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In this text-book, compiled at the request of the publishers, a rigid adherence to Robert Simson's well-known editions of Euclid's Elements has not been observed; but no change has been made on Euclid's sequence of propositions, and comparatively little on his modes of proof. Here and there useful corollaries and converses have been inserted, and a few of Simson's additions have been omitted. Intimation of such insertions and omissions has been given, when it was deemed necessary, in the proper place. Several changes, mostly, however, of arrangement, have been made on the definitions.

By a slight alteration of the lettering or the construction of the figure, an attempt has been made throughout, and particularly in the Second Book, to draw the attention of the reader to the analogy which exists between certain pairs of propositions. By Euclid this analogy is well-nigh ignored.

In the naming of both congruent and similar figures, care has been taken to write the letters which denote corresponding points in a corresponding order. This is a matter of minor importance, but it does not deserve to be neglected, as is too often the case.

The deductions or exercises appended to the various propositions (riders,' as they are sometimes termed) have been intentionally made easy and, in the First Book, numerous. It is hoped that beginners, who have little confidence in their own reasoning power, will thereby be encouraged to do more than merely learn the text of Euclid. It is hoped also that sufficient provision has been made for all classes of beginners, seeing that the questions, deductions, and corollaries to be

proved number considerably over fifteen hundred. It should be stated that when a deduction is repeated once or oftener, in the same words, a different mode of proof is expected in each



In the appendices, much curtailed from considerations of space, a few of the more useful and interesting theorems of elementary geometry have been given. It has not been thought expedient to introduce the signs + and to indicate opposite directions of measurement. The important advantages which result from this use of these signs are readily apprehended by readers who advance beyond the 'elements,' and it is only of the 'elements' that the present manual treats.

The historical notes, which are not specially intended for beginners, may save time and trouble to any one who wishes to investigate more fully certain of the questions which occur throughout the work. It would perhaps be well if such notes were more frequently to be found in mathematical text-books: the names of those who have extended the boundaries, or successfully cultivated any part of the domain, of science should not be unknown to those who inherit the results of their labour.

Though the utmost pains have been taken by all concerned in the production of this volume to make it accurate and workmanlike, a few errors may have escaped notice. Corrections of these will be gratefully received.

The editor desires to express his thanks to Mr J. R. PAIRMAN for the excellence of the diagrams, and to Mr DAVID TRAILL, M.A., B.Sc., and Mr A. Y. FRASER, M.A., for valuable hints while the work was going through the press.

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