PROP. 14. altitude C B likewise equal to the differIf within and about a semicircle there ence of AB, A C; and with respect to are inscribed and circumscribed the third, its altitude HE is likewise any two half-polygons, the one having the equal to the difference of AB, AC, and diameter of the semicircle for its diago- its base H L is a mean proportional benal, and the other the diameter pro- its adjacent sides are equal to AB and AC tween the squares of AB, AC, because duced; and if these figures are made to revolve together with the semicircle respectively, and (II. 35.) ABis to AB about the diameter ; the sphere gene- the difference of the cubes is less than XĀC as ABXAC to AC2. Therefore rated by the semicircle shall be greater than the inscribed solid of revolution, a parallelopiped, whose altitude is CB, and less than the circumscribed solid; and its base equal to three times the also the surface of the sphere shall be square of A B. But, because C B may greater than the surface of the inscribed be made less than any given line, this solid of revolution, and less than that of parallelopiped may be made less than the circumscribed solid. any given solid. Much more, therefore, For, with respect to may the difference of the cubes be made the solid contents, the less than any given solid, that is, than sphere contains the in any given difference. scribed solid, and is itself Therefore, &c. contained in the circum Cor. It appears from the demonstrascribed solid : and, with tion, that the difference of the cubes of respect to the surfaces, two straight lines is equal to the prothe surface of the sphere duct of the difference of the straight lines envelops the surface of by the sum of their squares, and a mean the inscribed solid, and proportional between those squares. is enveloped by that of PROP. 15. the circumscribed solid (Lemma 2.). Any sphere being given, two figures Therefore, &c. of revolution generated by similar half Lemma 3. polygons may be, the one inscribed in the sphere, and the other circumscribed If there be two straight lines, of which about it, such that the difference of their one is given, and the other may be made surfaces, or of their solid contents, shall to approach to it within any given dif- be less than any given difference. ference; the cube of the latter may also Let the given sphere be generated be made to approach to the cube of the by the revolution of former within any given difference. the semicircle ADB M Let AB, AC be the two straight lines, about the diameter A of which AB is given. Upon AB (I. 52.) AB. And first, let describe the square P be the given diffe T. BD; from A draw rence of surfaces; and AE perpendicular let Q be the surface of to the plane BAE H any solid circumscri(IV.37.); make AE bing the sphere. Then, IL equal to A B, and D since (as in prop. 7.) complete the cube a regular polygon K AF: and in like man- A CB may be inscribed in R ner upon A C describe the square CG the circle ABD, such in the same plane with B D, 'from AE that, CE being its cut off AH equal to AC, and com- apothem, CD2 CE2 shall be to plete the cube AK: and let the faces C E2 in a ratio less than that of P to HK and G K of the latter cube be pro- Q; let AFGHKB be the half of duced to meet the faces B F, EF, of the such a polygon, and let a similar half ormer Then the difference of the two polygon L MNORV be circumscribed, cubes is equal to the sum of the three so that one of its sides L M may touch parallelopipeds GF, C L, and L E. Of the circle in D (III. 27. Cor. 2.)' Then, these, the first has its base DF equal to if these inscribed and circumscribed the square of A B (IV. 22.), and its alti- half-polygons be made to revolve with tude D G equal to the difference of AB, the semicircle about the axis AB, they AC; the second has its base C K equal will generate, together with the sphere to the square of AC (IV. 22.), and its generated by the semicircle, two figures N D L G B a of revolution, the one inscribed in the For if this product be not equal to sphere, the other circumscribed about the surface of the sphere, it must either it. And because the surfaces of these be greater or less than it. figures are equal respectively (13.) to the If greater, it must be greater also rectangles under A B and the circum- than the surface of some circumscribed ference which has the radius C E, and solid of revolution (15. Cor.), greater, under L V and the circumference which that is (13.), than the product of the diahas the radius C D, and that these rect- gonal L V by the circumference which angles are to one another as the squares has for its radius the apothem C D (see of CE, CD (II. 37. Cor. 1.); the dif- the figure of Prop. 15.); which is imposference of the surfaces is to the surface sible, because the diameter is less than of the inscribed figure as C D — CE LV, and the circumference of the geneto C E2 (II. 20.); that is, in a less ratio rating circle is the same with the cir. than that of P to Q. But the surface of cumference which has the radius CD. the inscribed figure is less than Q: much If less, it must also be less than the more, therefore, is the difference of the surface of some inscribed solid of revosurfaces less than P (II. 18. Cor.). lution (15. Cor.)-less, that is (13.), than Next, let S be the given difference of the product of the diameter A B, which contents; and let T be the content of is the same with the diameter of the any solid circumscribing the sphere. generating circle, by the circumference Then, since a regular polygon may be which has for its radius the apothem inscribed in the circle ABD, such that, CE; which is impossible, because the CE being its apothem, CD–CE shall circumference of the generating circle is be less than any given difference, and greater than the circumference which therefore also such that CD3 - CE3 has the radius CE. shall be to C E3 in a ratio less than that Therefore the product in question is of Sto T (Lemma 3.); let A FGHKB neither greater nor less than the surface be the half of such a polygon, and let the of the sphere; that is, it is equal to it., figures of revolution be inscribed in the Therefore, &c. sphere, and circumscribed about it, as Cor. 1. The surface of a sphere is before. Then, because the contents of equal to four times the area of its genethese figures are (13.) equal respectively rating circle. For the area of this circle to the thirds of two parallelopipeds (IV. is equal to half the product of the radius 25. Schol.), having their bases equal to and circumference (III. 32.). the surfaces, and their altitudes equal Cor. 2. If a right cyto CE, CD, and that these parallelo- linder be circumscribed pipeds are to one another as the cubes about a sphere; the of CE, CD, for their bases are, as was surface of the sphere shown in the former part of the propo- shall be equal to the sition, as the squares of CE, CD; the convex surface of the difference of the contents is to the con- cylinder. For the latter tent of the inscribed figure as CDs— is equal to the proC Es to C E3 (II. 20.), that is, in a less duct of its altitude, and the circumference ratio than that of S to T. But the con- of its base (3.); and its base is equal tent of the inscribed figure is less than to the generating circle of the sphere, T: much more, therefore, is the dif- and its altitude to the diameter. ference of the contents less than S (II. Cor. 3. The surface of a sphere is 18. Cor.). equal to two-thirds of the whole surface Therefore, &c. of the circumscribing cylinder. Cor. Any sphere being given, a Cor. 4. If D is the diameter of a figure of revolution may be inscribed sphere, its whole surface is equal to - D2 (or circumscribed), which shall differ (III. 34. Schol.). from the sphere in surface or in content, by less than any given difference. For PROP. 17.' the difference between the sphere and The solid content of a sphere is equal either of the figures of revolution, whe- to one-third of the product of the radius ther in surface or in content, is less than by the surface. that of the two figures (14.). For the third part of this product PROP. 16. cannot be greater than the content of the sphere; since then it would be greater The surface of a sphere is equal to also than the content of some circumthe product of the circumference and scribed solid of revolution (15 Cor.) diameter of the generating circle. greater, that is, than one-third of the a product of the same radius by the sur- the radii), in the triplicate ratio of the face of that solid (13.); which is impos- radii, or (IV. 27. Cor. 2.) as the cubes of sible, because the surface of the sphere the radii. is less than that of the solid (14.). Otherwise ; Nor, on the other hand, can it be less Let D, d be the diameters of two than the content of the sphere, for then spheres, and R, r their radii. Then would it be less than some inscribed (16. Cor. 4.) “D”, ndo will represent solid of revolution (15.Cor.), less, that is, their surfaces, and (17. Cor. 4.) i «Da, than one-third of the product of the apo- and3 their solid contents; or, since D them by the surface of that solid (13.); is equal to 2R and d to 27, 4, R2, 4 7 12 which is impossible, because not only will represent their surfaces, and I x R3, is the radius greater than the apothem, p3 their solid contents. But (p. 47, but the surface of the sphere is likewise Rule ii.) 4. R2 is to 4 # po as Ro to r?, greater than the surface of the inscribed and I . R3 is to a p3 as R3 to rs. Theresolid (14.). fore, the surfaces are as the squares of Therefore the product in question is the radii, and the solid contents as the equal to the solid content of the sphere. cubes of the radii. Therefore, &c. Therefore, &c. Cor. 1. The solid content of a sphere is equal to one-third of the product of SECTION 3.-Surfaces and contents of the radius by four times the area of the certain portions of the sphere. generating circle (16. Cor. 1.). In order to have a clear apprehension Cor. 2. The solid content of a sphere of the figures intended in the following is two-thirds of the solid content of the definitions, it is necessary to keep in circumscribing cylinder. For the latter mind, that every section of a sphere is equal to twice the product of the which is made by a plane is a circle, the radius, and the area of the generating centre of which is the foot of the percircle (4) pendicular drawn to the plane from the Cor. 3. If any solid contained by centre of the sphere (IV. 8. Cor.). planes be circumscribed about a sphere, Def. 10. A segment of a sphere is any the content of the sphere will be to the portion of it which is cut off by a plane, content of the solid as the surface of the and the circle in which the plane cuts the sphere to the surface of the solid. For sphere is called the base of the segment. the solid may be divided into pyramids, When the plane having the centre of the sphere for their passes through the common vertex, and their altitudes equal centre, the two segeach to the radius of the sphere; and ments into which the since each of these pyramids is equal to sphere is divided are a third of the product of its base and equal to one another, altitude, their sum is equal to a third of and are therefore the product of the convex surface of the each of them called a hemisphere. solid and the radius of the sphere: also, The convex surface of a segment is the sphere is equal to a pyramid, liaving called a zone. the same altitude, and its base equal to 11. A double-based spherical segment the surface of the sphere (IV. 32.). is a portion of a sphere Cor. 4. If D is the diameter of a intercepted between two sphere, its whole solid content is equal parallel planes; and the to į + x D(16. Cor. 4.). circles in which these PROP. 18. (Euc. xii. 18.). planes cut the sphere are called the bases of The surfaces of spheres are as the the segment. The consquares of the radii, and their solid con- vex surface of a double-based segment tents are as the cubes of the radii. is likewise called a zone. For the surfaces are equal respectively 12. A sector of a sphere is the solid to four times the areas of the generating figure contained by circles (16. Cor. 1.), and these areas are the convex surface of as the squares of the radii (III. 33.). a segment, and that And the solid contents are to one of a right cone, which another in a ratio which is compounded has the same base of the ratios of the surfaces and of the with the segment, and radii ; that is (because the surfaces are for its vertex the cento one another in the duplicate ratio of tre of the sphere. 3 а O D F E N The convex surface of the segment is extremities of the arc: then, if the semicalled the base of the sector. circle be made to revolve about the dia13. A spherical orb meter A B, the zone which is generated is a portion of a sphere by the arc DF, shall be greater than the contained between its surface generated by the chord DF, and surface and that of a less than the surface generated by the lesser sphere, which is tangent GH. concentric (or has the From the points D, F draw the straight lines D d, Ff, each of them perpendicusame centre) with it. lar to A B (1. 45.). Then, in the sup14. A spherical wedge or ungula is a portion of a sphere intercepted between posed revolution of the figure, these two planes, each of which passes which have the points d, f for their cen straight lines will generate two circles through the centre of the sphere. The tres, and d D, f F for their radii respecconvex surface of an ungula is called a tively (IV. 3. Cor. 2.). And, because lune. Let ADB be a semicircle, and from the zone generated by the arc DEF, together with these two.circles, forms a the points D, E of convex surface which envelops, and the semicircumfe therefore (Lemma 2.) is greater than the rence, let thestraight lines DF, EG be convex surface consisting of the surface drawn at right an generated by the chord D F and the same two circles, the zone generated by the gles to the diameter arc DEF is greater than the surface A B; join C E, and let KNL be a se generated by the chord D F. In the next place, from the points cond semicircle, D, F draw the tangents DK, FL having the same (III. 56.) to meet GH in the points K, L centre C; then, if the whole figure re- respectively; bisect DK in M (I. 43.); volve about A B, the parts AEG, DE through MỈ draw MN parallel to CG GF, A E C, and ADB LNK will ge- (I. 48.) to meet G K in N, and from the nerate a spherical segment, a double, points M, N draw Mm, Nn perpendibased spherical segment, a spherical cular each of them to AB; and, lastly, sector, and a spherical orb respectively, through m draw mp parallel to M N to And if the semicircle A D B, instead of meet N nin p. Then, because the middle making a complete revolution, revolve point of D K, in the supposed revolution only through a certain angle, it will ge- of the figure about the axis AB, genenerate a spherical wedge or ungula. rates the circumference which has the radius M m, the surface generated by Prop. 19. DK is equal to the product of D K and If a semicircle be made to revolve the circumference which has the radius about its diameter, the zone which is Mm (11. Cor.). And, in like manner, generated by any arc of the semicircle since N is the middle point of GK shall be greater than the surface gene- (II. 29.), the surface generated by GK rated by the chord of that arc, and less is equal to the product of GK and the than the surface generated by the tan circumference which has the radius Nn. gent of the same arc, which is drawn But, because (III. 2. Cor. 1.) the angle parallel to the chord, and terminated by KDG is a right angle, and therefore the radii passing through its extremi- (I. 8.) the angle K G D less than a right ties. angle, that is, than KDG, DK is less Let ADB be a than GK (I. 9.); and, because (I. 22.) semicircle having the Mm is equal to N p, which is less than diameter AB, and Nn, the circumference which has the the centre C: let radius M m is less than the circumferDEF be any arc of ence which has the radius Nn (III.33.). the semicircle, DF Therefore, upon both accounts, the surits chord, and GH a face generated by D K is less than the straight line parallel surface generated by GK. And in the to DF, which touches same manner it may be shown that the the arc D E F in E, surface generated by L F is less than the and is terminated by surface generated by L H. Therefore, the radii CD, CF the whole convex surface generated by passing through the the three straight lines DK, KL, L Fis a A C H less than the whole surface generated by can it be less than the zone; for then it the tangent G H. But the zone gene- must be less also than 'some inscribed rated by the arc D E F is less than the surface-less, that is, than the product surface generated by DK, KL, LF of AL and the circumference which (Lemma 2.), for the zone together with has the radius CE; which is impossible, the circles which have the radii Dd, Ff, because the circumference AD B E is is enveloped by the latter surface toge- greater than that which has the radius ther with the same two circles. Much CE. Therefore, it must be equal to more, therefore, is the zone generated the zone ; that is, the zone is equal to by the arc D E F less than the surface the product of the circumference AD generated by the tangent GH. EB, and the part A L of the diameter. Therefore, &c. Next, let H K be any arc, by the rePROP. 20. volution of which about the diameter ; A spherical zone is equal to the pro- A B a double-based zone is generated; duct of the circumference of the let , generating circle and that portion of the cular to A B. Then, because the whole axis which is intercepted between its zone generated by the arc A K is equal convex surface and base; or, if it be to the product of AN and the cirdouble-based, between its two bases. Let ADB be a semicircle, and AK any arc, by the revolution of which A D P B cumference ADBE, and the part generated by the arc A H equal to the product of AL and the same circumfe rence, the remainder, that is, the about the diameter A B a spherical zone double-based zone in question, is equal is generated, and from K draw KL to the product of L N and the same cirperpendicular to AB: the zone shall be cumference. equal to the product of the whole cir- Therefore, &c. cumference ADBE by the part A L of Cor. 1. If a cylinder, having the axis the diameter. A B, be circumscribed about the sphere; The demonstration is in every respect any zone having the same axis, shall be similar to that of prop. 13. For, in the equal to that portion of the convex surfirst place, it is evident that the arc face of the cylinder which is intercepted AK may be divided into a number of between the base of the cylinder and equal arcs, such that, the chords AF, the plane of the base of the zone, or beFG, &c. being drawn, their common tween the planes of its two bases, if it distance CE from the centre C shall be double-based (3.). approach to the radius CD within any Cor. 2. In the same or in equal spheres, given difference; and hence it may be any two zones are to one another as the shown, as in prop. 15., that there may parts of the axis or axes which are inbe inscribed in the zone and circum- tercepted between their respective bases scribed about it, two surfaces of revo. (II. 35.). lution which differ from each other, and PROP. 21. therefore (19.) each of them from the Every spherical sector is equal to zone, by less than any given difference. one-third of the product of its base and Therefore, the product in question can- the radius of the sphere. not be greater than the zone; for then it must be greater also than some cir- is demonstrated are similar to those in The steps by which this proposition cumscribed surface-greater, that is, dicated in the preceding. In the first than the product of MQ and the circum- place, it may be shown that two solids ference AD BE; which is impossible, of revolution may be, one inscribed because AL is less than MQ.* Neither in the sector, the other circumscribed be the difference, not the sum, of M L, and LQ; but, them to the sector more nearly than by * If the point Q lies between A and C, MQ will about it, which approach each of in this case also, A L is less than M Q, because LQ is less than KP, that is, than M A. any given difference. Hence, the pro |