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POPULAR GEOMETRY;

CONTAINING

IN A FEW LESSONS SO MUCH OF

THE ELEMENTS OF EUCLID

AS IS

NECESSARY AND SUFFICIENT FOR A RIGHT UNDERSTANDING OF EVERY

ART AND SCIENCE

IN ITS LEADING TRUTHS AND GENERAL PRINCIPLES.

BY GEORGE DARLEY, A. B.

THE FOURTH EDITION.

LONDON:
PRINTED FOR TAYLOR AND WALTON,
BOOKSELLERS AND PUBLISHERS TO THE UNIVERSITY OF LONDON,

30, UPPER GOWER STREET.

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ADVERTISEMENT

TO THE

SECOND EDITION.

It is far more creditable to the public than to us, that a Second Edition of our Geometry has been so soon called for. Encouragement given to those who labour in facilitating the road to knowledge is but an involuntary effect of proportional ardour in those who pursue it. To promote the dissemination of mathematical reasoning, or rather a taste for mathematical reasoning, amongst the several classes of society, was the object of our publication, and this object appears to have been fully attained. Nothing shall be wanting on our part to keep alive the spirit of inquiry, and to cherish it on to greater things than it now dares contemplate even in imagination. This we can speak of as yet but darkly; the reader may be assured, however, that if he, with the moderate exertion of patience and industry which we require, is content to ascend through the scale of works prepared for him, he will reach a height as much beyond his present conceptions as his present abilities.

We have been still more gratified by private communications on the subject of our Work than by the public patronage of it, inasmuch as they denote that personal interest in the study which we were solicitous to create. It would be impossible and unnecessary for us to answer

individually the numerous letters with which we have been favoured; but we beg leave to return our sincerest thanks to all our correspondents, the suggestions and wishes of whom we have complied with in our Second Edition, as far as it was practicable or advisable. . We can truly declare, that in editing the present Series of Works, we have much more regard for the public good than our own mathematical reputation. Suggestions and objections were therefore, and will always be, acceptable; we feel no reluctance whatever to surrender our own opinion and preferences, if we can thereby accommodate our Treatises to the wants and wishes of those for whom they are compiled. It is in the power of few to become illustrious, but of every one to become useful; we are certainly not of the former class, but hope we may be of the latter.

The present edition of our Geometry has been carefully revised, and one or two oversights in the preceding corrected. Some of our Correspondents, habituated probably to the older system, complain of our having omitted the Axioms. To satisfy these, we subjoin each as it occurs, in a foot-note.

Upon the Doctrine of Ratio a fuller explanatory note is given; demonstrating its adequacy as well as simplicity. We did not give this explanation in our First Edition from an unwillingness to perplex a beginner, and lengthen our book with what then appeared needless. We thought that our readers would have been content with the general insight afforded them upon this subject, but are happy to find ourselves mistaken.

In order to render our work complete we have prefixed a table of Contents, which it will be often sufficient to consult when a reference is given, instead of examining the text itself for the Article or Definition specified.

The arrangement of the present Edition is in nowise different from the first, except that we have substituted a more serviceable element for one which is of little use in our system (Art. 18, 1st Ed.) This enables us also to simplify the demonstration of Art. 20, &c. Of. the suppressed element, which is the objectionable one in Euclid, those who desire it may see, in the section of Useful Results, a complete demonstration from our principles, that given in our first Edition being insufficient. It was suggested that the proofs of certain very obvious propositions (such as Arts. 33, 34, Part I., Arts. 48, 49, PART II., &c.) were unnecessary. In order to shorten and simplify our Elements we have merely preserved the enunciations of these Articles in the text, transferring the proofs to the Notes, which may be consulted by the reader.

An Author's satisfaction in his own work is not always a just criterion of the applause it will gain with the public. Our satisfaction in those Treatises we now publish will, it is confessed, mainly depend on the approbation of our readers, inasmuch as the real merit of such books is proportionate to their utility. We should have as little reason for self-congratulation in having written a scientifical treatise (however admirable) which was not read, as an artist in constructing a patent machine which was not handled: the present edition of our Geometry will, we hope, secure is in the continuance of public favour, this being the only solid proof of our having deserved it.

POSTSCRIPT TO FOURTH EDITION. The strictures upon Euclid in our Preface and Notes are, principally, as every scientific reader will be aware, those of able mathematicians, not our own. Whenever we found a necessity to deviate from that admirable work, we thought it but a due mark of respect to give our reasons for so doing. This motive should not be misconstrued.

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