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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!