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stimulate the learner to prosecute his inquiries beyond the mere elements, by showing him that there are algebraical investigations full of interest and abstract beauty, totally different from those of mere problem solving: they belong to a part of algebra marked by peculiar features; calling into operation more thought, more ingenuity, and more analytical address; but while it thus demands a keener intellectual effort, it rewards the extra expenditure of mental exertion with results of a higher practical utility, of a wider generality, and of greater intrinsic elegance. I could not prevail upon myself to close a book, even so elementary as this, without giving the learner, at least, a glimpse of the more attractive field of speculation which lies beyond.

The answers to all the unworked examples in the volume are either placed at the end, or they may be had separately, according to the option of the purchaser. This plan of collecting all the answers together in a few pages-a plan pretty generally adopted in works of the kind of late—is well worthy of imitation: it not only serves to economise space, but another important advantage is secured. It is easy to keep the type of the

ding, and by king at first a comparatively few impressions of them, to introduce whatever corrections may be afterwards found necessary; and thus to give to a large portion of a singie edition of the book the benefit of a revised edition of the answers. But I feel myself warranted in expressing myself with great confidence as to the general typographical accuracy of this work: the proofs have been examined and re-examined with the utmost care; and although the bulk of the examples are, of course, taken from a variety of sources, yet I have been at the trouble of solving every one of them anew from the revised proofs: the printed matter has thus been continuously before me for so long a period, that I think it scarcely possible that any error of consequence can have been overlooked.

I ought not to conclude this preface without noticing that the principle I have employed at page 186, for decomposing a polynomial into a pair of factors, with a view to abridge the labour of analysing an equation, is a principle that has been before applied to other purposes: first by Mr. Benjamin Gompertz, in his “ Tracts on Imaginary Quantities;” and afterwards by Mr. James Cockle, in the “ Philosophical Magazine.” The use of it, in reference to the researches in which I have endeavoured to make it available, is briefly hinted at in a note at the end of the book. Those who wish to see the application of an analagous principle to the curious subject of Porisms, must consult a recent tract on that topic by the venerable mathematician first mentioned.

J. R. YOUNG. April, 1851.

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