"Say that the blood-globule of one of our animalcules is a millionth of an inch in diameter. Fashion in thought a globe like our own, but so much larger that our globe is but a blood-globule in one of its animalcules: never mind the microscope which shows the creature being rather a bulky instrument. Call this the first globe _above_ us. Let the first globe above us be but a blood-globule, as to size, in the animalcule of a still larger globe, which call the second globe above us. Go on in this way to the twentieth globe above us. Now go down just as far on the other side. Let the blood-globule with which we started be a globe peopled with animals like ours, but rather smaller: {110} and call this the first globe below us. Take a blood-globule out of this globe, people it, and call it the second globe below us: and so on to the twentieth globe below us. This is a fine stretch of progression both ways. Now give the giant of the twentieth globe _above_ us the 607 decimal places, and, when he has measured the diameter of his globe with accuracy worthy of his size, let him calculate the circumference of his equator from the 607 places. Bring the little philosopher from the twentieth globe _below_ us with his very best microscope, and set him to see the small error which the giant must make. He will not succeed, unless his microscopes be much better for his size than ours are for ours. "Now it must be remembered by any one who would laugh at the closeness of the approximation, that the mathematician generally goes _nearer_; in fact his theorems have usually no error at all. The very person who is bewildered by the preceding description may easily forget that if there were _no error at all_, the Lilliputian of the millionth globe below us could not find a flaw in the Brobdingnagian of the millionth globe above. The three angles of a triangle, of perfect accuracy of form, are _absolutely_ equal to two right angles; no stretch of progression will detect _any_ error. "Now think of Mr. Lacomme's mathematical adviser (_ante_, Vol. I, p. 46) making a difficulty of advising a stonemason about the quantity of pavement in a circular floor!