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proofs of any theorems, in the proofs of which, in Euclid's text, the theorem of Proposition 4 is quoted.

It may of course be fairly objected that it would be more logical for a writer, who uses with freedom the method of superposition, to omit the first three Propositions of Book I. To this objection my reply must be that it is considered undesirable to alter the numbering of the Propositions in Books I. and II. at all events. No doubt a work written merely for the teaching of geometry, without immediate reference to the requirements of candidates preparing for examination, might well omit the first three Propositions and assume as a postulate that " a circle may be described with any point as centre, and with a length equal to any given straight line as radius," instead of the postulate of Euclid's text (Postulate 6 of the present edition), "a circle may be described with any point as centre and with any straight line drawn from that point as radius."

The use of the words "each to each" has been abandoned. The statement that two things are equal to two other things each to each, seems to imply, according to the natural meaning of the words, that all four things are equal to each other. Where we wish to state briefly that A has a certain relation to a, B has the same relation to b, and C has the same relation to c, we prefer to say that A, B, C have this relation to a, b, c respectively.

The enunciations of the Propositions in Books I. and II. have been, with some few slight exceptions, retained throughout, and the order of the Propositions remains unaltered, but different methods of proof have been adopted in many cases. The chief instances of alteration are to be found in Propositions 5 and 6 of Book I., and in Book II. The use of what may be called impossible figures, such as occurred in Euclid's text in the proofs of Propositions

6 and 7 of Book I. has been avoided. It seems better to prove that a line cannot be drawn satisfying a certain condition without making a pretence of doing what is impossible.

Two Propositions (10 A and 10 B), have been introduced to shew that, if the method of superposition be used, we need not take as a postulate "all right angles are equal to one another," but that we may deduce this theorem from other postulates which have been already assumed.

Another new Proposition introduced into the text is Proposition 26 A, "if two triangles have two sides equal to two sides, and the angles opposite to one pair of equal sides equal, the angles opposite to the other pair are either equal or supplementary," which may be described, with reference to Euclid's text, as the missing case of the equality of two triangles. It is intimately connected with what is called in Trigonometry "the ambiguous case" in the solution of triangles.

Another new Proposition (41 A) is the solution of the problem "to construct a triangle equal to a given rectilineal figure." It appears to be a more practical method of solving the general problem of Proposition 45 "to construct a parallelogram equal to a given rectilineal figure, having a side equal to a given straight line, and having an angle equal to a given angle," to begin with the construction of a triangle equal to the given figure rather than to follow the exact sequence of Euclid's propositions.

In the notes a few "Additional Propositions" have been introduced containing important theorems, which did not occur in Euclid's text, but with which it is desirable that the student should become familiar as early as possible. Also outlines have been given of some of the many different proofs which have been discovered of Pythagoras's Theorem. They may be found interesting and useful as exercises for the student.

Euclid's proofs of many of the Propositions of Book II. are unnecessarily long. His use of the diagonal of the square in his constructions in Propositions 4 to 8 can scarcely be considered elegant.

It is curious to notice that Euclid after giving a demonstration of Proposition 1 makes no use whatever of the theorem. It seems more logical to deduce from Proposition 1 those of the subsequent Propositions which can be readily so deduced.

In Book II. outlines of alternative proofs of several of the Propositions have been given, which may be developed more fully and used in examinations, in place of the proofs given in the text. Some of these proofs are not, so far as I know, to be found in English text books. The most interesting ones are those of Propositions 12 and 13. Some, which I thought at first were new, I have since found in foreign text books.

The Propositions in the text have not been distinguished by the words "Theorem" and "Problem." The student may be informed once for all that the word theorem is used of a geometrical truth which is to be demonstrated, and that the word problem is used of a geometrical construction which is to be performed.

Although Euclid always sums up the result of a Proposition by the words ὅπερ ἔδει δεῖξαι οι ὅπερ ἔδει ποιῆσαι, there seems to be no utility in putting the letters Q.E.D. or Q.E.F. at the end of a Proposition in an English textbook. The words "Quod erat demonstrandum" or "Quod erat faciendum" in a Latin text were not out of place.

When the book is opened, the reader will see as a rule on the left hand page a Proposition, and on the opposite page notes or exercises. The notes are either appropriate to the Proposition they face or introductory to the one next succeeding. The exercises on the right hand page are,

it is hoped, in all cases capable of being solved by means of the Proposition on the adjoining page and of preceding Propositions. They have been chosen with care and with the special view of inducing the student from the commencement of his reading to attempt for himself the solution of exercises.

For many Propositions it has been difficult to find . suitable exercises: consequently many of the exercises have been specially manufactured for the Propositions to which they are attached. Great pains have been taken to verify the exercises, but notwithstanding it can scarcely be hoped that all trace of error has been eliminated.

It is with pleasure that I record here my deep sense of obligation to many friends, who have aided me by valuable hints and suggestions, and more especially to A. R. Forsyth, M.A., Fellow and Assistant Tutor of Trinity College, Charles Smith, M. A., Fellow and Tutor of Sidney Sussex College, R. T. Wright, M.A., formerly Fellow and Tutor of Christ's College, my brother-in-law the Reverend T. J. Sanderson, M.A., formerly Fellow of Clare College, and my brother W. W. Taylor, M.A., formerly Scholar of Queen's College, Oxford, and afterwards Scholar of Trinity College, Cambridge. The time and trouble ungrudgingly spent by these gentlemen on this edition have saved it from many blemishes, which would otherwise have disfigured its pages.

I shall be grateful for any corrections or criticisms, which may be forwarded to me in connection either with the exercises or with any other part of the work.

TRINITY COLLEGE, CAMBRIDGE,

October 1, 1889.

H. M. TAYLOR.

PREFACE TO BOOKS III AND IV.

N Book III. the chief deviation from Euclid's text will

I be found in the first twelve Propositions, where a good

deal of rearrangement has been thought desirable. This rearrangement has led to some changes in the sequence of Propositions as well as in the Propositions themselves; but, even with these changes, the first twelve Propositions will be found to include the substance of the whole of the first twelve of Euclid's text.

The Propositions from 13 to 37 are, except in unimportant details, unchanged in substance and in order.

The enunciation of the theorem of Proposition 36 has been altered to make it more closely resemble that of the complementary theorem of Proposition 35.

An additional Proposition has been introduced on page 186 involving the principle of the rotation of a plane figure about a point in its plane. It is a principle of which extensive use might with advantage be made in the proof of some of the simpler properties of the circle. It has not however been thought desirable to do more in this edition than to introduce the student to this method and by a selection of exercises, which can readily be solved by its means, to indicate the importance of the method.

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