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whence they concluded, that the two ratios were the
Euclid has confined himself, in his Elements, to this strict method, sanctioned by the unanimous assent of the ancient geometricians. But for this very reason his demonstrations are sometimes long, indirect, complicated, and difficult to be followed by a beginner. This has induced several moderns, in the editions they have given us of Euclid's Elements, to employ more simple and easy demonstrations than those of the author. To this inconvenience attached to the ancient methods perhaps we must ascribe the difficulties, which Ptolemy Philadelphus, king of Egypt, in other respects a man of understanding, experienced in studying the mathematics. Weary of the extreme attention it required, he one day asked Euclid, whether he could not make the way smoother for him to which the philosophical geometrician ingenuously answered: no sire; there is no royal road
In Euclid's Elements we find all the principles necessary for determining the perimeters and areas of right lined polygons, and the superficies and solidities of polyedra terminated by plane rectilinear faces: it wants, however, the method of measuring the circumference of the circle, though the author has entered into several particulars respecting the properties of this curve, and it's different uses in determining and comparing angles. It is true, he demonstrates, that the peripheries of two circles are as their diameters; that their areas are as the squares of their diameters; that a cylinder is equal to the product of it's base multi
plied by it's altitude; and that a cone is equal to one third of a cylinder having the same altitude and base: but all these propositions are incomplete or steril, while we remain ignorant of the length of the periphery of the circle relative to it's radius or diameter. The knowledge of this, if we possessed it, would enable us to find the area of the circle, or in other words it's quadrature. In fact we see from Euclid himself, that by inscribing in a circle regular polygons, the number of the sides of which go on continually increasing ad infinitum, the area of the circle is equal to that of a triangle, the base of which would be the periphery drawn out into a straight line, and it's altitude the radius: whence it follows, that we should have a square equal to the area of the circle, by taking a mean geometrical proportional between the periphery and half the radius: but Euclid has not given this necessary supplement.
Archimedes, the greatest geometrician of antiquity, was the first who discovered the ratio which the periphery of the circle bears to it's diameter; not indeed with geometrical strictness, but by a method of approxi mation, admirable in it's kind, and the source and model of all the approximate quadratures of curvilinear spaces, when we are destitute of the means of determining them exactly, and without any omission. A. C. 250.
Having found, that, if we inscribe a regular polygon in a circle, and circumscribe another of the same number of sides about it, the periphery of the circle, being between their perimeters, will be greater than the one, and less than 1 other; and by going
on continually increasing the number of their sides, the difference will ultimately become less than any assignable quantity. He supposed the first two polygons had six sides each, the second twelve, and thus continuing the geometrical progression to the number of ninety-six, he perceived, that at this term, at which he stopped, the perimeters of the two polygons approached pretty near to equality. He took, in consequence, the arithmetical mean between them as the approximate value of the periphery of the circle: and the conclusion of his calculation was, that, if the diameter were represented by seven, the circumference would be comprised between the numbers twenty-one and twenty-two, but much nearer the latter than the former. The same method, if carried farther, gives the ratio of the circumference to the diameter more accurately, but that of seven to twen ty-two is sufficient in practical problems, which do not require very great precision.
Since the time of Archimedes a number of useless attempts have been made to assign the precise ratio of the circumference to the diameter: but adepts in geometry consider this problem, if not absolutely irresolvable in itself, at least as incapable of a perfect solution by any of the means the present state of geometry affords. If the hope of resolving it could be conceived for a moment, it was at the discovery of fluxions; for this method has rectified and squared curves, by which the ancient geometry was baffled : but the circle has resisted it, and there are now none but beginners, or persons altogether ignorant of geometry,
geometry, who seek for the absolute and rigorous quadrature of the circle.
The numerous discoveries, with which Archimedes enriched the mathematics, have placed him among the small number of those rare and inventive geniuses, who from time to time have given a great impulse to the whole body of science. Beside his work of the Dimension of the Circle, of which I have just given an abstract, we have his treatises of the Sphere and Cylinder, of Conoids and Spheroids, of Spiral Lines [De Spiralibus & Helicibus], of the Quadrature of the Parabola, of Equiponderants, of Bodies floating on a Fluid, and his Arenarius, or numbering of the Sand, with his Lemmata. In all these we admire the force of his genius. The titles of these works sufficiently indicate their subjects, and I shall not give an analysis of them here, but content myself with relating some of their principal results.
In the treatise on the Sphere and Cylinder Archimedes determines the ratio both of the superficies and solidities of these bodies to each other. He shows, that the superficies of the sphere is equal to the convex superficies of the circumscribed cylinder, or, which is the same thing, to the quadruple of one of it's great circles: that the superficies of a spherical segment is equal to the corresponding cylindrical superficies, or to that of the circle which has for it's radius the chord drawn from the summit to a point in the circumference of the base: that the solidity of the sphere is two thirds that of the cylinder : &c. The treatise on Conoids contains several properties of solids produced by the revolution of the conic sections
round their axes. with one another; and determines their ratios to the cylinder and the cone of the same base and altitude: he also demonstrates, for example, that the solidity of the paraboloid is only half that of the circumscribed cylinder: &c. In his work on the Quadrature of the Parabola he proves in two equally ingenious modes, that the area of the parabola is two thirds of that of the circumscribed rectangle: which is the first instance of an absolute and rigorous quadrature of a space comprised between right lines and a curve. The treatise on Spirals is built on a very profound geometry. Archimedes compares the length of these curves with arcs of corresponding circles, and the spaces they include with circular spaces; he also draws tangents and perpendiculars to them, &c. All these researches, so easy, since the invention of fluxions, were extremely difficult to the geometry of those times. We must not be surprised therefore, if the demonstrations of Archimedes be somewhat complex; on the contrary, we ought to admire that force of intellect, which was requisite to retain such a great number of propositions, and preserve the chain unbroken.
Archimedes compares these solids
This abstract is sufficient to give a general idea of the geometrical discoveries of Archimedes: to which I shall add, that he extended and clearly demonstrated the use of geometrical analysis, the principles of which were given by the school of Plato. We shall also see other proofs of the genius of this great man, when I come to speak of mechanics, hydrostatics, and optics.