Heron was earlier than Claudius Ptolemy (about 100-178 A.D.), and (2) from an apparent reference by Plutarch to a proposition about incidence and reflexion taking place at equal angles, proved by Heron in his Catoptrica, coupled with the facts that in that work Heron mentions Menelaus of Alexandria (about 100 A.D.) and that Plutarch died at a great age in 120 A.D.

Attempts have however been made in two recent tracts to overthrow almost the whole of these arguments1. (1) It is asserted that the olive-press of Mechanics III. 20 is not the same as that referred to by Pliny. (2) It is pointed out that Heron is mentioned with Archimedes and Ctesibius in a passage of Proclus which is supposed to be drawn from Geminus. But, as Geminus wrote about 70 B.C. and Posidonius not earlier than 90 B.C., while Heron quotes Posidonius and is therefore later, the intervals are all too short to make it probable that Heron would be mentioned in Geminus' historical work; and I think that the name of Heron may well have been inserted after that of Ctesibius by Proclus himself. (3) The view that Vitruvius did not use Heron's work is attacked, and the contrary sought to be proved, on the basis apparently of three passages. (a) Vitruvius' water-organ is held to be decidedly better than Heron's: therefore Vitruvius used Heron's in order to improve upon it. (b) Vitruvius, in a passage describing a certain use of the lever, takes a wrong point to be the fulcrum; and it is held that he cannot have made the mistake himself, but must necessarily have copied it from Heron. In order, however, to find the same error in Heron, Hoppe arbitrarily alters both the figure and the text. (c) Vitruvius describes the working of a certain crane in language less clear than that of Herons; therefore he used Heron but misunderstood him! All would appear to be grist which comes to the mill of such critics: but I doubt whether such arguments will convince those who hold to the second half of the first century as the date that their view is mistaken.

That Heron wrote a systematic commentary on the Elements might be inferred from Proclus, but it is rendered quite certain by references to the commentary in Arabian writers, and particularly in an-Nairizi's commentary on the first ten Books of the Elements. The Fihrist says, under Euclid, that "Heron wrote a commentary on this book [the Elements], endeavouring to solve its difficulties"; and under Heron, “He wrote: the book of explanation of the obscurities in Euclid"...." An-Nairīzī's commentary quotes Heron by name very frequently, and often in such a way as to leave no doubt that the author had Heron's work actually before him. Thus the extracts are

1 E. Hoppe, Ein Beitrag zur Zeitbestimmung Herons von Alexandrien, Hamburg, 1902; Rudolf Meier, De Heronis aetate, Leipzig, 1905. See the references to the arguments in Cantor, Gesch. d. Math. 13, PP. 365, 367, 545-7.

9 Proclus, p. 41, 10.

Vitruvius, X. 13; Heron, vol. I. p. 192 sqq. (Pneumatics, 1. 42, 43). Vitruvius, X. 3, 3; Heron, vol. 11. pp. 114-116 (Mechanics, 11. 8). Vitruvius, X. 2, 10; Heron, vol. II. pp. 202-4 (Mechanics, III. 2). 6 Das Mathematiker- Verzeichniss im Fihrist (tr. Suter), p. 16. 7 ibid. p. 22.

given in the first person introduced by " Heron says " (" Dixit Yrinus" or "Heron "); and in other places we are told that Heron " says nothing," or "is not found to have said anything," on such and such a proposition. The commentary of an-Nairīzi is being published by Besthorn and Heiberg from a Leiden MS. of the translation of the Elements by al-Hajjāj with the commentary attached'. But this MS. only contains six Books, and several pages in the first Book are missing, which contain the comments of Simplicius on the first twentytwo definitions of the first Book. Fortunately the commentary of an-Nairīzi has been discovered in a more complete form, in a Latin translation by Gherardus Cremonensis of the twelfth century, which contains the missing comments by Simplicius and an-Nairizi's comments on the first ten Books. This valuable work has recently been edited by Curtze".

Thus from the three sources, Proclus, and the two versions of an-Nairīzī, which supplement one another, we are able to form a very good idea of the character of Heron's commentary. In some cases observations given by Proclus without the name of their author are seen from an-Nairīzi to be Heron's; in a few cases notes attributed by Proclus to Heron are found in an-Nairīzi without Heron's name; and, curiously enough, one alternative proof (of I. 25) given as Heron's by Proclus is introduced by the Arab with the remark that he has not been able to discover who is the author.

Speaking generally, the comments of Heron do not seem to have contained much that can be called important. We find

(1) A few general notes, e.g. that Heron would not admit more than three axioms.

(2) Distinctions of a number of particular cases of Euclid's propositions according as the figure is drawn in one way or in another.

Of this class are the different cases of I. 35, 36, III. 7, 8 (where the chords to be compared are drawn on different sides of the diameter instead of on the same side), III. 12 (which is not Euclid's, but Heron's own, adding the case of external contact to that of internal contact in III. 11), VI. 19 (where the triangle in which an additional line is drawn is taken to be the smaller of the two), VII. 19 (where he gives the particular case of three numbers in continued proportion, instead of four proportionals).

(3) Alternative proofs. Of these there should be mentioned (a) the proofs of II. I-10 "without a figure," being simply the algebraic forms of proof, easy but uninstructive, which are so popular nowadays, the proof of III. 25 (placed after III. 30 and starting from the arc instead of the chord), III. 10 (proved by III. 9), III. 13 (a proof preceded by a lemma to the effect that a straight line cannot meet a circle in more than two points). Another class of alternative proof is

1 Codex Leidensis 399, 1. Euclidis Elementa ex interpretatione al-Hadschdschadschii cum commentariis al-Narizii. Two parts carrying the work to the end of Book 1. were issued in 1893 and 1897 respectively. Another part came out in 1905.

Anaritii in decem libros priores elementorum Euclidis commentarii ex interpretatione Gherardi Cremonensis...edidit Maximilianus Curtze (Teubner, Leipzig, 1899).

(b) that which is intended to meet a particular objection (evoTaσis) which had been or might be raised to Euclid's construction. Thus in certain cases he avoids producing a particular straight line, where Euclid produces it, in order to meet the objection of any one who should deny our right to assume that there is any space available1. Of this class are Heron's proofs of I. II, I. 20, and his note on I. 16. Similarly on I. 48 he supposes the right-angled triangle which is constructed to be constructed on the same side of the common side as the given triangle is. A third class (c) is that which avoids reductio ad absurdum. Thus, instead of indirect proofs, Heron gives direct proofs of I. 19 (for which he requires, and gives, a preliminary lemma), and of I. 25.

(4) Heron supplies certain converses of Euclid's propositions, e.g. converses of II. 12, 13, VIII. 27.

(5) A few additions to, and extensions of, Euclid's propositions are also found. Some are unimportant, e.g. the construction of isosceles and scalene triangles in a note on I. I, the construction of two tangents in III. 17, the remark that VII. 3 about finding the greatest common measure of three numbers can be applied to as many numbers as we please (as Euclid tacitly assumes in VII. 31). The most important extension is that of III. 20 to the case where the angle at the circumference is greater than a right angle, and the direct deduction from this extension of the result of III. 22. Interesting also are the notes on I. 37 (on I. 24 in Proclus), where Heron proves that two triangles with two sides of one equal to two sides of the other and with the included angles supplementary are equal, and compares the areas where the sum of the two included angles (one being supposed greater than the other) is less or greater than two right angles, and on I. 47, where there is a proof (depending on preliminary lemmas) of the fact that, in the figure of the proposition, the straight lines AL, BK, CF meet in a point. After IV. 16 there is a proof that, in a regular polygon with an even number of sides, the bisector of one angle also bisects its opposite, and an enunciation of the corresponding proposition for a regular polygon with an odd number of sides.

Van Pesch gives reason for attributing to Heron certain other notes found in Proclus, viz. that they are designed to meet the same sort of points as Heron had in view in other notes undoubtedly written by him. These are (a) alternative proofs of I. 5, I. 17, and I. 32, which avoid the producing of certain straight lines, (b) an alternative proof of 1.9 avoiding the construction of the equilateral triangle on the side of BC opposite to A; (c) partial converses of I. 35-38, starting from the equality of the areas and the fact of the parallelograms or triangles being in the same parallels, and proving that the bases are the same or equal, may also be Heron's. Van Pesch further supposes that it was in Heron's commentary that the proof by Menelaus of I. 25 and the proof by Philo of 1. 8 were given.

1 Cf. Proclus, 275, 7 εἰ δὲ λέγοι τις τόπον μὴ εἰδέναι..., 189, 18 λέγει οὖν τις ὅτι οὐκ ἔστι τόπος....

2 De Procli fontibus, Lugduni-Batavorum, 1900.

The last reference to Heron made by an-Nairīzī occurs in the note on VIII. 27, so that the commentary of the former must at least have reached that point.

II. Porphyry.

The Porphyry here mentioned is of course the Neo-Platonist who lived about 232-304 A.D. Whether he really wrote a systematic commentary on the Elements is uncertain. The passages in Proclus which seem to make this probable are two in which he mentions him (1) as having demonstrated the necessity of the words "not on the same side" in the enunciation of I. 141, and (2) as having pointed out the necessity of understanding correctly the enunciation of I. 26, since, if the particular injunctions as to the sides of the triangles to be taken as equal are not regarded, the student may easily fall into error'. These passages, showing that Porphyry carefully analysed Euclid's enunciations in these cases, certainly suggest that his remarks were part of a systematic commentary. Further, the list of mathematicians in the Fihrist gives Porphyry as having written "a book on the Elements." It is true that Wenrich takes this book to have been a work by Porphyry mentioned by Suidas and Proclus (Theolog. Platon.), περὶ ἀρχῶν libri II.

There is nothing of importance in the notes attributed to Porphyry by Proclus.

(1) Three alternative proofs of I. 20, which avoid producing a side of the triangle, are assigned to Heron and Porphyry without saying which belonged to which. If the first of the three was Heron's, I agree with van Pesch that it is more probable that the two others were both Porphyry's than that the second was Heron's and only the third Porphyry's. For they are similar in character, and the third uses a result obtained in the second1.

(2) Porphyry gave an alternative proof of 1. 18 to meet a childish objection which is supposed to require the part of AC equal to AB to be cut off from CA and not from AC.

Proclus gives a precisely similar alternative proof of I. 6 to meet a similar supposed objection; and it may well be that, though Proclus mentions no name, this proof was also Porphyry's, as van Pesch suggests".

Two other references to Porphyry found in Proclus cannot have anything to do with commentaries on the Elements. In the first a work called the Evμμixтá is quoted, while in the second a philosophical question is raised.

III. Pappus.

The references to Pappus in Proclus are not numerous; but we have other evidence that he wrote a commentary on the Elements. Thus a scholiast on the definitions of the Data uses the phrase "as

1 Proclus, pp. 297, 1-298, 10.

* ibid. p. 352, 13, 14 and the pages preceding.

8 Fihrist (tr. Suter), p. 9, 10 and p. 45 (note 5).

Van Pesch, De Procli fontibus, pp. 129, 130. Heiberg assigned them as above in his Euklid-Studien (p. 160), but seems to have changed his view later. (See Besthorn-Heiberg, Codex Leidensis, p. 93, note 2.)

5 Van Pesch, op. cit. pp. 130—1.

Pappus says at the beginning of his (commentary) on the 10th (book) of Euclid." Again in the Fihrist we are told that Pappus wrote a commentary to the tenth book of Euclid in two parts. Fragments of this still survive in a MS. described by Woepcke, Paris. No. 952. 2 (supplément arabe de la Bibliothèque impériale), which contains a translation by Abū 'Uthmān (beginning of 10th century) of a Greek commentary on Book x. It is in two books, and there can now be no doubt that the author of the Greek commentary was Pappus. Again Eutocius, in his note on Archimedes, On the Sphere and Cylinder I. 13, says that Pappus explained in his commentary on the Elements how to inscribe in a circle a polygon similar to a polygon inscribed in another circle; and this would presumably come in his commentary on Book XII., just as the problem is solved in the second scholium on Eucl. XII. I. Thus Pappus' commentary on the Elements must have been pretty complete, an additional confirmation of this supposition being forthcoming in the reference of Marinus (a pupil and follower of Proclus) in his preface to the Data to "the commentaries of Pappus on the book"."

The actual references to Pappus in Proclus are as follows:

(1) On the Postulate (4) that all right angles are equal, Pappus is quoted as saying that the converse, viz. that all angles equal to a right angle are right, is not true, since the angle included between the arcs of two semicircles which are equal, and have their diameters at right angles and terminating at one point, is equal to a right angle, but is not a right angle.

(2) On the axioms Pappus is quoted as saying that, in addition to Euclid's axioms, others are on record as well (ovvavaypápeσbai) about unequals added to equals and equals added to unequals"; these, says Proclus, follow from the Euclidean axioms, while others given by Pappus are involved by the definitions, namely those which assert that "all parts of the plane and of the straight line coincide with one another," that "a point divides a straight line, a line a surface, and a surface a solid," and that "the infinite is (obtained) in magnitudes both by addition and diminution."

1 Euclid's Data, ed. Menge, p. 262.

2 Fihrist (tr. Suter), p. 22.

• Mémoires présentés à l'académie des sciences, 1856, XIV. pp. 658-719.

4 Woepcke read the name of the author, in the title of the first book as B. los (the dot representing a missing vowel). He quotes also from other MSS. (e.g. of the Ta'rikh alHukamā and of the Fihrist) where he reads the name of the commentator as B. lis, B.n.s or B.1.s. Woepcke takes this author to be Valens, and thinks it possible that he may be the same as the astrologer Vettius Valens. This Heiberg (Euklid-Studien, pp. 169, 170) proves to be impossible, because, while one of the MSS. quoted by Woepcke says that "B.n.s, le Roumi" (late-Greek) was later than Claudius Ptolemy and the Fihrist says "B.l.s, le Roûmi" wrote a commentary on Ptolemy's Planisphaerium, Vettius Valens seems to have lived under Hadrian, and must therefore have been an elder contemporary of Ptolemy. But Suter shows (Fihrist, p. 22 and p. 54, note 92) that Banos is only distinguished from Babos by the position of a certain dot, and Balos may also easily have arisen from an original Babos (there is no P in Arabic), so that Pappus must be the person meant. This is further confirmed by the fact that the Fihrist gives this author and Valens as the subjects of two separate paragraphs, attributing to the latter astrological works only. Heiberg, Euklid-Studien, p. 173; Euclid's Data, ed. Menge, pp. 256, lii. 7 ibid. p. 197, 6-10.

Proclus, pp. 189, 190.

8 ibid. p. 198, 3-15.