Euclid, Elements (ed. Thomas L. Heath)

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[p. 64]

CHAPTER VI.

THE SCHOLIA.

Heiberg has collected scholia, to the number of about 1500, in Vol. v. of his edition of Euclid, and has also discussed and classified them in a separate short treatise, in which he added a few others.¹

These scholia cannot be regarded as doing much to facilitate the reading of the Elements. As a rule, they contain only such observations as any intelligent reader could make for himself. Among the few exceptions are XI. Nos. 33, 35 (where XI. 22, 23 are extended to solid angles formed by any number of plane angles), XII. No. 85 (where an assumption tacitly made by Euclid in XII. 17 is proved), IX. Nos. 28, 29 (where the scholiast has pointed out the error in the text of IX. 19).

Nor are they very rich in historical information; they cannot be compared in this respect with Proclus' commentary on Book I. or with those of Eutocius on Archimedes and Apollonius. But even under this head they contain some things of interest, e.g. II. No. 11 explaining that the gnomon was invented by geometers for the sake of brevity, and that its name was suggested by an incidental characteristic, namely that “from it the whole is known (gnôrizetai), either of the whole area or of the remainder, when it (the gnômôn) is either placed round or taken away”; II. No. 13, also on the gnomon; IV. No. 2 stating that Book IV. was the discovery of the Pythagoreans; V. No. 1 attributing the content of Book V. to Eudoxus; X. No. 1 with its allusion to the discovery of incommensurability by the Pythagoreans and to Apollonius' work on irrationals; X. No. 62 definitely attributing X. 9 to Theaetetus; XIII. No. 1 about the “Platonic” figures, which attributes the cube, the pyramid, and the dodecahedron to the Pythagoreans, and the octahedron and icosahedron to Theaetetus.

Sometimes the scholia are useful in connexion with the settlement of the text, (1) directly, e.g. III. No. 16 on the interpolation of the word “within” (entos) in the enunciation of III. 6, and X. No. 1 alluding to the discussion by “Theon and some others” of irrational “surfaces” and “solids,” as well as “lines,” from which we may [p. 65] conclude that the scholium at the end of Book X. is not genuine; (2) indirectly in that they sometimes throw light on the connexion of certain MSS.

Lastly, they have their historical importance as enabling us to judge of the state of mathematical science at the times when they were written.

Before passing to the classification of the scholia, Heiberg remarks that we must separate from them a number of additions in the nature of scholia which are found in the text of our MSS. but which can, in one way or another, be proved to be spurious. As they are found both in P and in the Theonine MSS., they must have been in the MSS. anterior to Theon (4th c.). But they are, in great part, only found in the margin of P and the Theonine MSS.; in V they are half in the text and half in the margin. This can hardly be explained except on the supposition that these additions were originally (in the MSS. before Theon) in the margin, and that Theon kept them there in his edition, but that they afterwards found their way gradually into the text of P as well as of the Theonine MSS., or were omitted altogether, while particular MSS. have in certain places preserved the old arrangement. Of such spurious additions Heiberg enumerates the following: the axiom about equals subtracted from unequals, the last lines of the porism to VI. 8, second porisms to V. 19 and to VI. 20, the porism to III. 31, VI. Def. 5, various additions in Book X., the analyses and syntheses of XIII. 1-5, and the proposition XIII. 6.

The two first classes of scholia distinguished by Heiberg are denoted by the convenient abbreviations “Schol. Vat.” and “Schol. Vind.”

I. Schol. Vat.

It is first necessary to set out the letters by which Heiberg denotes certain collections of scholia.

P=Scholia in P written by the first hand.

B=Scholia in B by a hand of the same date as the MS. itself, generally that of Arethas.

F=Scholia in F by the first hand.

Vat.=Scholia of the Vatican MS. 204 of the 10th c., which has these scholia on leaves 198-205 (the end is missing) as an independent collection. It does not contain the text of the Elements.

V^{c}=Scholia found on leaves 283-292 of V and written in the same hand as that part of the MS. itself which begins at leaf 235.

Vat. 192=a Vatican MS. of the 14th c. which contains, after (1) the Elements I.--XIII. (without scholia), (2) the Data with scholia, (3) Marinus on the Data, the Schol. Vat. as an independent collection and in their entirety, beginning with 1. No. 88 and ending with XIII. No. 44.

The Schol. Vat., the most ancient and important collection of scholia, comprise those which are found in PBF Vat. and, from VII. 12 to IX. 15, in PB Vat. only, since in that portion of the Elements F was restored by a later hand without scholia; they also include 1. [p. 66] No. 88 which only happens to be erased in F, and IX. Nos. 28, 29 which may be left out because F. here has a different text. In F and Vat. the collection ends with Book X.; but it must also include Schol. PB of Books XI.--XIII., since these are found along with Schol. Vat. to Books I.--X. in several MSS. (of which Vat. 192 is one) as a separate collection. The Schol. Vat. to Books X.--XIII. are also found in the collection V^{c} (where, curiously enough, XIII. Nos. 43, 44 are at the beginning). The Schol. Vat. accordingly include Schol. PBV^{c} Vat. 192, and doubtless also those which are found in two of these sources. The total number of scholia classified by Heiberg as Schol. Vat. is 138.

As regards the contents of Schol. Vat. Heiberg has the following observations. The thirteen scholia to Book I. are extracts made from Proclus by a writer thoroughly conversant with the subject, and cleverly recast (with some additions). Their author does not seem to have had the two lacunae which our text of Proclus has (at the end of the note on I. 36 and the beginning of the next note, and at the beginning of the note on I. 43), for the scholia I. Nos. 125 and 137 seem to fill the gaps appropriately, at least in part. In some passages he had better readings than our MSS. have. The rest of Schol. Vat. (on Books II.--XIII.) are essentially of the same character as those on Book I., containing prolegomena, remarks on the object of the propositions, critical remarks on the text, converses, lemmas; they are, in general, exact and true to tradition. The reason of the resemblance between them and Proclus appears to be due to the fact that they have their origin in the commentary of Pappus, of which we know that Proclus also made use. In support of the view that Pappus is the source, heiberg places some of the Schol. Vat. to Book X. side by side with passages from the commentary of Pappus in the Arabic translation discovered by Woepcke;² ; he also refers to the striking confirmation afforded by the fact that XII. No. 2 contains the solution of the problem of inscribing in a given circle a polygon similar to a polygon inscribed in another circle, which problem Eutocius says³ that Pappus gave in his commentary on the Elements.

But, on the other hand, Schol. Vat. contain some things which cannot have come from Pappus, e.g. the allusion in X. No. 1 to Theon and irrational surfaces and solids, Theon being later than Pappus; III. No. 10 about porisms is more like Proclus' treatment of the subject than Pappus', though one expression recalls that of Pappus about forming (schêmatizesthai) the enunciations of porisms like those of either theorems or problems.

The Schol. Vat. give us important indications as regards the text of the Elements as Pappus had it. In particular, they show that he could not have had in his text certain of the lemmas in Book X. For example, three of these are identical with what we find in Schol. [p. 67] Vat. (the lemma to X. 17=Schol. X. No. 106, and the lemmas to X. 54, 60 come in Schol. X. No. 328); and it is not possible to suppose. that these lemmas, if they were already in the text, would also be given as scholia. Of these three lemmas, that before X. 60 has already been condemned for other reasons; the other two, unobjectionable in themselves, must be rejected on the ground now stated. There were four others against which Heiberg found nothing to urge when writing his prolegomena to Vol. v., viz. the lemmas before X. 42, X. 14, X. 22 and X. 33. Of these, the lemma to X. 22 is not reconcilable with Schol. X. No. 161, which takes up the assumption in the text of Eucl. X. 22 as if no lemma had gone before. The lemma to X. 42, which, on account of the words introducing it (see p. 60 above), Heiberg at first hesitated to regard as an interpolation, is identical with Schol. X. No. 270. It is true that in Schol. X. No. 269 we find the words “this lemma has been proved before (en tois emprosthen), but it shall also be proved now for convenience' sake (tou hetoimou heneka),” and it is possible to suppose that “before” may mean in Euclid's text before X. 42; but a proof in that place would surely have been as “convenient” as could be desired, and it is therefore more probable that the proof had been given by Pappus in some earlier place. (It may be added that the lemma to X. 14, which is identical with the lemma to XI. 23, condemned on other grounds, is for that reason open to suspicion.)

Heiberg's conclusion is that all the lemmas are spurious, and that most or all of them have found their way into the text from Pappus' commentary, though at a time anterior to Theon's edition, since they are found in all our MSS. This enables us to fix a date for these interpolations, namely the first half of the 4th c.

Of course Pappus had not in his text the interpolations which, from the fact of their appearing only in some of our MSS., are seen to be later than those above-mentioned. Such are the lemmas which are found in the text of V only after X. 29 and X. 31 respectively and are given in Heiberg's Appendix to Book X. (numbered 10 and 11). On the other hand it appears from Woepcke's tract⁴ that Pappus already had X. 115 in his text: though it does not follow from this that the proposition is genuine but only that interpolations began very early.

Theon interpolated a proposition (or lemma) between X. 12 and X. 13 (No. 5 in Heiberg's Appendix). Schol. Vat. has the same thing (X. No. 125). The writer of the scholia therefore did not find this lemma in the text. Schol. Vat. IX. Nos. 28, 29 show that neither did he find in his text the alterations which Theon made in Eucl. IX. 19; the scholia in fact only agree with the text of P, not with Theon's. This suggests that Schol. Vat. were written for use with a MS. of the ante-Theonine recension such as P is. This probability is further confirmed by a certain independence which P shows in several places when compared with the Theonine MSS. Not only has P better readings in some passages, but more substantial divergences; and, [p. 68] in particular, the absence in P of three notes of a historical character which are added, wholly or partly from Proclus, in the Theonine MSS. attests an independent and more primitive point of view in P.

In view of the distinctive character of P, it is possible that some of the scholia found in it in the first hand, but not in the other sources of Schol. Vat., also belong to that collection; and several circumstances confirm this. Schol. XIII. No. 45, found in P only, which relates to a passage in Eucl. XIII. 13, shows that certain words in the text, though older than Theon, are interpolated; and, as the scholium is itself older than Theon, is headed “third lemma,” and follows a “second lemma” relating to a passage in the text immediately preceding, which “second lemma” belongs to Schol. Vat. and is taken from Pappus, the “third” in all probability came from Pappus also. The same is true of Schol. XII. No. 72 and XIII. No. 69, which are respectively identical with the propositions vulgo XI. 38 (Heiberg, App. to Book XI., No. 3) and XIII. 6; for both of these interpolations are older than Theon. Moreover most of the scholia which P in the first hand alone has are of the same character as Schol. Vat. Thus VII. No. 7 and XIII. No. I introducing Books VII. and XIII. respectively are of the same historical character as several of Schol. Vat.; that VII. No. 7 appears in the text of P at the beginning of Book VII. constitutes no difficulty. There are a number of converses, remarks on the relation of propositions to one another, explanations such as XII. No. 89 in which it is remarked that Ph, Ô in Euclid's figure to XII. 17 (Z, V in my figure) are really the same point but that this makes no difference in the proof. Two other Schol. P on XII. 17 are connected by their headings with XII. No. 72 mentioned above. XI. No. 10 (P) is only another form of XI. No. 11 (B); and B often, alone with P, has preserved Schol. Vat. On the whole Heiberg considers some 40 scholia found in P alone to belong to Schol. Vat.

The history of Schol. Vat. appears to have been, in its main outlines, the following. They were put together after 500 A.D., since they contain extracts from Proclus, to which we ought not to assign a date too near to that of Proclus' work itself; and they must at least be earlier than the latter half of the 9th c., in which B was written. As there must evidently have been several intermediate links between the archetype and B, we must assign them rather to the first half of the period between the two dates, and it is not improbable that they were a new product of the great development of mathematical studies at the end of the 6th c. (Isidorus of Miletus). The author extracted what he found of interest in the commentary of Proclus on Book I. and in that of Pappus on the rest of the work, and put these extracts in the margin of a MS. of the class of P. As there are no scholia to I. 1-22, the first leaves of the archetype or of one of the earliest copies must have been lost at an early date, and it was from that mutilated copy that partly P and partly a MS. of the Theonine class were taken, the scholia being put in the margin in both. Then the collection spread through the Theonine MSS., gradually losing some [p. 69] scholia which could not be read or understood, or which were accidentally or deliberately omitted. Next it was extracted from one of these MSS. and made into a separate work which has been preserved, in part, in its entirety (Vat. 192 etc.) and, in part, divided into sections, so that the scholia to Books X.--XIII. were detached (V^{c}). It had the same fate in the MSS. which kept the original arrangement (in the margin), and in consequence there are some MSS. where the scholia to the stereometric Books are missing, those Books having come to be less read in the period of decadence. It is from one of these MSS. that the collection was extracted as a separate work such as we find it in Vat. (10th c.).

II. The second great division of the scholia is Schol. Vind.

This title is taken from the Viennese MS. (V), and the letters used by Heiberg to indicate the sources here in question are as follows.

V^{a}=scholia in V written by the same hand that copied the MS. itself from fol. 235 onward.

q=scholia of the Paris MS. 2344 (q) written by the first hand.

l=scholia of the Florence MS. Laurent. XXVIII, 2 written in the 13th--14th c., mostly in the first hand, but partly in two later hands.

V^{b}=scholia in V written by the same hand as the first part (leaves 1-183) of the MS. itself; V^{b} wrote his scholia after V^{a}.

q^{1}=scholia of the Paris MS. (q) found here and there in another hand of early date.

Schol. Vind. include scholia found in V^{a}q. 1 is nearly related to q; and in fact the three MSS. which, so far as Euclid's text is concerned, show no direct interdependence, are, as regards their scholia, derived from one original. Heiberg proves this by reference to the readings of the three in two passages (found in Schol. 1. No. 109 and X. No. 39 respectively). The common source must have contained, besides the scholia found in the three MSS. V^{a}ql, those also which are contained in two of them, for it is more unlikely that two of the three should contain common interpolations than that a particular scholium should drop out of one of them. Besides V^{a} and q, the scholia V^{b} and q^{1} must equally be referred to Schol. Vind., since the greater part of their scholia are found in 1. There is a lacuna in q from Eucl. VIII. 25 to IX. 14, so that for this portion of the Elements Schol. Vind. are represented by Vl only. Heiberg gives about 450 numbers in all as belonging to this collection.

Schol. Vind. did not all come from one source; this is shown by differences of substance, e.g. between X. Nos. 36 and 39, and by differences of time of writing: e.g. VI. No. 52 refers at the beginning to No. 55 with the words “as the scholium has it” and is therefore later than that scholium; X. No. 247 is also later than X. No. 246.

The scholia to Book I. are here also extracts from Proclus, but more copious and more verbatim than in Schol. Vat. The author has not always understood Proclus; and he had a text as bad as that of our MSS., with the same lacunae. The scholia to the other [p. 70] Books are partly drawn (1) from Schol. Vat., the MSS. representing Schol. Vind. and Schol. Vat. in these cases showing nearly all possible combinations; but there is no certain trace in Schol. Vind. of the scholia peculiar to P. The author used a copy of Schol. Vat. in the form in which they were attached to the Theonine text; thus Schol. Vind. correspond to BF Vat., where these diverge from P, and especially closely to B. Besides Schol. Vat., the editors of Schol. Vind. used (2) other old collections of scholia of which we find traces in B and F; Schol. Vind. have also some scholia common with b. The scholia which Schol. Vind. have in common with BF come from two different sources, and were apparently afterwards introduced into the other MSS.; one result of this is that several scholia are reproduced twice.

But, besides the scholia derived from these sources, Schol. Vind. contain a large number of others of late date, characterised by incorrect language or by triviality of content (there are many examples in numbers, citations of propositions used, absurd aporiai, and the like). Unlike Schol. Vat., these scholia often quote words from Euclid as a heading (in one case a heading is inserted in Schol. Vind. where a scholium without the heading is quoted from Schol. Vat., see V. No. 14). The explanations given often presuppose very little knowledge on the part of the reader and frequently contain obscurities and gross errors.

Schol. Vind. were collected for use with a MS. of the Theonine class; this follows from the fact that they contain a note on the proposition vulgo VII. 22 interpolated by Theon (given in Heiberg's App. to Vol. 11. p. 430). Since the scholium to VII. 39 given in V and p in the text after the title of Book VIII. quotes the proposition as VII. 39, it follows that this scholium must have been written before the interpolation of the two propositions vulgo VII. 20, 22; Schol. Vind. contain (VII. No. 80) the first sentence of it, but without the heading referring to VII. 39. Schol. VII. No. 97 quotes VII. 33 as VII. 34, so that the proposition vulgo VII. 22 may have stood in the scholiast's text but not the later interpolation vulgo VII. 20 (later because only found in B in the margin by the first hand). Of course the scholiast had also the interpolations earlier than Theon.

For the date of the collection we have a lower limit in the date (12th c.) of MSS. in which the scholia appear. That it was not much earlier than the 12th c. is indicated (1) by the poverty of its contents, (2) by the quality of the MS. of Proclus which was used in the compilation of it (the Munich MS. used by Friedlein with which the scholiast's excerpts are essentially in agreement belongs to the 11th-- 12th c.), (3) by the fact that Schol. Vind. appear only in MSS. of the 12th c. and no trace of them is found in our MSS. belonging to the 9th--10th c. in which Schol. Vat. are found. The collection may therefore probably be assigned to the 11th c. Perhaps it may be in part due to Psellus who lived towards the end of that century: for in a Florence MS. (Magliabecch. XI, 53 of the 15th c.) containing a mathematical compendium intended for use in the reading of Aristotle [p. 71] the scholia 1. Nos. 40 and 49 appear with the name of Psellus attached.

Schol. Vind. are not found without the admixture of foreign elements in any of our three sources. In 1 there are only very few such in the first hand. In q there are several new scholia in the first hand, for the most part due to the copyist himself. The collection of scholia on Book X. in q (Heiberg's q^{c}) is also in the first hand; it is not original, and it may perhaps be due to Psellus (Maglb. has some definitions of Book X. with a heading “scholia of...Michael Psellus on the definitions of Euclid's 10th Element” and Schol. X. No. 9), whose name must have been attached to it in the common source of Maglb. and q; to a great extent it consists of extracts from Schol. Vind. taken from the same source as Vl. The scholia q^{1} (in an ancient hand in q), confined to Book II., partly belong to Schol. Vind. and partly correspond to b^{1} (Bologna MS.). q^{a} and q^{b} are in one hand (Theodorus Antiochita), the nearest to the first hand of q; they are doubtless due to an early possessor of the MS. of whom we know nothing more.

V^{a} has, besides Schol. Vind., a number of scholia which also appear in other MSS., one in BFb, some others in P, and some in v (Codex Vat. 1038, 13th c.); these scholia were taken from a source in which many abbreviations were used, as they were often misunderstood by V^{a}. Other scholia in V^{a} which are not found in the older sources--some appearing in V^{a} alone--are also not original, as is proved by mistakes or corruptions which they contain; some others may be due to the copyist himself.

V^{b} seldom has scholia common with the other older sources; for the most part they either appear in V^{b} alone or only in the later sources as v or F^{2} (later scholia in F), some being original, others not. In Book X. V^{b} has three series of numerical examples, (1) with Greek numerals, (2) alternatives added later, also mostly with Greek numerals, (3) with Arabic numerals. The last class were probably the work of the copyist himself. These examples (cf. p. 74 below) show the facility with which the Byzantines made calculations at the date of the MS. (12th c.). They prove also that the use of the Arabic numerals (in the East-Arabian form) was thoroughly established in the 12th c.; they were actually known to the Byzantines a century earlier, since they appear, in the first hand, in an Escurial MS. of the 11th c.

Of collections in other hands in V distinguished by Heiberg (see preface to Vol. v.), V^{1} has very few scholia which are found in other sources, the greater part being original; V^{2}, V^{3} are the work of the copyist himself; V^{4} are so in part only, and contain several scholia from Schol. Vat. and other sources. V^{3} and V^{4} are later than 13th --14th c., since they are not found in f (cod. Laurent. XXVIII, 6) which was copied from V and contains, besides V^{a} V^{b}, the greater part of V^{1} and VI. No. 20 of V^{2} (in the text).

In P there are, besides P^{3} (a quite late hand, probably one of the old Scriptores Graeci at the Vatican), two late hands (P^{2}), one of which has some new and independent scholia, while the other has [p. 72] added the greater part of Schol. Vind., partly in the margin and partly on pieces of leaves stitched on.

Our sources for Schol. Vat. also contain other elements. In P there were introduced a certain number of extracts from Proclus, to supplement Schol. Vat. to Book I.; they are all written with a different ink from that used for the oldest part of the MS., and the text is inferior. There are additions in the other sources of Schol. Vat. (F and B) which point to a common source for FB and which are nearly all found in other MSS., and, in particular, in Schol. Vind., which also used the same source; that they are not assignable to Schol. Vat. results only from their not being found in Vat. Of other additions in F, some are peculiar to F and some common to it and b; but they are not original. F^{2} (scholia in a later hand in F) contains three original scholia; the rest come from V. B contains, besides scholia common to it and F, b or other sources, several scholia which seem to have been put together by Arethas, who wrote at least a part of them with his own hand.

Heiberg has satisfied himself, by a closer study of b, that the scholia which he denotes by b, b and b^{1} are by one hand; they are mostly to be found in other sources as well, though some are original. By the same hand (Theodorus Cabasilas, 15th c.) are also the scholia denoted by b^{2}, B^{2}, b^{3} and B^{3}. These scholia come in great part from Schol. Vind., and in making these extracts Theodorus probably used one of our sources, l, mistakes in which often correspond to those of Theodorus. To one scholium is attached the name of Demetrius (who must be Demetrius Cydonius, a friend of Nicolaus Cabasilas, 14th c.); but it could not have been written by him, since it appears in B and Schol. Vind. Nor are all the scholia which bear the name of Theodorus due to Theodorus himself, though some are so.

As B^{3} (a late hand in B) contains several of the original scholia of b^{2}, B^{3} must have used b itself as his source, and, as all the scholia in B^{3} are in b, the latter is also the source of the scholia in B^{3} which are found in other MSS. B and b were therefore, in the 15th c., in the hands of the same person; this explains, too, the fact that b in a late hand has some scholia which can only come from B. We arrive then at the conclusion that Theodorus Cabasilas, in the 15th c., owned both the MSS. B and b, and that he transferred to B scholia which he had before written in b, either independently or after other sources, and inversely transferred some scholia from B to b. Further, B^{2} are earlier than Theodorus Cabasilas, who certainly himself wrote B^{3} as well as b^{2} and b^{3}.

An author's name is also attached to the scholia VI. No. 6 and X. No. 223, which are attributed to Maximus Planudes (end of 13th c.) along with scholia on I. 31, X. 14 and X. 18 found in 1 in a quite late hand and published on pp. 46, 47 of Heibėrg's dissertation. These seem to have been taken from lectures of Planudes on the Elements by a pupil who used l as his copy.

There are also in l two other Byzantine scholia, written by a late hand, and bearing the names Ioannes and Pediasimus respectively; [p. 73] these must in like manner have been written by a pupil after lectures of Ioannes Pediasimus (first half of 14th c.), and this pupil must also have used l.

Before these scholia were edited by Heiberg, very few of them had been published in the original Greek. The Basel editio princeps has a few (V. No. I, VI. Nos. 3, 4 and some in Book X.) which are taken, some from the Paris MS. (Paris. Gr. 2343) used by Grynaeus, others probably from the Venice MS. (Marc. 301) also used by him; one published by Heiberg, not in his edition of Euclid but in his paper on the scholia, may also be from Venet. 301, but appears also in Paris. Gr. 2342. The scholia in the Basel edition passed into the Oxford edition in the text, and were also given by August in the Appendix to his Vol. II.

Several specimens of the two series of scholia (Vat. and Vind.) were published by C. Wachsmuth (Rhein. Mus. XVIII. p. 132 sqq.) and by Knoche (Untersuchungen über die neu aufgefundenen Scholien des Proklus, Herford, 1865).

The scholia published in Latin were much more numerous. G. Valla (De expetendis et fugiendis rebus, 1501) reproduced apparently some 200 of the scholia included in Heiberg's edition. Several of these he obtained from two Modena MSS. which at one time were in his possession (Mutin. III B, 4 and II E, 9, both of the 15th c.); but he must have used another source as well, containing extracts from other series of scholia, notably Schol. Vind. with which he has some 87 scholia in common. He has also several that are new.

Commandinus included in his translation under the title “Scholia antiqua” the greater part of the Schol. Vat. which he certainly obtained from a MS. of the class of Vat. 192; on the whole he adhered closely to the Greek text. Besides these scholia Commandinus has the scholia and lemmas which he found in the Basel editio princeps, and also three other scholia not belonging to Schol. Vat., as well as one new scholium (to XII. 13) not included in Heiberg's edition, which are distinguished by different type and were doubtless taken from the Greek MS. used by him along with the Basel edition.

In Conrad Dasypodius' Lexicon mathematicum published in 1573 there is (on fol. 42-44) “Graecum scholion in definitiones Euclidis libri quinti elementorum appendicis loco propter pagellas vacantes annexum.” This contains four scholia, and part of two others, published in Heiberg's edition, with some variations of readings, and with some new matter added (for which see pp. 64-6 of Heiberg's pamphlet). The source of these scholia is revealed to us by another work of Dasypodius, Isaaci Monachi Scholia in Euclidis elementorum geometriae sex priores libros per C. Dasypodium in latinum sermonem translata et in lucem edita (1579). This work contains, besides excerpts from Proclus on Book I. (in part closely related to Schol. Vind.), some 30 scholia included in Heiberg's edition, several new scholia, and the above-mentioned scholia to the definitions of Book V. published in Greek in 1573. After the scholia follow “Isaaci Monachi [p. 74] prolegomena in Euclidis Elementorum geometriae libros” (two definitions of geometry) and “Varia miscellanea ad geometriae cognitionem necessaria ab Isaaco Monacho collecta” (mostly the same as pp. 252, 24-272, 27 in the Variae Collectiones included in Hultsch's Heron); lastly, a note of Dasypodius to the reader says that these scholia were taken “ex clarissimi viri Joannis Sambuci antiquo codice manu propria Isaaci Monachi scripto.” Isaak Monachus is doubtless Isaak Argyrus, 14th c.; and Dasypodius used a MS. in which, besides the passage in Hultsch's Variae Collectiones, there were a number of scholia marked in the margin with the name of Isaak (cf. those in b under the name of Theodorus Cabasilas). Whether the new scholia are original cannot be decided until they are published in Greek; but it is not improbable that they are at all events independent arrangements of older scholia. All but five of the others, and all but one of the Greek scholia to Book v., are taken from Schol. Vat.; three of the excepted ones are from Schol. Vind., and the other three seem to come from F (where some words of them are illegible, but can be supplied by means of Mut. III B, 4, which has these three scholia and generally shows a certain likeness to Isaak's scholia).

Dasypodius also published in 1564 the arithmetical commentary of Barlaam the monk (14th c.) on Eucl. Book II., which finds a place in Appendix IV. to the Scholia in Heiberg's edition.

Hultsch has some remarks on the origin of the scholia⁵ . He observes that the scholia to Book I. contain a considerable portion of Geminus' commentary on the definitions and are specially valuable because they contain extracts from Geminus only, whereas Proclus, though drawing mainly upon him, quotes from others as well. On the postulates and axioms the scholia give more than is found in Proclus. Hultsch conjectures that the scholium on Book v., No. 3, attributing the discovery of the theorems to Eudoxus but their arrangement to Euclid, represents the tradition going back to Geminus, and that the scholium XIII., No. 1, has the same origin.

A word should be added about the numerical illustrations of Euclid's propositions in the scholia to Book X. They contain a large number of calculations with sexagesimal fractions⁶ ; the fractions go as far as fourth-sixtieths (1/60^{4}). Numbers expressed in these fractions are handled with skill and include some results of surprising accuracy⁷

¹ Heiberg, Om Scholierne til Euklids Elementer, Kjøbenhavn, 1888. The tract is written in Danish, but, fortunately for those who do not read Danish easily, the author has appended (pp. 70-78) a résumé in French.

² Om Scholierne til Euklids Elementer, pp. 11, 12: cf. Euklid-Stulien, pp. 170, 171; Woepcke, Mémoires présent. à l' Acad. des Sciences, 1856, XIV. p. 658 sqq.

³ Archimedes, ed. Heiberg, III. P. 28, 19-22.

⁴ Woepcke, op. cit. p. 702.

⁵ Art. “Eukleides” in Pauly-Wissowa's Real-Encyclopädie.

⁶ Hultsch has written upon these in Bibliotheca Mathematica, V_{3}, 1904, pp. 225-233.

⁷ Thus \sgrt{(27)} is given (allowing for a slight correction by means of the context) as 5 II' 46'' 10''', which gives for \sgrt{3} the value 1 43' 55'' 23''', being the same value as that given by Hipparchus in his Table of Chords, and correct to the seventh decimal place. Similarly \sgrt{8} is given as 2 49' 42'' 20''' 10'''', which is equivalent to\sgrt{2}=1.41421335. Hultsch gives instances of the various operations, addition, subtraction, etc., carried out in these fractions, and shows how the extraction of the square root was effected. Cf. T. L. Heath, Históry of Greek Mathematics, 1., pp. 59-63.

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