In 1932 Euclid's... to appeal to every kind of reader: the worker, the student, the cultured man, the child, the man and the woman[so that]for a few shillings the reader may have a whole bookshelf of the immortals; for five pounds(which will procure him with a hundred volumes)a man may be intellectually rich for life.

*Elements*was published as No 891 in the Everyman's Library. Todhunter's

*Euclid,*published in 1862, was chosen to be the edition reprinted and an Introduction to the Everyman's Library book was written by

**Sir Thomas L Heath**.

Below we reproduce Heath's Introduction.

INTRODUCTION

It was with the greatest interest that I heard that the publishers thought of adding to "Everyman's Library" an edition of the *Elements* of Euclid; for what. book in the world could be more suitable for inclusion in the Library than this, the greatest textbook of elementary mathematics that there has ever been or is likely to be, a book which, ever since it was written twenty-two centuries ago., has been read and appealed to as authoritative by mathematicians great and small, from Archimedes and Apollonius of Perga onwards? No textbook, presumably, can ever be without flaw (especially in a subject like geometry, where some first principles, postulates or axioms, have to be assumed without proof, and any number of alternative systems are possible), and flaws there are in Euclid; but it is safe to say that no alternative to the *Elements* has yet been produced which is open to fewer or less serious objections. The only general criticism of it which is deserving of consideration is that it is unsuitable as a textbook for very young boys and girls who are just beginning to learn the first things about geometry. This can be admitted without detracting in the least from the greatness or the permanent value of the book. The simple truth is that it was not written for schoolboys or schoolgirls, but for the grown man who would have the necessary knowledge and judgment to appreciate the highly contentious matters which have to be grappled with in any attempt to set out the essentials of Euclidean geometry as a strictly logical system, and, in particular, the difficulty of making the best selection of unproved postulates or axioms to form the foundation of the subject. My advice would, therefore, be: if you must spoon-feed the very young, do so; but when they have shown a taste for the subject and attained the standard necessary for, passing honours examinations, let them *then* be introduced to Euclid in his original form as an antidote to the more or less feeble echoes of him that are to be found in the ordinary school textbooks of "geometry." I should be surprised if such qualified readers, making the acquaintance of Euclid for the first time, did not find it fascinating, a book to be read in bed or on a holiday, a book as difficult as any detective story to lay down when once begun. I know of one actual case, that of an undergraduate at Cambridge suddenly presented with a copy of Euclid, where this happened. This is the true test of such a book. Nor does the reading of it require the "higher mathematics." Any intelligent person with a fair recollection of school work in elementary geometry would find it (progressing as it does by gradual and nicely contrived steps) easy reading, and should feel a real thrill in following its development, always assuming that enjoyment of the book is not marred by any prospect of having to pass an examination in it! This is why I applaud the addition of this great classic to Everyman's Library; for everybody ought to read it who *can,* that is, all educated persons except the very few who are constitutionally incapable of mathematics.

When it came to the question of selecting the particular version of Euclid to be reproduced, this might have been difficult but for the fact that the format, size of page, etc. (which had to be copied exactly) had to be similar to those of the other volumes in the series. This restricted the choice considerably; and I think it was a fortunate circumstance that the most suitable edition, from the point of view of format, should be precisely that of Isaac Todhunter, which, as he says in his preface, substantially reproduces that of Robert Simson.

Robert Simson was born in 1687 and was Professor of Mathematics in the University of Glasgow from 1711 to 1761. He was an enthusiastic admirer of the Greek geometers and spent the best part of his life in studying and elucidating them. Steeped in the subject, he even made important attempts, with the aid of indications of content, etc., given in the *Collection* of Pappus of Alexandria, to restore three lost works, the Porisms of Euclid (a difficult treatise in higher geometry) and two minor works of Apollonius. The only textbook published by Simson besides his Euclid was on conic sections, *Sectionum conicarum libri quinque* (Edinburgh 1835); this, except for the definitions of the three conics, was based on Apollonius.

Simson's *Euclid,* which was published simultaneously in Latin and English, did not appear till 1756 (i.e., when he was about 69), so that the edition must have represented his most mature thought on the subject. The merits of Simson, both as interpreter and as critic of Euclid, are very great; and it was mainly due to the excellence of his edition that the words "Euclid" and "geometry" became almost synonymous terms in this country. Todhunter, Senior Wrangler in 1848, was the author of a series of mathematical textbooks quite unrivalled in their day; and his notes to Euclid, admirably concise and to the point, fully deserve re-impression.

Simson's own edition included the geometrical Books I-VI, XI and XII in full, omitting only Book X (on Irrationals), and Book XIII (on the five Regular Solids); Todhunter's edition includes only 21 propositions of Book XI (omitting 22-40), and two propositions of Book XII (out of 18).

Simson had, it is true, a "bee in his bonnet." The title-page of his first editions says that "in this edition the errors by which Theon or others have long ago vitiated these Books are corrected, and some of Euclid's Demonstrations are restored." Simson, however, was not in any real sense a competent textual critic; he acted on the simple but uncritical principle that whatever he found in the text which fell short of perfection, whether of form or content, must have been due to alterations made by Theon or "some unskilful editor. " Apart from this, the security for the general excellence of his version was the fact that it was made from the Latin translation of Commandinus (1572).

Unfortunately, the *editio princeps* of the Greek text by Grynaeus (Basel 1533) and the various translations from the Greek down to the end of the eighteenth century (including that of Commandinus) depended upon MSS. of the class now known as Theonine, i.e., containing the recension of Euclid by Theon of Alexandria in the fourth century A.D. They purport by their titles to be either "from the edition of Theon" or "from the school (i.e., lectures) of Theon." Moreover, Theon himself says in a passage of his commentary on the *Syntaxis* of Ptolemy:- "But that sectors in equal circles are to one another as the angles on which they stand has been proved by me in my edition of the Elements at the end of the sixth book," from which it is plain that the second part of Euclid VI, 33 containing this proof was added by Theon to the original. Now the great Vatican MS., Vat. Gr. 190, discovered by F Peyrard in or about 1808, was found to contain neither the words about Theon quoted above from the titles to the other MSS., nor the addition by Theon to VI, 33 referred to. Peyrard was, therefore, justified in concluding that in the Vatican MS. we have an edition more ancient than Theon's, and therefore a key to the changes made by Theon. Simson had not the advantage of knowing these facts, which were discovered after his time. But the Vatican MS. has now been given its proper weight, and it is among the main sources of the authoritative text of the *Elements* published, with Latin translation, critical notes, and prolegomena, in five volumes between 1883 and 1888 (Teubner), by J L Heiberg.

By way of addition to the bibliography of the subject in Todhunter's notes, pp. 250-251, the following references may be given. A faithful translation of Heiberg's text into English is contained in the present writer's *The Thirteen Books of Euclid's Elements, with Introduction and Commentary,* 3 vols., second edition, 1926 (Cambridge University Press). Those who wish to sample the original Greek text of Euclid, which is well worth while, may be referred to *Euclid in Greek, Book I, with Introduction and Notes* (Cambridge 1920). Of recent histories of Greek mathematics in general we may mention James Gow's *A Short History of Greek Mathematics,* 1884 (now out of print), and the present writer's *History of Greek Mathematics,* 2 vols., 1921, and *Manual of Greek Mathematics,* 1931 (Clarendon Press).

Although the edition of the *Elements* here presented, being Simson's edition as revised by Todhunter, rests on the Theonine recension, there is no objection to Euclid being read in this form. It is only necessary to bear certain things in mind, which shall be briefly noted here.

Simson's text itself suffered changes in later editions for which Simson himself was not responsible; some changes made by editors or publishers were not for the better. In this respect Simson's fate was not unlike Euclid's own. Simson died in 1768, and only two editions were issued under his own supervision, those of 1756 and 1762. To the edition of 1762 he added the *Data* of Euclid, a work recalling some of the subject matter of the *Elements* but in a different aspect; at the same time he revised somewhat his version of the *Elements* if we may judge by the sentence which he added to the preface of the same edition: "Besides, the translation is much amended by the friendly assistance of a learned gentleman."

There are some differences of substance between Simson's text and the genuine text of Euclid as restored in Heiberg's edition.

1. Perhaps the most important of these is the difference between the arrangement of the postulates and the axioms in the two editions. Simson's text contains three postulates and twelve axioms. The last two of these axioms Euclid gave, not as axioms (or "common notions" as he called them), but as postulates. Thus "Axiom 11," that all right angles are equal to one another, was Euclid's "Postulate 4," while "Axiom 12" (the well-known parallel-axiom) was "Postulate 5." Further, of the first ten "Axioms" only five can, with any probability, be attributed to Euclid himself (1, 2, 3, 8, and 9 in Todhunter's edition). Thus Euclid did not give, as a common notion, "Two straight lines cannot enclose a space." Where this is cited in the text of I, 4, the words are interpolated. Euclid inferred the coincidence of BC with EF immediately from the fact that the ends of the two straight lines coincide respectively; he seems to have regarded the fact that there is only one straight joining two given points as implicit in Postulate 1 that "a straight line can, be drawn from any one point to any other point."

2. In some cases where the text of Euclid as he found it gave two alternative proofs of one and the same proposition, e.g., III, 9 and 10, VI, 20 (and XI, 22), Simson happened to choose the alternative which had the lesser authority. There is nothing to wonder at in this, because the considerations in his mind might well be similar to those which influenced the author of the interpolated proof.

3. Simson is not responsible for the omission of VI, 27-29. Todhunter omitted them, "as they appear now to be never required and have been condemned as useless by various modem commentators: see Austin, Walker, and Lardner." Simson, however, had strongly protested against this view. He says of Props. 28 and 29: "These two problems, to the first of which the 27th proposition is necessary, are the most general and useful of all in the Elements, and are most frequently made use of by the ancient geometers in the solution of other problems; and, therefore, are very ignorantly left out by Tacquet and Dechales in their editions of the *Elements,* who pretend that they are scarce of any use." The important words here are those referring to the ancient geometers. The propositions embody, in fact, the general method known as the "application of areas," which was of vital consequence to the Greek geometers, being the geometrical equivalent of the solution of the general quadratic equations *ax* ± *bx*^{2}/*c* = *S* so far as they have real roots. Nowadays we solve such equations by algebra; the Greek geometers, however, had no algebraical notation, and hence they had to invent a sort of geometrical algebra. The simplest case of "application of areas," which is equivalent to the solution for *x* of the simple equation *ax* = *S*, can be read in this volume *(Eucl.* I, 44, 45); the general method is a marvel of geometrical ingenuity. No wonder that Plutarch mentions a doubt whether it was not the discovery of this method, rather than that of the theorem of the square of the hypotenuse *(Eucl.* I, 47), which was the occasion of the famous sacrifice of oxen supposed to have been made by Pythagoras. In order to make good the omission in Todhunter's edition, the three propositions in question, translated from the original text of Euclid, are here added in an Appendix.

One other note of Todhunter's should be mentioned. He says (p. 288) that the proposition VI, 32 "seems of no use." Now Euclid was not in the habit of giving any proposition except for use; and this one is, in fact, required and used by Euclid himself in Prop. 17 of Book XIII, containing the construction and "comprehension in a sphere" of a regular dodecahedron (the regular solid figure with twelve faces, all of which are equal regular pentagons).

T L HEATH.

November 1932.