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In the year 1853 Dr. Rutherford, of the Royal Military Academy, Woowich, sent a paper on the Computation of the value of π to the Royal Society, and the paper was published in the 'Proceedings’ of that learned dy. The value of π is there given to 607 decimals, the first 440 being the joint production of Dr. Rutherford and the author of this paper, and the remaining 167 decimals having been calculated by the present writer, the accuracy of which he alone is responsible. Subsequently, the Astronomer Royal, G. B. Airy, Esq., kindly presented the author’s paper the Calculation of the value of e, the base of Napier’s logarithms, to wards of 200 decimals; the aforesaid paper also contained the Napierian logarithms of 2, 3, and 5, as well as the modulus of the common system, to upwards of 200 places of decimals. This paper was not, however, published, but deposited in the Archives of the Royal Society; but an abstract, containing the numerical results, was printed in the ‘Proceedings’. In a paper sent by the author to the Astronomer Royal, and forwarded by m to the Royal Society, will, the author believes, be found the reciprocal the prime number 17389, consisting of a circulating period of no less an 17388 decimals, the largest on record. Some few remarks are also ven touching circulates generally, and the easiest modes of obtaining them. The writer now desires to supplement what he then did, by giving the numerical value of Euler’s constant, which is largely employed in "Infinitesimal Calculus,” to a greater extent than has hitherto been found, a free from error. In Crelle’s Journal for 1860, vol. lx. p. 375, M. Oettinger has contributed an article on Euler’s constant, and especially on “certain discrepancies ” in the value given by former mathematical writers. Adopt the formula there employed, as being well adapted for the purpose, the writer of this paper has both corrected and extended what has been previously done; and as very great care has been bestowed upon the calculations, so as to exclude error, he confidently believes that his results are as far as they go, absolutely correct. He may remark that, since the separate values of n in the formula (which, see below) produce identical results as far as they go, and the higher the value of n the more near we can approximate to the value of the constant, we thus have sufficient proof afforded of the correctness of the value found when n is 10, 20, 50 or 100. If the writer can command sufficient leisure, he may resume the calculation by and by, and, making n 1000, he may thus verify, as well extend, the value of Euler’s constant given in this paper. The number 10, 20, 50, 100, 200, and 1000, especially 10 and its integral powers, are more easily handled than others, particularly in those terms of the formula which contain Bernoulli’s numbers. The harmonic progression is he "summed” much further than was requisite for finding E to 50 or 55 decimals; but this was of some importance in ensuring correctness in the decimal expression of each of the higher terms of S100 and S200. It may be observed that the numbers of decimal places in E, obtained from n being 10, 20, 50, 100, and 200, are nearly proportional to 10⅓, 20⅓, 50⅓ 100⅓ and 200⅓— a rather curious coincidence.

I. On the calculation of the numerical value of Euler's constant, which Professor Price, of Oxford, calls E