poffible to avoid allowing this to have been Euclid's grand object. And accordingly he determined the chief properties in the menfuration of rectilineal plane and folid figures; and fquared all fuch planes, and cubed all fuch folids. The only curve figures which he attempted, are the circle and sphere; and when he could not accurately determine their measures, he gave an excellent method of approximating to them, by fhewing how in a circle to infcribe a regular polygon, which fhould not touch another circle, concentric with the former, although their circumferences fhould be ever fo near together; and, in like manner, between any two concentric fpheres to defcribe a polyhedron, which should not any-where touch the inner one: and approximations to their measures are all that have hitherto been given. But although he could not fquare the circle, nor cube the fphere, he determined the proportion of one circle to another, and of one fphere to another, as well as the proportions of all rectilineal fimilar figures to one another.

Archimedes took up menfuration where Euclid left it, and carried it a great length. He was the first who fquared a curvilineal space; unless Hypocrates must be excepted on account of his lunes. In his time the conic fections were admitted into geometry, and he applied himself clofely to the meafuring of them, as well as other figures. Accordingly he determined the relations of spheres, fpheroids, and conoids, to cylinders and cones; and the relations of parabolas to rectilineal planes whofe quadratures had long before been determined by Euclid. He hath left us alfo his attempts upon the circle: he proved that a circle is equal to a right-angled triangle, whofe bafe is equal to the circumference, and its altitude equal to the radius ; and confequently, that its area is equal to the rectangle of the radius, and half the circumference; and fo reduced the quadrature of the circle to the determination of the ratio of the diameter to the circumference; but which, however, hath not yet been done. Being difappointed of the exact quadrature of the circle, for want of the rectification of its circumference, which all his methods would not effect, he proceeded to affign an ufeful approximation to it; this he effected by the numeral calculation of the perimeters of the infcribed and circumfcribed polygons; from which calculation it appears, that the perimeter of the circumfcribed regular polygon of 192 fides, is to the diaineter, in a lefs ratio than that of 3 (3) to 1, and that the infcribed polygon of 96 fides, is to the diameter, in a greater ratio than that of 3 to 1; and confequently much more that the circumference of the circle is to the diameter, in a lefs ratio than that of 3 to 1, but greater than that of 3 to 1: The first ratio of 3 to 1, reduced to whole numbers, gives that of 22 to 7, for 3:1:: 22:7, which therefore is nearly the ratio of the circumference to the diameter. From this ratio of

the

the circumference to the diameter, he computed the approximate area of the circle, and found that it is to the fquare of the diameter, as 11 is to 14.-He likewife determined the relation between the circle and ellipfe, with that of their fimilar parts. It is highly probable that he likewife attempted the hyperbola; but it is not to be imagined that he met with any fuccefs, fince approximations to its area are all that can be given by the various methods that have fince been invented.

Befides these figures, he hath left us a treatife on the fpiral, described by a point moving uniformly along a right line, which at the fame time moves with an uniform angular motion; and he determined the proportion of its area to that of the circumfcribed circle; as alfo the proportion of their fectors.

Throughout the whole works of this great man, which are chiefly on menfuration, he every where discovers the deepest defign, and the finest invention; and feems to have been, with Euclid, exceedingly careful of admitting into his demonstrations nothing but principles perfectly geometrical and unexceptionable: and although his moft general method of demonftrating the relations of curved figures to straight ones, be by infcribing polygons in them; yet to determine thofe relations, he does not increafe the number, and diminish the magnitude, of the fides of the polygon in infinitum; but from this plain fundamental principle, allowed in Euclid's Elements, viz. that any quantity may be fo often multiplied, or added to itself, as that the refult fhould exceed any propofed finite quantity of the fame kind, he proves that to deny his figures to have the propofed relations, would involve an abfurdity.

He demonftrated alfo many properties, particularly in the parabola, by means of certain numeral progreffions, whofe terms are fimilar to the infcribed figures; but ftill without confidering such series as continued in infinitum, and then fumming up the terms of fuch infinite feries.

He had another very curious and fingular contrivance for determining the measures of figures, in which he proceeds as it were mechanically, by weighing them, or from the properties of the center of gravity.

Several other eminent men among the ancients wrote upon this fubject, both before and after Euclid and Archimedes; but their attempts were usually confined to particular parts of it, and made according to methods not effentially different from theirs. Among these are to be reckoned Thales, Anaxagoras, Pythagoras, Bryfon, Antiphon, Hypocrates of Chios, Plato, Apollonius, Philo, and Ptolemy; most of whom wrote of the quadrature of the circle; and thofe after Archimedes, by his method, ufually extended the approximation to a greater degree of accu

racy,

24

Many

3

Many of the moderns alfo have profecuted the fame problem of the quadrature of the circle, after the fame methods, to greater lengths; fuch are Vieta, and Metius, whofe proportion between the diameter and circumference, is that of 113 to 355, which is within about of the true ratio; but above all, Ludolph van Collen, or a Ceulen, who, with an amazing degree of induftry and patience, by the fame methods, extended the ratio to 36 places of figures, making the ratio to be that of 1 to 314159265358979323846264338327950288 + or 9-. And the fame was repeated and confirmed by his editor Snellius. The first material deviation from the principles ufed by the ancients, in geometrical demonftrations, was made by Cavalerius: the fides of their inferibed and circumfcribed figures, they always fuppofed to be of a finite and affignable number and length; he introduced the doctrine of indivifibles, a method which was very general and extenfive, and which, with great eafe and expedition, ferved to measure and compare geometrical figures. Very little new matter, however, was added to geometry by this method, its facility being its chief advantage. But there was great danger in ufing it, and it foon led the way to infinitely small elements, and infinitefimals of endle's orders; methods which were very ufeful in refolving difficult problems, and in inveftigating or demonftrating theories that are general. and extensive ; but fometimes led their incautious followers into errors and mistakes, which occafioned difputes and animofities amongst them. There were now, however, many excellent things performed in this fcience; not only many new properties were difcovered concerning the old figures, but new curves were measured; and although feveral of them could not be exactly fquared or cubed, yet general and infinite approximating feries were affigned, of which the laws of their continuation were manifeft, and in fome of which the terms were independent of each other. Dr. Wallis, Mr. Huygens, and Mr. James Gregory, performed wonders: Huygens in par ticular muft always be admired for his folid, accurate, and very masterly works.

During the preceding ftate of things, feveral men, whofe vas nity feemed to have overcome their regard for truth, afferted, that they had difcovered the quadrature of the circle, and pub lifhed their attempts in the form of strict geometrical demonftrations, with fuch affurance and ambiguity, as ftaggered and mifled many who could not fo well judge for themfelves, and perceive the fallacy of their principles and arguments. Among thofe were Longomontanus, and our countryman Hobbes, who obftinately refufed all conviction of his errors.

The ufe of infinites was, however, difliked by feveral people, and particularly by Sir Ifaac Newton, who, among his nume

rous

rous and great difcoveries, hath given us that of the method of fluxions; a difcovery of the greatest importance, both in philofophy and mathematics; being a method fo general and extenfive, as to include all inveftigations concerning magnitude, distance, motion, velocity, time, &c. with wonderful ease and brevity; a method established by its great author upon true and incontestable principles; principles perfectly confiftent with those of the ancients, and which were free from the imperfections and abfurdities attending fome that had lately been introduced by the moderns: he rejected no quantities as infinitely small, nor fuppofed any parts of curves to coincide with right lines; but propofed it in fuch a form as admits of a ftrict geometrical demonftration. Upon the introduction of this method, moft sciences affumed a different appearance, and the most abstruse problems became eafy and familiar to every one; things which before feemed to be infuperable, became eafy examples, or particular cafes, of theories still more general and extenfive; rectifications, quadratures, cubatures, tangencies, cafes de maximis & minimis, and many other fubjects, became general problems, and were delivered in the form of general theories, which included all particular cafes: thus, in quadratures, a formula was affigned, which would exprefs the areas of all poffible curves whatever, both known and unknown, and which, by proper fubftitutions, gave the area for any particular cafe, either in finite terms, or in infinite feries, of which any term, or any number of terms, could be eafily affigned; and the like in other things. And although no curve, whofe quadrature was unfuccessfully attempted by the ancients, became by this method perfectly quadrable, yet many general methods were difcovered. for approximating to their areas, of which in all probability the ancients had not the leaft idea or hope; and innumerable curves were fquared which were utterly unknown to them.

The excellency of this method revived fome hopes of fquaring the circle; and its quadrature was attempted with eagernefs. The quadrature of a space was now reduced to the finding of the fluent of a given fluxion; but this problem, however, was found to be incapable of a general folution in finite terms: the fluxion of every fluent was found to be always affignable, but the reverse of this problem could be effected only in particular cafes: among the exceptions, to the great mortification of geometricians, was included the cafe of the circle, with regard to all the forms of fluxions attending it.

Another method of obtaining the area was tried: of the quantity expreffing the fluxion of any area, in general, the fluent could always be affigned in the form of an infinite feries; which feries, therefore, defined all areas in general, and which, on fubftituting for particular cafes, was often found to break off and terminate, and to afford an area in finite terms: but here

again the cafe of the circle failed, its area being ftill an infinite feries.

All hopes of the quadrature of the circle being now at an end, the geometricians employed themfelves in difcovering and felecting the best forms of infinite feries for determining its area; among which it is evident, that thofe were to be preferred which were fimple, and would converge quickly; but it commonly happened that these two properties were divided, the fame feries very rarely including them both. The mathematicians in moft parts of Europe now applied themselves diligently to thefe new discoveries, and many feries were affigned on all hands, fome admired for their fimplicity, and others for their rate of convergency; thofe which converged the quickest, and were at the fame time fimpleft, which therefore were most ufeful in computing the area of the circle in numbers, were those in which, befides the radius, the tangent of fome certain arc of the circle, was the quantity by whofe powers the feries converged; and from fome of these feries, the area hath been computed to a great extent of figures. Dr. Edmund Halley gave a remarkable one, from the tangent of 30 degrees, by means of which the very induftrious Mr. Abraham Sharp computed the area of the circle to 72 places of figures; but even this was afterwards far exceeded by Mr. John Machin, who, by means hereafter defcribed in this book, compofed a feries fo fimple, and which converged fo quickly, that by it, in a very little time, he extended the quadrature of the circle to 100 places of figures; from which it appears, that if the diameter be 1, the circumference will be

31415926535,8979323846,2643383279, 5028841971,6939937510,

5820974944,5923078164,0628620899,8628034825,3421170679, and confequently the area will be 7853981633,9744830961,5660845819,8757210492,9234984377, 6455243736,1480769541,0157155224,9657008706,3355292669. And I have lately given, in the Philofophical Tranfactions, various other feries for the fame purpofe, which are ftill fimpler in their form, and converge more readily than those above mentioned.

Whilst I have been giving the preceding account of the progrefs of this fubject, I have at the fame time unawares been writing its panegyric; for, from hence it appears, that most of the material improvements or inventions in the fcience of geometry, have been principally made for the improvement of menfuration; which fufficiently fhews the dignity of this fubject; a fubject which, as Dr. Barrow fays, "deferves to be more curiously weighed, because from hence a name is impofed upon that mother and mistress of the rest of the mathematical fciences, which is employed about magnitudes, and which is