om the side.

"Write down the following rows of figures, and more, if you like, in the
way described:

1 2 5 12 29 70 169 408 985
1 3 7 17 41 99 239 577 1393

After the second, each number is made up of double the last increased by
the last but one: thus, 5 is 1 more than twice 2, 12 is 2 more than twice
5, 239 is 41 more than twice 99. Now, take out two adjacent numbers from
the upper line, and the one below the first from the lower: as

70 169
99.

Multiply together 99 and 169, giving 16,731. If, then, you will say that 70
diagonals are exactly equal to 99 sides, you are in error about the
diagonal, but an error the amount of which is not so great as the 16,731st
part of the diagonal. Similarly, to say that five diagonals make exactly
seven sides does not involve an error of the 84th part of the diagonal.

"Now, why has not the question of _crossing the square_ been as celebrated
as that of _squaring the circle_? Merely because Euclid demonstrated the
impossibility of the first {109} question, while that of the second was not
demonstrated, completely, until the last century.

"The mathematicians have many methods, totally different from each other,
of arriving at one and the same result, their celebrated approximation to
the circumference of the circle. An intrepid calculator has, in our own
time, carried his approximation to what they call 607 decimal places: this
has been done by Mr. Shanks,[204] of Houghton-le-Spring, and Dr.
Rutherford[205] has verified 441 of these places. But though 607 looks
large, the general public will form but a hazy notion of the extent of
accuracy acquired. We have seen, in Charles Knight's[206] _English
Cyclopaedia_, an account of the matter which may illustrate the
unimaginable, though rationally conceivable, extent of accuracy obtained.

"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