Note
See also the Executive Summary and Outline for Chapter III: Greek Commentators Other Than Proclus.
CHAPTER III — GREEK COMMENTATORS ON THE ELEMENTS OTHER THAN PROCLUS.¶
THAT there was no lack of commentaries on the Elements before the time of Proclus is evident from the terms in which Proclus refers to them; and he leaves us in equally little doubt as to the value which, in his opinion, the generality of them possessed. Thus he says in one place (at the end of his second prologue)[1]:
“Before making a beginning with the investigation of details, I warn those who may read me not to expect from me the things which have been dinned into our ears ad nauseam (διατεθρύληται) by those who have preceded me, viz. lemmas, cases, and so forth. For I am surfeited with these things and shall give little attention to them. But I shall direct my remarks principally to the points which require deeper study and contribute to the sum of philosophy, therein emulating the Pythagoreans who even had this common phrase for what I mean ‘a figure and a platform, but not a figure and sixpence[2].’”
In another place[3] he says: “Let us now turn to the elucidation of the things proved by the writer of the Elements, selecting the more subtle of the comments made on them by the ancient writers, while cutting down their interminable diffuseness, giving the things which are more systematic and follow scientific methods, attaching more importance to the working-out of the real subject-matter than to the variety of cases and lemmas to which we see recent writers devoting themselves for the most part.”
At the end of his commentary on Eucl. I. Proclus remarks[4] that the commentaries then in vogue were full of all sorts of confusion, and contained no account of causes, no dialectical discrimination, and no philosophic thought.
These passages and two others in which Proclus refers to “the commentators[5]” suggest that these commentators were numerous. He does not however give many names; and no doubt the only important commentaries were those of Heron, Porphyry, and Pappus.
Proclus alludes to Heron twice as Heron mechanicus[6], in another place[7] he associates him with Ctesibius, and in the three other passages[8] where Heron is mentioned there is no reason to doubt that the same person is meant, namely Heron of Alexandria. The date of Heron is still a vexed question. In the early stages of the controversy much was made of the supposed relation of Heron to Ctesibius. The best MS. of Heron’s Belopoeica has the heading Ἥρωνος Κτησιβίου βελοποιϊκά, and an anonymous Byzantine writer of the tenth century, evidently basing himself on this title, speaks of Ctesibius as Heron’s καθηγητής, “master” or “teacher.” We know of two men of the name of Ctesibius. One was a barber who lived in the time of Ptolemy Euergetes II, i.e. Ptolemy VII, called Physcon (died 117 B.C.), and who is said to have made an improved water-organ[9]. The other was a mechanician mentioned by Athenaeus as having made an elegant drinking-horn in the time of Ptolemy Philadelphus (285–247 B.C.)[10]. Martin[11] took the Ctesibius in question to be the former and accordingly placed Heron at the beginning of the first century B.C., say 126–50 B.C. But Philo of Byzantium[12], who repeatedly mentions Ctesibius by name, says that the first mechanicians had the advantage of being under kings who loved fame and supported the arts. Hence our Ctesibius is more likely to have been the earlier Ctesibius who was contemporary with Ptolemy II Philadelphus.
But, whatever be the date of Ctesibius, we cannot safely conclude that Heron was his immediate pupil. The title “Heron’s (edition of) Ctesibius’s Belopoeica” does not, in fact, justify any inference as to the interval of time between the two works.
We now have better evidence for a terminus post quem. The Metrica of Heron, besides quoting Archimedes and Apollonius, twice refers to “the books about straight lines (chords) in a circle” (ἐν τοῖς περὶ τῶν ἐν κύκλῳ εὐθειῶν). Now we know of no work giving a Table of Chords earlier than that of Hipparchus. We get, therefore, at once, 150 B.C. or thereabouts as the terminus post quem. But, again, Heron’s Mechanica quotes a definition of “centre of gravity” as given by “Posidonius, a Stoic”: and, even if this Posidonius lived before Archimedes, as the context seems to imply, it is certain that another work of Heron’s, the Definitions, owes something to Posidonius of Apamea or Rhodes, Cicero’s teacher (135–51 B.C.). This brings Heron’s date down to the end of the first century B.C., at least.
We have next to consider the relation, if any, between Heron and Vitruvius. In his De Architectura, brought out apparently in 14 B.C., Vitruvius quotes twelve authorities on machinationes including Archytas (second), Archimedes (third), Ctesibius (fourth) and Philo of Byzantium (sixth), but does not mention Heron. Nor is it possible to establish inter-dependence between Vitruvius and Heron; the differences between them seem on the whole more numerous and important than the resemblances (e.g. Vitruvius uses 3 as the value of $\pi$, while Heron always uses the Archimedean value $3\frac{1}{7}$). The inference is that Heron can hardly have written earlier than the first century A.D.
The most recent theory of Heron’s date makes him later than Claudius Ptolemy the astronomer (100–178 A.D.). The arguments are mainly these. (1) Ptolemy claims as a discovery of his own a method of measuring the distance between two places (as an arc of a great circle on the earth’s surface) in the case where the places are neither on the same meridian nor on the same parallel circle. Heron, in his Dioptra, speaks of this method as of a thing generally known to experts. (2) The dioptra described in Heron’s work is a fine and accurate instrument, much better than anything Ptolemy had at his disposal. (3) Ptolemy, in his work Περὶ ῥοπῶν, asserted that water with water round it has no weight and that the diver, however deep he dives, does not feel the weight of the water above him. Heron, strangely enough, accepts as true what Ptolemy says of the diver, but is dissatisfied with the explanation given by “some,” namely that it is because water is uniformly heavy—this seems to be equivalent to Ptolemy’s dictum that water in water has no weight—and he essays a different explanation based on Archimedes. (4) It is suggested that the Dionysius to whom Heron dedicated his Definitions is a certain Dionysius who was praefectus urbi in 301 A.D.
On the other hand Heron was earlier than Pappus, who was writing under Diocletian (284–305 A.D.), for Pappus alludes to and draws upon the works of Heron. The net result, then, of the most recent research is to place Heron in the third century A.D. and perhaps little earlier than Pappus. Heiberg[13] accepts this conclusion, which may therefore, perhaps, be said to hold the field for the present[14].
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-Nairīzī’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[15]”; and under Heron, “He wrote: the book of explanation of the obscurities in Euclid[16]…” 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 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-Nairīzī is in part edited by Besthorn and Heiberg from a Leiden MS. of the translation of the Elements by al-Ḥajjāj with the commentary attached[17]. But this MS. only contains six Books, and several pages in the first Book, which contain the comments of Simplicius on the first twenty-two definitions of the first Book, are missing. Fortunately the commentary of an-Nairīzī 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-Nairīzī’s comments on the first ten Books. This valuable work has recently been edited by Curtze[18].
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-Nairīzī to be Heron’s; in a few cases notes attributed by Proclus to Heron are found in an-Nairīzī 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
A few general notes, e.g. that Heron would not admit more than three axioms.
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).
Alternative proofs. Of these there should be mentioned (a) the proofs of II. 1—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
(b) that which is intended to meet a particular objection (ἔνστασις) 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 available[19]. Of this class are Heron’s proofs of I. 11, 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.
Heron supplies certain converses of Euclid’s propositions, e.g. converses of II. 12, 13, VIII. 27.
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. 1, 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[20] 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 I. 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 I. 8 were given.
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.
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. 14[21], 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[22]. 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.[23]
There is nothing of importance in the notes attributed to Porphyry by Proclus.
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 second[24].
Porphyry gave an alternative proof of I. 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[25].
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 Συμμικτά is quoted, while in the second a philosophical question is raised.
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 Pappus says at the beginning of his (commentary) on the 10th (book) of Euclid[26].” Again in the Fihrist we are told that Pappus wrote a commentary to the tenth book of Euclid in two parts[27]. Fragments of this still survive in a MS. described by Woepcke[28], 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[29]. 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. 1. 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[30].”
The actual references to Pappus in Proclus are as follows:
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[31], 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.
On the axioms Pappus is quoted as saying that, in addition to Euclid’s axioms, others are on record as well (συναναγράφεσθαι) about unequals added to equals and equals added to unequals[32]; 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[33].”
Pappus gave a pretty proof of I. 5. This proof has, I think, been wrongly understood; on this point see my note on the proposition.
On I. 47 Proclus says[34]: “As the proof of the writer of the Elements is manifest, I think that it is not necessary to add anything further, but that what has been said is sufficient, since indeed those who have added more, like Heron and Pappus, were obliged to make use of what is proved in the sixth book, without attaining any important result.” We shall see what Heron’s addition consisted of; what Pappus may have added we do not know, unless it was something on the lines of his extension of I. 47 found in the Synagoge (IV. p. 176, ed. Hultsch).
We may fairly conclude, with van Pesch[35], that Pappus is drawn upon in various other passages of Proclus where he quotes no authority, but where the subject-matter reminds us of other notes expressly assigned to Pappus or of what we otherwise know to have been favourite questions with him. Thus:
We are reminded of the curvilineal angle which is equal to but not a right angle by the note on I. 32 to the effect that the converse (that a figure with its interior angles together equal to two right angles is a triangle) is not true unless we confine ourselves to rectilineal figures. This statement is supported by reference to a figure formed by four semicircles whose diameters form a square, and one of which is turned inwards while the others are turned outwards. The figure forms two angles “equal to” right angles in the sense described by Pappus on Post. 4, while the other curvilineal angles are not considered to be angles at all, and are left out in summing the internal angles. Similarly the allusions in the notes on I. 4, 23 to curvilineal angles of which certain moon-shaped angles (μηνοειδεῖς) are shown to be “equal to” rectilineal angles savour of Pappus.
On I. 9 Proclus says[36] that “Others, starting from the Archimedean spirals, divided any given rectilineal angle in any given ratio.” We cannot but compare this with Pappus IV. p. 286, where the spiral is so used; hence this note, including remarks immediately preceding about the conchoid and the quadratrix, which were used for the same purpose, may very well be due to Pappus.
The subject of isoperimetric figures was a favourite one with Pappus, who wrote a recension of Zenodorus’ treatise on the subject[37]. Now on I. 35 Proclus speaks[38] about the paradox of parallelograms having equal area (between the same parallels) though the two sides between the parallels may be of any length, adding that of parallelograms with equal perimeter the rectangle is greatest if the base be given, and the square greatest if the base be not given etc. He returns to the subject on I. 37 about triangles[39]. Compare[40] also his note on I. 4. These notes may have been taken from Pappus.
Again, on I. 21, Proclus remarks on the paradox that straight lines may be drawn from the base to a point within a triangle which are (1) together greater than the two sides, and (2) include a less angle, provided that the straight lines may be drawn from points in the base other than its extremities. The subject of straight lines satisfying condition (1) was treated at length, with reference to a variety of cases, by Pappus[41], after a collection of “paradoxes” by Erycinus, of whom nothing more is known. Proclus gives Pappus’ first case, and adds a rather useless proof of the possibility of drawing straight lines satisfying condition (2) alone, adding that “the proposition stated has been proved by me without using the parallels of the commentators[42].” By “the commentators” Pappus is doubtless meant.
Lastly, the “four-sided triangle,” called by Zenodorus the “hollow-angled,”[43] is mentioned in the notes on I. Def. 24—29 and I. 21. As Pappus wrote on Zenodorus’ work in which the term occurred[44], Pappus may be responsible for these notes.
According to the Fihrist[45], Simplicius the Greek wrote “a commentary to the beginning of Euclid’s book, which forms an introduction to geometry.” And in fact this commentary on the definitions, postulates and axioms (including the postulate known as the Parallel-Axiom) is preserved in the Arabic commentary of an-Nairīzī[46]. On two subjects this commentary of Simplicius quotes a certain “Aganis,” the first subject being the definition of an angle, and the second the definition of parallels and the parallel-postulate. Simplicius gives word for word, in a long passage placed by an-Nairīzī after I. 29, an attempt by “Aganis” to prove the parallel-postulate. It starts from a definition of parallels which agrees with Geminus’ view of them as given by Proclus[47], and is closely connected with the definition given by Posidonius[48]. Hence it has been assumed that “Aganis” is none other than Geminus, and the historical importance of the commentary of Simplicius has been judged accordingly. But it has been recently shown by Tannery that the identification of “Aganis” with Geminus is practically impossible[49]. In the translation of Besthorn-Heiberg Aganis is called by Simplicius in one place “philosophus Aganis,” in another “magister noster Aganis,” in Gherard’s version he is “socius Aganis” and “socius noster Aganis.” These expressions seem to leave no doubt that Aganis was a contemporary and friend, if not master, of Simplicius; and it is impossible to suppose that Simplicius (fl. about 500 A.D.) could have used them of a man who lived four and a half centuries before his time. A phrase in Simplicius’ word-for-word quotation from Aganis leads to the same conclusion. He speaks of people who objected “even in ancient times” (iam antiquitus) to the use by geometers of this postulate. This would not have been an appropriate phrase had Geminus been the writer. I do not think that this difficulty can be got over by Suter’s suggestion[50] that the passages in question may have been taken out of Heron’s commentary, and that an-Nairīzī may have forgotten to name the author; it seems clear that Simplicius is the person who described “Aganis.” Hence we are driven to suppose that Aganis was not Geminus, but some unknown contemporary of Simplicius[51]. Considerable interest will however continue to attach to the comments of Simplicius so fortunately preserved.
Proclus tells us that one Aegaeas (? Aenaeas) of Hierapolis wrote an epitome of the Elements[52]; but we know nothing more of him or of it.