In a paper on the beats of organ-pipes and on temperament published some years ago, I said that equal temperament appeared to me insipid, and not so agreeable as the effect of the instrument when in progress towards being what is called out of tune, before it becomes offensively wrong. There is throughout that period unequal temperament, determined by accident. General Thompson, taking me one way, says I have launched a declaration which is likely to make an epoch in musical practice; a public musical critic, taking me another way, quizzes me for preferring music _out of tune_. I do not think I deserve either one remark or the other. My opponent critic, I suspect, takes _equally tempered_ and _in tune_ to be phrases of one meaning. But by equal temperament is meant equal distribution among all the keys of the error which an instrument _must_ have, which, with twelve sounds only in the octave, professes to be fit for all the keys. I am reminded of the equal temperament which was once applied to the postmen's jackets. The coats were all made for the average man: the {186} consequence was that all the tall men had their tails too short; all the short men had them too long. Some one innocently asked why the tall men did not change coats with the short ones. A diagram illustrating a discovery in the relation of circles to 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.