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!

"We will now, for our non-calculating reader, put the matter in another
way. We see that a circle-squarer can advance, with the utmost confidence,
the assertion that when the diameter is 1,000, the circumference is
accurately 3,125: the mathematician declaring that it is a trifle more than
3,141-1/2. If the squarer be right, the mathematician has erred by about a
200th part of the whole: or has not kept his accounts right by about 10s.
in every 100l. Of course, if he set out with such an error he will
accumulate blunder upon blunder. Now, if there be a process in which {111}
close knowledge of the circle is requisite, it is in the prediction of the
moon's place--say, as to the time of passing the meridian at Greenwich--on
a given day. We cannot give the least idea of the complication of details:
but common sense will tell us that if a mathematician cannot find his way
round the circle with