The possibility of solving the problem of the elementary division of a circumference into equal parts was known to the ancients for the cases

n = 2p, n = 3, n = 5, and their products.

Gauss in his Disquisitiones Arithmeticae proved the possibility of solving the problem by quadratics, and therefore geometrically with ruler and the compass, for all the prime numbers of the series

n = 22p + 1.

The objects of the present note are to exhibit an interesting property of some groups of subtraction and division, and to present a method of using such groups in the study of some elementary properties of the hyperbolic functions. The four finite groups of subtraction and division which transform every rational number into a rational number are of orders 4, 6, 8, and 12 respectively. With respect to each of these groups all the numbers, with the exception of two more than the order of the group, are divided into sets of conjugates, each set containing as many distinct numbers as the order of the group. The property to which we referred in the first sentence is that each of these sets contains two and only two negative numbers whenever the set is composed of real numbers and the number from which we subtract is positive.

Mr. Grant, in the last issue of this Gazette, having made several references to “Arithmetical Types and Examples,” perhaps a few remarks in reply will not be altogether out of place.

1. Multiplication. The only difference between the two methods first given is in the position of the various digits of the multiplicand.

We are told that the first rule (that “there are the same number of figures to the right of the decimal point in the first partial product as there are in the multiplicand,” and that therefore the first figure set down must be under the right-hand figure of the multiplicand) is mechanical; whereas the rule of “putting each first figure of a partial product underneath the figure to which that partial product is due” is sound and fundamental. It is not obvious to the writer that one is more mechanical than the other.

The current method of founding the teaching of Theoretical Statics on the experimental proof of the “Parallelogram of Forces” seems open to grave objection. The experiment is too remote from the facts of ordinary observation; and it is not so very easily performed. Moreover, the result, when accurately obtained, appears to the pupil to be so remarkable, that it is a pity not to reserve it for use as a verification of the soundness of pure reasoning.