The Lily-Pad Problem

Proposition – If the water lily is ten inches above the water, and disappears under the surface at a point distant twenty-one inches, what is the depth of the lake?

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The poet Longfellow was a fine mathematician who often spoke about the advantage of clothing our mathematical problems in such attractive or congenial garb as would appeal to the fancy of the student in place of following the dry, technical language of the textbooks. He would connect the proposition with some familiar subject which best explains the problems to be solved.

A clever kindergarten illustration of a mathematical theorem leaves a clearer and more lasting impression upon the mind of a student than a whole term of uncongenial study.

He always held mathematics to be the most important branch of knowledge taught in our colleges and high schools, for the reason that it enters so largely into all of the arts and sciences, and yet the average student graduates with such an undying aversion to figures that he speedily dismisses all recollections of them from his mind.

The water lily problem is one of several introduced in Longfellow's "Kavanah", written while occupying the Chair of Modern Languages in Harvard University, 1849. It is so simple that anyone, even without a knowledge of mathematics or geometry, could solve it with a pair of compasses or rule, and yet it illustrates an important geometrical truth in a never-to-be-forgotten way, which many graduates have never grasped at all.

I forget the exact language of the problem, as he described it to me personally during a discussion of the subject, but he told of a water lily growing in a lake; the flower was one span above the surface of the water, and when swayed by the breeze, would touch the surface at a distance of two cubits, from which data it was desired to compute the depth of the lake.

Now, let us suppose, as shown in the sketch, that the water lily is ten inches above the surface of the water, and that if it were pulled over to one side it would disappear under the surface at a point distant twenty-one inches from where it now stands, say just where the young lady is supposed to have drawn it, which shows that the two flowers are anchored to the same root at the bottom of the lake, what is the depth of the water?

Sam Loyd
"Cyclopedia of Puzzles", Lamb Publishing, New York, 1914