + \frac{1}{2})}; \quad \frac{3 \times 3 \times 5 \times 7}{2 \times 4 \times 6 \times 8} \sqrt{(1 + \frac{1}{2})}; \]

these values, alternately too great and too small, fall between the known limits.

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

\[ 1 + \frac{1}{2 + \frac{9}{2 + \frac{25}{2 + \frac{49}{2 + \cdots}}}} \]

which is supposed to go on to infinity, the numerators 1, 9, 25, 49, &c. 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, Newton himself showed that if \( s \) denote the sine, and \( v \) the versed sine of an arch, then the radius being unity, the arch is equal to either of the following series,

\[ s + \frac{1 \cdot 3^3}{2 \cdot 3} + \frac{1 \cdot 3 \cdot 5^3}{2 \cdot 4 \cdot 6} + \frac{1 \cdot 3 \cdot 5 \cdot 7^3}{2 \cdot 4 \cdot 6 \cdot 8} + \cdots \]

And James Gregory found that \( t \) being put for the tangent, the arch is expressed by the very simple series

\[ t - \frac{t^3}{3} + \frac{t^5}{5} - \frac{t^7}{7} + \frac{t^9}{9} + \cdots \]

We have investigated the first of these series at § 140, and the third at § 137, of the article Fluxions: the second is easily obtained from the first by considering that since the sine of an arch is half the chord of twice the arch, that is, half of a mean proportional between the diameter and versed sine of twice the arch; we have therefore only to multiply the first series by 2, and to substitute \( \frac{1}{\sqrt{2v}} \) instead of \( s \), and we get the second series.

By taking \( s = \frac{1}{2} \), then, because in this case the arch contains 30°, we have half the circumference to the radius 1, or the whole circumference to the diameter 1, expressed by the infinite series

\[ 3(1 + \frac{1}{2 \cdot 3 \cdot 2^2} + \frac{1 \cdot 3}{2 \cdot 4 \cdot 5 \cdot 2^2} + \frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6 \cdot 7 \cdot 2^2} + \frac{1 \cdot 3 \cdot 5 \cdot 7}{2 \cdot 4 \cdot 6 \cdot 8 \cdot 9 \cdot 2^2} + \cdots) \]

And by supposing that in the third series \( t = 1 \), in which case the arch is one-eighth of the circumference, we have the same things expressed by the series

\[ 4(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \frac{1}{11} + \cdots) \]

which was given by Leibnitz as a quadrature of the circle in the Leipsic Acts in the year 1682; but was discovered by him 1673. Gregory, however, had found the series under its general form several years before. By the first of these two numeral series we can readily compute the circumference of the circle to a tolerable degree of accuracy; but the second is altogether inapplicable in its present form on account 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 thousand 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 \( \sqrt{\frac{1}{3}} = \sqrt{3} \), and therefore half the circumference to radius unity, or the circumference to the diameter unity, which in this case is \( 6t(1 - \frac{t^3}{3} + \frac{t^5}{5} - \frac{t^7}{7} + \frac{t^9}{9} - \cdots) \) will be

\[ \sqrt{12}(1 - \frac{1}{3 \cdot 3} + \frac{1}{5 \cdot 3} - \frac{1}{7 \cdot 3} + \frac{1}{9 \cdot 3} - \cdots) \]

By means of this series, in an hour's time the circumference may be found to be nearly 3.141592653599, which is true to 11 decimal places, and is a very considerable degree of accuracy, considering the smallness of the labour. But 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 3.141592653589793, 2384626433832795028841971693993751058209749445923078164062862089986280348253421170685. After him, De Lagny continued it as far as 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 opose method employed by Van Ceulen, yet others have been since found which accomplish it with still greater ease. In Halley's series we have to compute the irrational quantity $\sqrt{12}$, because of the irrational value which it was necessary to give to $t$ in order to render it sufficiently small, and at the same time an exact part of the whole circumference; but Mr Machin contrived, by a very ingenious artifice, to reduce the computation of an arch of $45^\circ$, and consequently the length of the whole circumference, to two series which contain only rational quantities, and which at the same time converge with great rapidity. The nature of this artifice, and the manner in which it occurred to its author, is explained by Dr Hatton in his very excellent treatise on Mensuration, as follows: "Since the chief advantage (in the application of Gregory's series to the rectification of the circle) consists in taking small arches, whose tangents shall be numbers easy to manage, Mr Machin very properly considered, that since the tangent of $45^\circ$ is $1$, and that the tangent of any arch being given, the tangent of the double of that arch can easily be had; if there be assumed some small simple number as the tangent of an arch, and then the tangent of the double arch be continually taken, until a tangent be found nearly equal to $1$, which is the tangent of $45^\circ$, by taking the tangent answering to the small difference of $45^\circ$ and this multiple, there would be found two very small tangents, viz. the tangent first assumed, and the tangent of the difference between $45^\circ$ and the multiple arch; and that therefore the lengths of the arches corresponding to these two tangents being calculated, and the arch belonging to the tangent first assumed being so often doubled as the multiple directs, the result, increased or diminished by the other arch, according as the multiple should be below or above it, would be the arch of $45^\circ$.

Having thus thought of his method, by a few trials he was lucky enough to find a number (and perhaps the only one) proper for his purpose; viz. knowing that the tangent of $\frac{1}{2}$ of $45^\circ$ is nearly $\frac{1}{2}$, he assumed $\frac{1}{2}$ as the tangent of an arch. Then, since if $t$ be the tangent of an arch, the tangent of the double arch will be $\frac{2t}{1-t^2}$, the radius being $1$, the tangent of the double arch to that of which $\frac{1}{2}$ is the tangent will be $\frac{2}{\sqrt{3}}$, and the tangent of the double of this arch will be $\frac{4}{\sqrt{7}}$, which being very nearly equal to $1$, shews, that the arch which is equal to four times the first arch is very near $45^\circ$. Then, since the tangent of the difference between an arch of $45^\circ$, and an arch greater than $45^\circ$, whose tangent is $T$, is $T-1$, we shall have the tangent of the difference between $45^\circ$, and the arch whose tangent is $\frac{1}{\sqrt{3}}$, equal to $\frac{1}{\sqrt{7}}$. Now, by calculating from the general series the arches whose tangents are $\frac{1}{\sqrt{3}}$ and $\frac{1}{\sqrt{7}}$, (which may be quickly done by reason of the smallness and simplicity of the numbers), and taking the latter arch from four times the former, the remainder will be the arch of $45^\circ$."

If we substitute $\frac{1}{2}$ instead of $t$ in the general series, we shall have the arch whose tangent is $\frac{1}{2}$ expressed by the series $\frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} + \ldots$, &c.; and, in like manner, by substituting $\frac{1}{\sqrt{3}}$ for $t$, we get the arch whose tangent is $\frac{1}{\sqrt{3}}$ expressed by the series $\frac{1}{3} - \frac{1}{5} + \frac{1}{7} - \frac{1}{9} + \ldots$, &c. Now, since four times the arch to tan. $\frac{1}{2}$ diminished by the arch to tan. $\frac{1}{\sqrt{3}}$ is equal to the arch to tan. $1$, that is, to the arch of $45^\circ$ or $\frac{\pi}{4}$ of the semicircumference; therefore, half the circumference of a circle to rad. $= 1$, or the whole circumference, the diameter being $1$, is equal to

$$\frac{16}{3} \left( \frac{1}{3} - \frac{1}{5} + \frac{1}{7} - \frac{1}{9} + \ldots \right)$$

and this is Machin's series for the rectification of the circle.

The happy idea which Machin had conceived of reducing the rectification of the arch whose tangent is unity to that of two arches whose tangents are small rational fractions, having each unity for a numerator, appears also to have occurred to Euler; and the same thought has, since his time, been pursued by other mathematicians, who have contrived to resolve an arch of $45^\circ$ into three or more such arches. We shall show how this may be done, beginning with the investigation of the following problem.

**Problem.** Supposing $n$, $x$, and $y$, to denote three whole numbers, such, that the arch whose tangent is $\frac{1}{n}$ is equal to the sum of two arches whose tangents are $\frac{1}{x}$ and $\frac{1}{y}$, radius being unity, it is required to determine all possible values of the numbers $x$ and $y$ in terms of the number $n$.

**Solution.** It is manifest from the formula for the tangent of the sum of two arches (ALGEBRA, § 368.) that

$$\frac{1}{n} = \frac{1}{x} + \frac{1}{y}$$

hence we have $\frac{1}{n} = \frac{x+y}{xy}$, and $n(x+y)$

$$= xy - 1,$$

and $y(x-n) = nx + 1$; and, lastly, $y = \frac{nx+1}{x-n} = n + \frac{1}{x-n}$. Now, as by hypothesis, $y$ is a whole number, it is manifest that $\frac{n^2+1}{x-n}$ must be a whole number; therefore, $x-n$ must be a divisor of $n^2+1$. Let $p$ be any divisor of $n^2+1$, and $q$ the quotient, that is, let $pq = n^2+1$, then $x-n = p$, and $x = n+p$. And since $\frac{n^2+1}{x-n} = \frac{pq}{p} = q$, therefore $y = n+q$; thus the values of $x$ and $y$ are determined in terms of $n$ as required; and by giving to $p$ and $q$ all possible values, we shall have... Squaring have all the values of \( x \) and \( y \) that can exist. This solution affords us the following theorem.

**Theorem.** Let \( n \) denote any whole number, and let \( n^2 + 1 \) be resolved into any two factors \( p \) and \( q \), (one of which may be unity), that is, let \( pq = n^2 + 1 \); the arch whose tangent is \( \frac{1}{n} \) is equal to the sum of the arches whose tangents are \( \frac{1}{n+p} \) and \( \frac{1}{n+q} \) respectively.

For the sake of brevity, let \( A_1 = \frac{1}{n} \) be put to denote the arch, having for its tangent \( \frac{1}{n} \); then, according to this notation, our theorem will be expressed thus, \( A_1 = \frac{1}{n+p} + \frac{1}{n+q} \). Let us now suppose \( n = 1 \), then \( n^2 + 1 = 2 = 1 \times 2 \); therefore, the only values which we can give in this case to \( p \) and \( q \) are \( p = 1, q = 2 \), and these being substituted, we have

\[ A_1 = A_2 + A_3. \]

From which it appears, that the arch whose tangent is unity (that is, \( \frac{1}{\pi} \) of the circumference), is the sum of the arches whose tangents are \( \frac{1}{2} \) and \( \frac{1}{4} \). This is Euler's theorem, and by means of it, putting \( \frac{1}{2} \) and \( \frac{1}{4} \) for \( t \) in the general series \( t - \frac{t^3}{3!} + \frac{t^5}{5!} - \frac{t^7}{7!} + \ldots \), &c., we get half the circumference to radius 1 equal to

\[ \frac{1}{2} - \frac{1}{3 \cdot 2^3} + \frac{1}{5 \cdot 2^5} - \frac{1}{7 \cdot 2^7} + \frac{1}{9 \cdot 2^9}, \ldots \]

Let us now suppose \( n = 2 \), then \( n^2 + 1 = 5 = 1 \times 5 \); hence the only values which \( p \) and \( q \) can have are 1 and 5; and in this case our general formula gives \( A_1 = A_2 + A_3 \).

If now from the two equations

\[ A_1 = A_2 + A_3; \quad A_2 = A_3 + A_4, \]

we eliminate successively \( A_2 \) and \( A_3 \), we shall obtain the two following:

\[ A_1 = 2A_2 + A_3; \quad A_1 = 2A_2 - A_3. \]

From the first of these it appears that \( \frac{1}{\pi} \) of the circumference is equal to the sum of twice the arch to tan. \( \frac{1}{2} \) and once the arch to tan. \( \frac{1}{4} \); and from the second, that the same quantity is equal to the excess of twice the arch to tan. \( \frac{1}{2} \) above the arch to tan. \( \frac{1}{4} \); and from each of these, an expression for the whole circumference may be obtained analogous to that which we have found above from Euler's formula, but which will converge faster, and therefore is better.

The resolution of an arch of \( 45^\circ \) into three other arches, may be effected by means of our general formula, as follows: Put \( n = 3 \), then \( n^2 + 1 = 10 = 1 \times 10 = 2 \times 5 \), hence we have \( p = 1 \), and \( q = 10 \), and also \( p = 2 \), and \( q = 5 \); therefore, substituting, we get two different values of \( A_1 \), viz.

\[ A_1 = A_2 + A_3; \quad A_1 = A_2 + A_3. \]

From these, and the equation \( A_1 = 2A_2 + A_3 \), we get, by exterminating \( A_2 \), the two following expressions for \( A_1 \), an arch of \( 45^\circ \):

\[ A_1 = 2A_2 + A_3 + 2A_4; \quad A_1 = 2A_2 + A_3 + 2A_4. \]

These give such an expression for the circumference composed of three series. The labour, however, of computing by either of them, particularly the latter, will probably be less than by any of the formulas composed of two series, on account of the greater degree of quickness with which the series will converge. All the preceding formulas have been investigated in different ways by different mathematicians. That, however, which we are about to investigate, we believe, is new.

Let \( n \) in the general formula be taken equal to 5; then \( n^2 + 1 = 26 = 1 \times 26 = 2 \times 13 \), therefore \( p = 1, q = 26 \), also \( p = 2, q = 13 \), hence we find \( A_1 = A_2 + A_3 \), and also \( A_2 = A_3 + A_4 \). From this last equation, and the equation \( A_1 = 2A_2 + A_3 + 2A_4 \), let \( A_2 \) be eliminated, and the result is

\[ A_1 = 3A_2 + 2A_3 + 2A_4. \]

This appears to be the most convenient expression of any we have yet found, because the fractions are smaller, while at the same time two of the denominators consist of only one figure, and the third, which consists of two, admits of being resolved into factors. By the same mode of reasoning we have found this expression

\[ A_1 = 2A_2 + 3A_3 + 2A_4 + 3A_5 + 3A_6, \]

which consists of four terms; but for the sake of brevity we omit its investigation.

We shall now apply the formula \( A_1 = 3A_2 + 2A_3 + 2A_4 + 3A_5 + 3A_6 \) to the actual calculation of the arch of \( 45^\circ \), the radius of the circle being unity.

I. Calculation of the length of the arch whose tangent is \( \frac{1}{2} \).

In this case, because \( t = \frac{1}{2} \), we have

\[ A_1 = \frac{1}{2} - \frac{1}{3 \cdot 2^3} + \frac{1}{5 \cdot 2^5} - \frac{1}{7 \cdot 2^7} + \frac{1}{9 \cdot 2^9}, \ldots \]

II. Calculation of the length of the arch whose tangent is \( \frac{1}{8} \).

Here \( t = \frac{1}{8} \), therefore,

\[ A_1 = \frac{1}{8} - \frac{1}{3 \cdot 8^3} + \frac{1}{5 \cdot 8^5} - \frac{1}{7 \cdot 8^7} + \ldots \] Thus by a very easy calculation we have obtained one-fourth of the circumference true to 12 decimal places; and indeed by this method we may find an approximate value of the ratio of the diameter to the circumference to 200 places of figures, with perhaps as much ease as Vieta or Romansus found it to 10 or 15 figures. We have already observed that Van Ceulen desired that his quadrature, which extended only to 35 decimals, might be inscribed on his tomb; from which we may reasonably infer that the time and labour he had bestowed in the calculation must have been very great; but by an artifice of the kind we have been explaining, Euler in 18 hours verified Lagny's quadrature of 128 figures.

In concluding this article we shall briefly notice some series for the indefinite rectification of the circle, which have just appeared in the sixth volume of the Edinburgh Philosophical Transactions. They are given by Mr W. Wallace of the Royal Military College, in a paper entitled, New Series for the Quadrature of the Conic Sections, and the Computation of Logarithms. These series do not give the arch directly, but only its reciprocal, or the powers of that reciprocal; it is however evident, that any one of these being known, the arch itself becomes immediately known. The first series is as follows. Let \(a\) denote any arch of a circle, and let its tangent, the tangents of its half, &c. be briefly denoted by \(\tan_a\), \(\tan_{\frac{1}{2}a}\), &c. Then shall

\[ \tan_a = \frac{1}{\tan_a} + \frac{1}{4} \tan_{\frac{1}{2}a} + \frac{1}{4} \tan_{\frac{3}{4}a} + \cdots + T + T' + S. \]

Here the arches \(a\), \(\frac{1}{2}a\), \(\frac{3}{4}a\), &c. constitute a geometrical progression, having the number of its terms infinite, and their common ratio \(\frac{1}{2}\). The letters \(T\) and \(T'\) are put for any two adjoining terms (after the first) of the series, and \(S\) is put for the sum of all the terms following these; and this sum is always contained between two limits, one of which is \(\frac{1}{2}\) of the latter of the two terms \(TT'\), and the other is a third proportional to their difference; and the last of the two being always less than the first of these limits, but greater than the second. As a specimen of the way of applying this series, we shall give the calculation of the length of an arch of \(90^\circ\) to six decimal places. In this case

\[ \tan_a = \cotan_a = \tan_{\frac{1}{2}a} = \tan_{\frac{3}{4}a}, \text{the remaining quantities} \]

\[ \tan_{\frac{1}{2}a}, \tan_{\frac{3}{4}a}, \text{&c. are to be computed from } \tan_{\frac{1}{2}a}. \]

by this formula, \(\tan_{\frac{1}{2}A} = \sqrt{\left(\frac{1}{\tan^2 A} + 1\right)} - \frac{1}{\tan A}\).

Accordingly we find

\[ \tan_{\frac{1}{2}a} = 1, \quad \tan_{\frac{1}{2}a} = 0.984914, \quad \tan_{\frac{3}{4}a} = 0.491268, \quad \tan_{\frac{3}{4}a} = 0.245486, \]

\[ \tan_{\frac{1}{2}a} = 0.1989123, \quad \tan_{\frac{3}{4}a} = 0.061557, \quad \tan_{\frac{3}{4}a} = 0.0015352, \]

\[ \tan_{\frac{3}{4}a} = 0.0002836, \quad S = 0.0001278, \quad T = 0.0001278. \]

The second series given in this paper is expressed as follows. Let \(\cos_a\), \(\cos_{\frac{1}{2}a}\), &c. denote the cosine of the arch, the cosine of its half, &c. Then

\[ \cos_a = \frac{1}{4} \cos_a + \frac{1}{4} \cos_{\frac{1}{2}a} + \frac{1}{4} \cos_{\frac{3}{4}a} + \cdots + T + T' + S. \]

Here, as before, the letters \(T\), \(T'\) denote any two adjacent terms of the series in the parenthesis, and \(S\) is put for the sum of all the following terms, which in this case is always less than \(\frac{1}{2}T'\), but greater than a third proportional to \(T-T'\) and \(T'\). This second series converges. verges quicker than the first, and is besides better adapted to calculation, because the cosines of the series of arches \( \frac{1}{2}a, \frac{3}{4}a, \ldots \) are now easily deduced from the cosine of \( a \) and one another than the tangents. The formula in this case being \( \cos \frac{1}{2}A = \sqrt{\frac{1 + \cos A}{2}} \).

There are various other series for the rectification of any arch of a circle given in the same paper, some of which converge faster than either of the two we have here specified, and all have the property of being applicable to every possible case, and of having very simple limits, between which the sums of all their terms following any proposed term are always contained. It may also be observed that the principles from which they are deduced are of the most simple and elementary kind, insomuch that the author has stated it as his opinion, that their investigation might even be admitted into and form a part of the elements of geometry.

**SQUATINA.** See **SQUALUS**, **Ichthyology Index**.