Take P5, and divide it by Q5 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°.729578. We find that when x = .1, P6 = 1038.24021, Q6 = 10347.770999; whence P6 divided by Q6 gives .1003346711. Now 5°.729578 is 5°43′46½″; 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 = ¼π; in which case tan x = 1. If ¼π be commensurable with the unit, let it be (m/n), m and n being integers: we know that ¼π < 1. We have then

1=(m/n)

1- 		(m2/n2)

3- 		(m2/n2)

5- ... 		= m

n- 		m2

3n- 		m2

5n- 		m2

7n- ...

Now it is clear that m2/3n, m2/5n, m2/7n, etc. must at last become and continue severally less than unity. The continued fraction is therefore incommensurable, and cannot be unity. Consequently π2 cannot be commensurable: that is, π is an incommensurable quantity, and so also is π2.

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.

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 teacher and thinker on first principles. But he left the field immediately.

(The following is the passage referred to at Vol. II, page [54].)

Michael Stifelius[[679]] edited, in 1554, a second edition of the Algebra (Die Coss.), of Christopher Rudolff.[[680]] This is one of the earliest works in which + and - are used.