WorksOfArchimedes. Heath. Summary.

On the Sphere and Cylinder.

BookI.

  I first set out the axioms and the assumptions which I have used for the proofs of my propositions.

  Definitions.

  1. There are in a plane certain terminated bent lines, which either lie wholly on the same side of the straight lines joining their extremities, or have no part of them on the other side.

  2. I apply the term concave in the same direction to a line such that, if any two points on it are taken, either all the straightlines connecting the points fall on the same side of the line, or some fall on one and the same side while others fall on the line itself, but none on the other side.

  3. Similarly also there are certain terminated surfaces, not themselves being in a plane but having their extremities in a plane, and such that they will either be wholly on the same side of the plane containing their extremities, or have no part of them on the other side.

  4. I apply the term concave in the same direction to surfaces such that, if any two points on them are taken, the straightlines connecting the points either all fall on the same side of the surface, or some fall on one and the same side of it while some fall upon it, but none on the other side.

  5. I use the term solid sector, when a cone cuts a sphere, and has its apex at the centre of the sphere, to denote the figure comprehdned by the surface of the cone and the surface of the sphere included within the cone.

  6. I apply the term solid rhombus, when two cones with the same base have their apices on opposite sides of the plane of the base in such a position that their axes lie in a straightline, to denote the solid figure made up of both the cones.

  Assumptions.

  1. Of all lines which have the same extremities the straightline is the least.

  2. Of other lines in a plane and having the same extremities, [any two] such are unequal whenever both are concave in the same direction and one of them is either wholly included between the other and the straightline which has the same extremities with it, or is partly included by, and is partly common with, the other; and that [line] which is included is the lesser[ of the two].

  3. Similarly, of surfaces which have the same extremities, if those extremities are in a plane, the plane is the least [in area].

  4. Of other surfaces with the same extremities, the extremities being in a plane, [any two] such are unequal whenever both are concave in the same direction and one surface is either wholly included between the other and the plane which has the same extremities with it, or is partly included by, and partly common with, the other; and that [surface] which is included is the lesser[ of the two in area].

  5. Further, of unequal lines, unequal surfaces, and unequal solids, the greater exceeds the less by such a magnitude as, when added to itself, can be made to exceed any assigned magnitude among those which are comparable with [it and with] one another.

  These things being premised, if a polygon be inscribed in a circle, it is plain that the perimeter of the inscribed polygon is less than the circumference of the circle; for each of the sides of the polygon is less than that part of the circumference of the circle which is cut off by it."

  Prop.1. If a polygon be circumscribed about a circle, the perimeter of the circumscribed polygon is greater than the perimeter of the circle.

  Prop.2. Given two unequal magnitudes, it is possible to find two unequal straightlines such that the greater straightline has to the less a ratio than the greater magnitude has to the less.

  Prop.3. Given two unequal magnitudes and a circle, it is possible to inscribe a polygon in the circle and to describe another about it so that the side of the circumscribed polygon may have to the side of the inscribed polygon a ratio less than that of the greater magnitude to the less.

  Prop.4. Again, given two unequal magnitudes and a sector, it is possible to describe a polygon about the sector and to inscribe another in it so that the side of the circumscribed polygon may have to the side of the inscribed polygon a ratio less than the greater magnitude has to the less.

  Prop.5. Given a circle and two unequal magnitudes, to describe a polygon about the circle and inscribe another in it, so that the circumscribed polygon may have to the inscribed a ratio less than the greater magnitude has to the less.

  Prop.6. Similarly we  an show that, given two unequal magnitudes and a sector, it is possible to circumscribe a polygon about the sector and inscribe in it another similar one so that the circumscribed may have to the inscribed a ratio less than the greater magnitude has to the less.

  And it is likewise clear that, if a circle or a sector, as well as a certain area, be given, it is possible, by inscribing regular polygons in a circle or sector, and by continually inscribing such in the remaining segments, to leave segments of the circle or sector which are [together] less than the given area. For this is proved in theElements. [Eucl.XII.2.]

  But it is yet to be proved that, given a circle or sector and an area, it is possible to describe a polygon about the circle or sector, such that the area remaining between the circumference and the circumscribed figure is less than the given area."

  Prop.7. If in an isosceles cone [i.e. a right circular cone] a pyramid be inscribed having an equilateral base, the surface of the pyramid excluding the base is equal to a triangle having its base equal to the perimeter of the base of the pyramid and its height equal to the perpendicular drawn from the apex on one side of the base.

  Prop.8. If a pyramid be circumscribed about an isosceles cone, the surface of the pyramid excluding its base is equal to a triangle having its base equal to the perimeter of the base of the pyramid and its height equal to the side [i.e. a generator] of the cone.

  Prop.9. If in the circular base of an isosceles cone a chord be placed and from its extremities straightlines be drawn to the apex of the cone, the triangle so formed will be less than the portion of the surface of the cone intercepted between the lines drawn to the apex.

  Prop.10. If in the plane of the circulase base of an isosceles cone two tangents be drawn to the circle meeting in a point, and the points of contact and the point of concourse of the tangents be respectively joined to the apex of the cone, the sum of the two triangles formed by the joining lines and the two tangents are together greater than the included portion of the surface of the cone.

  Prop.11. If a plane parallel to the axis of a right cylinder cut the cylinder, the part of the surface of the cylinder cut off by the plane is greater than the area of the parallelogram in which the plane cuts it.

  Prop.12. If at the extremities of two generators of any right cylinder tangents be drawn to the circular bases in the planes of those bases respectively, and if the pairs of tangents meet, the parallelograms formed by each generator and the two corresponding tangents respectively are together greater than the included portion of the surface of the cylinder between the two generators.

  [The proofs of these two prop.s follow exactly the methods of Props.9,10 respectively, and it is therefore unnecessary to reproduce them.]

  From the properties thus proved it is clear (1) that, if a pyramid be inscribed in an isosceles cone, the surface of the pyramid excluding the base is less than the surface of the cone [excluding the base], (2) that, if a pyramid be circumscribed about an isosceles cone, the surface of the pyramid excluding the base is greater than the surface of the cone excluding the base.

  It is also clear from what has been proved both (1) that, if a prism be inscribed in a right cylinder, the surface of the prism made up of its parallelograms [i.e. excluding its bases] is less than the surface of the cylinder excluding its bases, and (2) that, if a prism be circumscribed about a right cylinder, the surface of the prism made up of its parallelograms is greater than the surface of the cylinder excluding its bases."

  Prop.13. The surface of any right cylinder excluding the bases is equal to a circle whose radius is a meanproportional between the side [i.e. a generator] of the cylidner and the diameter of its base.

  Prop.14. The surface of any isosceles cone excluding the base is equal to a circle whose radius is a meanproportional between the side of the cone [a generator] and the radius of the circle which is the base of the cone.

  Prop.15. The surface of any isosceles cone has the same ratio to its base as the side of the cone has to the radius of the base.

  Prop.16. If an isosceles cone be cut by a plane parallel to the base, the portion of the surface of the cone between the parallel planes is equal to a circle whose radius is a meanproportional between (1) the portion of the side of the cone intercepted by the parallel planes and (2) the line which is equal to the sum of the radii of the circles in the parallel planes.

  Lemmas.

  1. Cones having equal height have the same ratio as their bases; and those having equal bases have the same ratio as their heights. [Eucl.XII.11.]

  2. If a cylinder be cut by a plane parallel to the base, then, as the cylinder is to the cylinder, so is the axis to the axis. [Eucl.XII.14.]

  3. The cones which have the same bases as the cylinders [and equal height] are in the same ratio as the cylinders. [Eucl.XII.13.]

  4. Also the bases of equal cones are reciprocallyproportional to their heights; and those cones whose bases are reciprocallyproportional to their heights are equal. [Eucl.XII.15.]

  5. Also the cones, the diameters of whose bases have the same ratio as their axes, are to one another in the triplicateratio of the diameters of the bases. [Eucl.XII.12.]

  Prop.17. If there be two isosceles cones, and the surface of one cone be equal to the base of the other, while the perpendicular from the centre of the base [of the first cone] on the side of that cone is equal to the height [of the second], the cones will be equal.

  Prop.18. Any solid rhombus consisting of isosceles cones is equal to the cone which has its base equal to the surface of one of the cones composing the rhombus and its height equal to the perpendicular drawn from the apex of the second cone to one side of the first cone.

  Prop.19. If an isosceles cone be cut by a plane parallel to the base, and on the resulting circular section a cone be described having as its apex the centre of the base [of the first cone], and if the rhombus so formed be taken away from the whole cone, the part remaining will be equal to the cone with base equal to the surface of the portion of the first cone between the parallel planes and with height equal to the perpendicular drawn from the centre of the base of the first cone on one side of that cone.

  Prop.20. If one of the two isosceles cones forming a rhombus be cut by a plane parallel to the base and on the resullting circular section a cone be described having the same apex as the second cone, and if the resulting rhombus be taken from the whole rhombus, the remainder will be equal to the cone with base equal to the surface of the portion of the cone between the parallel planes and with height equal to the perpendicular drawn from the apex of the second cone to the side of the first cone.

  Prop.21. A regular polygon of an even number of sides being inscribed in a circle, As ABC...A'...C'B'A, so that AAj' is a diameter,

  if two angular points next but one to each other, as B, B', be joined, and the other lines parallel to BB' and joining pairs of angular points be drawn, as CC', DD'...,

  then (BB'+CC'+...):AA' = A'B:BA.

  Prop.22. If a polygon be described in a segment of a circle LAL' so that all its sides excluding the base are equal and their number even, as LK...A...K'L', A being the middle point of the segment, and if the lines BB', CC',... parallel to the base LL' and joining pairs of angular points be drawn,

  then (BB'+CC'+...+LM):AM=A'B:BA, where M is the middlepoint of LL' and AA' is the diameter throughM.

  Prop.23. The surface of the sphere is greater than the surface described by the revolution of the polygon inscribed in the great circle about the diameter of the great circle.

  Prop.24. If a reulgar polygon AB...A'...B'A, the number of whose sides is a multiple of four, be inscribed in a great circle of a sphere,

  andif BB' subtending two sides be joined, and all the other lines parallel to BB' and joining pairs of angular points be drawn,

  then the surface of the figure inscribed in the sphere by the revolution of the polygon about the diameter AA' is equal to a circle the square of whose radius is equal to the rectangle BA(BB'+CC'+...).

  Prop.25. The surface of the figure inscribed in a sphere as in the last prop., consisting of portions of conical surfaces, is less than fourtimes the greatest circle in the sphere.

  Prop.26. The figure inscribed as above in a sphere is equal [in volume] to a cone whose base is a circle equal to the surface of the figure inscribed in the sphere and whose height is equal to the perpendicular drawn from the centre of the sphere to one side of the polygon./

  Prop.27. The figure inscribed in the sphere as before is less than fourtimes the cone whose base is equal to a great circle of the sphere and whose height is equal to the radius of the sphere.

  Prop.28. The surface of the figure circumscribed to the given sphere is greater than that of the sphere itself.

  Prop.29. In a figure circumscribed to a sphere in the manner shown in the previous prop., the surface is equal to a circle the square on whose radius is equal to AB(BB'+CC'+...).

  Prop.30. The surface of a figure circumscribed as before about a sphere is greater than fourtimes the great circle of the sphere.

  Prop.31. The solid of revolution circumscribed as before about a sphere is equal to a cone whose base is equal to the surface of the solid and whose height is equal to the radius of the sphere.

  Cor. The solid circumscribed about the smaller sphere is greater than four times the cone whose base is a great circle of the sphere and whose height is equal to the radius of the sphere.

  Prop.32. If a regular polygon with 4n sides be inscribed in a great circle of a sphere, as ab...a'...b'a, and a similar polygon AB...A'...B'A be described about the great circle,

  andif the polygons resolve with the great circle about the diameters aa', AA' respectively, so that they describe the surfaces of solid figures inscribed in and circumscribed to the sphere respectively

  then (1) the surfaces of the circumscribed and inscribed figures are to one another in the duplicateratio of their sides,

  and (2) the figures themselves [i.e. their volumes] are in the triplicateratio of their sides.

  Prop.33. The surface of any sphere is equal to fourtimes the greatest circle in it.

  Prop.34. Any sphere is equal to fourtimes the cone which has tis base equal to the greatest circle in the sphere and it sheight equal to the radius of the sphere.

  Cor. From what has been proved it follows that every cylinder whose base is the greatest circle in a sphere and whose height is equal to the diameter of the sphere is 3/2 of the sphere, and its surface together with it sbases is 3/2 of the surface of the sphere.

  Prop.35. If in a segment of a circle LAL', where A is the middle point of the arc, a polygon LK...A...K'L' be inscribed of which LL' is one side, while the other sides are 2n in number and all equal, and if the polygon resolve with the segment about the diameter AM, generating a solidfigure inscribed in a segment of a sphere, then the surface of the inscribed solid is equal to a circle the sphere on whose radius is equal to the rectangle

  AB(BB'+CC'+...+KK'+(LL'/2).

  Prop.36. The surface of the figure inscribed as before in the segment of a sphere is less than that of the segment of the sphere.

  Prop.37. The surface of the solid figure inscribed in the segment of the sphere by the revolution of LK...A...K'L' about AM is less than a circle with radius equal toAL.

  Prop.38. Thed solidfigure described as before in a segment of a sphere less than a hemisphere, together with the cone whose base is the base of the segment and whose apex is the centre of the sphere, is equal to a cone whose base is equal to the surface of the inscribed solid and whose height is equal to the perpendicular from the centre of the sphere on any side of the polygon.

  Cor. The cone whose base is a circle with radius equal to AL and whose height is equal to the radius of the sphere is greater than the sum of the inscribed solid and the cone OLL'.

  Prop.39. The surface of the solid figure so circumscribed about the sector of the sphere [excluding its base] will be greater than that of the segment of the sphere whose base is the circle on ll' as diameter.

  Cor. The surface of the figure so described about the sector of the sphere is equal to a circle the square on whose radius is equal to the rectangle

  AB(BB'+CC'+...+KK'+(1/2)LL').

  Prop.40. The surface of the figure circumscribed to the sector as before is greater than a circle whose radius is equal to al.

  Cor.1. The volume of the figure circumscribed about the sector together with the cone whose apex is O and base the circle on LL' as diameter, is equal to the volume of a cone whose base is equal to the surface of the circumscribed figure and whose height is ON.

  Cor.2. The volume of the circumscribed figure with the cone OLL' is greater than the cone whose base is a circle with radius equal to al and whose height is equal to the radius (Oa) of the inner sphere.

  Prop.41. Let lal' be a segment of a great circle of a sphere which is less than a semicircle.

  Suppose a polygon inscribed in the sector Olal' such that the sides lk,...ba, ab',lllk'l' are 2n in number and all equal. Let a similar polygon be circumscribed about the sector so that its sides are parallel to those of the first polygon; and draw the circle circumscribing the outer polygon.

  Now let the polygons and circles revolve together about OaA, the radius bisecting the segment lal'.

  Then (1) the surfaces of the outer and inner solids of revolution so described are in the ratio of A*Bsquare to a*bsquare,

  and (2) their volumes together with the corresponding cones with the same base and with apex O in each case are as A*Bsquare to a*bsquare.

  Prop.42. If lal' be a segment of a sphere less than a hemisphere and Oa the radius perpendicular to the base of the segment, the surfaces of the segment is equal to a circle whose radius is equal to al.

  Prop.43. Even if the segment of the sphere is greater than a hemisphere, its surface is still equal to a circle whose radius is equal to al.

  Prop.44. The volume of any sector of a sphere is equal to a cone whose base is equal to the surface of the segment of the sphere included in the sector, and whose height is equal to the radius of the sphere.

  On the sphere and cylinder. BookII.

  Prop.1. Problem. Given a cone or a cylinder, to find a sphere equal to the cone or to the cylinder.

  Prop.2. If BAB' be a segment of a sphere, BB' a diameter of the base of the segment, and O the centre of the sphere, and if AA' be the diameter of the sphere bisecting BB' in M,

  then the volume of the segment is equal to that of a cone whose base is the same as that of the segment and whose height is h, where h:AM = OA'+A'M:A'M.

  Cor. The segment BAB' is to a cone with the same base and equal height in the ratio of OA'+A'M to A'M.

  Prop.3. Problem. to cut a given sphere by a plane so that the surfaces of the segments may have to one another a given ratio.

  Prop.4. Problem. To cut a given sphere by a plane so that the volumes of the segments are to one another in a given ratio.

  Note. The solution of the subsidiary problem to which the original problem ofProp.4 is reduced, and of which Archimedes promises a discussion, is given in a highlyinteresting and important note byEutocius, who introduces the subject with the following explanation.

  The investigation which follows may be thus reproduced. The general problem is,

  Given two straightlines AB, AC and an area D, to divide AB at M so that AM:AC = D:M*Bsquare.

  Prop.5. Problem. To construct a segment of a sphere similar to one segment and equal in volume to another.

  Prop.6. Problem. Given two segments of spheres, to find a third segment of a sphere similar to one of the given segments and having its surface equal to that of the other.

  Prop.7. From a given sphere to cut off a segment by a plane so that the segment may have a given ratio to the cone which has the same base as the segment and equal height.

  Prop.8. If a sphere be cut by a plane not passing through the centre into two segments A'BB', ABB', of which A'BB' is the greater,

  then the ratio (segment A'BB'):(Segment ABB') < (surface of A'BB')square:(surface of ABB')square,

  but >(surface of A'BB')power3/2 : (surface of ABB')power3/2.

  [The text of Archimedes adds an alternative proof of this prop., which is here omitted, because it is in fact neither clearer nor shorter than the above.]

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

  Measurement of a circle.

  Prop.1. The area of any circle is equal to a rightangledtriangle in which one of the sides about the rightangle is equal to the radius, and the other to the circumference, of the circle.

  Prop.2. The area of a circle is to the square on its diameter as 11 to 14.

  Prop.3. The ratio of the circumference of any circle to its diameter is less than 3*(1/7), but greater than 3*(10/71).

  On Conoids and spheroids.

  Introduction.

  Definitions.

  Lemma. If in an ascending arithmeticalprogression consisting of the magnitudes A1, A2, ... An the common difference be equal to the least term A1,

  then n.An < 2(A1 + A2 + ... + An)

  and > 2(A1 + A2 + ... + An-1)

  [The proof of this is given incidentally in the treatise OnSpirals.Prop.11. By placing lines side by side to represent the terms of the progression and then producing each so as to make it equal to the greatest term, Archimedes gives the equivalent of the following proof.]

  Prop.1. If A1, B1, C1, ... K1 and A2, B2, C2, ... K2 be twoseries of magnitudes such that

  A1:B1 = A2:B2,

  B1:C1 = B2:C2, and so on.

  andif A3, B3, C3, ... K3 and A4, B4, C4, ... K4 be two other series such that

  A1:A3 = A2:A4,

  B1:B3 = B2:B4, and so on,

  then (A1+B1+C1+...+K1):(A3+B3+C3+...+K3) = (A2+B2+C2+...+K2):(A4+B4+...+K4).

  Cor. If any terms in the third and fourth series corresponding to terms in the first and second be left out, the result is the same. For example, if the last terms K3, K4 are absent,

  (A1+B1+C1+...+K1):(A3+B3+C3+...+I3)

  = (A2+B2+C2+...+K2):(A4+B4+C4+...+I4), where I immediatelyprecedes K in each series.

  Lemma toProp.2. [OnSpirals, Prop.10] If A1, A2, A3, ... An be n lines forming an ascending arithmeticalprogression in which the common difference is equal to the least term A1,

  then (n+1)An-square + A1(A1+A2+A3+...+An)

  = 3(A1-square + A2-square + A3-square + .... + An-square.)

  Cor.1. From this it is evident that

  n*An-square < 3(A1-square + A2-square + ... + An-square).

  Also An-square = A1{An + 2(An-1 + An-2 + ... + A1)}, as above,

  so that An-square > A1(An + An-1 + ... + A1),

  and therefore An-square + A1(A1+A2+...+An)<2An-square.

  It follows from the prop. that

  n*An-square > 3(A1-square + A2-square + ... + An-1-square).

  Cor.2. All these results will hold if we substitude similar figures for squares on all the lines; for similar figures are in the duplicateratio of their sides.

  Prop.2. If A1, A2, ... An be any number of areas such that

  A1 = Ax + x-square

  A2 = a*2x + (2x)-square

  A3 = a*3x + (3x)-square

  ...

  An = a*nx + (nx)-square,

  then n*An:(A1+A2+...+An) < (a+nx):(u/2+nx/3),

  and n*An:(A1+A2+...+An-1) > (a+nx):(a/2+nx/3).

  Prop.3. (1) If TP, TP' be two tangents to any conic meeting in T,

  andif Qq, Q'q' be any two chords parallel respectively to TP, TP' and meeting in O,

  then QO*Oq:Q'O*Oq' = TP-square:TP'-square.

  And this is proved in the elements of conics. i.e. in the treatises on conics byAristaeus and Euclid.

  (2) If QQ' be a chord of a prabola bisected in V by the diameter PV,

  and if PV be of constant length,

  then the areas of the triangle PQQ' and of the segment PQQ' are both constant whatever be the direction of QQ'.

  Prop.4. The area of any ellipse is to that of the auxiliary circle as the minor axis to the major.

  Prop.5. If AA', BB' be the major and minor axis of an ellipse respectively,

  and if d be the diameter of any circle,

  then (area of ellipse):(area of circle) = AA'*BB':d-square.

  Prop.6. The area of ellipses are as the rectangle under their axes. This follows at once from Props.4.5.

  Cor. The areas of similar ellipses are as the squares of corresponding axes.

  Prop.7. Given an ellipse with centre C, and a line CO drawn perpendicular to its plane, it is possible to find a circular cone with vertex O and such that the given ellipse is a section of it [or, in other words, to find the circular sections of the cone with vertex O passing through the circumference of the ellipse.]

  Prop.8. Given an ellipse, a plane through one of its axes AA' and perpendicular to the plane of the llipse, and a line CO drawn from C, the centre, in the given plane through AA' but not perpendicular to AA', it is possible to find a cone with vertex O such that the given ellipse is a section of it [or, in other words, to find the circular sections of the cone with vertex O whose surface passes through the circumference of the ellipse.]

  Prop.9. Given an ellipse, a plane through one of its axes and perpendicular to that of the ellipse, and a straightline CO drawn from the centre C of the ellipse in the given plane through the axis but not perpendicular to that axis, it is possible to find a cylinder with axis OC such that the ellipse is a section of it [or, in other words, to find the circular sections of the cylinder with axis OC whose surface passes through the circumference of the given ellipse.]

  Prop.10. It was proved by the earlier geometers that any two cones have to one another the ratio compounded of the ratios of their bases and of their heights. [This follows fromEucl.XII.11. and .14. taken together.] The same method of proof will show that any segments of cone have to one another the ratio compounded of the ratios of their bases and of their heights.

  The prop. that any 'frustum' of a cylinder is triple of the conical segment which has the same base as the frustum and equal height is also proved in the same manner as the prop. that the cylinder is triple of the cone which has the same base as the cylinder and equal height. [This prop. was prroved by Eudocux, as stated in the preface to OnTheSphereAndCylinder.I. Cf. Eucl.XII.10.]

  Prop.11. (1) If a praboloid of revolution be cut by a plane through, or parallel to, the axis, the section will be a parabola equal to the original parabola which by its revolution generates the paraboloid. And the axis of the section will be the intersection between the cutting plane and the plane through the axis of the paraboloid at right angles to the cutting plane.

  If the paraboloid be cut by a plane at rightangles to its axis, the section will be a circle whose centre is on the axis.

  (2) If a hyperboloid of revolution be cut by a plane through the axis, parallel to the axis, or through the centre, the section will be a hyperbola,

  (a) if the section be through the axis, equal,

  (b) if parallel to the axis, similar,

  (c) if through the centre, not similar, to the original hyperbola which by its revolution generates the hyperboloid. And the axis of the section will be the intersection of the cutting plane and the plane through the axis of the hyperboloid at rightangles to the cutting plane.

  Any section of the hyperboloid by a plane at rightangles to the axis will be a circle whose centre is on the axis.

  (3) If any of the spheroidal figures be cut by a plane through the axis or parallel to the axis, the section will be an ellipse,

  (a) if the section be through the axis, equal,

  (b) if parallel to the axis, similar, to the ellipse which by its revolution generates the figure. And the axis of the section will be the intersection of the cutting plane and the plane through the axis of the spheroid at rightangles to the cutting plane.

  If the section be by a plane at rightangles to the axis of the spheroid, it will be a circle whose centre is on the axis.

  (4) If any of the said figures be cut by a plane through the axis,

  and if a perpendicular be drawn to the plane of section from any point on the surface of the figure by not on the section, that perpendicular will fall within the section.

  And the proofs of all these props. are evident.

  Prop.12. If a paraboloid of revolution be cut by a plane neither parallel nor perpendicular to the axis,

  and if the plane through the axis perpendicular to the cutting plane intersect it in a straightline of which the portion intercepted within the paraboloid is RR',

  the section of the paraboloid will be an ellipse whose major axis is RR' and whose minor axis is equal to the perpendicular distance between the lines through R, R' parallel to the axis of the paraboloid.

  Prop.13.14. If a hyperboloid of revolution be cut by a plane meeting all the generators of the enveloping cone,

  or if an 'oblong' spheroid be cut by a plane not perpendicular to the axis,

  and if a plane through the axis intersect the cutting plane at right angles in a straightline on which the hyperboloid or spheroid intercepts a length RR',

  then the section by the cuttingp lane will be an ellipse whose major axis is RR'.

  [Archimedes begins Prop.14 for the spheroid with the remark that, when the cutting plane passes through or is parallel to the axis, the case is clear.]

  Cor.1. If the spheroid be a 'flat' spheroid,

  then section will be an ellipse, and everything will proceed as before except that RR' will in this case be the minor axis.

  Cor.2. In all conoids or spheroid parallel sections will be similar, since the ratio O*A-square:O*P-square the same for all the parallel sections.

  Prop. 15. (1) If from any point on the surface of a conoid a line be drawn,

  in the case of the paraboloid, parallel to the axis,

  and, in the case of the hyperboloid, parallel to any line passing through the vertex of the enveloping cone,

  the part of the straightline which is in the same direction as the convexity of the surface will fall without it,

  and the part which is in the other direction within it.

  (2) If a plane touch a conoid without cutting it,

  it will touch it at one point only,

  and the plane drawn through the point of contact and the axis of the concoid will be at rightangles to the plane which touches it.

  Prop.16. (1) If a plane touch any of the spheroidal figures without cutting it,

  it will touch at one point only,

  and the plane through the point of contact and the axis will be at rightangles to the tangent plane.

  This is proved by the same method as the last prop.

  (2) If any conoid or spheroid be cut by a plane through the axis,

  and if through any tangent to the resulting conic a plane be erected at right angles to the plane of section,

  the plane so erected will touch the conoid or spheroid in the same point as that in which the line touches the conic.

  Prop.17. If two parallel planes touch any of the spheroidal figures, and another plane be drawn parallel to the tangent planes and passing through the centre,

  the line through any point of the circumference of the resulting section parallel to the chord of contact of the tangent planes will fall outside the spheroid.

  This is proved at once by reduction to a plane prop.

  Prop.18. Any spheroid figure which is cut by a plane through the centre is divided, both as regards its surface and its volume, into two equal parts by that plane.

  [To prove this, Archimedes takes another equal and similar spheroid, divides it similarly by a plane through the centre, and then uses the method of application.]

  Prop.19.20. Given a segment cut off by a plane from a praboloid or hyperboloid of revolution, or a segment of a spheroid less than half the spheroid also cut off by a plane,

  it is possible to inscribe in the segment one solid figure and to circumscribe about it another solid figure, each made up of cylinders or 'frusta' of cylinders of equal height, and such that the circumscribed figure exceeds the inscribed figure by a volume less than that of any given solid.

  Prop.21.22. 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 the same axis.

  Prop.23. If from a paraboloid of revolution two segments be cut off,

  then one by a plane perpendicular to the axis, the other by a plane not perpendicular to the axis,

  and if the axes of the segments are euqla,

  then the segments will be equal in volume.

  Prop.24.  If from a paraboloid of revolution two segments be cut off by planes drawn in any manner,

  then the segments will be to one another as the squares on their axes.

  Prop.25.26. In any hyperboloid of revolution,

  if A be the vertex and AD the axis of any segment cut off by a plane,

  and if CA be the semidiameter of the hyperboloid through A (CA being of course in the same straight line with AD),

  then (segment):(cone with same base and axis) = (AD+3CA):(AD+2CA).

  Prop.27.28.29.30. (1) In any spheroid whose centre is C,

  if a plane meeting the axis cut off a segment not greater than half the spheroid and having A for its vertex and AD for its axis,

  and if A'D be the axis of the remaining segment of the spheroid,

  then (first segment):(cone or segment of cone with the same base and axis)

  = CA+A'D:A'D

  [=3CA-AD:2CA-AD].

  (2) As a particular case, if the plane passes through the centre,

  so that the segment is half the spheroid, half the spheroid is double of the cone or segment of a cone which has the same vertex and axis.

  Prop.31.32. If a plane divide a spheroid into two unequal segments,

  and if AN, A'N be the axes of the lesser and greater segments respectively,

  while C is the centre of the spheroid,

  then (greater segment):(cone or segment of cone with same base and axis) = CA+AN:AN.

  On Spirals.

  Prop.1. If a point move at a uniform rate along any line, and two lengths be taken on it,

  then they will be proportional to the times of describing them.

  Prop.2. If each of two points on different lines respectively move along them each at a uniform rate,

  and if lengths be taken, one on each line, forming pairs,

  such that each pair are described in equal times,

  then the lengths will be proportionals.

  Prop.3. Given any number of circles, it is possible to find a straightline greater than the sum of all their circumferences.

  Prop.4. Given two unequal lines, viz. a straightline and the circumference of a circle, It is possible to find a straightline less than the greater of the two lines and greater than the less.

  Prop.5. Given a circle with centre O, and the tangent to it at a point A, it is possible to draw from O a straightline OPF, meeting the circle in P and the tangent in F,

  such that, if c be the circumference of any given circle whatever,

  then FP:OP < (arcAP):c.

  Prop.6. Given a circle with centre O, a chord AB less than the diameter, and OM the perpendicular on AB from O, it is possible to draw a straightline OFP, meeting the chord AB in F and the circle in P,

  such that FP:PB = D:E,

  where D:E is any given ratio less than BM:MO.

  Prop.7. Given a circle with centre O, a chord AB less than the diameter, and OM the perpendicular on it from O, it is possible to draw from O a straightline OPF, meeting the circle in P and AB produced in F,

  such that FP:PB = D:E, where D:E is any given ratio greater than BM:MO.

  Prop.8. Given a circle with centre O, a chord AB less than the diameter, the tangent at B, and the perpendicular OM from O on AB, it is possible to draw from O a straightline OFP, meeting the chord AB in F, the circle in P and the tangent in G,

  such that FP:BG = D:E, where D:E is any given ratio less than BM:MO.

  Prop.9. Given a circle with centre O, a chord AB less than the diameter, the tangent at B, and the perpendicular OM from O on AB, it is possbiel to draw from O a straightline OPGF', meeting the circle in P, the tangent in G, and AB produced in F,

  such that FP:BG = D:E, where D:E is any given ratio greater than BM:MO.

  Prop.10. If A1, A2, A3, ..., An be n lines forming an ascending arithmeticalprogression in which the common difference is equal to A1, the least term,

  then (n+1)*An-square + A1(A1+A2+...+An)

  = 3(A1-square + A2-square + ... + An-square).

  Prop.11. If A1, A2, ..., An be n lines forming an ascending arithmeticalprogression [in which the common difference is equal to A1, the least term],

  then (n-1)*An-square : (An-square + An-1-square + ... + A2-square) < An-square : {An*A1 + 1/3*(An-A1)-square}

  but (n-1)+An-square : (An-1-square + An-2-square + ... + A1-square) > An-square : {An*A1 + 1/3*(An-A1)square).

  Cor. The results in the above prop. are equallytrue if similar figures be substituted for square on the several lines.

  Def.s.

  1. Spiral.

  2. Origin of the spiral.

  3. Initial line in the revolution.

  4. The first distance. The second distance.

  5. The first area. The second area.

  6. The forward revolution. The backward revolution.

  7. The first cicle. The second circle.

  Prop.12. If any number of straightlines drawn from the origin to meet the spiral make equal angles with one another,

  then the lines will be in arithmetical progression.

  [The proof is obvious.]

  Prop.13. If a straightline touch the spiral,

  then it will touch it in onepointonly.

  Prop.14. If O be the origin, and P, Q two points on the first turn of the spiral,

  and if OP, OQ produced meet the first circle AKP'Q' in P', Q' respectively, OA being the initial line,

  then OP:OQ = (arc AKP'):(arc AKQ').

  Prop.15. If P,Q be points on the second turn of the spiral, and OP, OQ meet the first circle AKP'Q' in P', Q' as in the last prop.

  and if c be the circumference of the first circle,

  then OP:OQ = c+(arc AKP'):c+(arc AKQ').

  Prop.16.17. If BC be the tangent at P, any point on the spiral, PC being the forward part of BC,

  and if OP be joined,

  then the angle OPC is obtuse

  while the angle OPB is acute.

  Prop.18.19. I. If OA be the initial line, A the end of the first turn of the spiral,

  and if the tangent to the spiral at A be drawn,

  then the straightline OB drawn from O perpendicular to OA will meet the said tangent in some point B,

  and OB will be equal to the circumference of the first circle.

  II. If A' be the end of the second turn,

  then the perpendicular OB will meet the tangent at A' in some point B',

  and OB' will be equal to 2 (circumference of second circle).

  III. Generally, if An be the end of the nth turn,

  and OB meet the tangent at An in Bn,

  then OBn = ncn, where cn is the circumference of the nth circle.

  Prop.20. I. If P be any point on the first turn of the spiral and OT be drawn perpendicular to OP,

  then OT will meet the tangent at P to the spiral in some point in T,

  and if the circle drawn with centre O and radius OP meet the initial line in K,

  then OT is equal to the arc this circle between K and P measured in the forward direction of the spiral.

  II. Generally, if P be a point on the nth turn,

  and the notation be as before,

  while p represents the circumference of the circle with radius OP,

  then OT = (n-1)p + arcKP (measured forward).

  Prop.21.22.23. Given an area bounded by any arc of a spiral and the lines joining the extremities of the arc to the origin, it is possible to circumscribe about the area one figure, and to inscribe in it another figure, each consisting of similar sectors of circles, and such that the circumscribed figure exceeds the inscribed by less than any assigned area.

  Cor. Since the area bounded by the spiral is intermediate in magnitude between the circumscribed and inscribed figures, it follows that

  (1) a figure can be circumscribed to the area such that it exceeds the area by less than any assigned space,

  (2) a figure can be inscribed such that the area exceeds it by less than any assigned space.

  Prop.24. The area bounded by the first turn of the spiral and the initial line is equal to onethird of the first circle [=1/3*pie*(2piea)-square, where the spiral is r=a*thetha].

  [The same proof shows equally that, if OP be anyradius vector in the first turn of the spiral, the arae of the portion of the spiral bound thereby is equal to onethird of that sector of the circle drawn with radius OP which is bounded by the initial line and OP, measured in the forward direction from the initial line.]

  Prop.25.26.27.

  [25.] If A2 be the end of the second turn of the spiral, the area bounded by the second turn and OA-square is to the area of the second circle in the ratio of 7 to 12, being the ratio of {r2r1 + (1/3)(r2-r1)-square} to r2-square, where r1, 42 are the radii of the first and second circles respectively.

  [26.] If BC be any arc measured in the forward direction on any turn of a spiral, not being greater than the complete turn,

  and if a circle be drawn with O as centre and OC as radius meeting OB in B',

  then (area of spiral between OB, OC):(sector OB'C) = {OC*OB + (1/3)(OC-OB)-square : OC-square}.

  [27.] 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 that of the ring added by the third turn, and so on,

  then R3=2R2, R4=3R2, R5=4R2, ..., Rn=(n-1)R2.

  Also R2=6R1.

  [Archimedes's proof of Prop.25 is, mutatismutandis, the same as his proof of the more general Prop.26. The latter will accordingly be given here, and applied toProp.25 as a particular case.]

  Prop.28. If O be the origin and BC any arc measured in the forward direction on any turn of the spiral,

  let two circles be drawn (1) with centre O, and radius OB, meeting OC in C',

  and (2) with centre O and radius OC, meeting OB produced in B'.

  Then, if E denote the area bounded by the larger circular arc B'C, the line B'B, and the spiral BC,

  while F denotes the area bounded by the smaller arc BC', the line CC' and the spiral BC,

  E:F = {OB+(2/3)(OC-OB)} : {OB+(1/3)(OC-OB)}.

  On the equilibrium of planes, or The centres of Gravity of planes.

  BookI.

  I postulate the following.

  1. Equal weights at equal distances are in equilibrium, and equal weights at unequal distances are not in equilibrium but incline towards the weight which is at the greater distance.

  2. If, when weights at certain distances are in equilibrium, something be added to one of the weights, they are not in equilibrium but incline towards that weight to which the addition was made.

  3. Similarly, if anything be taken away from one of the weights, they are not in equilibrium but incline towards the weight from which nothing was taken.

  4. When equal and similar plane figures coincide if applied ot one another, their centres of gravity similarlycoincide.

  5. In figures which are unequal but similar the centres of gravity will be similarlysituated. By points similarlysituated in relation to similar figure I mean points such that, if straightlines be drawn from them to the equal angles, they make equal angles with the corresponding sides.

  6. If magnitudes at certain distances be in equilibrium, (other) magnitudes equal to them will also be in equilibrium at the same distances.

  7. In any figure whose perimeter is concave in (one and) the same direction the centre of gravity must be within the figure.

  Prop.1. Weights which balance at equal distances are equal.

  Prop.2. Unequal weights at equal distances will not balance but will incline towards the greater weight.

  Prop.3. Unequal weights will balance at unequal distances, the greater weight being at the lesser distance.

  Prop.4. If two equal weights have not the same centre of gravity, the centre of gravity of both taken together is at the middle point of the line joining their centre of gravity.

  [Proved fromProp.3. by reductioadabsurdum. Archimedes assumes that the centre of gravity of both together is on the straightline joining the centres of gravity of each, saying that this had been proved before. The allusion is no doubt to the lost treatise, OnLevers.]

  Prop.5. If three equal magnitudes have their centres of gravity on a straightline at equal distances, the centre of gravity of the system will coincide with that of the middle magnitude.

  [This followsimmediately fromProp.4.]

  Cor.1. The same is true of any odd number of magnitudes if those which are at equal distances from the middle one are equal, while the distances between their centres of gravity are equal.

  Cor.2. If there be an even number of magnitudes which their centres of gravity situated at equal distances on one straightline,

  and if the two middles ones be equal,

  while those which are equidistant from them (on each side) are equal respectively,

  then the centre of gravity of the system is the middle point of the line joining the centres of gravity of the two middle ones.

  Prop.6.7. Two magnitudes, whether commensurable [Prop.6.] or incommensurable [Prop.7.], balance at distances reciprocallyproportional to the magnitudes.

  Prop.8. If AB be a magnitude whose centre of gravity is C,

  and AD a part of it whose centre of gravity is F,

  then the centre of gravity of the remaining part will be a point G on FC produced such that

  GC:CF = (AD):(DE).

  Prop.9. The centre of gravity of any parallelogram lies on the straightline joining the middle points of opposite sides.

  Prop.10. The centre of gravity of a parallelogram is the point of intersection of its diagnoals.

  Prop.11. If abc, ABC be two similar triangles, and g, G two points in them similarlysituated with respect to them respectively,

  then, if g be the centre of gravity of the triangle abc,

  G must be the centre of gravity of the triangle ABC.

  Prop.12. Given two similar triangles abc, ABC, and d, D the middle points of bc, BC respectively,

  then, if the centre of gravity of abc lie on ad,

  that of ABC will lie on AD.

  Prop.13. In any triangle the centre of gravity lies on the straightline joining any angle to the middle point of the opposite side.

  Prop.14. It follows at once from the last prop. that the centre of gravity of any triangle is at the intersection of the lines drawn from any two angles to the middle points of the opposite sides respectively.

  Prop.15. If AD, BC be the two parallel sides of a trapezium ABCD, AD being the smaller,

  and if AD, BC be bisected at E, F respectively,

  then the centre of gravity of the trapezium is at a point G on EF

  such that GE:GF = (2BC+AD):(2AD+BC).

  On the equilibrium of planes, BookII.

  Prop.1. If P, P' be two parabolic segments and D, E their centres of gravity respectively,

  then the centre of gravity of the two segments taken together will be at a point C on DE determined by the relation P:P' = CE:CD.

  Definitions and lemmas preliminary toProp.2.

  If in a segment bounded by a straightline and a section of a rightangled cone [a parabola] a triangle be inscribed having the same base as the segment and equal height,

  if again triangles be inscribed in the remaining segments having the same bases as the segments and equal height,

  and if in the remaining segments triangles be isncribed in the same manner,

  let the resulting figure be said to be inscribed in the recognised manner in the segment.

  And it is plain,

  (1) that the lines joining the two angles of the figure so inscribed which are nearest to the vertex of the segment, and the next pairs of angles in order, will be parallel to the base of the segment,

  (2) that the said lines will be bisected by the diameter of the segment, and

  (3) that they will cut the diameter in the proportions of the successive odd numbers, the number one having reference to [the length adjacent to] the vertex of the segment.

  And these properties will have to be proved in their proper places.

  [The last words indicate an intention to give these prop.s in their proper connexion with systematic proofs; but the intention does not appear to have been carried out, or at least we know of no lost work ofArchimedes in which they could have appeared. The results can however be easilyderived from prop.s given in theQuadratureOfTheParabola as follows.]

  Prop.2. If a figure be inscribed in the recognised manner in a parabolic segment,

  then the centre of gravity of the figure so inscribed will lie on the diameter of the segment.

  Prop.3. If BAB', bab' be two similar parabolic segments whose diameters are AO, ao respectively,

  and if a figure be inscribed in each segment "in the recognised manner, the number of sides in each figure being equal,

  then the centres of gravity of the inscribed figures will divide AO, ao in the same ratio.

  [Archimedes enunciates this prop. as true of similar segments, but it is equally true of segments which are not similar, as the course of the proof will show.]

  Prop.4. The centre of gravity of any parabolic segment cut off by a straightline lies on the diameter of the segment.

  Prop.5. If in a parabolic segment a figure be inscribed in the recognised manner,

  then the centre of gravity of the segment is nearer to the vertex of the segment than the centre of gravity of the inscribed figure is.

  Prop.6. Given a segment of a parabola cut off by a straightline, it is possible to inscribe in it in the recognised manner a figure such that the distance between the centres of gravity of the segment and of the inscribed figure is less than any assigned length.

  Prop.7. If there be two similar parabolic segments,

  then the centres of gravity divide their diameter in the same ratio.

  [This prop., though enunciated of similar segments only, like Prop.3 on which it depends, is equallytrue of any segments. This fact did not escapeARchimedes, who uses the prop. in its more general form for the proof of Prop.8 immediately following.]

  Prop.8. If AO be the diameter of a parabolic segment,

  and G its centre of gravity,

  then AG = (3/2)GO.

  Prop.9.Lemma. If a, b, c, d be fourlines in continuedproportion and in descending order of magnitude,

  and if d:(a-d) = x:(3/5)(a-c),

  and (2a+4b+6c+3d):(5a+10b+10c+5d) = y:(a-c),

  it is required to prove that x+y = (2/5)a.

  [The following is the proof given byArchimedes, with the only difference that it is set out in algebraical isntead of geometricalnotation. This is done in the particular case simply in order to make the proof easier to follow. Archimedes exhibits his lines in the figure reproduced in the margin, but, now that it is possible to use algebraicalnotation, there is no advantage in using the figure and the more cumbrous notation which only obscures the course of the proof.]

  Prop.10. If PP'B'B be the portion of a parabola intercepted between two parallel chords PP', BB' bisected respectively in N, O by the diameter ANO (N being nearer than O to A, the vertex of the segments),

  and if NO be divided into five equal parts of which LM is the middle one (L being nearer than M to N),

  then, if G be a point on LM such that

  LG:GM = B*O-square*(2PN+BO):P*N-square*(2BO+PN),

  G will be the centre of gravity of the area PP'B'B.

  The Sandreckoner.

  I say then that, even if a sphere were made up of the sand, as great as Aristarchus supposes the sphere of the fixed stars to be, I shall still prove that, of the numbers named in the Principles [** was apparently the title of the work sent toZeuxippus. Cf. the note attached to the enumeration of lost works ofArchimedes in the Introduction, ChapterII., ad fin.], some exceed in multitude the number of the sand which is equal in magnitude to the sphere referred to, provided that the following assumptions be made.

  1. the perimeter of the earth is about 3 000 000 stadia and not greater.

  2. The diameter of the earth is greater than the diameter of the moon, and the diameter of the sun is greater than the diameter of the earth.

  3. The diameter of the sun is about 30 times the diameter of the moon and not greater.

  4. The diameter of the sun is greater than the side of the chiliagon inscribed in the greatest circle in the (sphere of the) universe.

  [Up to this point the treatise has been literally translated because of the historial interest attaching to the ipsissima verba ofArchimedes on such a subject. The rest of the work can now be morefreelyreproduced, and, before proceeding to the math.contents of it, it is only necessary to remark that Archimedes next describes how he arrived at a higher and a lower limit for the angle subtended by the sun. This he did by taking a long rod or ruler, fastening on the end of it a small cylinder or disc, pointing the rod in the direction of the sun just after its rising (so that it was possible to look directly at it), then putting the cylinder at such a distance that it just concealed, and just failed to conceal, the sun, and lastly measuring the angles subtended by the cylinder. He explains also the correction which he thought it necessary to make because "the eye does not see from one point but from a certain area".]

  To prove that (on this assumption) the diameter of the sun is greater than the side of a chiliagon, or figure with 1000 equal sides, inscribed in a great circle of the 'universe'.

  5. Suppose a quantity of sand taken not greater than a poppyseed, and suppose that it contains not more than 10 000 grains.

  Next suppose that the diameter of the poppyseed to be not less than (1/40)th of a fingerbreath.

  Orders and periods of numbers.

  Octads.

  Theorem.

  Application to the number of the sand.

  Conclusion.

  Quadrature of the parabola.

  Prop.1. If from a point on a parabola a straightline be drawn which is either itself the axis or parallel to the axis, as PV,

  and if QQ' be a chord parallel to the tangent to the parabola at P and meeting PV in V,

  then QV = VQ'.

  Conversely, if QV = VQ',

  then the chord QQ' will be parallel to the tangent at P.

  Prop.2. If in a parabola QQ' be a chord parallel to the tangent at P,

  and if a straightline be drawn through P which is either itself the axis or parallel to the axis, and which meets QQ' in V and the tangent at Q to the parabola in T,

  then PV = PT.

  Prop.3. If from a point on a parabola a straightline be drawn which is either itself the axis or parallel to the axis, as PV,

  and if from two other points Q, Q' on the parabola straightlines be drawn parallel to the tangent at P and meeting PV in V, V' respectively,

  then PV:PV' = Q*V-square:Q'*V'-square.

  And these prop.s are proved in the elements of conics. [i.e. in the treatises on conics by Euclid and Aristaeus.]

  Prop.4. If Qq be the base of any segment of a parabola, and P the vertex of the segment,

  and if the diameter through any other point R meet Qq in O and QP (produced if necessary) in F,

  then QV:VO = OF:FR.

  Prop.5. If Qq be the base of any segment of a parabola, P the vertex of the segment, and PV its diameter,

  and if the diameter of the parabola through any other point R meet Qq in O and the tangent in Q in E,

  then QO:Oq = ER:RO.

  Prop.6.7. Suppose a lever AOB placedhorizontally and supported at its middle point O.

  Let a triangle BCD in which the angle C is right or obtuse be suspended from B and O,

  so that C is attached to O and CD is in the same vertical line with O.

  Then, if P be such an area as, when suspended from A, will keep the system in equilibrium,

  P = (1/3)*TriangleBCD.

  Prop.8.9. Suppose a lever AOB placedhorizontally and supported at its middle point O.

  Let a triangle BCD, rightangled or obtuseangled at C, be suspended from the points B, E on OB, the angular point C being so attached to E that the side CD is in the some vertical line with E.

  Let Q be an area such that AO:OE = TriangleBCD:Q.

  Then, if an area P suspended from A keep the system in equilibrium,

  P<TriangleBCD, but >Q.

  Prop.10.11 Suppose a lever AOB placed horizontally and supported at O, its middle point.

  Let CDEF be a trapezium which can be so placed that its parallel sides CD, FE are vertical,

  while C is verticallybelow O, and the other sides CF, DE meet B.

  Let EF meet BO in H,

  and let the trapezium be suspended by attaching F to H and C to O.

  Further, suppose Q to be an area such that

  AO:OH = (trapeziumCDEF):Q.

  Prop.12.13. If the trapezium CDEF be placed as in the last prop.s,

  except that CD is vertically below a point L on OB instead of being below O,

  and the trapezium is suspended from L, H,

  suppose that Q, R are areas such that

  AO:OH = (trapezium CDEF): Q,

  and AO:OL = (trapezium CDEF):R.

  If then an area P suspended from A keep the system in equilibrium,

  P>R, but <Q.

  Prop.14.15. Let Qq be the base of any segment of a parabola.

  Then, if two lines be drawn from Q,q, each parallel to the axis of the parabola and on the same side of Qq as the segment is,

  either (1) the angles so formed at Q,q are both rightangles,

  or (2) one is acute and the other obtuse.

  In the latter case let the angle at q be the obtuseangle.

  Prop.16. Suppose Qq to be the base of a parabolic segment,

  q being not more distant than Q from the vertex of the parabola.

  Draw through q the straightline qE parallel to the axis of the parabola to meet the tangent at Q in E.

  It is required to prove that

  (area of segment)=(1/3)*TriangleEqQ.

  Prop.17. It is now manifest that the area of any segment of a parabola is fourthirds of the triangle which has the same base as the segment and equal height.

  Def. In segments bounded by a straightline and any curve. Base. Height. Vertex.

  Prop.18. If Qq be the base of a segment of a parabola, and V the middle point of Qq,

  and if the diameter through V meet the curve in P,

  then P is the vertex of the segment.

  Prop.19. If Qq be a chord of a parabola bisected in V by the diameter PV,

  and if RM be a diameter bisecting QV in M, and RW be the ordinate from R to PV,

  then PV = (4/3)RM.

  Prop.20. If Qq be the base, and P the vertex, of a parabolic segment,

  then the triangle PQq is greater than the half the segment PQq.

  Cor. It follows that it is possible to isncribe in the segment a polygon such that the segments left over are together less than any assigned area.

  Prop.21. If Qq be the base, and P the vertex, of any parabolic segment,

  and if R be the vertex of the segment cut off by PQ,

  then TrianglePQq = 8*TrianglePRQ.

  Prop.22. If there be a series of areas A, B, C, D,... each of which is fourtimes the next in order,

  and if the largest, A, be equal to the triangle PQq inscribed in a parabolic segment PQq and having the same base with it and equal height,

  then (A+B+C+D+...) < (area of segment PQq).

  Prop.23. Given a series of areas A, B, C, D ... Z, of which A is the greatest,

  and each is equal to four times the next in order,

  then A+B+C+...+Z+(1/3)Z = (4/3)A.

  Prop.24. Every segment bounded by a parabola and a chord Qq is equal to fourthirds of the triangle which has the same base as the segment and equal height.

  On Floating bodies. BookI.

  Postulate1. Let it be supposed that a fluid is of such a character that, its parts lying evenly and being continuous, that part which is thrust the less is driven along by that which is thrust the more; and that each of its parts is thrust by the fluid which is above it in a perpendicular direction if the fluid be sunk in anything and compressed by anything else.

  Prop.1. If a surface be cut by a plane alwayspassing through a certain point,

  and if the section be always a circumference [of a circle] whose centre is the aforesaid point,

  then the surface is that of a sphere.

  Prop.2. The surface of any fluid at rest is the surface of a sphere whose centre is the same as that of the earth.

  Prop.3. Of solids those which, size for size, are of equal weight with a fluid will, if let down into the fluid, be immersed

  so that they do not project above the surface but do not sink lower.

  Prop.4. A solid lighter than a fluid will, if immersed in it, not be completelysubmerged, but part of it will project above the surface.

  Prop.5. Any solid lighter than a fluid will, if placed in the fluid, be so far immersed that the weight of the solid will be equal to the weight of the fluid displaced.

  Prop.6. If a solid lighter than a fluid be forciblyimmersed in it,

  then the solid will be driven upwards by a force equal to the difference between its weight and the weight of the fluid displaced.

  Prop.7. A solid heavier than a fluid will, if placed in it, descend to the bottom of the fluid,

  and the solid will, when weighted in the fluid, be lighter than its true weight by the weight of the fluid displaced.

  [This prop. may, I think, safely be regarded as decisive of the question how ARchimedes determined the prop.s of gold and silver contained in the famous crown. the prop. suggests in fact the following method.]

  Prop.8. If a solid in the form of a segment of a sphere, and of a substance lighter than a fluid, be immersed in it so that its base does not touch the surface,

  then the solid will rest in such a position that its axis is perpendicular to the surface;

  and, if the solid be forced into such a position that its base touches the fluid on one side and be then set free,

  then it will not remain in that position but will return to the symmetrical position.

  [The proof of this prop. is wanting in the latin version ofTarTaglia. Commandinus supplied a proof of his own in his edition.]

  Prop.9. If a solid in the form of a segment of a sphere, and of a substance lighter than a fluid, be immersed in it so that its base is completely below the surface,

  then the solid will rest in such a position that its axis is perpendicular to the surface.

  [The proof of this prop. has only survived in a mutilated form. It deals moreover with only one case out of three which are distinguished at the beginning, viz. that in which the segment is greater than a hemisphere, while figures only are given for the cases where the segment is equal to, or less than, a hemisphere.]

  On floating bodies. BookII.

  Prop.1. If a solid lighter than a fluid be at rest in it,

  then the weight of the solid will be to that of the same volume of the fluid as the immersed portion of the solid is to the whole.

  Prop.2. If a right segment of paraboloid of revolution whose axis is not greater than (3/4)p, where p is the principal parameter of the generating parabola, and whose specific gravity is less than that of a fluid, be placed in the fluid with is axis inclined to the vertical at any angle,

  but so that the base of the segment does not touch the surface of the fluid,

  then the segment of the paraboloid will not remain in that position but will return to the position in which its axis is vertical.

  Prop.3. If a right segment of a paraboloid of revolution whose axis is not greater than (3/4)p, where p is the parameter, and whose specific gravity is less than that of a fluid, be placed in the fluid with its axis inclined at any angle to the vertical,

  but so that is base is entirely submerged,

  then the solid will not remain in that position but will return to the position in which the axis is vertical.

  Prop.4. Given a right segment of a paraboloid of revolution whose axis AN is greater than (3/4)p, where p is the parameter, and whose specific gravity is less than that of a fluid but bears to it a ratio not less than (AN-(3/4)p)-square:A*N-square,

  if the segment of the paraboloid be placed in the fluid with its axis at any inclination to the vertical,

  but so that its base does not touch the surface of the fluid,

  then it will not remain in that position but will return to the position in which its axis is vertical.

  Prop.5. Givena right segment of a paraboloid of revolution such that its axis AN is greater than (3/4)p, where p is the parameter, and its specific gravity is less than that of a fluid but in a ratio to it not greater than the ratio {A*N-square - (AN - (3/4)p)-square}:A*N-square,

  if the segment be placed in the fluid with its axis inclined at any angle to the vertical,

  but so that its base is completely submerged,

  then it will not remain in that position but will return to the position in which AN is vertical.

  Prop.6. If a right segment of a paraboloid lighter than a fluid be such that its axis AM is greater than (3/4)p, but AM:(1/2)p < 15:4,

  and if the segment be placed in the fluid with its axis so inclined to the vertical that its base touches the fluid,

  then it will never remain in such a position that the base touches the surface in one point only.

  Prop.7. Given a right segment of a praboloid of revolution lighter than a fluid

  and such that its axis AM is greater than (3/4)p, but AM:(1/2)p < 15:4,

  if the segment be placed in the fluid

  so that its base is entirely submerged,

  then it will never rest in such a position that the base touches the surface of the fluid at one point only.

  Prop.8. Given a solid in the form of a right segment of a paraboloid of revolution whose axis AM is greater than (3/4)p,

  but such that AM:(1/2)P < 15:4, and whose gravity bears to that of a fluid a ratio less than (AM-(3/4)p)-square : A*M-square,

  then, if the solid be placed in the fluid so that its base does not touch the fluid and its axis is inclined at an angle to the vertical,

  then the solid will not return to the position in which its axis is vertical and will not remain in any position except that in which its axis makes with the surface of the fluid a certain angle to be described.

  Prop.9. Given a solid in the form of a right segment of a paraboloid of revolution whose axis AM is greater than (3/4)p, but such that AM:(1/2)p < 15:4, and whose specific travity bears to that of a fluid a ratio greater than {A*M-square-(AM-(3/4)p)-square}:A*M-square,

  then, if the solid be placed in the fluid with its axis inclined at an angle to the vertical but so that its base is entirely below the surface,

  then the solid will not return to the position in which its axis is vertical and will not remain in any position except that in which its axis makes with the surface of the fluid an angle equal to that described in the last prop.

  Prop.10. Given a solid in the form fo a right segment of a paraboloid of revolution in which the axis AM is of a length such that AM:(1/2)p > 15:4,

  and supposing the solid placed in a fluid of greater specific gravity so that its base is entirely above the surface of the fluid,

  to investigate the positions of rest.

  Preliminary.

  (Enunciation.)

  (Proof.)

  Book of lemmas.

  Prop.1. If two circles touch at A,

  and if BD, EF be parallel diameters in them,

  then ADF is a straightline.

  Prop.2. Let AB be the diameter of a semicircle,

  and let the tangents to it at B and at any other point D on it meet in T.

  If now DE be drawn perpendicular to AB,

  and if AT, DE meet in F,

  then DF = FE.

  Prop.3. Let P be any point on a segment of a circle whose base is AB,

  and let PN be perpendicular to AB.

  TAke D on AB so that AN = ND.

  If now pQ be an arc equal to the arc PA,

  and BQ be joined,

  then BQ, BD shall be equal.

  Prop.4. If AB be the diameter of a semicircle and N any point on AB,

  and if semicircles be described within the first semicircle and having AN, BN as diameters respectively,

  then the figure included between the circumferences of the three semicircles is "what Archimedes called an **";

  and its area is equal to the circle on PN as diameter, where PN is perpendicular to AB and meets the original semicircle in P.

  Prop.5. Let AB be the diameter of a semicircle, C any point on AB, and CD perpendicular to it,

  and let semicircles be described within the first semicircle and having AC, CB as diameters.

  Then, if two circles be drawn touching CD on different sides and each touching two of the semicircles,

  then the circls so drawn will be equal.

  [As pointed out by an arabian scholiat Al-Kau-Hi, this prop. may be stated moregenerally. If, instead of one point on C on AB, we have two points C, D, and semicircles be described on AC, BD as diameters,

  and if, instead of the perpendicular to AB through C, we take the radical axis of the twosemicircles,

  then the circles described on different sides of the radical axis and each touching it as well as two of the semicircles are equal.

  The proof is similar and presents no difficulty.]

  Prop.6. Let AB, the diameter of a semicircle, be divided at C

  so that AC = (3/2)CB [or in any ratio].

  Describe semicircles within the first semicircle and on Ac, CB as diameters,

  and suppose a circle drawn touching all three semicircles.

  If GH be the diameter of this circle,

  to find the relation betwen GH and AB.

  Prop.7. If circles be circumscribed about and inscribed in a square,

  then the circumscribed circle is double of the inscribed circle.

  Prop.8. If AB be any chord of a circle whose centre is O,

  and if AB be produced to C

  so that BC is equal to the radius;

  if further CO meet the circle in D and be produced to meet the circle a second time in E,

  then the arc AE will be equal to threetimes the arc BD.

  Prop.9. If in a circle two chords AB, CD which do not pass through the centre intersect at right angles,

  then (arc AD)+(arc CB) = (arc AC)+(arc DB).

  Prop.10. Suppose that TA, TB are two tangents to a circle,

  while TC cuts it.

  Let BD be the chord through B parallel to TC,

  and let AD meet TC in E.

  Then, if EG be drawn perpendicular to BD,

  it will bisect it in H.

  Prop.11. If two chords AB, CD in a circle intersect at right angles in a point O, not being the centre,

  then A*O-square + B*O-square + C*O-square + D*O-square = (diamter)-square.

  Prop.12. If AB be the diameter of a semicircle,

  and TP, TQ the tangents to it from any point T,

  and if AQ, BP be joined meeting in R,

  then TR is perpendicular to AB.

  Prop.13. If a diameter AB of a circle meet any chord CD, not a diameter, in E,

  and if AM, BN be drawn perpendicular to CD,

  then CN = DM.

  Prop.14. Let ACB be a semicircle on AB as diameter,

  and let AD, BE be equal lengths measured along AB from A, B respectively.

  On AD, BE as diameters describe semicircles on the side towards C,

  and on DE as diameter a semicircle on the opposite side.

  Let the perpendicular to AB through O, the centre of the first semicircle, meet the opposite semicircles in C, F respectively.

  Then shall the area of the figure bounded by the circumference of all the semicircles ("which archimedes calls 'Salinon;") be equal to the area of the circle on CF as diameter.

  Prop.15. Let AB be the diameter of a circle, AC a side of an inscribed regular polygon, D the middle point of the arc AC.

  Join AC and produce it to meet BA produced in E;

  join AC, DB meeting in F,

  and draw FM perpendicular to AB.

  Then EM = (radius of circle).

  The Cattleproblem.

  It is required to find the number of bulls and cows of each of fourcolours, or to find eight unknown qualities. The first part of the problem connects the unknowns by seven simple equations; and the second part adds two more conditions to which the unknowns must be subject.

  Wunn's problem.

  The Method ofArchimedes traeting of mechanical problems.

  First then I will set out the very first theorem which became known to me by means ofMechanics, namely that, Any segment of a section of a rightangled cone (i.e. a parabola) is fourthires 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 prop.s]...

  [I premise the following prop.s which I shall use int he course of the work.]

  1. If from [one magnitude another magnitude be subtracted which has not the same centre of gravity,

  then the centre of gravity of the remainder is found by] producing [the straightline joining the centres of gravity of the whole magnitude and of the subtracted part in the direction of the centre of gravity of the whole] and cutting off from it alength which has to the distance between the said centres of gravity the ratio which the weight of the subtracted magnitude has to the weight of the remainder. [OnTheQuilibriumOfPlanes, I.8.]

  2. If the centres of gravity of any number of magnitudes whatever be on the same straightline,

  then the centre of gravity of the magnitude made up of all of them will be on the same straightline. [Ibid. I.5.]

  3. The centre of gravity of any straightline is the point of bisection of straightline. [Ibid. I.4.]

  4. The centre of gravity of any triangle is the point in which the straightlines drawn from the angular points of the triangle to the middle points of the (opposite) sides cut one another. [Ibid. I.13,14.]

  5. The centre of gravity of any parallelogram is the point in which the diagonals meet. [I.10.]

  6. The centre of gravity of a circle is the point which is also the centre [of the circle].

  7. The centre of gravity of any cylinder is the point of bisection of the axis.

  8. The centre of gravity of any cone is [the point which divides its axis so that] the portion [adjacent to the vertex is] triple [of the portion adjacent to the base].

  [All these prop.s have already been] proved. [The problem of finding the centre of gravity of a cone is notsolved in any extant work ofArchimedes. It may have been solved either in a separate treatise, such as the **, which is lost, or perhaps in a larger mechanical work of which the extant books OnTheQuilibriumOfplaned formed only a part.

  [Besides these I require also the following prop., which is easily proved:

  If in two series of magnitudes those of the first series are, in order, proportional to those of the second series and further] the magnitudes [of the first series], either all or some of them, are in any ratio whatever [to those of a third series],

  and if the magnitudes of the second series are in the same ratio to the corresponding magnitudes [of a fourth series],

  then the sum of the magnitudes of the first series has to the sum of the selected magnitudes of the third series the same ratio which the sum of the magnitudes of the second series has to the sum of the (corresponding) selected magnitudes of the fourth series. [OnConoidsAndSpheroids, Prop.1.]"

  Prop.1.  Let ABC be a segment of a parabola bounded by the straightline AC and the parabola ABC,

  and let D be the middle point of AC.

  Draw the straightline DBE parallel to the axis of the parabola and join AB, BD.

  Then shall the segment ABC be 4/3 of the triangle ABC.

  Prop.2. We can investigate by the same method the prop.s that

  (1) Any sphere is (in respect of solid content) four times the cone with base equal to a great circle of the sphere and height equal to its radius; and

  (2) the cylinder with base equal to a great circle of the sphere and height equal to the diameter is 1*(1/2) times the sphere.

  Prop.3. By this method we can also investigate the theorem that

  A cylinder with base equal to the greatest circle in a spheroid and height equal to the axis of the spheroid

  is 1*(1/2) times the spheroid;

  and, when this is established, it is plain that

  If any spheroid be cut by a plane through the centre and at right angles to the axis,

  then the half of the spheroid is double of the cone which has the same base and the same axis as the segment (i.e. the half of the spheroid).

  Prop.4. Any segment of a rightangled conoid (i.e. a paraboloid of revolution)  cut off by a plane at rightangles to the axis is 1*(1/2) times the cone which has the same base and the same axis as the segment.

 Prop.5. The centre of gravity of a segment of a rightangled conoid (i.e. a paraboloid of revolution) cut off by a plane at rightangles to the axis

  is on the straightline which is the axis of the segment,

  and divides the said straightline in wuch a way that the portion of it adjacent to the vertex is double of the remaining portion.

  Prop.6. The centre of gravity of any hemisphere [is on the straightline which] is its axis,

  and divides the said straightline in such a way that the portion of it adjacent to the surface of the hemisphere has to the remaining portion the ratio which 5 has to 3.

  Prop.7. We can also investigate by the same method the thoerem that

  [Any segment of a sphere has] to a cone [with the same base and height the ratio which the sum of the radius of the sphere and the height of the complementary segment has to the height of the complementary segment.]

  [There is a lacuna here; but all that is missing is the construction, and the construction is easily understood by means of the figure. BAD is of course the segment of the sphere the volume of which is to be compared with the volume of a cone with the same base and height.]

  Prop.8. [The nunciation, the settingout, and a few words of the construction are missing.

  The enunciation, however, can be supplied from that of Prop.9, with which it must be identical except that it cannot refer to "any segment", and the presumption therefore is that the prop. was enunciated with reference to one kind of segment only, i.e. either a segment greater than a hemisphere of a segment less than a hemisphere.

  Heiberg's figure corresponds to the case of a segment greater than a hemisphere. The segment investigated is of course the segment BAD. The settingout and construction are selfevident from the figure.]

  Prop.9. In the same way we can investigate the theorem that

  The centre of gravity of any segment of a sphere is on the straightline which is the axis of the segment, and divides this straightline in such a way that the part of it adjacent to the vertex of the segment has to the remaining part the ratio which the sum of the axis of the segment and fourtimes the axis of the complementary segment has to the sum of the axis of the segment and double the axis of the complementary segment.

  [As this theorem relates to "any segment" but states the same result as that proved in the preceding prop., it follows that Prop.8. must have related to one kind of segment, either a segment greater than a semicircle (as in Heiberg's figure of Prop.8) or a segment less than semicircle; and the present prop. completed the proof for both kinds of segments. It would only require a slight change in the figure, in any case.]

  Prop.10. By this method too we can investigate the theorem that

  [A segment of an obtuseangled conoid (i.e. a hyperboloid of revolution) has to the cone which has] the same base [as the segment and equal height the same ratio as the sum of the axis of the segment and three times] the "annex to the axis" (i.e. half the transverse axis of the hyperbolic section through the axis of the hyperboloid or, in other words, the distance between the vertex of the segment and the vertex of the enveloping cone) has to the sum of the axis of the segment and double of the "annex". [The text has "triple" (**) in the last line instead of "double". As there is a considerable lacuna before the last few lines, a theorem about the centre of gravity of a segment of a hyperboloid of revolution may have fallen out.] [this is the theorem proved inOnConoidsAndSpheroids.25.], "and also many other theorems, which as the method has been made clear by means of the foregoing examples, I will omit, in order that I may now proceed to compass the proofs of the theorems mentioned above."

  Prop.11. If in a right prism with square bases a cylinder be inscribed having its bases in opposite square faces and touching with is surface the remaining four parallelogrammic faces,

 and if through the centre of the circle which is the base of the cylinder and one side of the opposite square face a plane be drawn,

  then the figure cut off by the plane so drawn is onesixthpart of the whole prism.

  "This can be investigated by the method, and, when it is set out, I will go back to the proof of it by geometrical considerations."

  [The investigations by the mechanical method is contained in the two props., 11,12. Prop.13 gives another solution which, although it contains no mechanics, is still of the character which Archimedes regards as inconclusive, since it assumes that the solid is actually made up of parallel plane sections and that an auxiliary parabola is actually made up of parallel straightlines in it. Prop.14 added the conclusive geometrical proof.]

  Prop.12.

  [The rest of the proof is missing, but, as Zeuthen says, the result obtained and the method of arrving at it are plainlyindicated by the above.

  Archimedes wishes to prove that the half cylinder PQR, in the place where it is, balances the prism GHM, in the place where it is, about H as fixed point.]

  Prop.13. Let there be a right prism with square bases, one of which is ABCD;

  in the prism let a cylinder be inscribed, the base of which is the circle EFGH touching the sides of the square ABCD in E, F, G, H.

  [the above prop. and the next are peculiarly interesting for the fact that the parabola is an auxiliary curve introduced for the sole purpose of analyticallyreducing the required cubature to the known quadrature of the parabola.]

  Prop.14. [There are large gaps in the exposition of this geometrical proof, but the way in which the methodofexhaustion was applied, and the parallelism between this and other applications of it, are clear. The first fragment shows that solid figures made up of prisms were circumscribed and inscribed to the portion of the cylinder. The parallel triangular faces of these prisms were perpendicular to GE in the figure of Prop.13; they divided GE into equal portions of the requisite smallness; each section of the portion of the cylinder by such a plane was a triangular face common to an inscribed and a circumscribed right prism. The planes also produced prisms in the prism cut off by the same oblique plane as cuts off the portion of the cylinder and standing on GD as base.

  The number of parts into which the parallel planes divided Ge was made great enough to secure that the circumscribed figure exceeded the inscribed figure by less than a small assigned magnitude.

  The second part of the proof began with the assumption that the portion of the cylinder is > (2/3) of the prism cut off; and this was proved to be impossible, by means of the use of the auxiliary parabola and the prop.

  MN:ML = M*N-square:M*O-square, which are employed inProp.13.

  We may supply the missing proof as follows.