Archimedes

s of Euclid, namely, Archimedes and Apollonius of Perga (the "great geometer," and author of the classical treatise on Conics). The works of the

ing with the quadrature of curvilinear plane figures and with the quadrature and cubature of curved surfaces, investigations whic

red, and had to invert the second half of the reductio ad absurdum to enable approximation from below (so to speak) to be applied in that case also. Archimedes, on the other hand, approximates from above as well as from below; he approaches the area or volume to be measured by taking closer and closer circumscribed figures, as well as inscribed, and thereby compressing, as it were, the insc

ose. The method suggests the tactics of some master of strategy who foresees everything, eliminates everything not immediately conducive to the execution of his plan, masters every position in its order, and then suddenly (when the very elaboration of the scheme has almost obscured, in the mind of the onlooker, its ultimate object) strikes the final blow. Thus we read in Archimedes proposition after proposition the bearing of which is not immediately obvious but which we find infallibly used later on; and we are led on by such easy stages that the difficulty of the original problem, as presented at the outset, is scarcely appreciated. As Plutarch says, "It is not possible to find in geometry more difficult and troublesome questions, or more simple an

d certain theorems in quadrature and cubature, and he is at the same time careful to insist on the difference between (1) the means which may serve to suggest the truth of theorems, althou

geometry, because their investigation by the said method did not furnish an actual demonstration. But it is of course easier, when we have previously acquired by the method some knowledge of the questions, to supply the proof than it is to find the proof without any previous knowledge. This is a reason why, in the case of the theorems the proof of which Eudoxus was the first to discover, namely, that the cone is a third part of the cylinder, and the pyramid a third part of the prism, having the same base and equal height, we should give no small share of the credit to Democritus, who was the first to assert this truth with regard to the sai

right-angled cone [i.e. a parabola] is four-thirds of the triangle which has the same base and equal height; and after this I will give each

straight lines and in the case of a solid figure as parallel planes, and the aggregate of the infinite number of sections is said to make up the whole figure X. (Although the sections are so spoken of as straight lines or planes, they are really indefinitely narrow plane strips or indefinitely thin laminae respectively.) The diameter or axis is produced in the direction away from the figure to be measured, and the diameter or axis as produced is imagined to be the bar or lever of a balance. The object is now to apply all the separate elements of X at one point on the lever, while the corresponding elements of the known figure B operate at different points, namely, where they actually are in the first instance. Archimedes contrives, therefore, to move the elements of X away from their original position and to concentrate them at one point on the lever, such that each of the elements balances, about the

= CG

or content o

his case the elements of X, and X itself, have to be applied where they are, and the elements of the known figure or fi

= CG

ere G is the cent

of a sphere and the volume of a right segment of each of the three conicoids of revolution, (4) the centre of gravity (a) of a hemisphere, (b) of any se

two solid figures, which are the special subje

es, suppose a plane drawn through one side of the square containing one base of the cylinder and through the parallel diameter of the opposite base of the cylinder. The plane cu

es. Another cylinder is inscribed which is similarly related to another pair of opposite faces. The two cylinders include between t

rigorous geometrical proof of both propositions by the method of exhaustion. The MS. is unfor

he other treatises of Archimedes in the order in

here and

s since discovered, and which seem to him to be worthy of comparison with Eudoxus's propositions about the volumes of a pyramid and a cone. The theorems are (1) that the surface of a sphere is equal to four times its greatest circle (i.e. what we call a "great circle" of the sphere); (2) that the surface of any segment of a sphere is equal to a circle with radius equal to

owed by certain Assumptions, two

straight line is the least (this has been made the b

as, when (continually) added to itself, can be made to exceed any assigned magnitude among those which are

ded by the same extremities, the outer is greater than the inner. These assumptions are fundamental to his investigation, whic

of propositions he finds expressions for (a) the surfaces, (b) the volumes, of the figures so inscribed and circumscribed to the sphere. Next he proves (Prop. 32) that, if the inscribed and circumscribed polygons which, by their revolution, generate the figures are similar, the surfaces of the figures are in the duplicate ratio, and their volumes in the triplicate ratio, of their sides. Then he proves that the surfaces and volumes of the inscribed and circumscribed figures respectively are less and greater than the surface and volume respectively to which the main propositions declare the surface an

ces, by means of 1., 44, an expression for the volume of a segment of a sphere, and Props. 3, 4 solve the important problems of cutting a given sphere by a plane so that (a) the surfaces, (b) the volumes, of the

ength) = (a giv

of two conics, a parabola and a rectangular hyperbola. Three problems of construction follow, the first two of which are to construct a segment of a sphere similar to one given segment, and having (a) its volume, (b) its surface, equal to that of ano

V : V′ > S

f spheres which have equal surfaces, t

rement of

nly three propositions; the second, being an easy deduction from Props

bes and circumscribes successive regular polygons, beginning with hexagons, and doubling the number of sides continually, until he arrives at inscribed and circumscribed regular polygons with 96 sides; seeing then that the length of the circumference of the circle is intermediate between the perimeters of the two polygons, he calculates the two perimeters in terms of the diameter of the circle. His calculation is based on two close approximations (an upper and a lower) to the value of √3, that being the cotangent of the angle of 30°, from which he begins to work. He assumes as known that 265/153 < √3 < 1351/780. In the text, as we have it, only the results of the steps

ds and S

oid is therefore a paraboloid of revolution. The second is the obtuse-angled conoid, which is a hyperboloid of revolution described by the revolution of a hyperbola (a "section of an obtuse-angled cone") about its transverse axis. The spheroids are two, being the solids of revolution described by the revolution of an ellipse (a "section of an acute-angled cone") about (1) its major axis and (2) its minor axis; the first is called the "oblong" (or

operties of conics (Prop. 3), then the determination by the method of exhaustion of the area of an ellipse (Prop. 4). Three propositions follow, the first two of which (Props. 7, 8) show that the conical figure above referred to is really a segment of an oblique circular cone; this is done by actually finding the circular sections. Prop. 9 gives a similar proof that each elliptic section of a conoid or spheroid is a section of a

arallel faces and presenting the appearance of the steps of a staircase. Adding the elements of the inscribed and circumscribed figures respectively and using the method of exhaustion, Archimedes finds the volumes of the respective segments of the solids in the approved manner (Props. 21, 22 for the paraboloid, Props. 25, 26 for the hyperboloid, and Props. 27-30 for the spheroids). The results are stated in this form: (1) Any segment of a paraboloid of revolution is half as large ag

Spi

origin as a fixed point. Props. 1-11 are preliminary, the last two amounting to the summation of certain series required for the final addition of an indefinite number of element-areas, which again amounts to integration, in order to find the area of the figure cut off between any portion of the curve and the two radii vectores drawn to its extremities. Props. 13-20 are interesting and difficult propositions establishing the properties of tangents to the spiral. Props. 21-23 show how to inscribe and circumscribe to any portion of the spiral figures consisting of a multitude of elements which are narrow sectors of circles with the origin as centre; the area of the spiral is intermediate between the areas of the inscribed and circumscribed f

e of the

thod of exhaustion (the Postulate of Archimedes, or the theorem of Euclid X., 1). He mentions as having been proved by means of it (1) the theorems that the areas of circles are to one another in the duplicate ratio of their diameters, and that the volumes of spheres are in the triplicate ratio of their diameters, and (2) the propositions proved by Eudoxus about the volumes of a cone and a pyramid. No one, he says, so far as he is aware, has yet tried to square the segment bounded by a straight line and a section of a right-angle

t is more elaborate in that the elements of the area of the parabola to be measured are not straight lines but narrow strips. The figures inscribed and circumscribed to the segment are made up of such narrow strips and have a saw-like edge; all the elements are trapezia except two, which are triangles, one in each figure. Each trapezium (or triangle) is wei

er, and proves that the sum of the two new triangles is ? of the original inscribed triangle. Again, drawing triangles inscribed in the same way in the four segments left over, he proves that their sum is ? of the sum of t

(?)2 + (?)3

infinite series is 4?3Δ, and then confirms by reductio ad absurdu