Then, the difference between either variable and S', that is, S + 2 (B − B') - S' and S' - s − 2 (b − B'), Now if the number of faces of the prisms be indefinitely increased, Bb, and therefore V- v, approaches the limit 0. But V' is evidently >v, and < V. Then, VV' and V' - v approach the limit 0. Whence, V and v approach the limit V'. SEX L and sex L. (§ 363, II) 3. We have, (§ 484) e, approaches the limit 0. Now if the number of faces of the prisms be indefinitely increased, Ss, and therefore E (§ 640, 1) But E', the perimeter of a rt. section of the cylinder, is <E; for the theorem of § 683 is evidently true when for the O is taken any closed curve whose tangents do not intersect its surface; also, E' is >e. Then, E E' and E' e approach the limit 0. Whence, E and e approach the limit E'. (Ax. 4) PROOFS OF THE LIMIT STATEMENTS OF § 650. 687. Theorem. The total surface of a circular cone is less than the total surface of any circumscribed pyramid. Given pyramid S-ABCD circumscribed about circular cone S-EF. To Prove total surface S-EF< total surface S-ABCD. Proof. Of the total surfaces of the cone and of its circumscribed pyramids, there must be one total surface such that the area of every other is either equal to or > it. D S B E But no circumscribed pyramid can have this total surface. For suppose pyramid S-ABCD to have this total surface; and let S-BCDFE be a circumscribed pyramid, whose face SEF intersects faces SAB and SAD in lines SE and SF, respectively. Now, face SEF is <sum of faces SAE, SAF, and AEF. (§ 684) Whence, total surface of pyramid S-BCDFE is total surface of pyramid S-ABCD. Then, total surface of cone S-EF is < total surface of any circumscribed pyramid. PROOFS OF THE LIMIT STATEMENTS OF § 650. 688. Let H denote the altitude, S and s the lateral areas, V and v the volumes, and B and b the areas of the bases, of the circumscribed and inscribed pyramids, respectively; also, S' the lateral area of the cone, V' its volume, and B' the area of its base. 1. We have,. S+B>S' + B'. S+ (B - B')>S'. (§ 687) Again, the total surface of the inscribed pyramid is < the total surface of the cone. Then, :. S' + B' > s + b, or S' >s+ (b − B'). S+ (B-B')> S'>s+ (b − B'). (§ 684) Now if the number of faces of the pyramids be indefinitely increased, BB' and b - B' approach the limit 0. (§ 363, II) Also, the difference between the perimeters of the bases of the pyramids approaches the limit 0. (§ 363, I) Then, S+B continually decreases, and s + b continually increases; and the difference between them can be made less than any assigned value, however small. Then, S-s+ (B - b) approaches the limit 0. But Bb approaches the limit 0. Whence, Ss approaches the limit 0. (§ 684) (§ 363, II) Then, S' is intermediate in value between two variables, the difference between which approaches the limit 0. Whence, the difference between either variable and S', that is, S + (B − B') - S' and S" - s − (b − B'), approaches the limit 0. Then, S S' and S' s approach the limit 0. Whence, S and s approach the limit S'. 2. We have, Whence, V=B x H and v = b x H. (§ 521) Now if the number of faces of the pyramids be indefinitely increased, Bb, and therefore V- v, approaches the limit 0. But, Vis evidently > v, and < V. Then, V-V and V v approach the limit 0. Whence, V and v approach the limit V. (§ 363, II) PROOF OF THE LIMIT STATEMENT IN NOTE FOOT OF PAGE 374. 689. Theorem. If a regular broken line, inscribed in an arc, be revolved about a diameter, not intersecting the arc, as an axis, and the subdivisions of the arc be bisected indefinitely, the area of the surface generated by the broken line approaches the area of the surface generated by the arc as a limit. Given regular broken line ABCD, inscribed in arc AD, revolving about diameter OM as an axis. To Prove that, if the subdivisions of arc AD be bisected indefinitely, area of surface generated by ABCD approaches area of surface generated by arc AD as a limit. B' A M B Proof. Let A'B', B'C', and C'D' be tangents || to AB, BC, and CD, respectively, points A', B', C', and D' being in radii OA, OB, OC, and OD, respectively, produced; and let S, s, and S' denote the areas of the surfaces generated by A'B'C'D', and ABCD, and arc AD, respectively. Of the surfaces generated by arc AD, by ABCD, and by regular inscribed broken lines obtained by bisecting the subdivisions of the arc indefinitely, there must be one surface such that the areas of all the others are either equal to or < it. But no regular inscribed broken line can generate this surface. For if this were the case, by bisecting the subdivisions of the arc, a regular inscribed broken line would be obtained having the same projection on the axis; but the from 0 to each line would be greater, and hence the surface generated would be greater. (§ 665, and Note foot of p. 374.) Hence, surface generated by arc AD is > surface generated by ABCD; that is, S' is > s. Again, of the surfaces generated by arc AD, by A'B'C'D', and by regular circumscribed broken lines obtained by bisecting the subdivisions of the arc indefinitely, there must be one surface such that the areas of all the others are either equal to or > it. But no regular circumscribed broken line can generate this surface. For if this were the case, by bisecting the subdivisions of the arc, a regular circumscribed broken line would be obtained in which the from 0 to each line would be the same; but the projection on the axis would be smaller, and hence the surface generated would be smaller. Hence, surface generated by arc AD is < surface generated by A'B'C'D'; that is, S' is < S. Then, S S' and S -s are < S 8. Now if the subdivisions of arc AD be bisected indefinitely, the difference between broken lines A'B'C'D' and ABCD approaches the limit 0. (Note foot p. 374.) Then, the difference between the projections on OM of A'B'C'D' and ABCD approaches the limit 0. Also, the difference between the s from 0 to A'B' and AB approaches the limit 0. (Note foot p. 374.) Then, the difference between the areas of the surfaces generated by A'B'C'D' and ABCD, that is, S-s approaches the limit 0. (§ 665) Then, SS' and S s approach the limit 0. Whence, S and s approach the limit S'. PROOF OF THE LIMIT STATEMENT IN NOTE FOOT OF PAGE 377. 690. Theorem. If a regular polygonal sector, inscribed in a sector of a circle, be revolved about a diameter, not crossing the sector, as an axis, and the subdivisions of the arc be bisected indefinitely, the volume of the solid generated by the polygonal sector approaches the volume of the solid generated by the sector as a limit. Given regular polygonal sector OABCD, inscribed in sector OAD, revolved about diameter OM as an axis. (Fig. of § 689.) To Prove that, if the subdivisions of arc AD be bisected indefinitely, volume of solid generated by OABCD approaches volume of solid generated by sector OAD as a limit. Proof. Let A'B', B'C', and C'D' be tangents || to AB, BC, and CD, respectively, points A', B', C', and D' being in radii OA, OB, OC, and OD, respectively, produced; and let V, v, and V' denote the volumes of the solids generated by OA'B'C'D', OABCD, and sector OAD, respectively. Then, V' is evidently > v, and < V. Whence, V VI and VI v are V- v. Now if the subdivisions of arc AD be bisected indefinitely, the difference between the areas of OA'B'C'D' and OABCD, and therefore Vv, approaches the limit 0. (Note foot p. 377.) Then, VV' and V-v approach the limit 0. |