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IN bringing out a new edition of the ELEMENTS OF EUCLID, three chief objects have been kept in view-to render the study of Geometry more easy and attractive--to adopt a mode of treatment suggestive of other truths-and to introduce such improvements as should tend to remove the desire that Euclid should be replaced by any other text-book. Attempts to supplant it have been made from time to time; but they have not met with much acceptance in this country. The purpose, however, is still entertained; but for many reasons the superseding of Euclid is to be most strongly deprecated. Not only has this great work received the approval of many successive ages, and served to connect the science of the present with that of the past, but is even now, with all the progress of modern times in Pure and Applied Mathematics, open to criticism on very few points. It is an object of paramount importance that the work should be retained, not only as a common standard of reference, and a guide in public examinations, but as a standard in a much higher sense-one by which the purity and rigorous character of geometrical demonstrations shall be maintained, and a true logical sequence kept up in the order in which these are presented to the mind of the student. To retain as a textbook a work of such surpassing excellence and high sanction, is of inestimable importance, whether it is regarded as a means of intellectual discipline, or as a standard of reference in regard to geometrical truth. "There never has been, and till we see it we never shall believe there can be, a system of Geometry worthy of the name which has any material departures (we do not speak of corrections, or extensions, or developments) from the plan laid down by Euclid."*

* De Morgan, Brit. Alm. and Comp, for 1848, 2nd Part, p. 20.

Though thus approaching their task with a profound respect for the work of Euclid, the Editors were not unaware of its defects, and its want of adaptation in some respects to geometrical methods, which have an important relation to those of the modern analysis; and their constant aim has been to incorporate such in the work, while preserving the strict methods of Euclid. Abandoning in the definitions the restriction in regard to the meaning of an angle, which Euclid himself ultimately gives up in the thirty-third Proposition of the Sixth Book, they have been enabled to shorten the proofs of several Propositions in the First Book, and of many more in the Third, and to give easier proofs of others; and thus early to familiarize the student with an idea which he usually acquires only when finishing the Elements and entering on Trigonometry.

The order of Euclid's Propositions is in some respects defective, as separating those relating to the same subject, so that the connection among them is less easily seen. This defect it has been attempted to remedy, so far as the exigencies of demonstration would permit. Some examples will be seen in the First, Third, and Sixth Books, more especially in the Third and Sixth. Several of the changes made in this direction are in conformity with hints first thrown out by Professor De Morgan in the article already quoted. In carrying out these changes, the Third and Sixth Books have been greatly shortened, while Euclid's Propositions have been preserved. It is of great importance to have the related. truths thus placed together; viewed in their connection, they take a firmer hold of the mind, and much smaller demands are thus made for separate efforts of memory.

The length and prolixity of Euclid's proofs, and the repetitions which occur in them, are subjects of complaint, and in some cases justly. These defects, it is hoped, have been removed, yet without omitting any step in the process, or link of the chain. To do this would be no real shortening of a proof; if the mind has to go through a process of reasoning in order to connect two of the steps, no advantage is gained by withholding the explicit statement. The better to distinguish the parts of a Proposition, and to render the apprehension of the proof and sequence of the steps

more easy, the demonstrations are printed in a smaller type, the steps on separate lines, and the Propositions referred to, placed in one vertical line along the outer margin of the page. The signs used in Algebra and Arithmetic, and such symbols and abbreviations as an Examiner permits when a student is writing against time, are unsuitable and inelegant in a work on Geometry, while they sadly mar the look of a page.

All the remarks relating to the connection of Geometry with Arithmetic and Algebra, and other matters foreign to Euclid's mode of treatment, are given in Appendices to the various books. In these also are placed Supplementary Propositions, and a series of Theorems and Problems, for the exercise of the student, gradually increasing in difficulty, and related, as far as possible, to the order of the Propositions, especially those under the First Book. These, as well as the various Scholia and Corollaries, are intended to make the work more suggestive to the student. It may, however, be found necessary to omit many of them on a first reading; or merely to refer to such of them as bear most directly on applications of the Propositions in the solution of Problems.

The treatment of the subject of Rectangles in the Second Book is somewhat different from that of Euclid, though the Propositions are retained; the main purpose being to shorten the proofs of those which were the longest and most prolix, and to make the method used in the others consistent with them, that all the proofs might be upon the same plan. The method of projection has been sometimes employed, and two Propositions demonstrated in this way are brought from the Second Book into the First.

In the Third and Fourth Books, some important extra Propositions have been introduced into the text, as they are very closely connected with those of Euclid, and were considered of too much importance to place in an Appendix. The subject of Tangents has some novelty, and that of Loci is frequently referred to.

The treatment of Proportion has given the Editors much anxious consideration. Though Euclid's method of delivering the doctrine in the series of Propositions given in his Fifth Book is the only one free from objection, and no other

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