§ 1. The Buckets

Problem. — Lardner states that a solid, immersed in a fluid, displaces an amount equal to itself in bulk. How can this be true of a small bucket floating in a larger one?

Solution. — Lardner means, by “displaces”, “occupies a space which might be filled with water without any change in the surroundings.” If the portion of the floating bucket, which is above the water, could be annihilated, and the rest of it transformed into water, the surrounding water would not change its position: which agrees with Lardner’s statement.

Five answers have been received, none of which explains the difficulty arising from the well-known fact that a floating body is the same weight as the displaced fluid. Hecla says that “Only that portion of the smaller bucket which descends below the original level of the water can be properly said to be immersed, and only an equal bulk of water is displaced.” Hence, according to Hecla, a solid whose weight was equal to that of an equal bulk of water, would not float till the whole of it was below “the original level” of the water: but, as a matter of fact, it would float as soon as it was all under water. Magpie says the fallacy is “the assumption that one body can displace another from a place where it isn’t”, and that Lardner’s assertion is incorrect, except when the containing vessel “was originally full to the brim”. But the question of floating depends on the present state of things, not on past history. Old King Cole takes the same view as Hecla. Tympanum and Vindex assume that “displaced” means “raised above its original level”, and merely explain how it comes to pass that the water, so raised, is less in bulk than the immersed portion of bucket, and thus land themselves — or rather set themselves floating — in the same boat as Hecla.

I regret that there is no Class List to publish for this Problem.

§ 2. Balbus’s Essay

Problem. — Balbus states that if a certain solid be immersed in a certain vessel of water, the water will rise through a series of distances, two inches, one inch, half an inch, etc., which series has no end. He concludes that the water will rise without limit. Is this true?

Solution. — No. This series can never reach 4 inches, since, however many terms we take, we are always short of 4 inches by an amount equal to the last term taken.

Three answers have been received — but only two seem to me worthy of honours.

Tympanum says that the statement about the stick “is merely a blind, to which the old answer may well be applied, solvitur ambulando, or rather mergendo”. I trust Tympanum will not test this in his own person, by taking the place of the man in Balbus’s Essay! He would infallibly be drowned.

Old King Cole rightly points out that the series, 2, 1, etc., is a decreasing geometrical progression: while Vindex rightly identifies the fallacy as that of “Achilles and the Tortoise”.

Class List.

I.

Old King Cole. Vindex.

§ 3. The Garden

Problem. — An oblong garden, half a yard longer than wide, consists entirely of a gravel walk, spirally arranged, a yard wide and 3630 yards long. Find the dimensions of the garden.

Answer. — 60, 60½.

Solution. — The number of yards and fractions of a yard traversed in walking along a straight piece of walk, is evidently the same as the number of square yards and fractions of a square yard contained in that piece of walk: a............