right-lined geometrical figures. London, 1863, 12mo.

The circle is divided into equal sectors, which are joined head and tail:
but a property is supposed which is not true.

An attempt to assign the square roots of negative powers; or what is
[sqrt] -1? By F.H. Laing.[321] London, 1863, 8vo.

If I understand the author, -a and +a are the square roots of -a^2, as
proved by multiplying them together. The author seems quite unaware of what
has been done in the last fifty years.

BYRNE'S DUAL ARITHMETIC.

Dual Arithmetic. A new art. By Oliver Byrne.[322] London, 1863, 8vo.

The plan is to throw numbers into the form a(1.1)^{b} (1.01)^{c}
(1.001)^{d}... and to operate with this form. This is an ingenious and
elaborate speculation; and I have no doubt the author has practised his
method until he could surprise any one else by his use of it. But I doubt
if he will persuade others to use it. As asked of Wilkins's universal
language, Where is the second man to come from?

An effective predecessor in the same line of invention {187} was the late
Mr. Thomas Weddle,[323] in his "New, simple, and general method of solving
numeric equations of all orders," 4to, 1842. The Royal Society, to which
this paper was offered, declined to print it: they ought to have printed an
organized method, which, without subsidiary tables, showed them, in six
quarto pages, the solution (x=8.367975431) of the equation

1379.664 x^{622} + 2686034 x 10^{432} x^{152} - 17290224 x 10^{518}
x^{60} + 2524156 x 10^{574} = 0.

The method proceeds by successive factors of the form, a being the first
approximation, a x 1.b x 1.0c x 1.00d.... In my copy I find a few
corrections made by me at the time in Mr. Weddle's announcement. "It was
read before that learned body [the R. S.] and they were pleased [but] to
transmit their thanks to the author. The en[dis]couragement which he
received induces [obliges] him to lay the result of his enquiries in this
important branch of mathematics before the public [, at his own expense; he
being an usher in a school at Newcastle]." Which is most satirical, Mr.
Weddle or myself? The Society, in the account which it gave of this paper,
described it as a "new and remarkably simple method" possessing "several
important advantages." Mr. Rutherford's[324] extended value of [pi] was
read at the very next meeting, and was printed in the _Transactions_; and
very properly: Mr. Weddle's paper was exc