Chapter 4 GEOMETRY IN ARCHIMEDES.

The famous French geometer, Chasles, drew an instructive distinction between the predominant features of the geometry of the two great successors of Euclid, namely, Archimedes and Apollonius of Perga (the "great geometer," and author of the classical treatise on Conics). The works of these two men may, says Chasles, be regarded as the origin and basis of two great inquiries which seem to share between them the domain of geometry.

Apollonius is concerned with the Geometry of Forms and Situations, while in Archimedes we find the Geometry of Measurements, dealing with the quadrature of curvilinear plane figures and with the quadrature and cubature of curved surfaces, investigations which gave birth to the calculus of the infinite conceived and brought to perfection by Kepler, Cavalieri, Fermat, Leibniz and Newton.

In geometry Archimedes stands, as it were, on the shoulders of Eudoxus in that he applied the method of exhaustion to new and more difficult cases of quadrature and cubature. Further, in his use of the method he introduced an interesting variation of the procedure as we know it from Euclid. Euclid (and presumably Eudoxus also) only used inscribed figures, "exhausting" the figure to be measured, 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 inscribed and circumscribed figure into one, so that they ultimately coincide with one another and with the figure to be measured. But he follows the cautious method to which the Greeks always adhered; he never says that a given curve or surface is the limiting form of the inscribed or circumscribed figure; all that he asserts is that we can approach the curve or surface as nearly as we please.

The deductive form of proof by the method of exhaustion is apt to obscure not only the way in which the results were arrived at but also the real character of the procedure followed. What Archimedes actually does in certain cases is to perform what are seen, when the analytical equivalents are set down, to be real integrations; this remark applies to his investigation of the areas of a parabolic segment and a spiral respectively, the surface and volume respectively of a sphere and a segment of a sphere, and the volume of any segments of the solids of revolution of the second degree. The result is, as a rule, only obtained after a long series of preliminary propositions, all of which are links in a chain of argument elaborately forged for the one purpose. 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 and lucid explanations". But it is decidedly a rhetorical exaggeration when Plutarch goes on to say that we are deceived by the easiness of the successive steps into the belief that any one could have discovered them for himself. On the contrary, the studied simplicity and the perfect finish of the treatises involve at the same time an element of mystery. Although each step depends upon the preceding ones, we are left in the dark as to how they were suggested to Archimedes. There is, in fact, much truth in a remark of Wallis to the effect that he seems "as it were of set purpose to have covered up the traces of his investigation as if he had grudged posterity the secret of his method of inquiry while he wished to extort from them assent to his results".

A partial exception is now furnished by the Method; for here we have (as it were) a lifting of the veil and a glimpse of the interior of Archimedes's workshop. He tells us how he discovered 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, although not furnishing scientific proofs of them, and (2) the rigorous demonstrations of them by approved geometrical methods which must follow before they can be finally accepted as established.

Writing to Eratosthenes he says: "Seeing in you, as I say, an earnest student, a man of considerable eminence in philosophy and an admirer of mathematical inquiry when it comes your way, I have thought fit to write out for you and explain in detail in the same book the peculiarity of a certain method, which, when you see it, will put you in possession of a means whereby you can investigate some of the problems of mathematics by mechanics. This procedure is, I am persuaded, no less useful for the proofs of the actual theorems as well. For certain things which first became clear to me by a mechanical method had afterwards to be demonstrated by 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 said figures, though he did not prove it. I am myself in the position of having made the discovery of the theorem now to be published in the same way as I made my earlier discoveries; and I thought it desirable now to write out and publish the method, partly because I have already spoken of it and I do not want to be thought to have uttered vain words, but partly also because I am persuaded that it will be of no little service to mathematics; for I apprehend that some, either of my contemporaries or of my successors, will, by means of the method when once established, be able to discover other theorems in addition, which have not occurred to me.

"First then I will set out the very first theorem which became known to me by means of mechanics, namely, that Any segment of a section of a 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 of the other theorems investigated by the same method. Then, at the end of the book, I will give the geometrical proofs of the propositions."

The following description will, I hope, give an idea of the general features of the mechanical method employed by Archimedes. Suppose that X is the plane or solid figure the area or content of which is to be found. The method in the simplest case is to weigh infinitesimal elements of X against the corresponding elements of another figure, B say, being such a figure that its area or content and the position of its centre of gravity are already known. The diameter or axis of the figure X being drawn, the infinitesimal elements taken are parallel sections of X in general, but not always, at right angles to the axis or diameter, so that the centres of gravity of all the sections lie at one point or other of the axis or diameter and their weights can therefore be taken as acting at the several points of the diameter or axis. In the case of a plane figure the infinitesimal sections are spoken of as parallel 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 point of suspension of the lever, the corresponding element of B acting at its centre of gravity. The elements of X and B respectively balance about the point of suspension in accordance with the property of the lever that the weights are inversely proportional to the distances from the fulcrum or point of suspension. Now the centre of gravity of B as a whole is known, and it may then be supposed to act as one mass at its centre of gravity. (Archimedes assumes as known that the sum of the "moments," as we call them, of all the elements of the figure B, acting severally at the points where they actually are, is equal to the moment of the whole figure applied as one mass at one point, its centre of gravity.) Moreover all the elements of X are concentrated at the one fixed point on the bar or lever. If this fixed point is H, and G is the centre of gravity of the figure B, while C is the point of suspension,

X : B = CG : CH.

Thus the area or content of X is found.

Conversely, the method can be used to find the centre of gravity of X when its area or volume is known beforehand. In this case the elements of X, and X itself, have to be applied where they are, and the elements of the known figure or figures have to be applied at the one fixed point H on the other side of C, and since X, B and CH are known, the proportion

B : X = CG : CH

determines CG, where G is the centre of gravity of X.

The mechanical method is used for finding (1) the area of any parabolic segment, (2) the volume of a sphere and a spheroid, (3) the volume of a segment 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 segment of a sphere, (c) of any right segment of a spheroid and a paraboloid of revolution, and (d) of a half-cylinder, or, in other words, of a semicircle.

Archimedes then proceeds to find the volumes of two solid figures, which are the special subject of the treatise. The solids arise as follows:-

(1) Given a cylinder inscribed in a rectangular parallelepiped on a square base in such a way that the two bases of the cylinder are circles inscribed in the opposite square faces, 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 cuts off a solid with a surface resembling that of a horse's hoof. Archimedes proves that the volume of the solid so cut off is one sixth part of the volume of the parallelepiped.

(2) A cylinder is inscribed in a cube in such a way that the bases of the cylinder are circles inscribed in two opposite square faces. Another cylinder is inscribed which is similarly related to another pair of opposite faces. The two cylinders include between them a solid with all its angles rounded off; and Archimedes proves that the volume of this solid is two-thirds of that of the cube.

Having proved these facts by the mechanical method, Archimedes concluded the treatise with a rigorous geometrical proof of both propositions by the method of exhaustion. The MS. is unfortunately somewhat mutilated at the end, so that a certain amount of restoration is necessary.

I shall now attempt to give a short account of the other treatises of Archimedes in the order in which they appear in the editions. The first is-

On the Sphere and Cylinder.

Book I. begins with a preface addressed to Dositheus (a pupil of Conon), which reminds him that on a former occasion he had communicated to him the treatise proving that any segment of a "section of a right-angled cone" (i.e. a parabola) is four-thirds of the triangle with the same base and height, and adds that he is now sending the proofs of certain theorems which he has 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 the straight line drawn from the vertex of the segment to a point on the circle which is the base of the segment; (3) that, if we have a cylinder circumscribed to a sphere and with height equal to the diameter, then (a) the volume of the cylinder is 1? times that of the sphere and (b) the surface of the cylinder, including its bases, is 1? times the surface of the sphere.

Next come a few definitions, followed by certain Assumptions, two of which are well known, namely:-

1. Of all lines which have the same extremities the straight line is the least (this has been made the basis of an alternative definition of a straight line).

2. Of unequal lines, unequal surfaces and unequal solids the greater exceeds the less by such a magnitude as, when (continually) added to itself, can be made to exceed any assigned magnitude among those which are comparable [with it and] with one another (i.e. are of the same kind). This is the Postulate of Archimedes.

He also assumes that, of pairs of lines (including broken lines) and pairs of surfaces, concave in the same direction and bounded by the same extremities, the outer is greater than the inner. These assumptions are fundamental to his investigation, which proceeds throughout by means of figures inscribed and circumscribed to the curved lines or surfaces that have to be measured.

After some preliminary propositions Archimedes finds (Props. 13, 14) the area of the surfaces (1) of a right cylinder, (2) of a right cone. Then, after quoting certain Euclidean propositions about cones and cylinders, he passes to the main business of the book, the measurement of the volume and surface of a sphere and a segment of a sphere. By circumscribing and inscribing to a great circle a regular polygon of an even number of sides and making it revolve about a diameter connecting two opposite angular points he obtains solids of revolution greater and less respectively than the sphere. In a series 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 and volume of the sphere to be equal (Props. 25, 27, 30, 31 Cor.). He has now all the material for applying the method of exhaustion and so proves the main propositions about the surface and volume of the sphere. The rest of the book applies the same procedure to a segment of the sphere. Surfaces of revolution are inscribed and circumscribed to a segment less than a hemisphere, and the theorem about the surface of the segment is finally proved in Prop. 42. Prop. 43 deduces the surface of a segment greater than a hemisphere. Prop. 44 gives the volume of the sector of the sphere which includes any segment.

Book II begins with the problem of finding a sphere equal in volume to a given cone or cylinder; this requires the solution of the problem of the two mean proportionals, which is accordingly assumed. Prop. 2 deduces, 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 segments may have to one another a given ratio. The solution of the second problem (Prop. 4) is difficult. Archimedes reduces it to the problem of dividing a straight line AB into two parts at a point M such that

MB : (a given length) = (a given area) : AM2.

The solution of this problem with a determination of the limits of possibility are given in a fragment by Archimedes, discovered and preserved for us by Eutocius in his commentary on the book; they are effected by means of the points of intersection 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 another given segment of a sphere. The last two propositions are interesting. Prop. 8 proves that, if V, V′ be the volumes, and S, S′ the surfaces, of two segments into which a sphere is divided by a plane, V and S belonging to the greater segment, then

S2 : S′ 2 > V : V′ > S3/2 : S′ 3/2.

Prop. 9 proves that, of all segments of spheres which have equal surfaces, the hemisphere is the greatest in volume.

The Measurement of a Circle.

This treatise, in the form in which it has come down to us, contains only three propositions; the second, being an easy deduction from Props. 1 and 3, is out of place in so far as it uses the result of Prop. 3.

In Prop. 1 Archimedes inscribes and circumscribes to a circle a series of successive regular polygons, beginning with a square, and continually doubling the number of sides; he then proves in the orthodox manner by the method of exhaustion that the area of the circle is equal to that of a right-angled triangle, in which the perpendicular is equal to the radius, and the base equal to the circumference, of the circle. Prop. 3 is the famous proposition in which Archimedes finds by sheer calculation upper and lower arithmetical limits to the ratio of the circumference of a circle to its diameter, or what we call π; the result obtained is 31?7 > π > 310?71. Archimedes inscribes 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 in the calculation are given, but they involve the finding of approximations to the square roots of several large numbers: thus 11721?8 is given as the approximate value of √(137394333?64), 3013? as that of √(9082321) and 18389?11 as that of √(3380929). In this way Archimedes arrives at 14688 / 4673? as the ratio of the perimeter of the circumscribed polygon of 96 sides to the diameter of the circle; this is the figure which he rounds up into 31?7. The corresponding figure for the inscribed polygon is 6336 / 2017?, which, he says, is > 310?71. This example shows how little the Greeks were embarrassed in arithmetical calculations by their alphabetical system of numerals.

On Conoids and Spheroids.

The preface addressed to Dositheus shows, as we may also infer from internal evidence, that the whole of this book also was original. Archimedes first explains what his conoids and spheroids are, and then, after each description, states the main results which it is the aim of the treatise to prove. The conoids are two. The first is the right-angled conoid, a name adapted from the old name ("section of a right-angled cone") for a parabola; this conoid 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 oblate) spheroid, the second the "flat" (or prolate) spheroid. As the volumes of oblique segments of conoids and spheroids are afterwards found in terms of the volume of the conical figure with the base of the segment as base and the vertex of the segment as vertex, and as the said base is thus an elliptic section of an oblique circular cone, Archimedes calls the conical figure with an elliptic base a "segment of a cone" as distinct from a "cone".

As usual, a series of preliminary propositions is required. Archimedes first sums, in geometrical form, certain series, including the arithmetical progression, a, 2a, 3a, ... na, and the series formed by the squares of these terms (in other words the series 12, 22, 32, ... n2); these summations are required for the final addition of an indefinite number of elements of each figure, which amounts to an integration. Next come two properties 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 certain oblique circular cylinder (with axis parallel to the axis of the segment of the conoid or spheroid cut off by the said elliptic section). Props. 11-18 show the nature of the various sections which cut off segments of each conoid and spheroid and which are circles or ellipses according as the section is perpendicular or obliquely inclined to the axis of the solid; they include also certain properties of tangent planes, etc.

The real business of the treatise begins with Props. 19, 20; here it is shown how, by drawing many plane sections equidistant from one another and all parallel to the base of the segment of the solid, and describing cylinders (in general oblique) through each plane section with generators parallel to the axis of the segment and terminated by the contiguous sections on either side, we can make figures circumscribed and inscribed to the segment, made up of segments of cylinders with parallel 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 again as the cone or segment of a cone which has the same base and axis; (2) Any segment of a hyperboloid of revolution or of a spheroid is to the cone or segment of a cone with the same base and axis in the ratio of AD + 3CA to AD + 2CA in the case of the hyperboloid, and of 3CA ? AD to 2CA ? AD in the case of the spheroid, where C is the centre, A the vertex of the segment, and AD the axis of the segment (supposed in the case of the spheroid to be not greater than half the spheroid).

On Spirals.

The preface addressed to Dositheus is of some length and contains, first, a tribute to the memory of Conon, and next a summary of the theorems about the sphere and the conoids and spheroids included in the above two treatises. Archimedes then passes to the spiral which, he says, presents another sort of problem, having nothing in common with the foregoing. After a definition of the spiral he enunciates the main propositions about it which are to be proved in the treatise. The spiral (now known as the Spiral of Archimedes) is defined as the locus of a point starting from a given point (called the "origin") on a given straight line and moving along the straight line at uniform speed, while the line itself revolves at uniform speed about the 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 figures, and by the usual method of exhaustion Archimedes finds the areas required. Prop. 24 gives the area of the first complete turn of the spiral (= 1?3π (2πa)2, where the spiral is r = aθ), and of any portion of it up to OP where P is any point on the first turn. Props. 25, 26 deal similarly with the second turn of the spiral and with the area subtended by any arc (not being greater than a complete turn) on any turn. Prop. 27 proves the interesting property that, if R1 be the area of the first turn of the spiral bounded by the initial line, R2 the area of the ring added by the second complete turn, R3 the area of the ring added by the third turn, and so on, then R3 = 2R2, R4 = 3R2, R5 = 4R2, and so on to Rn = (n ? 1) R2, while R2, = 6R1.

Quadrature of the Parabola.

The title of this work seems originally to have been On the Section of a Right-angled Cone and to have been changed after the time of Apollonius, who was the first to call a parabola by that name. The preface addressed to Dositheus was evidently the first communication from Archimedes to him after the death of Conon. It begins with a feeling allusion to his lost friend, to whom the treatise was originally to have been sent. It is in this preface that Archimedes alludes to the lemma used by earlier geometers as the basis of the method 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-angled cone (a parabola); but he has succeeded in proving, by means of the same lemma, that the parabolic segment is equal to four-thirds of the triangle on the same base and of equal height, and he sends the proofs, first as "investigated" by means of mechanics and secondly as "demonstrated" by geometry. The phraseology shows that here, as in the Method, Archimedes regarded the mechanical investigation as furnishing evidence rather than proof of the truth of the proposition, pure geometry alone furnishing the absolute proof required.

The mechanical proof with the necessary preliminary propositions about the parabola (some of which are merely quoted, while two, evidently original, are proved, Props. 4, 5) extends down to Prop. 17; the geometrical proof with other auxiliary propositions completes the book (Props. 18-24). The mechanical proof recalls that of the Method in some respects, but 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 weighed where it is against another area hung at a fixed point of an assumed lever; thus the whole of the inscribed and circumscribed figures respectively are weighed against the sum of an indefinite number of areas all suspended from one point on the lever. The result is obtained by a real integration, confirmed as usual by a proof by the method of exhaustion.

The geometrical proof proceeds thus. Drawing in the segment the inscribed triangle with the same base and height as the segment, Archimedes next inscribes triangles in precisely the same way in each of the segments left over, 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 the preceding pair of triangles and therefore (?)2 of the original inscribed triangle. Proceeding thus, we have a series of areas exhausting the parabolic segment. Their sum, if we denote the first inscribed triangle by Δ, is

Δ {1 + ? + (?)2 + (?)3 + . . . .}

Archimedes proves geometrically in Prop. 23 that the sum of this infinite series is 4?3Δ, and then confirms by reductio ad absurdum the equality of the area of the parabolic segment to this area.

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