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NOTE.

The Special Board for Mathematics in the University of Cambridge in a Report on Geometrical Teaching dated May 10, 1887, state as follows:

"The majority of the Board are of opinion that the rigid adherence

to Euclid's texts is prejudicial to the interests of education, and that greater freedom in the method of teaching Geometry is desirable. As it appears that this greater freedom cannot be attained while a knowledge of Euclid's text is insisted upon in the examinations of the University, they consider that such alterations should be made in the regulations of the examinations as to admit other proofs besides those of Euclid, while following however his general sequence of propositions, so that no proof of any proposition occurring in Euclid should be accepted in which a subsequent proposition in Euclid's order

is assumed.' On March 8, 1888, Amended Regulations for the Previous Examination, which contained the following provision, were approved by the Senate : · Euclid's definitions will be required, and no axioms or postulates

except Euclid's may be assumed. The actual proofs of propositions as given in Euclid will not be required, but no proof of any proposition occurring in Euclid will be admitted in which use is made of any proposition which in Euclid's order occurs

subsequently.' And in the Regulations for the Local Examinations conducted by the University of Cambridge it is provided that: •Proofs other than Euclid's will be admitted, but Euclid's Axioms

will be required, and no proof of any proposition will be accepted which assumes anything not proved in preceding propositions in Euclid.'

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PREFACE TO BOOKS I AND II.

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T was with extreme diffidence that I accepted an invi

Press to undertake for them a new edition of the Elements of Euclid. Though I was deeply sensible of the honour, which the invitation conferred, I could not but recognise the great responsibility, which the acceptance of it would entail.

The invitation of the Syndics was in itself, to my mind, a sign of a widely felt conviction that the editions in common use were capable of improvement. Now improvement necessitates change, and every change made in a work, which has been a text book for centuries, must run the gauntlet of severe criticism, for while some will view every alteration with aversion, others will consider that every change demands an apology for the absence of more and greater changes.

I will here give a short account of the chief points, in which this edition differs from the best known editions of the Elements of Euclid at present in use in England.

While the texts of the editions of Potts and Todhunter are confessedly little more than reprints of Simson's English version of the Elements published in 1756, the text of the present edition does not profess to be a translation from the Greek. I began by retranslating the First Book: but

there proved to be so many points, in which I thought it Ć

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desirable to depart from the original, that it seemed best to give up all idea of simple translation and to retain merely the substance of the work, following closely Euclid's sequence of Propositions in Books I. and II. at all events.

Some of the definitions of Euclid, for instance trapezium, rhomboid, gnomon are omitted altogether as unnecessary. The word trapezium is defined in the Greek to mean four sided figure other than those already defined,” but in many modern works it is defined to be “a quadrilateral,

s which has one pair of parallel sides." The first of these definitions is obsolete, the second is not universally accepted. On the other hand definitions are added of several words in general use, such as perimeter, parallelogram, diagonal, which do not occur in Euclid's list.

The chief alteration in the definitions is in that of the word figure, which is in the Greek text defined to be “that which is enclosed by one or more boundaries.I have preferred to define a figure as "a combination of points, lines and surfaces.That Euclid's definition leads to difficulty is seen from the fact that, though Euclid defines a

a figure contained by one line...", he demands in his postulate that "

a circle

may

be described...". Now it is the circumference of a circle which is described and not the surface. Again, when two circles intersect, it is the circumferences which intersect and not the surfaces.

I have rejected the ordinarily received definition of a square as a quadrilateral, whose sides are equal, and whose angles are right angles.There is no doubt that, when we define any geometrical figure, we postulate the possibility of the figure; but it is useless to embrace in the definition more properties than are requisite to determine the figure.

The word axiom is used in many modern works as applicable both to simple geometrical propositions, such as

two straight lines cannot enclose a space," and to proposi

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tions, other than geometrical, accepted without demonstration and true universally, such as “the whole of a thing is greater than a part.These two classes of propositions are often distinguished by the terms "geometrical axioms” and “ general axioms.” I prefer to use the word axiom as applicable to the latter class only, that is, to simple propositions, true of magnitudes of all kinds (for instance "things which are equal to the same thing are equal to one another”), and to use the term postulate for a simple geometrical proposition, whose truth we assume.

When a child is told that A weighs exactly as much as B, and B weighs exactly as much as C, he without hesitation arrives at the conclusion that A weighs exactly as much as C. His conviction of the validity of his conclusion would not be strengthened, and possibly his confidence in his conclusion might be impaired, by his being directed to appeal to the authority of the general proposition "things which are equal to the same thing are equal to one another.I have therefore, as a rule, omitted in the text all reference to the general statements of axioms, and have only introduced such a statement occasionally, where its introduction seemed to me the shortest way of explaining the nature of the next step in the demonstration.

If it be objected that all axioms used should be clearly stated, and that their number should not be unnecessarily extended, my reply is that neither the Greek text nor any edition of it, with which I am acquainted, has attempted to make its list of axioms perfect in either of these respects. The lists err in excess, inasmuch as some of the axioms therein can be deduced from others : they err in defect, inasmuch as in the demonstrations of Propositions conclusions are often drawn, to support the validity of which no appeal can be made to any axiom in the lists.

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Under the term postulate I have included not only what may be called the postulates of geometrical operation, such as “it is assumed that a straight line may be drawn from any point to any other point," but also geometrical theorems, the truth of which we assume, such as “two straight lines cannot have a common part.

The postulates of this edition are nine in number.

Postulates 3, 4, 6 are the postulates of geometrical operation, which are common to all editions of the Elements of Euclid. Postulates 1, 5, 9 are the Axioms 10, 11, 12 of modern editions. Postulates 2, 7, 8 do not appear under the head either of axioms or of postulates in Euclid's text, but the substance of them is assumed in the demonstrations of his propositions.

Postulate 9 has been postponed until page 51, as it seemed undesirable to trouble the student with an attempt to unravel its meaning, until he was prepared to accept it as the converse of a theorem, with the proof of which he had already been made acquainted.

It may be mentioned that a proof of Postulate 5, "all right angles are equal" is given in the text (Proposition 10 B), and that therefore the number of the Postulates might have been diminished by one : it was however thought necessary to retain this Postulate in the list, so that it might be used as a postulate by any person who might prefer to adhere closely to the original text of Euclid.

One important feature in the present edition is the greater freedom in the direct use of “the method of superposition " in the proofs of the Propositions. The method is used directly by Euclid in his proof of Proposition 4 of Book I., and indirectly in his proofs of Proposition 5 and of every other Proposition, in which the theorem of Proposition 4 is quoted. It seems therefore but a slight alteration to adopt the direct use of this method in the

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