May I be permitted to make some comments on the interesting paper by Dr. Robb in the July number of the Gazette? The point he raises is indeed interesting; though, without his having said so, I should have been surprised to find that it was not generally known; that the Time-Space Theory may, in a limiting instance, be incompatible with Euclid’s (I, 4). My comments must in a sense be critical, though I have nothing to say against his mathematics, so far as it is “pure,” or merely symbolic; for even though we might not, with Macaulay, say that it is well known to every schoolboy, or was even known to Macaulay himself, it would be obvious enough to every modern-side schoolboy who has got as far as “conic sections”; provided he was willing to take some of Dr. Robb’s technical terms in a more or less Pickwickian sense.

1. The method given by Euler * for the transformation of series was used by him to obtain “sums” for various divergent series, and recently Knopp † has written extensively on the “Euler summation process” derived from this transformation. But it does not seem to have been remarked that an interesting variation of this method is given by Hutton in the first volume of his Tracts, published in 1812. In Tract 8, “The Valuation of Infinite Series,” written in 1780, Hutton gives an account of his method, claiming to have arrived at it independently of Euler’s work, and employing it to approximate to the sums of series whose terms alternate in sign. When the proposed series converges, we have an easy and rapid way of determining its sum, and in the case of a divergent series, a number is determined which would nowadays be recognized as a conventional sum. Very little attention seems to have been paid to Hutton’s paper and his name is apparently not mentioned by modern writers on this subject.‡ De Morgan § refers to the “remarkable method” of Hutton, but he applies it only to convergent series, and makes no reference to its application to oscillating series. This is noteworthy in view of the remarks made by De Morgan a t the end of his Chapter 19 on “The Transformation of Divergent Developments,” but it might be conjectured that even so acute a logician as De Morgan had not clearly perceived, at any rate at the time of writing his “Calculus,” that the word “sum” applied to any infinite series is being used in a conventional sense.

From the earliest times the movements of the heavenly bodies have excited the interest of the inhabitants of the earth. The ancients believed that the earth was a flat disc floating in a great ocean, and they thought the sun was a god who drove his chariot across the sky every day. During the night he travelled back to the east across the great ocean, but his light was shut off by a range of high mountains which existed in the northern part of the earth. At night the majority of stars were noticed to move across the sky as if they were all fixed in some spherical shell which revolved. Several stars, however, seemed to move irregularly and were called planets or wanderers. Seven of these bodies were known—Moon, Mercury, Venus, Sun, Mars, Jupiter—and it is probably to these seven bodies that we owe our week of seven days.

If a pack of playing cards is shuffled systematically and the operation of shuffling repeated exactly, then after a certain number of repetitions of the operation the original order of the pack will be restored.

The method of shuffling in ordinary card-games is a rough and ready inter-leaving of two parts of the pack. If this method were carried out in simple perfection, one half of the pack would be interleaved with the other, so that no two cards adjacent before the shuffling would be together after.