hat i/k, l/m,... are all less than unity.
Then the fraction i/k+ l/m+ ... is incommensurable, as proved: let it be
[kappa]. Then g/(h + [kappa]) is incommensurable, say [lambda]; e/(f +
[lambda]) is the same, say [mu]; also c/(d + [mu]), say [nu], and a/(b +
[nu]), say [rho]. But [rho] is the fraction a/b+ c/d+ ... itself; which is
therefore incommensurable.

Let [phi]z represent

a a^2 a^3
1 + - + ------- + -------------- + ....
z 2z(z+1) 2.3.z(z+1)(z+2)

{370} Let z be positive: this series is convergent for all values of a, and
approaches without limit to unity as z increases without limit. Change z
into z + 1, and form [phi]z - [phi](z+1): the following equation will
result--

a
[phi]z-[phi](z+1) = ------([phi](z+2))
z(z+1)

a [phi](z+1) a [phi](z+1) a [phi](z+2)
or a = - ---------- . z + - ---------- . --- ----------
z [phi]z z [phi]z z+1 [phi](z+1)

a = [psi]z(z+[psi](z+1))

[psi]z being (a/z)([phi](z+1)/[phi]z); of which observe that it diminishes
without limit as z increases without limit. Accordingly, we have

[psi]z = a/z+ [psi](z+1) = a/z+ a/(z+1)+ [psi](z+2)
= a/z+ a/(z+1)+ a/(z+2)+ [psi](z+3), etc.

And, [psi](z + n) diminishing without limit, we have

a/z . [phi](z+1)/[phi]z = (a/z+) (a/(z+1)+) (a/(z+2)+) (a/((z+3)+ ...))

Let z = 1/2; and let 4a = -x^2. Then (a/z)[phi](z+1) is -(x^2/2) ( 1 -
x^2/(2.3) + x^4/(2.3.4.5...)) or -(x/2) sin x. Again [phi]z is 1 - x^2/2 +
x^4/(2.3.4) or cos x: and the continued fraction is

(1/4)x^2/(1/2)+ (1/4)x^2/(3/2)+ (1/4)x^2/(5/2)+ ...
or -x/2 x/1+ -x^2/3+ -x^2/5+
...

{371} whence tan x = x/1+ -x^2/3+ -x^2/5+ -x^2/7+ ...

Or, as written in the usual way,

tan x = x
-------
1 - x^2
-------
3 - x^2
-------
5 - x^2
-------
7 - ...

This result may be proved in various ways: it may also be verified by
calculation. To do this, remember that if

a_1/b_1+ a_2/b_2+ a_3/b_3+ ... a_n/b_n = P_n/Q_n; then

P_1=a_1, P_2=b_2 P_1, P_3=b_3 P_2+a_3 P_1, P_4=b_4 P_3+a_4 P_2, etc.
Q_1=b_1, Q_2=b_2 Q_1+a_2, Q_3=b_3 Q_2+a_3 Q_1, Q_4=b_4 Q_3+a_4 Q_2, etc.

in the case before us we have

a_1=x, a_2=-x^2, a_3=-x^2, a_4=-x^2, a_5=-x^2, etc.
b_1=1, b_2=3,