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as I was well aware that the profits of the book would not afford him the means of rewarding my pains.
I have in the first place, therefore, used all the means in my power to render the work correct. I began by collating the 3d, or best edition of Sherwin's tables, with some others of the most perfect works of the same kind, as Briggs's, Vlacq's, Gardiner's quarto book, &c: by which means I detected many errors in each of them, which had not before been discovered ; and of these between twenty and thirty were in the two editions of Gardiner's work, printed at London in 1742, and at Avignon in 1770; the errata of which two books are here printed at the end of our tables. But, besides detecting many unknown errors in my copy of the said 3d Edition of Sherwin, which was no more than what I expected, I discovered, with no small surprize, that the last figures in the table of logarithms were not uniformly true to the nearest unit, except in a very few pages at the beginning and end of the table; although Mr. Gardiner, the editor of that edition, had made the table correct in that respect in his own quarto work before-mentioned, which was also printed in the same year 1742 with the said third edition of Sherwin! The errors from this cause amounted to several thousands; and they have continued to run through all the editions of Sherwin ever since that time! But I have here corrected them. Nor have I employed less attention in correcting the press, than in previously correcting the copy; every proof having been several times read over, and compared with the best of the books before
But in giving this edition to the world, I was not satisfied with barely making it correct. I was aware that the materials themselves might be much improved ; and I have accordingly enlarged, or otherwise greatly amended them, in various respects. Among the improvements of the old materials, may be reckoned the following: namely, in the large table of logarithms, the proportional parts, near the beginning, are more conveniently arranged, being now all placed in the same opening of the book where their corresponding differences occur: The logarithms to fixty one figures are brought to their proper place in the book, and more conveniently disposed all in one page: The large table of fines, tangents and secants, is more commodiously arranged, and rendered more distinct and convenient for use; the natural lines, tangents, secants and versed sines, being all separated from the others, and placed all together on the left-hand pages, and the logarithmic ones facing them on the right-hand pages ; the common differences, in both, set between the two columns to which each of them answers; and the versed lines here introduced into their proper place in the same pages with the fines, tangents and secants. Besides these, there are some other alterations in the new tables here given, and the reader will find a number of very important improvements in the description and use of the whole; especially in the arithmetic of logarithms, and in the resolution of plane and spherical triangles, according to the present improved methods
of calculation used by the Astronomer Royal, and other persons the most experienced in these matters.
The improvements in the tables, by the introduction of new matter, are both great and numerous. The tables numbered 2, 3, and 4 are here added, being an entire new set, with their differences, for finding numbers and logarithms to twenty places. The columns of common differences, in the pages of natural fines &c, are now first introduced : As are also the tables of hyperbolic and logistic logarithms; the logarithmic fines and tangents
second, in the first two degrees of the quadrant ; together with a table of the lengths of arcs; a table to change common and hyperbolic logarithms from the one to the other; &c; the uses and exemplifications of the whole being very amply des tailed.
But the greatest alteration of all, is the very extenfive and new introduction here given, instead of the former inadequate and heterogeneous one, consisting of about 180 pages of new matter, on a methodical plan, containing the historical account and description of all trigonometrical writings, and the tables relating to that subject, both natural and logarithmic ; besides the compleat use of our own tables. Inventions are here ascribed to the proper authors, and their methods and improvements described and compared. This historical description will evidently appear to be the result of immense labour and reading. And indeed I
have painfully gone over all the books which are here so minutely described ; and that description with a detail in some degree adequate to their great merits; especially the works of Napier, Briggs, Kepler, &c; which was the more necessary, as the writings and methods of those great masters had not been any
where properly described and discriminated, although they are in themselves highly curious and important:
These readings and commentaries have been carried on to an extent far beyond what was at first intended. But the tables having been in the press for the space of seven or eight years, I had thereby an opportunity of collecting and examining a still greater number of books; so that I was gradually led on, and my views and plans rendered still more extensive and compleat. This delay, therefore, though in many respects it proved very inconvenient and disagreeable, has at length given the occasion of rendering these commentaries more perfect and satisfactory.
Besides what immediately relates to trigonometrical subjects, the reader will here find many other curious and uncommon articles, relating to the several authors and their discoveries, which have occurred in the course of my reading, and which appeared of too much consequence to be passed over unnoticed, in the analysis of their several compositions. Among these is the discovery of the first author of the binomial theorem, and the differential method, which are due to Mr. Henry Briggs,
whose writings are replete with ingenious and original matter, and are well deserving to be more generally known and studied than they have been for some time past.
This long course of examination and description, however, having been carried on for so many years, at different intervals, and interrupted by various avocations, and by business of different kinds, it will be no wonder if this circumstance may have occasioned some inequalities in the stile and composition of this history ; and for which therefore, should any such appear, it is hoped the occasion will plead an apology.
P. S. Since my History of Trigonometrical tables, in the following in. troduction, was printed, there has been published, in the Philosophical Transactions for the year 1784, a paper of mine concerning a project for the trigonometrical tables to be constructed on a new plan, namely, in which the arc of the quadrant is divided into aliquot parts of the radius, or according to the real lengths of the arcs. Which construction is now in some degree of forwardness, as myself and several assistants have been closely engaged in the execution of it ever since.
Royal Military Academy, 2