Euclid, Elements (ed. Thomas L. Heath)

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[p. 46]

CHAPTER V.

THE TEXT¹ .

It is well known that the title of Simson's edition of Euclid (first brought out in Latin and English in 1756) claims that, in it, “the errors by which Theon, or others, have long ago vitiated these books are corrected, and some of Euclid's demonstrations are restored”; and readers of Simson's notes are familiar with the phrases used, where anything in the text does not seem to him satisfactory, to the effect that the demonstration has been spoiled, or things have been interpolated or omitted, by Theon “or some other unskilful editor.” Now most of the MSS. of the Greek text prove by their titles that they proceed from the recension of the Elements by Theon; they purport to be either “from the edition of Theon” (ek tês Theônos ekdoseôs) or “from the lectures of Theon” (apo sunousiôn tou Theônos). This was Theon of Alexandria (4th c. A.D.) who also wrote a commentary on Ptolemy, in which there occurs a passage of the greatest importance in this connexion² : “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.” Thus Theon himself says that he edited the Elements and also that the second part of VI. 33, found in nearly all the MSS., is his addition.

This passage is the key to the whole question of Theon's changes in the text of Euclid; for, when Peyrard found in the Vatican the MS. 190 which contained neither the words from the titles of the other MSS. quoted above nor the interpolated second part of VI. 33, he was justified in concluding, as he did, that in the Vatican MS. we have an edition more ancient than Theon's. It is also clear that the copyist of P, or rather of its archetype, had before him the two recensions and systematically gave the preference to the earlier one; for at XIII. 6 in P the first hand has added a note in the margin: “This theorem is not given in most copies of the new edition, but is found in those of the old.” Thus we are more fortunate than Simson, since our judgment of Theon's recension can be formed on the basis, not of mere conjecture, but of the documentary evidence afforded by a comparison of the Vatican MS. just mentioned with what we may conveniently call, after Heiberg, the Theonine MSS. [p. 47]

The MSS. used for Heiberg's edition of the Elements are the following:

(1) P = Vatican MS. numbered 190, 4to, in two volumes (doubtless one originally); 10th c.

This is the MS. which Peyrard was able to use; it was sent from Rome to Paris for his use and bears the stamp of the Paris Imperial Library on the last page. It is well and carefully written. There are corrections some of which are by the original hand, but generally in paler ink, others, still pretty old, by several different hands, or by one hand with different ink in different places (P m. 2), and others again by the latest hand (P m. rec.). It contains, first, the Elements I.--XIII. with scholia, then Marinus' commentary on the Data (without the name of the author), followed by the Data itself and scholia, then the Elements XIV., XV. (so called), and lastly three books and a part of a fourth of a commentary by Theon eis tous procheirous kanonas Ptolemaiou.

The other MSS. are “Theonine.”

(2) F = MS. XXVIII, 3, in the Laurentian Library at Florence, 4to; 10th c.

This MS. is written in a beautiful and scholarly hand and contains the Elements I.--XV., the Optics and the Phaenomena, but is not well preserved. Not only is the original writing renewed in many places, where it had become faint, by a later hand of the 16th c., but the same hand has filled certain smaller lacunae by gumming on to torn pages new pieces of parchment, and has replaced bodily certain portions of the MS., which had doubtless become illegible, by fresh leaves. The larger gaps so made good extend from Eucl. VII. 12 to IX. 15, and from XII. 3 to the end; so that, besides the conclusion of the Elements, the Optics and Phaenomena are also in the later hand, and we cannot even tell what in addition to the Elements I.--XIII. the original MS. contained. Heiberg denotes the later hand by ph and observes that, while in restoring words which had become faint and filling up minor lacunae the writer used no other MS., yet in the two larger restorations he used the Laurentian MS. XXVIII, 6, belonging to the 13th--14th c. The latter MS. (which Heiberg denotes by f) was copied from the Viennese MS. (V) to be described below.

(3) B = Bodleian MS., D'Orville X. 1 inf. 2, 30, 4to; A.D. 888.

This MS. contains the Elements I.--XV. with many scholia. Leaves 15-118 contain I. 14 (from about the middle of the proposition) to the end of Book VI., and leaves 123-387 (wrongly numbered 397) Books VII.--XV. in one and the same elegant hand (9th c.). The leaves preceding leaf 15 seem to have been lost at some time, leaves 6 to 14 (containing Elem. I. to the place in I. 14 above referred to) being carelessly written by a later hand on thick and common parchment (13th c.). On leaves 2 to 4 and 122 are certain notes in the hand of Arethas, who also wrote a two-line epigram on leaf 5, the greater part of the scholia in uncial letters, a few notes and corrections, and two sentences on the last leaf, the first of which states that the MS. was written by one Stephen clericus in the year of the world 6397 [p. 48] (= 888 A.D.), while the second records Arethas' own acquisition of it. Arethas lived from, say, 865 to 939 A.D. He was Archbishop of Caesarea and wrote a commentary on the Apocalypse. The portions of his library which survive are of the greatest interest to palaeography on account of his exact notes of dates, names of copyists, prices of parchment etc. It is to him also that we owe the famous Plato MS. from Patmos (Cod. Clarkianus) which was written for him in November 895³ .

(4) V = Viennese MS. Philos. Gr. No. 103; probably 12th c.

This MS. contains 292 leaves, Eucl. Elements I.--XV. occupying leaves 1 to 254, after which come the Optics (to leaf 271), the Phaenomena (mutilated at the end) from leaf 272 to leaf 282, and lastly scholia, on leaves 283 to 292, also imperfect at the end. The different material used for different parts and the varieties of handwriting make it necessary for Heiberg to discuss this MS. at some length⁴ . The handwriting on leaves 1 to 183 (Book I. to the middle of X. 105) and on leaves 203 to 234 (from XI. 31, towards the end of the proposition, to XIII. 7, a few lines down) is the same; between leaves 184 and 202 there are two varieties of handwriting, that of leaves 184 to 189 and that of leaves 200 (verso) to 202 being the same. Leaf 235 begins in the same handwriting, changes first gradually into that of leaves 184 to 189 and then (verso) into a third more rapid cursive writing which is the same as that of the greater part of the scholia, and also as that of leaves 243 and 282, although, as these leaves are of different material, the look of the writing and of the ink seems altered. There are corrections both by the first and a second hand, and scholia by many hands. On the whole, in spite of the apparent diversity of handwriting in the MS., it is probable that the whole of it was written at about the same time, and it may (allowing for changes of material, ink etc.) even have been written by the same man. It is at least certain that, when the Laurentian MS. XXVIII, 6 was copied from it, the whole MS. was in the condition in which it is now, except as regards the later scholia and leaves 283 to 292 which are not in the Laurentian MS., that MS. coming to an end where the Phaenomena breaks off abruptly in V. Hence Heiberg attributes the whole MS. to the 12th c.

But it was apparently in two volumes originally, the first consisting of leaves 1 to 183; and it is certain that it was not all copied at the same time or from one and the same original. For leaves 184 to 202 were evidently copied from two MSS. different both from one another and from that from which the rest was copied. Leaves 184 to the middle of leaf 189 (recto) must have been copied from a MS. similar to P, as is proved by similarity of readings, though not from P itself. The rest, up to leaf 202, were copied from the Bologna MS. (b) to be mentioned below. It seems clear that the content of leaves 184 to 202 was supplied from other MSS. because there was a lacuna in the original from which the rest of V was copied. [p. 49]

Heiberg sums up his conclusions thus. The copyist of V first copied leaves 1 to 183 from an original in which two quaterniones were missing (covering from the middle of Eucl. X. 105 to near the end of XI. 31). Noticing the lacuna he put aside one quaternio of the parchment used up to that point. Then he copied onwards from the end of the lacuna in the original to the end of the Phaenomena. After this he looked about him for another MS. from which to fill up the lacuna; finding one, he copied from it as far as the middle of leaf 189 (recto). Then, noticing that the MS. from which he was copying was of a different class, he had recourse to yet another MS. from which he copied up to leaf 202. At the same time, finding that the lacuna was longer than he had reckoned for, he had to use twelve more leaves of a different parchment in addition to the quaternio which he had put aside. The whole MS. at first formed two volumes (the first containing leaves 1 to 183 and the second leaves 184 to 282); then, after the last leaf had perished, the two volumes were made into one to which two more quaterniones were also added. A few leaves of the latter of these two have since perished.

(5) b = MS. numbered 18-19 in the Communal Library at Bologna, in two volumes, 4to; 11th c.

This MS. has scholia in the margin written both by the first hand and by two or three later hands; some are written by the latest hand, Theodorus Cabasilas (a descendant apparently of Nicolaus Cabasilas, 14th c.) who owned the MS. at one time. It contains (a) in 14 quaterniones the definitions and the enunciations (without proofs) of the Elements I.--XIII. and of the Data, (b) in the remainder of the volumes the Proem to Geometry (published among the Variae Collectiones in Hultsch's edition of Heron, pp. 252, 24 to 274, 14) followed by the Elements I.--XIII. (part of XIII. 18 to the end being missing), and then by part of the Data (from the last three words of the enunciation of Prop. 38 to the end of the penultimate clause in Prop. 87, ed. Menge). From XI. 36 inclusive to the end of XII. this MS. appears to represent an entirely different recension. Heiberg is compelled to give this portion of b separately in an appendix. He conjectures that it is due to a Byzantine mathematician who thought Euclid's proofs too long and tiresome and consequently contented himself with indicating the course followed⁵ . At the same time this Byzantine must have had an excellent MS. before him, probably of the ante-Theonine variety of which the Vatican MS. 190 (P) is the sole representative.

(6) p = Paris MS. 2466, 4to; 12th c.

This manuscript is written in two hands, the finer hand occupying leaves 1 to 53 (recto), and a more careless hand leaves 53 (verso) to 64, which are of the same parchment as the earlier leaves, and leaves 65 to 239, which are of a thinner and rougher parchment showing traces of writing of the 8th--9th c. (a Greek version of the Old Testament). The MS. contains the Elements I.--XIII. and some scholia after Books XI., XII. and XIII. [p. 50]

(7) q = Paris MS. 2344, folio; 12th c.

It is written by one hand but includes scholia by many hands. On leaves 1 to 16 (recto) are scholia with the same title as that found by Wachsmuth in a Vatican MS. and relied upon by him to prove that Proclus continued his commentaries beyond Book I.⁶ Leaves 17 to 357 contain the Elements I.--XIII. (except that there is a lacuna from the middle of VIII. 25 to the ekthesis of IX. 14); before Books VII. and X. there are some leaves filled with scholia only, and leaves 358 to 366 contain nothing but scholia.

(8) Heiberg also used a palimpsest in the British Museum (Add. 17211). Five pages are of the 7th--8th c. and are contained (leaves 49-53) in the second volume of the Syrian MS. Brit. Mus. 687 of the 9th c.; half of leaf 50 has perished. The leaves contain various fragments from Book X. enumerated by Heiberg, Vol. III., p. v, and nearly the whole of XIII. 14.

Since his edition of the Elements was published, Heiberg has collected further material bearing on the history of the text⁷ . Besides giving the results of further or new examination of MSS., he has collected the fresh evidence contained in an-Nairīzī's commentary, and particularly in the quotations from Heron's commentary given in it (often word for word), which enable us in several cases to trace differences between our text and the text as Heron had it, and to identify some interpolations which actually found their way into the text from Heron's commentary itself; and lastly he has dealt with some valuable fragments of ancient papyri which have recently come to light, and which are especially important in that the evidence drawn from them necessitates some modification in the views expressed in the preface to Vol. V. as to the nature of the changes made in Theon's recension, and in the principles laid down for differentiating between Theon's recension and the original text, on the basis of a comparison between P and the Theonine MSS. alone.

The fragments of ancient papyri referred to are the following.

1. Papyrus Herculanensis No. 1061.⁸

This fragment quotes Def. 15 of Book I. in Greek, and omits the words hê kaleitai periphereia, “which is called the circumference,” found in all our MSS., and the further addition pros tên tou kuklou periphereian also found in practically all the MSS. Thus Heiberg's assumption that both expressions are interpolations is now confirmed by this oldest of all sources.

2. The Oxyrhynchus Papyri I. p. 58, No. XXIX. of the 3rd or 4th c. This fragment contains the enunciation of Eucl. II. 5 (with figure, apparently without letters, immediately following, and not, as usual in our MSS., at the end of the proof) and before it the part of a word periechome belonging to II. 4 (with room for -nôi orthogôniôi: hoper edei [p. 51] deixai and a stroke to mark the end), showing that the fragment had not the Porism which appears in all the Theonine MSS. and (in a later hand) in P, and thereby confirming Heiberg's assumption that the Porism was due to Theon.

3. A fragment in Fayum towns and their papyri, p. 96, No. IX. of 2nd or 3rd c.

This contains I. 39 and I. 41 following one another and almost complete, showing that I. 40 was wanting, whereas it is found in all the MSS. and is recognised by Proclus. Moreover the text of the beginning of I. 39 is better than ours, since it has no double diorismos but omits the first (“I say that they are also in the same parallels”) and has “and” instead of “for let AD be joined” in the next sentence. It is clear that I. 40 was interpolated by someone who thought there ought to be a proposition following I. 39 and related to it as I. 38 is related to I. 37 and I. 36 to I. 35, although Euclid nowhere uses I. 40, and therefore was not likely to include it. The same interpolator failed to realise that the words “let AD be joined” were part of the ekthesis or setting-out, and took them for the kataskeuê or “construction” which generally follows the diorismos or “particular statement” of the conclusion to be proved, and consequently thought it necessary to insert a diorismos before the words.

The conclusions drawn by Heiberg from a consideration of particular readings in this papyrus along with those of our MSS. will be referred to below.

We now come to the principles which Heiberg followed, when preparing his edition, in differentiating the original text from the Theonine recension by means of a comparison of the readings of P and of the Theonine MSS. The rules which he gives are subject to a certain number of exceptions (mostly in cases where one MS. or the other shows readings due to copyists' errors), but in general they may be relied upon to give conclusive results.

The possible alternatives which the comparison of P with the Theonine MSS. may give in particular passages are as follows:

I. There may be agreement in three different degrees.

(1) P and all the Theonine MSS. may agree.

In this case the reading common to all, even if it is corrupt or interpolated, is more ancient than Theon, i.e. than the 4th c.

(2) P may agree with some (only) of the Theonine MSS.

In this case Heiberg considered that the latter give the true reading of Theon's recension, and the other Theonine MSS. have departed from it.

(3) P and one only of the Theonine MSS may agree.

In this case too Heiberg assumed that the one Theonine MS. which agrees with P gives the true Theonine reading, and that this rule even supplies a sort of measure of the quality and faithfulness of the Theonine MSS. Now none of them agrees alone with P in preserving the true reading so often as F. Hence F must be held to have preserved Theon's recension more faithfully than the other Theonine MSS.; and it would follow that in those portions where F fails us P must [p. 52] carry rather more weight even though it may differ from the Theonine MSS. BVpq. (Heiberg gives many examples in proof of this, as of his main rules generally, for which reference must be made to his Prolegomena in Vol. v.) The specially close relation of F and P is also illustrated by passages in which they have the same errors; the explanation of these common errors (where not due to accident) is found by Heiberg in the supposition that they existed, but were not noticed by Theon, in the original copy in which he made his changes.

Although however F is by far the best of the Theonine MSS., there are a considerable number of passages where one of the others (B, V, p or q) alone with P gives the genuine reading of Theon's recension.

As the result of the discovery of the papyrus fragment containing I. 39, 41, the principles above enunciated under (2) and (3) are found by Heiberg to require some qualification. For there is in some cases a remarkable agreement between the papyrus and the Theonine MSS. (some or all) as against P. This shows that Theon took more trouble to follow older MSS., and made fewer arbitrary changes of his own, than has hitherto been supposed. Next, when the papyrus agrees with some of the Theonine MSS. against P, it must now be held that these MSS. (and not, as formerly supposed, those which agree with P) give the true reading of Theon. If it were otherwise, the agreement between the papyrus and the Theonine MSS. would be accidental: but it happens too often for this. It is clear also that there must have been contamination between the two recensions; otherwise, whence could the Theonine MSS. which agree with P and not with the papyrus have got their readings? The influence of the P class on the Theonine F is especially marked.

II. There may be disagreement between P and all the Theonine MSS.

The following possibilities arise.

(1) The Theonine MSS. differ also among themselves.

In this case Heiberg considered that P nearly always has the true reading, and the Theonine MSS. have suffered interpolation in different ways after Theon's time.

(2) The Theonine MSS. all combine against P.

In this case the explanation was assumed by Heiberg to be one or other of the following.

(a) The common reading is due to an error which cannot be imputed to Theon (though it may have escaped him when putting together the archetype of his edition); such error may either have arisen accidentally in all alike, or (more frequently) may be referred to a common archetype of all the MSS.

(b) There may be an accidental error in P; e.g. something has dropped out of P in a good many places, generally through homoioteleuton

(g) There may be words interpolated in P.

(d) Lastly, we may have in the Theonine MSS. a change made by Theon himself.

(The discovery of the ancient papyrus showing readings agreeing [p. 53] with some, or with all, of the Theonine MSS. against P now makes it necessary to be very cautious in applying these criteria.)

It is of course the last class (d) of changes which we have to investigate in order to get a proper idea of Theon's recension.

Heiberg first observes, as regards these, that we shall find that Theon, in editing the Elements, altered hardly anything without some reason, often inadequate according to our ideas, but still some reason which seemed to him sufficient. Hence, in cases of very slight differences where both the Theonine MSS. and P have readings good and probable in themselves, Heiberg is not prepared to put the differences down to Theon. In those passages where we cannot see the least reason why Theon, if he had the reading of P before him, should have altered it, Heiberg would not at once assume the superiority of P unless there was such a consistency in the differences as would indicate that they were due not to accident but to design. In the absence of such indications, he thinks that the ordinary principles of criticism should be followed and that proper weight should be attached to the antiquity of the sources. And it cannot be denied that the sources of the Theonine version are the more ancient. For not only is the British Museum palimpsest (L), which is intimately connected with the rest of our MSS., át least two centuries older than P, but the other Theonine MSS. are so nearly allied that they must be held to have had a common archetype intermediate between them and the actual edition of Theon; and, since they themselves are as old as, or older than P, their archetype must have been much older. Heiberg gives (pp. xlvi, xlvii) a list of passages where, for this reason, he has followed the Theonine MSS. in preference to P.

It has been mentioned above that the copyist of P or rather of its archetype wished to give an ancient recension. Therefore (apart from clerical errors and interpolations) the first hand in P may be relied upon as giving a genuine reading even where a correction by the first hand has been made at the same time. But in many places the first hand has made corrections afterwards; on these occasions he must have used new sources, e.g. when inserting the scholia to the first Book which P alone has, and in a number of passages he has made additions from Theonine MSS.

We cannot make out any “family tree” for the different Theonine MSS. Although they all proceeded from a common archetype later than the edition of Theon itself, they cannot have been copied one from the other; for, if they had been, how could it have come about that in one place or other each of them agrees alone with P in preserving the genuine reading? Moreover the great variety in their agreements and disagreements indicates that they have all diverged to about the same extent from their archetype. As we have seen that P contains corrections from the Theonine family, so they show corrections from P or other MSS. of the same family. Thus V has part of the lacuna in the MS. from which it was copied filled up from a MS. similar to P, and has corrections apparently derived from the same; the copyist, however, in correcting V, also used another MS. to which [p. 54] he alludes in the additions to IX. 19 and 30 (and also on X. 23 Por.): “in the book of the Ephesian (this) is not found.” Who this Ephesian of the 12th c. was, we do not know.

We now come to the alterations made by Theon in his edition of the Elements. I shall indicate classes into which these alterations may be divided but without details (except in cases where they affect the mathematical content as distinct from form or language pure and simple).⁹ .

I. Alterations made by Theon where he found, or thought he found, mistakes in the original.

1. Real blots in the original which Theon saw and tried to remove.

(a) Euclid has a porism (corollary) to VI. 19, the enunciation of which speaks of similar and similarly described figures though the proposition itself refers only to triangles, and therefore the porism should have come after VI. 20. Theon substitutes triangle for figure and proves the more general porism after VI. 20.

(b) In IX. 19 there is a statement which is obviously incorrect. Theon saw this and altered the proof by reducing four alternatives to two, with the result that it fails to correspond to the enunciation even with Theon's substitution of “if” for “when” in the enunciation.

(c) Theon omits a porism to IX. II, although it is necessary for the proof of the succeeding proposition, apparently because, owing to an error in the text (kata ton corrected by Heiberg into epi to), he could not get out of it the right sense.

(d) I should also put into this category a case which Heiberg classifies among those in which Theon merely fancied that he found mistakes, viz. the porism to V. 7 stating that, if four magnitudes are proportional, they are proportional inversely. Theon puts this after V. 4 with a proof, which however has no necessary connexion with V. 4 but is obvious from the definition of proportion.

(e) I should also put under this head XI. 1, where Euclid's argument to prove that two straight lines cannot have a common segment is altered.

2. Passages which seemed to Theon to contain blots, and which he therefore set himself to correct, though more careful consideration would have shown that Euclid's words are right or at least may be excused and offer no difficulty to an intelligent reader. Under this head come:

(a) an alteration in III. 24.

(b) a perfectly unnecessary alteration, in Vi. 14, of “equiangular parallelograms” into “parallelograms having one angle equal to one angle,” where Theon followed the false analogy of VI. 15.

(c) an omission of words in V. 26, owing to his having been misled by a wrong figure.

(d) an alteration of the order of XI. Deff. 27, 28.

(e) the substitution of “parallelepipedal solid” for “cube” in XI. [p. 55] 38, because Theon observed, correctly enough, that it was true of the parallelepipedal solid in general as well as of the cube, but failed to give weight to the fact that Euclid must have given the particular case of the cube for the simple reason that that was all he wanted for use in XIII. 17.

(f) the substitution of the letter Ph for Ô (V for Z in my figure) because he saw that the perpendicular from K to BPh would fall on Ph itself, so that Ph, Ô coincide. But, if the substitution is made, it should be proved that Ph, Ô coincide. Euclid can hardly have failed to notice the fact, but it may be that he deliberately ignored it as unnecessary for his purpose, because he did not want to lengthen his proposition by giving the proof.

II. Emendations intended to improve the form or diction of Euclid.

Some of these emendations of Theon affect passages of appreciable length. Heiberg notes about ten such passages; the longest is in Eucl. XII. 4 where a whole page of Heiberg's text is affected and Theon's version is put in the Appendix. The kind of alteration may be illustrated by that in IX. 15 where Euclid uses successively the propositions VII. 24, 25, quoting the enunciation of the former but not of the latter; Theon does exactly the reverse. In a few of the cases here quoted by Heiberg, Theon shortened the original somewhat.

But, as a rule, the emendations affect only a few words in each sentence. Sometimes they are considerable enough to alter the conformation of the sentence, sometimes they are trifling alterations “more magistellorum ineptorum” and unworthy of Theon. Generally speaking, they were prompted by a desire to change anything which was out of the common in expression or in form, in order to reduce the language to one and the same standard or norm. Thus Theon changed the order of words, substituted one word for another where the latter was used in a sense unusual with Euclid (e.g. epeidêper, “since,” for hoti in the sense of “because”), or one expression for another in like circumstances (e.g. where, finding “that which was enjoined would be done” in a theorem, VII. 31, and deeming the phrase more appropriate to a problem, he substituted for it “that which is sought would be manifest”; probably also and for similar reasons he made certain variations between the two expressions usual at the end of propositions hoper edei deixai and hoper edei poiêsai, quod erat demonstrandum and quod erat faciendum). Sometimes his alterations show carelessness in the use of technical terms, as when he uses haptesthai (to meet) for ephaptesthai (to touch) although the ancients carefully distinguished the two words. The desire of keeping to a standard phraseology also led Theon to omit or add words in a number of cases, and also, sometimes, to change the lettering of figures.

But Theon seems, in editing the Elements, to have bestowed the most attention upon

III. Additions designed to supplement or explain Euclid.

First, he did not hesitate to interpolate whole propositions where he thought there was room or use for them. We have already [p. 56] mentioned the addition to VI. 33 of the second part relating to sectors, for which Theon himself takes credit in his commentary on Ptolemy. Again, he interpolated the proposition commonly known as VII. 22 (ex aequo in proportione perturbata for numbers, corresponding to V. 23), and perhaps also VII. 20, a particular case of VII. 19 as VI. 17 is of VI. 16. He added a second case to VI. 27, a porism to II. 4, a second porism to III. 16, and a lemma after X. 12; perhaps also the porism to V. 19 and the first porism to VI. 20. He also inserted alternative proofs here and there, e.g. in II. 4 (where the alternative differs little from the original) and in VII. 31; perhaps also in X. 1, 6, and 9.

Secondly, he sometimes repeats an argument where Euclid had said “For the same reason,” adds specific references to points, straight lines etc. in the figures in order to exclude the possibility of mistake arising from Euclid's reference to them in general terms, or inserts words to make the meaning of Euclid more plain, e.g. componendo and alternately, where Euclid had left them out. Sometimes he thought to increase by his additions the mathematical precision of Euclid's language in enunciations or elsewhere, sometimes to make smoother and clearer things which Euclid had expressed with unusual brevity and harshness or carelessness, in reliance on the intelligence of his readers.

Thirdly, he supplied intermediate steps where Euclid's argument seemed too rapid and not easy enough to follow. The form of these additions varies; they are sometimes placed as a definite intermediate step with “therefore” or “so that,” sometimes they are additions to the statement of premisses, sometimes phrases introduced by “since,” “for” and the like, after the inference.

Lastly, there is a very large class of additions of a word, or one or two words, for the sake of clearness or consistency. Heiberg gives a number of examples of the addition of such nouns as “triangle,” “square,” “rectangle,” “magnitude,” “number,” “point,” “side,” “circle,” “straight line,” “area” and the like, of adjectives such as “remaining,” “right,” “whole,” “proportional,” and of other parts of speech, even down to words like “is” (esti) which is added 600 times, dê, ara, men, gar, kai and the like.

IV. Omissions by Theon.

Heiberg remarks that, Theon's object having been, as above shown, to amplify and explain Euclid, we should not natuially have expected to find him doing much in the contrary process of compression, and it is only owing to the recurrence of a certain sort of omissions so frequently (especially in the first Books) as to exclude the hypothesis of their being all due to chance that we are bound to credit him with alterations making for greater brevity. We have seen, it is true, that he made omissions as well as additions for the purpose of reducing the language to a certain standard form. But there are also a good number of cases where in the enunciation of propositions, and in the exposition (the re-statement of them with reference to the figure), he has left out words because, apparently, he regarded Euclid's language as being too careful and precise. [p. 57] Again, he is apparently responsible for the frequent omission of the words hoper edei deixai (or poiêsai), Q.E.D. (or F.), at the end of propositions. This is often the case at the end of porisms, where, in omitting the words, Theon seems to have deliberately departed from Euclid's practice. The MS. P seems to show clearly that, where Euclid put a porism at the end of a proposition, he omitted the Q.E.D. at the end of the proposition but inserted it at the end of the porism, as if he regarded the latter as being actually a part of the proposition itself. As in the Theonine MSS. the Q.E.D. is generally omitted, the omission would seem to have been due to Theon. Sometimes in these cases the Q.E.D. is interpolated at the end of the proposition.

Heiberg summed up the discussion of Theon's edition by the remark that Theon evidently took no pains to discover and restore from MSS. the actual words which Euclid had written, but aimed much more at removing difficulties that might be felt by learners in studying the book. His edition is therefore not to be compared with the editions of the Alexandrine grammarians, but rather with the work done by Eutocius in editing Apollonius and with an interpolated recension of some of the works of Archimedes by a certain Byzantine, Theon occupying a position midway between these two editors, being superior to the latter in mathematical knowledge but behind Eutocius in industry (these views now require to be somewhat modified, as above stated). But however little Theon's object may be approved by those of us who would rather know the ipsissima verba of Euclid, there is no doubt that his work was approved by his pupils at Alexandria for whom it was written; and his edition was almost exclusively used by later Greeks, with the result that the more ancient text is only preserved to us in one MS.

As the result of the above investigation, we may feel satisfied that, where P and the Theonine MSS. agree, they give us (except in a few accidental instances) Euclid as he was read by the Greeks of the 4th c. But even at that time the text had been passed from hand to hand through more than six centuries, so that it is certain that it had already suffered changes, due partly to the fault of copyists and partly to the interpolations of mathematicians. Some errors of copyists escaped Theon and were corrected in some MSS. by later hands. Others appear in all our MSS. and, as they cannot have arisen accidentally in all, we must put them down to a common source more ancient than Theon. A somewhat serious instance is to be found in III. 8; and the use of haptesthô for ephaptesthô in the sense of “touch” may also be mentioned, the proper distinction between the words having been ignored as it was by Theon also. But there are a number of imperfections in the ante-Theonine text which it would be unsafe to put down to the errors of copyists, those namely where the good MSS. agree and it is not possible to see any motive that a copyist could have had for altering a correct reading. In these cases it is possible that the imperfections are due to a certain degree of carelessness on the part of Euclid himself; for it [p. 58] is not possible “Euclidem ab omni naevo vindicare,” to use the words of Saccheri¹⁰ , and consequently Simson is not right in attributing to Theon and other editors all the things in Euclid to which mathematical objection can be taken. Thus, when Euclid speaks of “the ratio compounded of the sides” for “the ratio compounded of the ratios of the sides,” there is no reason for doubting that Euclid himself is responsible for the more slip-shod expression. Again, in the Books XI.--XIII. relating to solid geometry there are blots neither few nor altogether unimportant which can only be attributed to Euclid himself¹¹ ; and there is the less reason for hesitation in so attributing them because solid geometry was then being treated in a thoroughly systematic manner for the first time. Sometimes the conclusion (sumperasma) of a proposition does not correspond exactly to the enunciation, often it is cut short with the words kai ta hexês “and the rest” (especially from Book X. onwards), and very often in Books VIII., IX. it is omitted. Where all the MSS. agree, there is no ground for hesitating to attribute the abbreviation or omission to Euclid; though, of course, where one or more MSS. have the longer form, it must be retained because this is one of the cases where a copyist has a temptation to abbreviate.

Where the true reading is preserved in one of the Theonine MSS. alone, Heiberg attributes the wrong reading to a mistake which arose before Theon's time, and the right reading of the single MS. to a successful correction.

We now come to the most important question of the Interpolations introduced before Theon's time.

I. Alternative proofs or additional cases.

It is not in itself probable that Euclid would have given two proofs of the same proposition; and the doubt as to the genuineness of the alternatives is increased when we consider the character of some of them and the way in which they are introduced. First of all, we have those of VI. 20 and XII. 17 introduced by “we shall prove this otherwise more readily (procheiroteron)” or that of X. 90 “it is possible to prove more shortly (suntomôteron).” Now it is impossible to suppose that Euclid would have given one proof as that definitely accepted by him and then added another with the express comment that the latter has certain advantages over the former. Had he considered the two proofs and come to this conclusion, he would have inserted the latter in the received text instead of the former. These alternative proofs must therefore have been interpolated. The same argument applies to alternatives introduced with the words “or even thus” (ê kai houtôs), “or even otherwise” (ê kai allôs). Under this head come the alternatives for the last portions of III. 7, 8; and Heiberg also compares the alternatives for parts of III. 31 (that the angle in a semicircle is a right angle) and XIII. 18, and the alternative proof of the lemma after X. 32. The alternatives to X. 105 and 106, [p. 59] again, are condemned by the place in which they occur, namely after an alternative proof to X. 115. The above alternatives being all admitted to be spurious, suspicion must necessarily attach to the few others which are in themselves unobjectionable. Heiberg instances the alternative proofs to III. 9, III. 10, VI. 30, VI. 31 and XI. 22, observing that it is quite comprehensible that any of these might have occurred to a teacher or editor and seemed to him, rightly or wrongly, to be better than the corresponding proofs in Euclid. Curiously enough, Simson adopted the alternatives to III. 9, 10 in preference to the genuine proofs. Since Heiberg's preface was written, his suspicion has been amply confirmed as regards III. 10 by the commentary of an-Nairīzī (ed. Curtze) which shows not only that this alternative is Heron's, but also that the substantive proposition III. 12 in Euclid is also Heron's, having been given by him to supplement III. II which must originally have been enunciated of circles “touching one another” simply, i.e. so as to include the case of external as well as internal contact, though the proof covered the case of internal contact only. “Euclid, in the 11th proposition,” says Heron, “supposed two circles touching one another internally and wrote the proposition on this case, proving what it was required to prove in it. But I will show how it is to be proved if the contact be external.¹² .” This additional proposition of Heron's is by way of adding another case, which brings us to that class of interpolation. It was the practice of Euclid and the ancients to give only one case (generally the most difficult one) and to leave the others to be investigated by the reader for himself. One interpolation of a second case (VI. 27) is due, as we have seen, to Theon. The two extra cases of XI. 23 were manifestly interpolated before Theon's time, for the preliminary distinction of three cases, “(the centre) will either be within the triangle LMN, or on one of the sides, or outside. First let it be within,” is a spurious addition (B and V only). Similarly an unnecessary case is interpolated in III. 11.

II. Lemmas.

Heiberg has unhesitatingly placed in his Appendix to Vol. III. certain lemmas interpolated either by Theon (on X. 13) or later writers (on X. 27, 29, 31, 32, 33, 34, where V only has the lemmas). But we are here concerned with the lemmas found in all the MSS., which however are, for different reasons, necessarily suspected. We will deal with the Book X. lemmas last.

(1) There is an a priori ground of objection to those lemmas which come after the propositions to which they relate and prove properties used in those propositions; for, if genuine, they would be a sign of faulty arrangement such as would not be likely in a systematic work so carefully ordered as the Elements. The lemma to VI. 22 is one of this class, and there is the further objection to it that in VI. 28 Euclid makes an assumption which would equally require a lemma though none is found. The lemma after XII. 4 is open to the further objections that certain altitudes are used but are not drawn in the [p. 60] figure (which is not in the manner of Euclid), and that a peculiar expression “parallelepipedal solids described on (anagraphomena apo) prisms” betrays a hand other than Euclid's. There is an objection on the score of language to the lemma after XIII. 2. The lemmas on XI. 23, XIII. 13, XIII. 18, besides coming after the propositions to which they relate, are not very necessary in themselves and, as regards the lemma to XIII. 13, it is to be noticed that the writer of a gloss in the proposition could not have had it, and the words “as will be proved afterwards” in the text are rightly suspected owing to differences between the MS. readings. The lemma to XII. 2 also, to which Simson raised objection, comes after the proposition; but, if it is rejected, the words “as was proved before” used in XII. 5 and 18, and referring to this lemma, must be struck out.

(2) Reasons of substance are fatal to the lemma before X. 60, which is really assumed in X. 44 and therefore should have appeared there if anywhere, and to the lemma on X. 20, which tries to prove what is already stated in X. Def. 4.

We now come to the remaining lemmas in Book X., eleven in number, which come before the propositions to which they relate and remove difficulties in the way of their demonstration. That before X. 42 introduces a set of propositions with the words “that the said irrational straight lines are uniquely divided ... we will prove after premising the following lemma,” and it is not possible to suppose that these words are due to an interpolator; nor are there any objections to the lemmas before X. 14, 17, 22, 33, 54, except perhaps that they are rather easy. The lemma before X. 10 and X. 10 itself should probably be removed from the Elements; for X. 10 really uses the following proposition X. 11, which is moreover numbered 10 by the first hand in P, and the words in X. 10 referring to the lemma “for we learnt (how to do this)” betray the interpolator. Heiberg gives reason also for rejecting the lemmas before X. 19 and 24 with the words “in any of the aforesaid ways” (omitted in the Theonine MSS.) in the enunciations of X. 19, 24 and in the exposition of X. 20. Lastly, the lemmas before X. 29 may be genuine, though there is an addition to the second of them which is spurious.

Heiberg includes under this heading of interpolated lemmas two which purport to be substantive propositions, XI. 38 and XIII. 6. These must be rejected as spurious for reasons which will be found in detail in my notes on XI. 37 and XIII. 6 respectively. The latter proposition is only quoted once (in XIII. 17); probably the words quoting it (with grammê instead of eutheia) are themselves interpolated, and Euclid thought the fact stated a sufficiently obvious inference from XIII. 1.

III. Porisms (or corollaries).

Most of the porisms in the text are both genuine and necessary; but some are shown by differences in the MSS. not to be so, e.g. those to I. 15 (though Proclus has it), III. 31 and VI. 20 (Por. 2). Sometimes parts of porisms are interpolated. Such are the last few lines in the porisms to IV. 5, VI. 8; the latter addition is proved later by [p. 61] means of VI. 4, 8, so that the writer of these proofs could not have had the addition to VI. 8 Por. before him. Lastly, interpolators have added a sort of proof to some porisms, as though they were not quite obvious enough; but to add a demonstration is inconsistent with the idea of a porism, which, according to Proclus, is a by-product of a proposition appearing without our seeking it.

IV. Scholia.

Several interpolated scholia betray themselves by their wording, e.g. those given by Heiberg in the Appendix to Book X. and containing the words kalei, ekalese (“he calls” or “called”); these scholia were apparently written as marginal notes before Theon's time, and, being adopted as such by Theon, found their way into the text in P and some of the Theonine MSS. The same thing no doubt accounts for the interpolated analyses and syntheses to XIII. 1-5, as to which see my note on XIII. 1.

V. Interpolations in Book X.

First comes the proposition “Let it be proposed to us to show that in square figures the diameter is incommensurable in length with the side,” which, with a scholium after it, ends the tenth Book. The form of the enunciation is suspicious enough and the proposition, the proof of which is indicated by Aristotle and perhaps was Pythagorean, is perfectly unnecessary when X. 9 has preceded. The scholium ends with remarks about commensurable and incommensurable solids, which are of course out of place before the Books on solids. The scholiast on Book X. alludes to this particular scholium as being due to “Theon and some others.” But it is doubtless much more ancient, and may, as Heiberg conjectures have been the beginning of Apollonius' more advanced treatise on incommensurables. Not only is everything in Book X. after X. 115 interpolated, but Heiberg doubts the genuineness even of X. 112-115, on the ground that X. 111 rounds off the theory of incommensurables as we want it in the Books on solid geometry, while X. 112-115 are not really connected with what precedes, nor wanted for the later Books, but seem to form the starting-point of a new and more elaborate theory of irrationals.

VI. Other minor interpolations are found of the same character as those above attributed to Theon. First there are two places (XI. 35 and XI. 26) where, after “similarly we shall prove” and “for the same reason,” an actual proof is nevertheless given. Clearly the proofs are interpolated; and there are other similar interpolations. There are also interpolations of intermediate steps in proofs, unnecessary explanations and so on, as to which I need not enter into details.

Lastly, following Heiberg's order, I come to

VII. Interpolated definitions, axioms etc.

Apart from VI. Def. 5 (which may have been interpolated by Theon although it is found written in the margin of P by the first hand), the definition of a segment of a circle in Book I. is interpolated, as is clear from the fact that it occurs in a more appropriate place in Book III. and Proclus omits it. VI. Def. 2 (reciprocal figures) is rightly condemned by Simson--perhaps it was taken from Heron--and [p. 62] Heiberg would reject VII. Def. 10, as to which see my note on that definition. Lastly the double definition of a solid angle (XI. Def. 11) constitutes a difficulty. The use of the word epiphaneia suggests that the first definition may have been older than Euclid, and he may have quoted it from older elements, especially as his own definition which follows only includes solid angles contained by planes, whereas the other includes other sorts (cf. the words grammôn, grammais) which are also distinguished by Heron (Def. 22). If the first definition had come last, it could have been rejected without hesitation: but it is not so easy to reject the first part up to and including “otherwise” (allôs). No difficulty need be felt about the definitions of “oblong,” “rhombus,” and “rhomboid,” which are not actually used in the Elements; they were no doubt taken from earlier elements and given for the sake of completeness.

As regards the axioms or, as they are called in the text, common notions (koinai ennoiai), it is to be observed that Proclus says¹³ that Apollonius tried to prove “the axioms,” and he gives Apollonius' attempt to prove Axiom I. This shows at all events that Apollonius had some of the axioms now appearing in the text. But how could Apollonius have taken a controversial line against Euclid on the subject of axioms if these axioms had not been Euclid's to his knowledge? And, if they had been interpolated between Euclid's time and his own, how could Apollonius, living so comparatively short a time after Euclid, have been ignorant of the fact? Therefore some of the axioms are Euclid's (whether he called them common notions, or axioms, as is perhaps more likely since Proclus calls them axioms): and we need not hesitate to accept as genuine the first three discussed by Proclus, viz. (1) things equal to the same equal to one another, (2) if equals be added to equals, wholes equal, (3) if equals be subtracted from equals, remainders equal. The other two mentioned by Proclus (whole greater than part, and congruent figures equal) are more doubtful, since they are omitted by Heron, Martianus Capella, and others. The axiom that “two lines cannot enclose a space” is however clearly an interpolation due to the fact that I. 4 appeared to require it. The others about equals added to unequals, doubles of the same thing, and halves of the same thing are also interpolated; they are connected with other interpolations, and Proclus clearly used some source which did not contain them.

Euclid evidently limited his formal axioms to those which seemed to him most essential and of the widest application; for he not unfrequently assumes other things as axiomatic, e.g. in VII. 28 that, if a number measures two numbers, it measures their difference.

The differences of reading appearing in Proclus suggest the question of the comparative purity of the sources used by Proclus, Heron and others, and of our text. The omission of the definition of a segment in Book I. and of the old gloss “which is called the circumference” in I. Def. 15 (also omitted by Heron, Taurus, Sextus [p. 63] Empiricus and others) indicates that Proclus had better sources than we have; and Heiberg gives other cases where Proclus omits words which are in all our MSS. and where Proclus' reading should perhaps be preferred. But, except in these instances (where Proclus may have drawn from some ancient source such as one of the older commentaries), Proclus' MS. does not seem to have been among the best. Often it agrees with our worst MSS., sometimes it agrees with F where F alone has a certain reading in the text, so that (e.g. in I. 15 Por.) the common reading of Proclus and F must be rejected, thrice only does it agree with P alone, sometimes it agrees with P and some Theonine MSS., and once it agrees with the Theonine MSS. against P and other sources.

Of the other external sources, those which are older than Theon generally agree with our best MSS., e.g. Heron, allowing for the difference in the plan of his definitions and the somewhat free adaptation to his purpose of the Euclidean definitions in Books X., XI.

Heiberg concludes that the Elements were most spoiled by interpolations about the 3rd c., for Sextus Empiricus had a correct text, while Iamblichus had an interpolated one; but doubtless the purer text continued for a long time in circulation, as we conclude from the fact that our MSS. are free from interpolations already found in Iamblichus' MS.

¹ The material for the whole of this chapter is taken from Heiberg's edition of the Elements, introduction to vol. v., and from the same scholar's Litterargeschichtliche Studien über Euklid, p. 174 sqq. and Paralipomena zu Euklid in Hermes, XXXVIII., 1903.

² I. p. 201 ed. Halma=p. 50 ed. Basel.

³ See Pauly-Wissowa, Real-Encyclopädie der class. Altertumswissenschaft, vol. II., 1896, p. 675.

⁴ Heiberg, vol. v. pp. xxix--xxxiii.

⁵ Zeitschrift fiir Math. u. Physik, XXIX., hist.-litt. Abtheilung, p. 13.

⁶ [eis t]a tou Eukleidou stoicheia prolambanomena ek tôn Proklou sporadên kai kat epitomên. Cf. p. 32, note 8, above.

⁷ Heiberg, Paralipomena zu Euklid in Hermes, XXXVIII., 1903, pp. 46-74, 161-201, 321-356.

⁸ Described by Heiberg in Oversigt over det kngl. danske Videnskabernes Selskabs Forhandlinger, 1900, p. 161.

⁹ Exhaustive details under all the different heads are given by Heiberg (Vol. v. pp. lii--lxxv).

¹⁰ Euclides ab omni naevo vindicatus, Mediolani, 1733.

¹¹ Cf. especially the assumption, without proof or definition, of the criterion for equal solid angles, and the incomplete proof of XII. 17.

¹² An-Nairīzī, ed. Curtze, p. 121.

¹³ Proclus, pp. 194, 10 sqq.

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