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Problems from Another Time

Individual problems from throughout mathematics history, as well as articles that include problem sets for students.

In a certain lake, swarming with red geese, the tip of a lotus bud was seen to extend a span [9 inches] above the surface of the water.
Of a collection of mango fruits, the king took 1/6; the queen 1/5 the remainder, and the three princes took 1/4, 1/3 and 1/2 (of the same remainder); and the youngest child took the remaining 3 mangoes.
Three vertical posts along a straight canal, each rising to the same height above the surface of the water. By looking at the line of vision, determine, to the nearest mile, the radius of the earth.
What number which being increased by 1/2, 1/3, and 1/4 of itself, the sum shall be 75?
A gentleman has bought a rectangular piece of land whose perimeter is to be 100 rods.
Prove that if the sums of the square opposite sides of any quadrilateral are equal, its diagonals interect at right angles.
A copper water tank in the form of a rectangular parallelopiped is to made. If its length is to be a times its breadth, how high should it be that for a given capacity it should cost as little as possible?
A powerful, unvanquished, excellent black snake, 80 angulas in length, enters into a hole at the rate of 7 1/2 angulas in 5/14 of a day, and in the course of a day its tail grows 11/4 of an angula.
A ladder is placed perpendicular to the plane of the horizon, and in coincidence with the plane of an upright wall.
A series of circles have their centers on an equilateral hyperbola and pass through its center. Show that their envelope is a lemniscate.