Sources page biographical material

6.Y. ROPE ROUND THE EARTH
The first few examples illustrate what must be the origin of the idea in more straightforward situations.
Lucca 1754. c1330. F. 8r, pp. 31 32. This mentions the fact that a circumference increases by 44/7 times the increase in the radius.

Muscarello. 1478.

Ff. 932-93v, p. 220. A circular garden has outer circumference 150 and the wall is 3½ thick. What is the inner circumference? Takes π as 22/7.

F. 95r, p. 222. The internal circumference of a tower is 20 and its wall is 3 thick. What is the outer circumference? Again takes π as 22/7.

Pacioli. Summa. 1494. Part II, f. 55r, prob. 33. Florence is 5 miles around the inside. The wall is 3½ braccia wide and the ditch is 14 braccia wide -- how far is it around the outside? Several other similar problems.

William Whiston. Edition of Euclid, 1702. Book 3, Prop. 37, Schol. (3.). ??NYS -- cited by "A Lover" and Jackson, below.

"A Lover of the Mathematics." A Mathematical Miscellany in Four Parts. 2nd ed., S. Fuller, Dublin, 1735. The First Part is: An Essay towards the Probable Solution of the Forty five Surprising PARADOXES, in GORDON's Geography, so the following must have appeared in Gordon. Part I, no. 73, p. 56. "'Tis certainly Matter of Fact, that three certain Travellers went a Journey, in which, Tho' their Heads travelled full twelve Yards more than their Feet, yet they all return'd alive, with their Heads on."

Carlile. Collection. 1793. Prob. XXV, p. 17. Two men travel, one upright, the other standing on his head. Who "sails farthest"? Basically he compares the distance travelled by the head and the feet of the first man. He notes that this argument also applies to a horse working a mill by walking in a circle; the outside of the horse travels about six times the thickness of the horse further than the inside on each turn.

Jackson. Rational Amusement. 1821. Geographical Paradoxes, no. 54, pp. 46 & 115-116. "It is a matter of fact, that three certain travellers went on a journey, in which their heads travelled full twelve yards more than their feet; and yet, they all returned alive with their heads on." Solution says this is discussed in Whiston's Euclid, Book 3, Prop. 37, Schol. (3.). [This first appeared in 1702.]

K. S. Viwanatha Sastri. Reminiscences of my esteemed tutor. In: P. K. Srinivasan, ed.; Ramanujan Memorial Volumes: 1: Ramanujan -- Letters and Reminiscences; 2: Ramanujan -- An Inspiration; Muthialpet High School, Number Friends Society, Old Boys' Committee, Madras, 1968. Vol. 1, pp. 89-93. On p. 93, he relates that this was a favourite problem of his tutor, Srinivasan Ramanujan. Though not clearly dated, this seems likely to be c1908-1910, but may have been up to 1914. "Suppose we prepare a belt round the equator of the earth, the belt being 2π feet longer, and if we put the belt round the earth, how high will it stand? The belt will stand 1 foot high, a substantial height."

Dudeney. The paradox party. Strand Mag. 38 (No. 228) (Dec 1909) 673 674 (= AM, p. 139).

Anon. Prob. 58. Hobbies 30 (No. 773) (6 Aug 1910) 405 & (No. 776) (27 Aug 1910) 448. Double track circular railway, five miles long. Move all rails outward one foot. How much more material is needed? Solution notes the answer is independent of the length.

Ludwig Wittgenstein was fascinated by the problem and used to pose it to students. Most students felt that adding a yard to the rope would raise it from the earth by a negligible amount -- which it is, in relation to the size of the earth, but not in relation to the yard. See: John Lenihan; Science in Focus; Blackie, 1975, p. 39.

Ernest K. Chapin. Loc. cit. in 5.D.1. 1927. Prob. 5, p. 87 & Answers p. 7. A yard is added to a band around the earth. Can you raise it 5 inches? Answer notes the size of the earth is immaterial.

Collins. Book of Puzzles. 1927. The globetrotter's puzzle, pp. 68 69. If you walk around the equator, how much farther does your head go?

Abraham. 1933. Prob. 33 -- A ring round the earth, pp. 12 & 24 (9 & 112).

Perelman. FMP. c1935?? Along the equator, pp. 342 & 349. Same as Collins.

Sullivan. Unusual. 1943.

Prob. 20: A global readjustment. Take a wire around the earth and insert an extra 40 ft into it -- how high up will it be?

Prob. 23: Getting ahead. If you walk around the earth, how much further does your head go than your feet?

W. A. Bagley. Puzzle Pie. Op. cit. in 5.D.5. 1944. Things are seldom what they seem -- No. 42a, 43, 44, pp. 50-51. 42a and 43 ask how much the radius increases for a yard gain of circumference. No. 44 asks if we add a yard to a rope around the earth and then tauten it by pulling outward at one point, how far will that point be above the earth's surface?

Richard I Hess. Puzzles from Around the World. The author, 1997. (This is a collection of 117 puzzles which he published in Logigram, the newsletter of Logicon, in 1984-1994, drawn from many sources. With solutions.) Prob. 28. Consider a building 125 ft wide and a rubber band stretched around the earth. If the rubber band has to stretch an extra 10 cm to fit over the building, how tall is the building? He takes the earth's radius as 20,902,851 ft. He gets three trigonometric equations and uses iteration to obtain 85.763515... ft.

Erwin Brecher & Mike Gerrard. Challenging Science Puzzles. Sterling, 1997. [Reprinted by Goodwill Publishing House, New Delhi, India, nd [bought in early 2000]]. Pp. 38-39 & 77. The M25 is a large ring road around London. A man commutes from the south to the north and finds the distance is the same if goes by the east or the west, so he normally goes to the east in the morning and to the west in the evening. Recalling that the English drive on the left, he realised that his right wheels were on the outside in both journeys and he worried that they were wear out sooner. So he changed and drove both ways by the east. But he then worried whether the wear on the tires was the same since the evening trip was on the outer lanes of the Motorway.
6.Z. LANGLEY'S ADVENTITIOUS ANGLES
Let ABC be an isosceles triangle with  B =  C = 80^o. Draw BD and CE, making angles 50^o and 60^o with the base. Then  CED = 20^o.
JRM 15 (1982 83) 150 cites Math. Quest. Educ. Times 17 (1910) 75. ??NYS

Peterhouse and Sidney Entrance Scholarship Examination. Jan 1916. ??NYS.

E. M. Langley. Note 644: A Problem. MG 11 (No. 160) (Oct 1922) 173.

Thirteen solvers, including Langley. Solutions to Note 644. MG 11 (No. 164) (May 1923) 321 323.

Gerrit Bol. Beantwoording van prijsvraag No. 17. Nieuw Archief voor Wiskunde (2) 18 (1936) 14 66. ??NYS. Coxeter (CM 3 (1977) 40) and Rigby (below) describe this. The prize question was to completely determine the concurrent diagonals of regular polygons. The 18 gon is the key to Langley's problem. However Bol's work was not geometrical.

Birtwistle. Math. Puzzles & Perplexities. 1971. Find the angle, pp. 86-87. Short solution using law of sines and other simple trigonometric relations.

Colin Tripp. Adventitious angles. MG 59 (No. 408) (Jun 1975) 98 106. Studies when  CED can be determined and all angles are an integral number of degrees. Computer search indicates that there are at most 53 cases.

CM 3 (1977) 12 gives 1939 & 1950 reappearances of the problem and a 1974 variation.

D. A. Q. [Douglas A. Quadling]. The adventitious angles problem: a progress report. MG 61 (No. 415) (Mar 1977) 55-58. Reports on a number of contributions resolving the cases which Tripp could not prove. All the work is complicated trigonometry -- no further cases have been demonstrated geometrically.

CM 4 (1978) 52 53 gives more references.

D. A. Q. [Douglas A. Quadling]. Last words on adventitious angles. MG 62 (No. 421) (Oct 1978) 174-183. Reviews the history, reports on geometric proofs for all cases and various generalizations.

J[ohn]. F. Rigby. Adventitious quadrangles: a geometrical approach. MG 62 (No. 421) (Oct 1978) 183-191. Gives geometrical proofs for almost all cases. Cites Bol and a long paper of his own to appear in Geom. Dedicata (??NYS). He drops the condition that ABC be isosceles. His adventitious quadrangles correspond to Bol's triple intersections of diagonals of a regular n-gon.

MS 27:3 (1994/5) 65 has two straightforward letters on the problem, which was mentioned in ibid. 27:1 (1994/5) 7. One letter cites 1938 and 1955 appearances. P. 66 gives another solution of the problem. See next item.

Douglas Quadling. Letter: Langley's adventitious angles. MS 27:3 (1994/5) 65 66. He was editor of MG when Tripp's article appeared. He gives some history of the problem and some life of Langley (d. 1933). Edward Langley was a teacher at Bedford Modern School and the founding editor of the MG in 1894-1895. E. T. Bell was a student of Langley's and contributed an obituary in the MG (Oct 1933) saying that Langley was the finest expositor he ever heard -- ??NYS. Langley also had botanical interests and a blackberry variety is named for him.

6.AA. NETS OF POLYHEDRA
Albrecht Dürer. Underweysung der messung mit dem zirckel uň [NOTE: ň denotes an n with an overbar.] richtscheyt, in Linien ebnen unnd gantzen corporen. Nürnberg, 1525, revised 1538. Facsimile of the 1525 edition by Verlag Dr. Alfons Uhl, Nördlingen, 1983. German facsimile with English translation of the 1525 edition, with notes about the 1538 edition: The Painter's Manual; trans. by Walter L. Strauss; Abaris Books, NY, 1977. Figures 29 43 (erroneously printed 34) (pp. 316-347 in The Painter's Manual, Dürer's 1525 ff. M-iii-v - N-v-r) show nets and pictures of the regular polyhedra, an approximate sphere (16 sectors by 8 zones), truncated tetrahedron, truncated cube, cubo-octahedron, truncated octahedron, rhombi cubo-octahedron, snub cube, great rhombi-cubo-octahedron, truncated cubo octahedron (having a pattern of four triangles replacing each triangle of the cubo octahedron -- not an Archimedean solid) and an elongated hexagonal bipyramid (not even regular faced). (See 6.AT.3 for more details.) (Panofsky's biography of Dürer asserts that Dürer invented the concept of a net -- this is excerpted in The World of Mathematics I 618 619.) In the revised version of 1538, figure 43 is replaced by the icosi-dodecahedron and great rhombi-cubo-octahedron (figures 43 & 43a, pp. 414 419 of The Painter's Manual) to make 9 of the Archimedean polyhedra.

Albrecht Dürer. Elementorum Geometricorum (?) -- the copy of this that I saw at the Turner Collection, Keele, has the title page missing, but Elementorum Geometricorum is the heading of the first text page and appears to be the book's title. This is a Latin translation of Unterweysung der Messung .... Christianus Wechelus, Paris, 1532. This has the same figures as the 1525 edition, but also has page numbers. Liber quartus, fig. 29-43, pp. 145-158 shows the same material as in the 1525 edition.

Cardan. De Rerum Varietate. 1557, ??NYS = Opera Omnia, vol. III, pp. 246-247. Liber XIII. Corpora, qua regularia diei solent, quomodo in plano formentur. Shows nets of the regular solids, except the two halves of the dodecahedron have been separated to fit into one column of the text.

Barbaro, Daniele. La Practica della Perspectiva. Camillo & Rutilio Borgominieri, Venice, (1569); facsimile by Arnaldo Forni, 1980, HB. [The facsimile's TP doesn't have the publication details, but they are given in the colophon. Various catalogues say there are several versions with dates on the TP and colophon varying independently between 1568 and 1569. A version has both dates being 1568, so this is presumed to be the first appearance. Another version has an undated title in an elaborate border and this facsimile must be from that version.] Pp. 45-104 give nets and drawings of the regular polyhedra and 11 of the 13 Archimedean polyhedra -- he omits the two snub solids.

E. Welper. Elementa geometrica, in usum geometriae studiosorum ex variis Authoribus collecta. J. Reppius, Strassburg, 1620. ??NYS -- cited, with an illustration of the nets of the octahedron, icosahedron and dodecahedron, in Lange & Springer Katalog 163 -- Mathematik & Informatik, Oct 1994, item 1350 & illustration on back cover, but the entry gives Trassburg.

Athanasius Kircher. Ars Magna, Lucis et Umbrae. Rome, 1646. ??NX. Has net of a rhombi-cuboctahedron.

Pike. Arithmetic. 1788. Pp. 458-459. "As the figures of some of these bodies would give but a confused idea of them, I have omitted them; but the following figures, cut out in pasteboard, and the lines cut half through, will fold up into the several bodies." Gives the regular polyhedra.

Dudeney. MP. 1926. Prob. 146: The cardboard box, pp. 58 & 149 (= 536, prob. 316, pp. 109 & 310). All 11 nets of a cube.

Perelman. FMP. c1935? To develop a cube, pp. 179 & 182 183. Asserts there are 10 nets and draws them, but two "can be turned upside down and this will add two more ...." One shape is missing. Of the two marked as reversible, one is symmetric, hence equal to its reverse, but the other isn't.

C. Hope. The nets of the regular star faced and star pointed polyhedra. MG 35 (1951) 8 11. Rather technical.

H. Steinhaus. One Hundred Problems in Elementary Mathematics. (As: Sto Zadań, PWN -- Polish Scientific Publishers, Warsaw, 1958.) Pergamon Press, 1963. With a Foreword by M. Gardner; Basic Books, NY, 1964. Problem 34: Diagrams of the cube, pp. 20 & 95 96. (Gives all 11 nets.) Gardner (pp. 5 6) refers to Dudeney and suggests the four dimensional version of the problem should be easy.

M. Gardner. SA (Nov 1966) c= Carnival, pp. 41 54. Discusses the nets of the cube and the Answers show all 11 of them. He asks what shapes these 11 hexominoes will form -- they cannot form any rectangles. He poses the four dimensional problem; the Addendum says he got several answers, no two agreeing.

Charles J. Cooke. Nets of the regular polyhedra. MTg 40 (Aut 1967) 48 52. Erroneously finds 13 nets of the octahedron.

Joyce E. Harris. Nets of the regular polyhedra. MTg 41 (Winter 1967) 29. Corrects Cooke's number to 11.

A. Sanders & D. V. Smith. Nets of the octahedron and the cube. MTg 42 (Spring 1968) 60 63. Finds 11 nets for the octahedron and shows a duality with the cube.

Peter Turney. Unfolding the tesseract. JRM 17 (1984 85) 1 16. Finds 261 nets of the 4 cube. (I don't believe this has ever been confirmed.)

Peter Light & David Singmaster. The nets of the regular polyhedra. Presented at New York Acad. Sci. Graph Theory Day X, 213 Nov 1985. In Notes from New York Graph Theory Day X, 23 Nov 1985; ed. by J. W. Kennedy & L. V. Quintas; New York Acad. Sci., 1986, p. 26. Based on Light's BSc project in 1984-1984 under my supervision. Shows there are 43,380 nets for the dodecahedron and icosahedron. I may organize this into a paper, but several others have since verified the result.
6.AB. SELF RISING POLYHEDRA
H. Steinhaus. Mathematical Snapshots. Stechert, NY, 1938. (= Kalejdoskop Matematyczny. Książnica Atlas, Lwów and Warsaw, 1938, ??NX.) Pp. 74-75 describes the dodecahedron and says to see the model in the pocket at the end, but makes no special observation of the self-rising property. Described in detail with photographs in OUP, NY, eds: 1950: pp. 161-164; 1960: pp. 209 212; 1969 (1983): pp. 196-198.

Donovan A. Johnson. Paper Folding for the Mathematics Class. NCTM, 1957, p. 29, section 66: Pop-up dodecahedron.

M. Kac. Hugo Steinhaus -- a reminiscence and a tribute. AMM 81 (1974) 572 581. Material is on pp. 580 581, with picture on p. 581.

A pop up octahedron was used by Waddington's as an advertising insert in a trade journal at the London Toy Fair about 1981. Pop-up cubes have also been used.

6.AC. CONWAY'S LIFE
There is now a web page devoted to Life run by Bob Wainwright -- address is:

http://members.aol.com/life1ine/life/lifepage.htm [sic!].

M. Gardner. Solitaire game of "Life". SA (Oct 1970). On cellular automata, self reproduction, the Garden of Eden and the game of "Life". SA (Feb 1971). c= Wheels, chap. 20-22. In the Oct 1970 issue, Conway offered a $50 prize for a configuration which became infinitely large -- Bill Gosper found the glider gun a month later. At G4G2, 1996, Bob Wainwright showed a picture of Gosper's telegram to Gardner on 4 Nov 1970 giving the coordinates of the glider gun. I wasn't clear if Wainwright has this or Gardner still has it.

Robert T. Wainwright, ed. (12 Longue Vue Avenue, New Rochelle, NY, 10804, USA). Lifeline (a newsletter on Life), 11 issues, Mar 1971 -- Sep 1973. ??NYR.

John Barry. The game of Life: is it just a game? Sunday Times (London) (13 Jun 1971). ??NYS -- cited by Gardner.

Anon. The game of Life. Time (21 Jan 1974). ??NYS -- cited by Gardner.

Carter Bays. The Game of Three dimensional Life. Dept. of Computer Science, Univ. of South Carolina, Columbia, South Carolina, 29208, USA, 1986. 48pp.

A. K. Dewdney. The game Life acquires some successors in three dimensions. SA 256:2 (Feb 1987) 8 13. Describes Bays' work.

Bays has started a quarterly 3 D Life Newsletter, but I have only seen one (or two?) issues. ??get??

Alan Parr. It's Life -- but not as we know it. MiS 21:3 (May 1992) 12-15. Life on a hexagonal lattice.

6.AD. ISOPERIMETRIC PROBLEMS
There is quite a bit of classical history which I have not yet entered. Magician's Own Book notes there is a connection between the Dido version of the problem and Cutting a card so one can pass through it, Section 6.BA. There are several relatively modern surveys of the subject from a mathematical viewpoint -- I will cite a few of them.
Virgil. Aeneid.  19. Book 1, lines 360 370. (p. 38 of the Penguin edition, translated by W. F. Jackson Knight, 1956.) Dido came to a spot in Tunisia and the local chiefs promised her as much land as she could enclose in the hide of a bull. She cut it into a long strip and used it to cut off a peninsula and founded Carthage. This story was later adapted to other city foundations. John Timbs; Curiosities of History; With New Lights; David Bogue, London, 1857, devotes a section to Artifice of the thong in founding cities, pp. 49-50, relating that in 1100, Hengist, the first Saxon King of Kent, similarly purchased a site called Castle of the Thong and gives references to Indian, Persian and American versions of the story as well as several other English versions.

Pappus. c290. Synagoge [Collection]. Book V, Preface, para. 1 3, on the sagacity of bees. Greek and English in SIHGM II 588 593. A different, abridged, English version is in HGM II 389 390.

The Friday Night Book (A Jewish Miscellany). Soncino Press, London, 1933. Mathematical Problems in the Talmud: Arithmetical Problems, no. 2, pp. 135-136. A Roman Emperor demanded the Jews pay him a tax of as much wheat as would cover a space 40 x 40 cubits. Rabbi Huna suggested that they request to pay in two instalments of 20 x 20 and the Emperor granted this. [The Talmud was compiled in the period -300 to 500. This source says he is one of the few mathematicians mentioned in the Talmud, but gives no dates and he is not mentioned in the EB. From the text, the problem would seem to be sometime in the 1-5 C.]

The 5C Saxon mercenary, Hengist or Hengest, is said to have requested from Vortigern: "as much land as can be encircled by a thong". He "then took the hide of a bull and cut it into a single leather thong. With this thong he marked out a certain precipitous site, which he had chosen with the greatest possible cunning." This is reported by Geoffrey of Monmouth in the 12C and this is quoted by the editor in: The Exeter Book Riddles; 8-10C (the book was owned by Leofric, first Bishop of Exeter, who mentioned it in his will of 1072); Translated and edited by Kevin Crossley-Holland; (As: The Exeter Riddle Book, Folio Society, 1978, Penguin, 1979); Revised ed., Penguin, 1993; pp. 101-102.

Lucca 1754. c1330. Ff. 8r 8v, pp. 31 33. Several problems, e.g. a city 1 by 24 has perimeter 50 while a city 8 by 8 has perimeter 32 but is 8/3 as large; stitching two sacks together gives a sack 4 times as big.

Calandri. Arimethrica. 1491. F. 97v. Joining sacks which hold 9 and 16 yields a sack which holds 49!!

Pacioli. Summa. 1494. Part II, ff. 55r-55v. Several problems, e.g. a cord of length 4 encloses 100 ducats worth, how much does a cord of length 10 enclose? Also stitching bags together.

Buteo. Logistica. 1559. Prob. 86, pp. 298-299. If 9 pieces of wood are bundled up by 5½ feet of cord, how much cord is needed to bundle up 4 pieces? 5 pieces?

Pitiscus. Trigonometria. Revised ed., 1600, p. 223. ??NYS -- described in: Nobuo Miura; The applications of trigonometry in Pitiscus: a preliminary essay; Historia Scientarum 30 (1986) 63-78. A square of side 4 and triangle of sides 5, 5, 3 have the same perimeter but different areas. Presumably he was warning people not to be cheated in this way.

J. Kepler. The Six Cornered Snowflake, op. cit. in 6.AT.3. 1611. Pp. 6 11 (8 19). Discusses hexagons and rhombic interfaces, but only says "the hexagon is the roomiest" (p. 11 (18 19)).

van Etten. 1624. Prob. 90 (87). Pp. 136 138 (214 218). Compares fields 6 x 6 and 9 x 3. Compares 4 sacks of diameter 1 with 1 sack of diameter 4. Compares 2 water pipes of diameter 1 with 1 water pipe of diameter 2.

Ozanam. 1725.

Question 1, 1725: 327. Question 3, 1778: 328; 1803: 325; 1814: 276; 1840: 141. String twice as long contains four times as much asparagus.

Question 2, 1725: 328. If a cord of length 10 encloses 200, how much does a cord of length 8 enclose?

Question 3, 1725: 328. Sack 5 high by 4 across versus 4 sacks 5 high by 1 across. c=  Q. 2, 1778: 328; 1803: 324; 1814: 276; 1840: 140-141, which has sack 4 high by 6 around versus two sacks 4 high by 3 around.

Question 4, 1725: 328 329. How much water does a pipe of twice the diameter deliver?

Les Amusemens. 1749.

Prob. 211, p. 376. String twice as long contains four times as much asparagus.

Prob. 212, p. 377. Determine length of string which contains twice as much asparagus.

Prob. 223-226, pp. 386-389. Various problems involving changing shape with the same perimeter. Notes the area can be infinitely small.

Ozanam Montucla. 1778.

Question 1, 1778: 327; 1803: 323-324; 1814: 275-276; 1840: 140. Square versus oblong field of the same circumference.

Prob. 35, 1778: 329-333; 1803: 326-330; 1814: 277-280; 1840: 141-143. Les alvéoles des abeilles (On the form in which bees construct their combs).

Jackson. Rational Amusement. 1821. Geometrical Puzzles.

No. 30, pp. 30 & 90. Square field versus oblong (rectangular?) field of the same perimeter.

No. 31, pp. 30 & 90-91. String twice as long contains four times as much asparagus.

Magician's Own Book (UK version). 1871. To cut a card for one to jump through, p. 124, says: "The adventurer of old, who, inducing the aborigines to give him as much land as a bull's hide would cover, and made it into one strip by which acres were enclosed, had probably played at this game in his youth." See 6.BA.

M. Zacharias. Elementargeometrie und elementare nicht-Euklidische Geometrie in synthetischer Behandlung. Encyklopädie der Mathematischen Wissenschaften. Band III, Teil 1, 2te Hälfte. Teubner, Leipzig, 1914-1931. Abt. 28: Maxima und Minima. Die isoperimetrische Aufgabe. Pp. 1118-1128. General survey, from Zenodorus (-1C) and Pappus onward.

6.AD.1. LARGEST PARCEL ONE CAN POST
New section. I have just added the problem of packing a fishing rod as the diagonal of a box. Are there older examples?
Richard A. Proctor. Greatest content with parcels' post. Knowledge 3 (3 Aug 1883) 76. Height + girth  6 ft. States that a cylinder is well known to be the best solution. Either for a cylinder or a box, the optimum has height = 2, girth = 4, with optimum volumes 2 and 8/π = 2.54... ft³.

R. F. Davis. Letter: Girth and the parcel post. Knowledge 3 (17 Aug 1883) 109-110, item 897. Independent discussion of the problem, noting that length  3½ ft is specified, though this doesn't affect the maximum volume problem.

H. F. Letter: Parcel post problem. Knowledge 3 (24 Aug 1883) 126, item 905. Suppose 'length' means "the maximum distance in a straight line between any two points on its surface". By this he means the diameter of the solid. Then the optimum shape is the intersection of a right circular cylinder with a sphere, the axis of the cylinder passing through the centre of the sphere, and this has the 'length' being the diameter of the sphere and the maximum volume is then 2⅓ ft³.

Algernon Bray. Letter: Greatest content of a parcel which can be sent by post. Knowledge 3 (7 Sep 1883) 159, item 923. Says the problem is easily solved without calculus. However, for the box, he says "it is plain that the bulk of half the parcel will be greatest when [its] dimensions are equal".

Pearson. 1907. Part II, no. 20: Parcel post limitations, pp. 118 & 195. Length  3½ ft; length + girth  6 ft. Solution is a cylinder.

M. Adams. Puzzle Book. 1939. Prob. B.86: Packing a parcel, pp. 79 & 107. Same as Pearson, but first asks for the largest box, then the largest parcel.

Philip Kaplan. More Posers. (Harper & Row, 1964); Macfadden-Bartell Books, 1965. Prob. 18, pp. 27 & 89. Ship a rifle about 1½ yards long when the post office does not permit any dimension to be more than 1 yard.

T. J. Fletcher. Doing without calculus. MG 55 (No. 391) (Feb 1971) 4 17. Example 5, pp. 8 9. He says only that length + girth  6 ft. However, the optimal box has length 2, so the maximal length restriction is not critical.

I have looked at the current parcel post regulations and they say length  1.5m and length + girth  3m, for which the largest box is 1 x ½ x ½, with volume 1/4 m³. The largest cylinder has length 1 and radius 1/π with volume 1/π m³.

I have also considered the simple question of a person posting a fishing rod longer than the maximal length by putting it diagonally in a box. The longest rod occurs at a boundary maximum, at 3/2 x 3/4 x 0 or 3/2 x 0 x 3/4, so one can post a rod of length 35/4  =  1.677... m, which is about 12% longer than 1.5m. In this problem, the use of a cylinder actually does worse!