## Contributions to mathematics, comprising chiefly the rectification of the circle to 607 places of decimals |

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Page v - Author first formed the design of rectifying the circle upwards of 300 places of decimals. He was fully aware at that time, that the accomplishment of his purpose would add little or nothing to his fame as a Mathematician though it might as a Computer: nor would it be productive of anything in the shape of pecuniary recompense. Shanks actually attempted to hand-calculate 707 digits but a mistake crept in at the 527th digit. This went unnoticed until 1945, when D. Ferguson, in one of the last "nondigital"...

Page v - Computer ; nor would it be productive of anything in the shape of pecuniary recompense at all adequate to the labour of such lengthy computations. He was anxious to fill up scanty intervals of leisure with the achievement of something original, and which, at the same time, should not subject him either to great tension of thought, or to consult books.

Page xv - Subsequently it was found that the last 56 figures were incorrect, an error having crept into the value of one of the terms of the series employed. But of this talented writer we shall have to speak afterwards. The Mathematician who seems next to have engaged in the solution of this Problem is M. Dase, about the year 1846, then a young man of great promise, who used the formula f = tan J-r+ tan "'4-+ tan -'-f.

Page xvi - The values of tan ~' 3-, and of tan ~'¿ are given to 609 places. These values, then, have been carefully collated, as far as 441 decimals, with Dr. Rutherford's results, and may be pronounced free from errors, inasmuch as each party worked independently of the other. The Author, therefore, is alone responsible for the accuracy of the additional 166 and 168 respective places of decimals. Whether any other Mathematician will appear, possessing...

Page iii - ... rarely met with. This however is not all. We seldom indeed find profundity united with great facility of computation; — but you happily combine both in a very eminent degree. I regret my inability adequately to convey to you the heart-felt sentiments of esteem, gratitude, and respect with which I am, My dear Sir, Your sincere and obliged Friend, THE AUTHOR.

Page vii - Author in separately calculating and then collating the value of т to 441 places of decimals, so that accuracy is ensured at least to that extent ; and it is confidently hoped no error has crept into the remaining 86 places. It is proper to state, that the talented writer just mentioned lately sent a " Paper on determining the value of т...

Page 1 - The value of each term of the above series employed in finding the two arcs is given separately, so that its accuracy may readily be tested; and no difficulty can possibly arise to Mathematicians, for whose perusal chiefly the following pages are intended, in comprehending all that follows. 25.5a l 49.5...

Page xiv - Previous to 1831, the value of т, as the late Professor Thomson of the University of Glasgow writes, in his work on the Differential and Integral Calculus, had been calculated " to the extraordinary extent of 140 figures !" We may here be permitted to indulge a smile at the learned writer's...

Page iii - I know of no person to whom I can, with so great pleasure, so much propriety, and such deep gratitude, inscribe this small volume as to you, from whom I received my earliest lessons in numbers; and I earnestly wish it had been something more worthy of notice that the Pupil was presenting to the Master, than the present "CONTRIBUTIONS то MATHEMATICS

Page xv - Nachrichten," published in 1847. In the earlier portions of the year 1851, the Author of this Volume, not then aware of M. Dase's, or of Dr. Clausen's labours, employed Machin's formula, given above, and calculated the value of т to 315 decimals.