b_3=5, b_4=7, b_5=9, etc.

P_1=x Q_1=1
P_2=3x Q_2=3-x^2
P_3=15x-x^3 Q_3=15-6x^2
P_4=105x-10x^3 Q_4=105-45x^2+x^4
P_5=945x-105x^3+x^5 Q_5=945-420x^2+15x^4
P_6=10395x-1260x^3+21x^5 Q_6=10395-4725x^2+210x^4-x^6

We can use this algebraically, or arithmetically. If we divide P_n by Q_n,
we shall find a series agreeing with the known series for tan x, _as far
as_ n _terms_. That series is

x + x^3/3 + 2x^5/15 + 17x^7/315 + 62x^9/2835 + ...

{372} Take P_5, and divide it by Q_5 in the common way, and the first five
terms will be as here written. Now take _x_ = .1, which means that the
angle is to be one tenth of the actual unit, or, in degrees 5 deg..729578.
We find that when x = .1, P_6 = 1038.24021, Q_6 = 10347.770999; whence P_6
divided by Q_6 gives .1003346711. Now 5 deg..729578 is 5 deg.43'46-1/2";
and from the old tables of Rheticus[675]--no modern tables carry the
tangents so far--the tangent of this angle is .1003347670.

Now let x = (1/4)[pi]; in which case tan x = 1. If (1/4)[pi] be
commensurable with the unit, let it be (m/n), m and n being integers:
we know that (1/4)[pi] < 1. We have then

1=(m/n)/1- (m^2/n^2)/3- (m^2/n^2)/5- ... = m/n- m^2/3n- m^2/5n- m^2/7n-
...

Now it is clear that m^2/3n, m^2/5n, m^2/7n, etc. must at last become and
continue severally less than unity. The continued fraction is therefore
incommensurable, and cannot be unity. Consequently [pi]^2 cannot be
commensurable: that is, [pi] is an incommensurable quantity, and so also
is [pi]^2.

I thought I should end with a grave bit of appendix, deeply mathematical:
but paradox follows me wherever I go. The foregoing is--in my own
language--from Dr. (now Sir David) Brewster's[676] English edition of
Legendre's Geometry, (Edinburgh, 1824, 8vo.) translated by some one who is
not named. I picked up a notion, which others had at Cambridge in 1825,
that the translator was the late Mr. Galbraith,[677] then known at
Edinburgh as a writer and teacher.

{373} But it turns out that it was by a very different person, and one
destined to shine in quite another walk; it was a young man named Thomas
Carlyle.[678] He prefixed, from his own pen, a thoughtful and ingenious
essay on Proportion, as good a substitute for the fifth Book of Euclid as
could have been given in the space; and quite enough to show that he would
have been a distinguished tea