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'N the investigation of this problem I have avoided the usual method by which its solution has been hitherto attempted, viz., by increasing the number of the sides of a polygon enclosed within, and touching the boundary of a circle of any given diameter, as I consider that this mode of enquiry could never lead to any satisfactory or very certain results, even to the limited extent that they are assumed to be perfectly correct in their decimals, the absolute solution being admittedly unattainable by this mode of research. For when we take into consideration the great number and extent of the calculations required, and the great care necessary in their preparation in the forms of squares and square roots, to procure forty or fifty decimals perfectly correct—while the slightest error existing in any stage of the work must necessarily vitiate all that follows-we need not think it strange when we find that these decimals, even to the limited extent that they are assumed to be perfectly correct, are found differing from each other in the statements of different calculators. And who can be certain which of them, or if any one of them be correct, without proof?—a test which would require the reproduction of the whole work, which, say to obtain fifty decimals true, would require the most careful and incessant labour of a good calculator for some years; and even then, where is the absolute certainty of the results so found? With these considerations, I firmly believe that much valuable time and labour have been wasted in fruitless attempts to accomplish in this way what even to a limited extent is doubtful and uncertain; whilst the ascertainment of the absolute contents of a circle by this means must ever remain an absolute impossibility, so long as the right line forming a side of the polygon, however minute, remains unparallel with the curved line forming its circular boundary. And I speak from experience, as few persons have carried this mode of enquiry farther

than I have done myself; and in confirmation of my views, I would here notice that the above-stated mode was that adopted by the late Van Cullen in obtaining his well-known series of thirty-six figures, assumed to be absolute to that extent in giving the boundary of a circle of two inches diameter. I give the figures said to be engraved on his tomb at St. Peter's Church, Leyden, viz., 6-28318530717958647692528676655900576. Yet we find these figures differing materially from those of his contemporaries. I need not give names, which could serve no purpose where all are differing.

And notwithstanding that I have since discovered a much more simple and certain mode of obtaining more perfect results with less than half the calculations required by the former method (which I hope to explain in the course of this enquiry), yet, as it is in some measure subject to the same objections as the former method, I do not intend pursuing my enquiry by it further, for the reasons already stated. And with my past experience, and a strong unwillingness to be defeated in obtaining the object of my research, I have again determined to pursue my enquiry by a new method altogether, and in this investigation I have directed my thoughts solely to the nature and relative value of certain circular rings, with a view to ascertain their absolute value, and that of the complete contents of the circle which they collectively constitute; and by this means I hope to show that I have been successful in obtaining all that can be reasonably desired or hoped for as a true and very simple solution of this very abstruse problem, as far as it relates to the circular area.

I have since been induced, however, to add some further very important proofs, procured from a distinctly different mode of research by angular measurement. I have also described a new mode of converting the angular boundary of the hexagon into a circular boundary, while it still retains the same linear extent, the area alone being thus proportionably increased; to all of which matters I have added, in the form of an Appendix, a short statement of my views regarding the nature of the errors existing in the present systems of circular and spherical geometry. I now beg to offer this little work-which has been the product of much time and mental labour-with the fullest confidence to the kind perusal and consideration of all that class of intelligent readers who would desire to see an improvement in that department of mathematical science to which the subject relates.

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