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possible to avoid allowing this to have been Euclid's grand object. And accordingly he determined the chief properties in the mensuration of rectilineal plane and folid figures; and squared all such planes, and cubed all such solids. 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 shewing how in a circle to infcribe a regular polygon, which should not touch another circle, concentric with the former, although their circumferences should be ever so near together; and, in like manner, between any two concentric spheres to describe 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 square the circle, nor cube the sphere, he determined the proportion of one circle to another, and of one sphere to another, as well as the proportions of all rectilineal fimilar figures to one another.
Archimedes took up mensuration where Euclid left it, and carried it a great length. He was the first who squared a curvilineal space; unless Hypocrates must be excepted on account of his lunes. In his time the conic sections were admitted into geometry, and he applied himself closely to the measuring of them, as well as other figures. Accordingly he determined the relations of spheres, spheroids, and conoids, to cylinders and cones ; and the relations of parabolas to rectilineal planes whose quadratures had long before been determined by Euclid. . He hath left us also his attempts upon the circle: he proved that a circle is equal to a right-angled triangle, whose base is equal to the circumference, and its altitude equal to the radius; and consequently, that its area is equal to the rectangle of the radius, and half the circumference; and so 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 disappointed of the cxact quadrature of the circle, for want of the rectification of its circumference, which all his methods would not effect, he proceeded to assign an useful approximation to it; this he effected by the numeral calculation of the perimeters of the inscribed and circumscribed polygons; from which calculation it appears, that the perimeter of che circumscribed regular polygon of 192 fides, is to the diaineter, in a less ratio than that of 31 (318) to 1, and that the infcribed polygon of 96 fides, is to the diameter, in a greater ratio than that of 31 0,1; and consequently much more that the circumference of the circle is to the diameter, in a less ratio than that of 3} to , 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 37:1::22:7, which therefore is nearly the ratio of the circumference to the diameter. Froin this ratio of
the circumference to the diameter, he computed the approximate area of the circle, and found that it is to the square of the diameter, as 11 is to 14.--He likewise determined the relation between the circle and ellipse, with that of their fimilar parts. It is highly probable that he likewise attempted the hyperbola ; but it is not to be imagined that he met with any success, since approximations to its area are all that can be given by the various methods that have since been invented.
Besides these figures, he hath left us a treatise on the spiral, described by a point moving uniformly along a right line, which at the same time moves with an uniform angular motion; and he determined the proportion of its area to that of the circumscribed circle; as also the proportion of their sectors.
Throughout the whole works of this great man, which are chiefly on mensuration, he every where discovers the deepest design, and the finest invention; and seems to have been, with Euclid, exceedingly careful of admitting into his demonstrations nothing but principles perfectly geometrical and unexceptionable: and although his most general method of demonstrating the relations of curved figures to straight ones, be by inscribing polygons in them ; yet to determine those relations, he does not increase the number, and diminish the magnitude, of the sides of the polygon in infinitum ; but from this plain fundamental principle, allowed in Euclid's Elements, viz. that any quantity may be so often multiplied, or added to itself, as that the result should exceed any proposed finite quantity of the fame kind, he proves that to deny his figures to have the proposed relations, would involve an absurdity.
He demonstrated also many properties, particularly in the parabola, by means of certain numeral progressions, whose terms are similar to the inscribed figures ; but still without confidering such series as continued in infinitum, and then summing up the terms of such infinite series.
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 subject, both before and after Euclid and Archimedes; but their attempts were usually confined to particular parts of it, and made according to methods not essentially different from theirs. Among these are to be reckoned Thales, Anaxagoras, Pythagoras, Bryson, Antiphon, Hypocrates of Chios, Plato, Apollonius, Philo, and Ptolemy; most of whom wrote of the quadrature of the circle; and those after Archimedes, by his method, usually extended the approximation to a greater degree of accuracy.
this Many of the moderns also have prosecuted the same problem of the quadrature of the circle, after the same methods, to greater lengths; such are Vieta, and Metius, whose proportion between the diameter and circumference, is that of 113 to 355, which is within about joróboro of the true ratio; but above all, Ludolph van Collen, or a Ceulen, who, with an amazing degree of industry and patience, by the same methods, extendcd the ratio to 36 places of figures, making the ratio to be that of 1 to 3•14159265358979323846264338327950288 + or 9-, And the same was repeated and confirmed by his editor Snellius,
The first material deviation from the principles used by the ancients, in geometrical demonstrations, was made by Cavales rius: the sides of their inseribed and circumscribed figures, they always supposed to be of a finite and affignable number and length; he introduced the doctrine of indivisibles, a method which was very general and extensive, and which, with great ease and expedition, served 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 using it, and it soon led the way to infinitely small elements, and infinitesimals of endless orders; methods which were very useful in refolving difficult problems, and in investigating or demonstrating theories that are general and extensive ; but sometimes led their incautious followers into errors and mistakes, which occafioned disputes and animosities amongst them. There were now, however, many excellent things performed in this science; not only many new properties were discovered concerning the old figures, but new curves were measured ; and although several of them could not be exactly squared or cubed, yet general and infinite approximating series were assigned, of which the laws of their continuation were manifest, and in some of which the terms were independent of each other. Dr. Wallis, Mr. Huygens, and Mr. James Gregory, performed wonders: Huygens in para ticular must always be admired for his solid, accurate, and very masterly works.
During the preceding state of things, several men, whose va, nity seemed to have overcome their regard for truth, asserted, that they had discovered the quadrature of the circle, and published their attempts in the form of strict geometrical demon ftrations, with such assurance and ambiguity, as staggered and misled many who could not so well judge for themselves, and perceive the fallacy of their principles and arguments. Among those were Longomontanus, and our countryman Hobbes, who obstinately refused all conviction of his errors.
The use of infinites was, however, disliked by several people, and particularly by Sir Isaac Newton, who, among his numerous and great discoveries, hath given us that of the method of Auxions; a discovery of the greatest importance, both in philosophy and mathematics ; being a method fo general and extensive, as to include all investigations 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 consistent with those of the ancients, and which were free from the imperfe&tions and absurdities attending some that had lately been introduced by the moderns : he rejected no quantities as infinitely small, nor supposed any parts of curves to coincide with right lines; but proposed it in such a form as admits of a strict geometrical demonstration. Upon the introduction of this method, most sciences assumed a different appearance, and the most abstruse problems became easy and familiar to every one ; things which before seemed to be infuperable, became easy examples, or particular cases, of theories still more general and extensive; rectifications, quadratures, cubatures, tangencies, cases de maximis & minimis, and many other subjects, became general problems, and were delivered in the form of general theories, which included all particular cases: thus, in quadratures, a formula was affigned, which would express the areas of all possible curves whatever, both known and unknown, and which, by proper substitutions, gave the area for any particular case, either in finite terms, or in infinite series, of which any term, or any number of terms, could be easily aligned; and the like in other things. And although no curve, whose quadrature was unsuccessfully attempted by the ancients, became by this method perfectly quadrable, yet many general methods were discovered for approximating to their areas, of which in all probability the ancients had not the least idea or hope ; and innumerable curves were squared which were utterly unknown to them.
The excellency of this method revived some hopes of squaring the circle; and its quadrature was attempted with eagerness. 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 solution in finite terms: the fluxion of every fuent was found to be always assignable, but the reverse of this problem could be effected only in particular cases: among the exceptions, to the great mortification of geometricians, was included the case of the circle, with regard to all the forins of fluxions attending it.
Another method of obtaining the area was tried: of the quantity expressing the fluxion of any area, in general, the fluent could always be assigned in the form of an intinite feries; which series, therefore, defined all areas in general, and which, on substituting for particular cases, was often found to break off and terminate, and to attord an area in inite terins : but here Ċ E. again the case of the circle failed, its area being still an infinite series.
All hopes of the quadrature of the circle being now at an end, the geometricians employed themselves in discovering and selecting the best forms of infinite series for determining its area; among which it is evident, that those were to be preferred which were fimple, and would converge quickly ; but it commonly happened that these two properties were divided, the fame series very rarely including them both. The mathema. ticians in most parts of Europe now applied themselves diligently to these new discoveries, and many series were assigned on all hands, some admired for their fimplicity, and others for their rate of convergency; those which converged the quickest, and were at the same time: simplest, which therefore were most useful in computing the area of the circle in numbers, were those in which, besides the radius, the tangent of some certain arc of the circle, was the quantity by whose powers the feries converged; and from some of these series, 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 industrious 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 described in this book, composed a series so simple, and which converged so 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 3*1415926535,897932 38 46,2643383279,5028841971,6939937510, 5820974944, 5923078164,0628620899,8628034825,3421170679,
and consequently the area will be •7853981633,9744830961,5660845819,8757210492,9234984377, 6455243736,1480769541,0157155224, 9657008706,3355292669.
And I have lately given, in the Philosophical Transactions, various other series for the same purpose, which are still timpler in their form, and converge more readily than those above mentioned.
Whilst I have been giving the preceding account of the progress of this subject, I have at the same time unawares been writing its panegyric; for, from hence it appears, that most of the material improvements or inventions in the science of geometry, have been principally made for the improvement of mensuration; which sufficiently shews the dignity of this subject; a subject which, as Dr. Barrow fays, " deserves to be more curiously weighed, because from hence a name is imposed upon that mother and mistress of the rest of the mathematical feiences, which is employed about magnitudes, and which is