The subject of this essay may be described as an application of the Theory of Number to Geometry, its principal or characteristic problem being to determine under what conditions the sides or surfaces of figures may be represented by integer numbers.

Like all other inquiries into the properties of integers, it is rather speculative than practical, and yet, perhaps on that very account, is more apt to engross the attention of its cultivators than almost any other department of pure mathematics. It seems, indeed to be of very little moment whether we can demonstrate that the sum of two cubes never can be a cube; and yet one after another of the most eminent mathematicians have tried, and, though foiled, have again and again essayed the proof of Fermat's negation.