In 427 B.C., a plague ravaged Athens, killing a quarter of its population, including the great leader Pericles. In desperation, the Athenians sent a delegation to Delos, to entreat Apollo's oracle to beg the god to spare their lives. The oracle returned with the god's demand: Apollo wanted the Athenians to double the size of his temple on the island. The Athenians quickly set to work. They doubled the length, the width, and the height of the Athenian temple to Apollo. They decorated it opulently and lavished it with gifts, and soon the Athenian temple on Delos was the most magnificent on the island, or perhaps anywhere. The delegation returned to Athens with great hope, expecting that the god would now lift the curse. But the plague continued to ravage the city. So a second delegation left Athens for Delos. When its members met with the oracle, he surprised them by saying: "You have not followed Apollo's instructions!" The oracle continued: "You have not doubled the size of the god's temple, as he demanded of you. Go back and do as he had commanded you to do!"

Again, the Athenians set to work. They understood their mistake: they had doubled each of the dimensions of the old temple—the length, the width, and the height—and a calculation they now made showed them that they had actually increased the volume of the temple eightfold (2x2x2=8). Apparently the god wanted the volume to be doubled, not the dimensions.

Ancient Greek draftsmanship and geometry were always carried out using only a straightedge and a compass, so the Athenians architects did their best with these two tools. But they failed. As hard as they tried, they could not double the volume of the cubic structure that was Apollo's original temple—or for that matter, double the volume of any cube—with straightedge and compass alone.

According to Theon of Smyrna (early second century A.D.), the Athenian architects went to ask Plato for his help. Plato, who had established the Academy in Athens, in which the best mathematicians of his age worked, enlisted the help of the two great mathematicians Eratosthenes and Eudoxus in trying to solve this difficult problem.

Eratosthenes was such a superb mathematician that he had been able to estimate with excellent precision the circumference of the earth by measuring the angle rays of light from the sun made at two different locations separated by a known distance. Eudoxus, on the other hand, was the great genius who could compute areas and volumes using methods that anticipated the calculus, which would only be codisovered over two thousand years later by Leibniz and Newton. But neither Eratosthenes nor Eudoxus could solve the mystery of doubling the cube with straightedge and compass. Nor could anyone else, however fervently urged by Plato, who was desperate to help his countrymen rid themselves of the plague. Plato, who was not a mathematician himself but was called the Maker of Mathematicians because the best mathematicians studied and worked in his Academy, had such great interest in the cube and in other three–dimensional objects of perfect symmetry that such objects would eventually be named after him.

Why was it impossible to double the size of Apollo's temple? If the volume of the original temple was, say, 1,000 cubic meters (having length, width, and height 10 meters each), then the new volume should be 2x1,000=2,000 cubic meters (and not 8,000 cubic meters, as they had obtained on their first attempt by doubling the length, width, and height to 20 meters each). So in order to double the cube—in this case, to obtain a structure with volume 2,000 starting with a temple of volume 1,000— they would need to increase the length, the width, and the height by the cube root of 2 each. This is so because the cube root of 2, when cubed, gives us 2—the factor needed to multiply the volume. This way, each measurement would have to change from 10 meters to 10x (cube root of 2), or approximately 12.6 meters. It turns out that no finite sequence of operations with straightedge and compass can turn a given length to a number that is the product of that length by the cube root of 2—or the cube root of any number that is not a perfect cube. The problem Apollo's oracle gave the Athenians was an impossible one to solve. It is important to note that the new temple had to remain in the shape of a cube: otherwise, simply doubling one of its dimensions, say the length, would have done the trick.

The Greeks of antiquity did not know that the Delian problem, as it has come to be known, is mathematically impossible, given their tools. An understanding of this problem would have to wait many centuries. The ancient Greeks—Pythagoras, Euclid, and other great mathematicians of antiquity—were excellent geometers. But they did not have a well–developed theory of algebra. And algebra is needed in order to understand and properly address complex problems of geometry such as doubling the cube, squaring the circle, and trisecting an angel, collectively called the three classical problems of antiquity.