Note
See also the Executive Summary and Outline for Chapter IV: Proclus and His Sources.
CHAPTER IV — PROCLUS AND HIS SOURCES[1].¶
IT is well known that the commentary of Proclus on Eucl. Book I. is one of the two main sources of information as to the history of Greek geometry which we possess, the other being the Collection of Pappus. They are the more precious because the original works of the forerunners of Euclid, Archimedes and Apollonius are lost, having probably been discarded and forgotten almost immediately after the appearance of the masterpieces of that great trio.
Proclus himself lived 410–485 A.D., so that there had already passed a sufficient amount of time for the tradition relating to the pre-Euclidean geometers to become obscure and defective. In this connexion a passage is quoted from Simplicius[2] who, in his account of the quadrature of certain lunes by Hippocrates of Chios, while mentioning two authorities for his statements, Alexander Aphrodisiensis (about 220 A.D.) and Eudemus, says in one place[3], “As regards Hippocrates of Chios we must pay more attention to Eudemus, since he was nearer the times, being a pupil of Aristotle.”
The importance therefore of a critical examination of Proclus’ commentary with a view to determining from what original sources he drew need not be further emphasised.
Proclus received his early training in Alexandria, where Olympiodorus was his instructor in the works of Aristotle, and mathematics was taught him by one Heron[4] (of course a different Heron from the “mechanicus Hero” of whom we have already spoken). He afterwards went to Athens where he was imbued by Plutarch, and by Syrianus, with the Neo-Platonic philosophy, to which he then devoted heart and soul, becoming one of its most prominent exponents. He speaks everywhere with the highest respect of his masters, and was in turn regarded with extravagant veneration by his contemporaries, as we learn from Marinus his pupil and biographer. On the death of Syrianus he was put at the head of the Neo-Platonic school. He was a man of untiring industry, as is shown by the number of books which he wrote, including a large number of commentaries, mostly on the dialogues of Plato. He was an acute dialectician, and pre-eminent among his contemporaries in the range of his learning[5]; he was a competent mathematician; he was even a poet. At the same time he was a believer in all sorts of myths and mysteries and a devout worshipper of divinities both Greek and Oriental.
Though he was a competent mathematician, he was evidently much more a philosopher than a mathematician[6]. This is shown even in his commentary on Eucl. I., where, not only in the Prologues (especially the first), but also in the notes themselves, he seizes any opportunity for a philosophical digression. He says himself that he attaches most importance to “the things which require deeper study and contribute to the sum of philosophy[7]”; alternative proofs, cases, and the like (though he gives many) have no attraction for him; and, in particular, he attaches no value to the addition of Heron to I. 47[8], which is of considerable mathematical interest. Though he esteemed mathematics highly, it was only as a handmaid to philosophy. He quotes Plato’s opinion to the effect that “mathematics, as making use of hypotheses, falls short of the non-hypothetical and perfect science[9]”…”Let us then not say that Plato excludes mathematics from the sciences, but that he declares it to be secondary to the one supreme science[10].” And again, while “mathematical science must be considered desirable in itself, though not with reference to the needs of daily life,” “if it is necessary to refer the benefit arising from it to something else, we must connect that benefit with intellectual knowledge (νοερὰν γνῶσιν), to which it leads the way and is a propaedeutic, clearing the eye of the soul and taking away the impediments which the senses place in the way of the knowledge of universals (τῶν ὅλων)[11].”
We know that in the Neo-Platonic school the younger pupils learnt mathematics; and it is clear that Proclus taught this subject, and that this was the origin of the commentary. Many passages show him as a master speaking to scholars. Thus “we have illustrated and made plain all these things in the case of the first problem, but it is necessary that my hearers should make the same inquiry as regards the others as well[12],” and “I do not indicate these things as a merely incidental matter but as preparing us beforehand for the doctrine of the Timaeus[13].” Further, the pupils whom he was addressing were beginners in mathematics; for in one place he says that he omits “for the present” to speak of the discoveries of those who employed the curves of Nicomedes and Hippias for trisecting an angle, and of those who used the Archimedean spiral for dividing an angle in any given ratio, because these things would be too difficult for beginners (δυσθεωρήτους τοῖς εἰσαγομένοις)[14]. Again, if his pupils had not been beginners, it would not have been necessary for Proclus to explain what is meant by saying that sides subtend certain angles[15], the difference between adjacent and vertical angles[16] etc., or to exhort them, as he often does, to work out other particular cases for themselves, for practice (γυμνασίας ἕνεκα)[17].
The commentary seems then to have been founded on Proclus’ lectures to beginners in mathematics. But there are signs that it was revised and re-edited for a larger public; thus he gives notice in one place[18] “to those who shall come upon” his work (τοῖς ἐντευξομένοις). There are also passages which could not have been understood by the beginners to whom he lectured, e.g. passages about the cylindrical helix[19], conchoids and cissoids[20]. These passages may have been added in the revised edition, or, as van Pesch conjectures, the explanations given in the lectures may have been much fuller and more comprehensible to beginners, and they may have been shortened on revision.
In his comments on the propositions of Euclid, Proclus generally proceeds in this way: first he gives explanations regarding Euclid’s proofs, secondly he gives a few different cases, mainly for the sake of practice, and thirdly he addresses himself to refuting objections raised by cavillers to particular propositions. The latter class of note he deems necessary because of “sophistical cavils” and the attitude of the people who rejoiced in finding paralogisms and in causing annoyance to scientific men[21]. His commentary does not seem to have been written for the purpose of correcting or improving Euclid. For there are very few passages of mathematical content in which Proclus can be supposed to be propounding anything of his own; nearly all are taken from the works of others, mostly earlier commentators, so that, for the purpose of improving on or correcting Euclid, there was no need for his commentary at all. Indeed only in one place does he definitely bring forward anything of his own to get over a difficulty which he finds in Euclid[22]; this is where he tries to prove the parallel-postulate, after first giving Ptolemy’s attempt and then pointing out objections to it. On the other hand, there are a number of passages in which he extols Euclid; thrice[23] also he supports Euclid against Apollonius where the latter had given proofs which he considered better than Euclid’s (I. 10, 11, and 23).
Allusion must be made to the debated question whether Proclus continued his commentaries beyond Book I. His intention to do so is clear from the following passages. Just after the words above quoted about the trisection etc. of an angle by means of certain curves he says, “For we may perhaps more appropriately examine these things on the third book, where the writer of the Elements bisects a given circumference[24].” Again, after saying that of all parallelograms which have the same perimeter the square is the greatest “and the rhomboid least of all,” he adds: “But this we will prove in another place; for it is more appropriate to the (discussion of the) hypotheses of the second book[25].” Lastly, when alluding (on I. 45) to the squaring of the circle, and to Archimedes’ proposition that any circle is equal to the right-angled triangle in which the perpendicular is equal to the radius of the circle and the base to its perimeter, he adds, “But of this elsewhere[26]”; this may imply an intention to treat of the subject on Eucl. XII., though Heiberg doubts it[27]. But it is clear that, at the time when the commentary on Book I. was written, Proclus had not yet begun to write on the other Books and was uncertain whether he would be able to do so: for at the end he says[28], “For my part, if I should be able to discuss the other books[29] in the same manner, I should give thanks to the gods; but, if other cares should draw me away, I beg those who are attracted by this subject to complete the exposition of the other books as well, following the same method, and addressing themselves throughout to the deeper and better defined questions involved” (τὸ πραγματειῶδες πανταχοῦ καὶ εὐδιαίρετον μεταδιώκοντας).
There is in fact no satisfactory evidence that Proclus did actually write any more commentaries than that on Book I.[30] The contrary view receives support from two facts pointed out by Heiberg, viz. (1) that the scholiast’s copy of Proclus was not so much better than our MSS. as to suggest that the scholiast had further commentaries of Proclus which have vanished for us[31]; (2) that there is no trace in the scholia of the notes which Proclus promised in the passages quoted above.
Coming now to the question of the sources of Proclus, we may say that everything goes to show that his commentary is a compilation, though a compilation “in the better sense” of the term[32]. He does not even give us to understand that we shall find in it much of his own; “let us,” he says, “now turn to the exposition of the theorems proved by Euclid, selecting the more subtle of the comments made on them by the ancient writers, and cutting down their interminable diffuseness…[33]”: not a word about anything of his own. At the same time, he seems to imply that he will not necessarily on each occasion quote the source of each extract from an earlier commentary; and, in fact, while he quotes the name of his authority in many places, especially where the subject is important, in many others, where it is equally certain that he is not giving anything of his own, he mentions no authority. Thus he quotes Heron by name six times; but we now know, from the commentary of an-Nairīzī, that a number of other passages, where he mentions no name, are taken from Heron, and among them the not unimportant addition of an alternative proof to I. 19. Hence we can by no means conclude that, where no authority is mentioned, Proclus is giving notes of his own. The presumption is generally the other way; and it is often possible to arrive at a conclusion, either that a particular note is not Proclus’ own, or that it is definitely attributable to someone else, by applying the ordinary principles of criticism. Thus, where the note shows an unmistakable affinity to another which Proclus definitely attributes to some commentator by name, especially when both contain some peculiar and distinctive idea, we cannot have much doubt in assigning both to the same commentator[34]. Again, van Pesch finds a criterion in the form of a note, where the explanation is so condensed as to be only just intelligible; the note is that in which a converse of I. 32 is proved[35] the proposition namely that a rectilineal figure which has all its interior angles together equal to two right angles is a triangle.
It is not safe to attribute a passage to Proclus himself because he uses the first person in such expressions as “I say” or “I will prove” —for he was in the habit of putting into his own words the substance of notes borrowed from others—nor because, in speaking of an objection raised to a particular proposition, he uses such expressions as “perhaps someone may object” (ἴσως δ’ ἄν τινες ἐνσταῖεν…): for sometimes other words in the same passage indicate that the objection had actually been taken by someone[36]. Speaking generally, we shall not be justified in concluding that Proclus is stating something new of his own unless he indicates this himself in express terms.
As regards the form of Proclus’ references to others by name, van Pesch notes that he very seldom mentions the particular work from which he is borrowing. If we leave out of account the references to Plato’s dialogues, there are only the following references to books: the Bacchae of Philolaus[37], the Symmikta of Porphyry[38], Archimedes On the Sphere and Cylinder[39], Apollonius On the cochlias[40], a book by Eudemus On the Angle[41], a whole book of Posidonius directed against Zeno of the Epicurean sect[42], Carpus’ Astronomy[43], Eudemus’ History of Geometry[44], and a tract by Ptolemy on the parallel-postulate[45].
Again, Proclus does not always indicate that he is quoting something at second-hand. He often does so, e.g. he quotes Heron as the authority for a statement about Philippus, Eudemus as attributing a certain theorem to Oenopides etc.; but he says on I. 12 that “Oenopides first investigated this problem, thinking it useful for astronomy” when he cannot have had Oenopides’ work before him.
It has been said above that Proclus was in the habit of stating in his own words the substance of the things which he borrowed. We are prepared for this when we find him stating that he will select the best things from ancient commentaries and “cut short their interminable diffuseness,” that he will “briefly describe” (συντόμως ἱστορῆσαι) the other proofs of I. 20 given by Heron and Porphyry and also the proofs of I. 25 by Menelaus and Heron. But the best evidence is of course to be found in the passages where he quotes works still extant, e.g. those of Plato, Aristotle and Plotinus. Examination of these passages shows great divergences from the original; even where he purports to quote textually, using the expressions “Plato says,” or “Plotinus says,” he by no means quotes word for word[46]. In fact, he seems to have had a positive distaste for quoting textually from other works. He cannot conquer this even when quoting from Euclid; he says in his note on I. 22, “we will follow the words of the geometer” but fails, nevertheless, to reproduce the text of Euclid unchanged[47].
We now come to the sources themselves from which Proclus drew in writing his commentary. Three have already been disposed of, viz. Heron, Porphyry and Pappus, who had all written commentaries on the Elements[48]. We go on to
Eudemus, the pupil of Aristotle, who, among other works, wrote a history of arithmetic, a history of astronomy, and a history of geometry. The importance of the last mentioned work is attested by the frequent use made of it by ancient writers. That there was no other history of geometry written after the time of Eudemus seems to be proved by the remark of Proclus in the course of his famous summary: “Those who compiled histories bring the development of this science up to this point. Not much younger than these is Euclid[49]….” The loss of Eudemus’ history is one of the gravest which fate has inflicted upon us, for it cannot be doubted that Eudemus had before him a number of the actual works of earlier geometers, which, as before observed, seem to have vanished completely when they were superseded by the treatises of Euclid, Archimedes and Apollonius. As it is, we have to be thankful for the fragments from Eudemus which such writers as Proclus have preserved to us.
I agree with van Pesch[50] that there is no sufficient reason for doubting that the work of Eudemus was accessible to Proclus at first hand. For the later writers Simplicius and Eutocius refer to it in terms such as leave no room for doubt that they had it before them. I have already quoted a passage from Simplicius’ account of the lunes of Hippocrates to the effect that Eudemus must be considered the best authority since he lived nearer the times[51]. In the same place Simplicius says[52], “I will set out what Eudemus says word for word (κατὰ λέξιν λεγόμενα), adding only a little explanation in the shape of reference to Euclid’s Elements owing to the memorandum-like style of Eudemus (διὰ τὸν ὑπομνηματικὸν τρόπον τοῦ Εὐδήμου) who sets out his explanations in the abbreviated form usual with ancient writers. Now in the second book of the history of geometry he writes as follows.” It is not possible to suppose that Simplicius would have written in this way about the style of Eudemus if he had merely been copying certain passages second-hand out of some other author and had not the original work itself to refer to. In like manner, Eutocius speaks of the paralogisms handed down in connexion with the attempts of Hippocrates and Antiphon to square the circle[53], “with which I imagine that those are accurately acquainted who have examined (ἐπεσκεμμένους) the geometrical history of Eudemus and know the Ceria Aristotelica.” How could the contemporaries of Eutocius have examined the work of Eudemus unless it was still extant in his time?
The passages in which Proclus quotes Eudemus by name as his authority are as follows:
On I. 26 he says that Eudemus in his history of geometry
referred this theorem to Thales, inasmuch as it was necessary to Thales’ method of ascertaining the distance of ships from the shore[54].
Eudemus attributed to Thales the discovery of Eucl. I. 15[55], and
Eudemus referred the discovery of the theorem in I. 32 to the Pythagoreans, and gave their proof of it, which Proclus reproduces[57].
On I. 44 Proclus tells us[58] that Eudemus says that “these things are ancient, being discoveries of the Pythagorean muse, the application (παραβολή) of areas, their exceeding (ὑπερβολή) and their falling short (ἔλλειψις).” The next words about the appropriation of these terms (parabola, hyperbola and ellipse) by later writers (i.e. Apollonius) to denote the conic sections are of course not due to Eudemus.
Coming now to notes where Eudemus is not named by Proclus, we may fairly conjecture, with van Pesch, that Eudemus was really the authority for the statements (1) that Thales first proved that a circle is bisected by its diameter[59] (though the proof by reductio ad absurdum which follows in Proclus cannot be attributed to Thales[60]), (2) that “Plato made over to Leodamas the analytical method, by means of which it is recorded (ἱστόρηται) that the latter too made many discoveries in geometry[61],” (3) that the theorem of I. 5 was due to Thales, and that for equal angles he used the more archaic expression “similar” angles[62], (4) that Oenopides first investigated the problem of I. 12, and that he called the perpendicular the gnomonic line (κατὰ γνώμονα)[63], (5) that the theorem that only three sorts of polygons can fill up the space round a point, viz. the equilateral triangle, the square and the regular hexagon, was Pythagorean[64]. Eudemus may also be the authority for Proclus’ description of the two methods, referred to Plato and Pythagoras respectively, of forming right-angled triangles in whole numbers[65].
We cannot attribute to Eudemus the beginning of the note on I. 47 where Proclus says that “if we listen to those who like to recount ancient history, we may find some of them referring this theorem to Pythagoras and saying that he sacrificed an ox in honour of his discovery[66].” As such a sacrifice was contrary to the Pythagorean tenets, and Eudemus could not have been unaware of this, the story cannot rest on his authority. Moreover Proclus speaks as though he were not certain of the correctness of the tradition; indeed, so far as the story of the sacrifice is concerned, the same thing is told of Thales in connexion with his discovery that the angle in a semi-circle is a right angle[67], and Plutarch is not certain whether the ox was sacrificed on the discovery of I. 47 or of the problem about application of areas[68]. Plutarch’s doubt suggests that he knew of no evidence for the story beyond the vague allusion in the distich of Apollodorus “Logisticus” (the “calculator”) cited by Diogenes Laertius also[69]; and Proclus may have had in mind this couplet with the passages of Plutarch.
We come now to the question of the famous historical summary given by Proclus[70]. No one appears to maintain that Eudemus is the author of even the early part of this summary in the form in which Proclus gives it. It is, as is well known, divided into two distinct parts, between which comes the remark, “Those who compiled histories[71] bring the development of this science up to this point. Not much younger than these is Euclid, who put together the Elements, collecting many of the theorems of Eudoxus, perfecting many others by Theaetetus, and bringing to irrefragable demonstration the things which had only been somewhat loosely proved by his predecessors.” Since Euclid was later than Eudemus, it is impossible that Eudemus can have written this. Yet the style of the summary after this point does not show any such change from that of the former portion as to suggest different authorship. The author of the earlier portion recurs frequently to the question of the origin of the elements of geometry in a way in which no one would be likely to do who was not later than Euclid; and it must be the same hand which in the second portion connects Euclid’s Elements with the work of Eudoxus and Theaetetus[72].
If then the summary is the work of one author, and that author not Eudemus, who is it likely to have been? Tannery answers that it is Geminus[73]; but I think, with van Pesch, that he has failed to show why it should be Geminus rather than another. And certainly the extracts which we have from Geminus’ work suggest that the sort of topics which it dealt with was quite different; they seem rather to have been general questions of the content of mathematics, and even Tannery admits that historical details could only have come incidentally into the work[74].
Could the author have been Proclus himself? Circumstances which seem to suggest this possibility are (1) that, as already stated, the question of the origin of the Elements is kept prominent, (2) that there is no mention of Democritus, whom Eudemus would not be likely to have ignored, while a follower of Plato would be likely enough to do him the injustice, following the example of Plato who was an opponent of Democritus, never once mentions him, and is said to have wished to burn all his writings[75], and (3) the allusion at the beginning to the “inspired Aristotle” (ὁ δαιμόνιος Ἀριστοτέλης)[76], though this may easily have been inserted by Proclus in a quotation made by him from someone else. On the other hand there are considerations which suggest that Proclus himself was not the writer. (1) The style of the whole passage is not such as to point to him as the author. (2) If he wrote it, it is hardly conceivable that he would have passed over in silence the discovery of the analytical method, the invention of Plato to which he attached so much importance[77].
There is nothing improbable in the conjecture that Proclus quoted the summary from a compendium of Eudemus’ history made by some later writer: but as yet the question has not been definitely settled. All that is certain is that the early part of the summary must have been made up from scattered notices found in the great work of Eudemus.
Proclus refers to another work of Eudemus besides the history, viz. a book on The Angle (βιβλίον περὶ γωνίας)[78]. Tannery assumes that this must have been part of the history, and uses this assumption to confirm his idea that the history was arranged according to subjects, not according to chronological order[79]. The phraseology of Proclus however unmistakably suggests a separate work; and that the history was chronologically arranged seems to be clearly indicated by the remark of Simplicius that Eudemus “also counted Hippocrates among the more ancient writers” (ἐν τοῖς παλαιοτέροις)[80].
The passage of Simplicius about the lunes of Hippocrates throws considerable light on the style of Eudemus’ history. Eudemus wrote in a memorandum-like or summary manner (τὸν ὑπομνηματικὸν τρόπον τοῦ Εὐδήμου)[81] when reproducing what he found in the ancient writers; sometimes it is clear that he left out altogether proofs or constructions of things by no means easy[82].
Geminus. The discussions about the date and birthplace of Geminus form a whole literature, as to which I must refer the reader to Manitius and Tittel[83]. Though the name looks like a Latin name (Gemĭnus), Manitius concluded that, since it appears as Γεμῖνος in all Greek MSS. and as Γεμεῖνος in some inscriptions, it is Greek and possibly formed from γεμ as Ἐργῖνος is from ἐργ and Ἀλεξῖνος from ἀλεξ (cf. also Ἰκτῖνος, Κρατῖνος). Tittel is equally positive that it is Gemĭnus and suggests that Γεμῖνος is due to a false analogy with Ἀλεξῖνος etc. and Γεμεῖνος wrongly formed on the model of Ἀντωνεῖνος, Ἀγριππεῖνα. Geminus, a Stoic philosopher, born probably in the island of Rhodes, was the author of a comprehensive work on the classification of mathematics, and also wrote, about 73–67 B.C., a not less comprehensive commentary on the meteorological textbook of his teacher Posidonius of Rhodes.
It is the former work in which we are specially interested here. Though Proclus made great use of it, he does not mention its title, unless we may suppose that, in the passage (p. 177, 24) where, after quoting from Geminus a classification of lines which never meet, he says, “these remarks I have selected from the φιλοκαλία of Geminus,” φιλοκαλία is a title or an alternative title. Pappus however quotes a work of Geminus “on the classification of the mathematics” (ἐν τῷ περὶ τῆς τῶν μαθημάτων τάξεως)[84], while Eutocius quotes from “the sixth book of the doctrine of the mathematics” (ἐν τῷ ἕκτῳ τῆς τῶν μαθημάτων θεωρίας)[85]. Tannery[86] pointed out that the former title corresponds well enough to the long extract[87] which Proclus gives in his first prologue, and also to the fragments contained in the Anonymi variae collectiones published by Hultsch at the end of his edition of Heron[88]; but it does not suit most of the other passages borrowed by Proclus. The correct title was therefore probably that given by Eutocius, The Doctrine, or Theory, of the Mathematics; and Pappus probably refers to one particular portion of the work, say the first Book. If the sixth Book treated of conics, as we may conclude from Eutocius, there must have been more Books to follow, because Proclus has preserved us details about higher curves, which must have come later. If again Geminus finished his work and wrote with the same fulness about the other branches of mathematics as he did about geometry, there must have been a considerable number of Books altogether. At all events it seems to have been designed to give a complete view of the whole science of mathematics, and in fact to be a sort of encyclopaedia of the subject.
I shall now indicate first the certain, and secondly the probable, obligations of Proclus to Geminus, in which task I have only to follow van Pesch, who has embodied the results of Tittel’s similar inquiry also[89]. I shall only omit the passages as regards which a case for attributing them to Geminus does not seem to me to have been made out.
First come the following passages which must be attributed to Geminus, because Proclus mentions his name:
matical sciences into arithmetic, geometry, mechanics, astronomy, optics, geodesy, canonic (science of musical harmony), and logistic (apparently arithmetical problems);
(in the note on the definition of a straight line) on the classification of lines (including curves) as simple (straight or circular) and mixed, composite and incomposite, uniform (ὁμοιομερεῖς) and non-uniform (ἀνομοιομερεῖς), lines “about solids” and lines produced by cutting solids, including conic and spiric sections[91];
(in the note on the definition of a plane surface) on similar distinctions extended to surfaces and solids[92];
(in the note on the definition of parallels) on lines which do not meet (ἀσύμπτωτοι) but which are not on that account parallel, e.g. a curve and its asymptote, showing that the property of not meeting does not make lines parallel—a favourite observation of Geminus—and, incidentally, on bounded lines or those which enclose a figure and those which do not[93];
(in the same note) the definition of parallels given by Posidonius[94];
on the distinction between postulates and axioms, the futility of trying to prove axioms, as Apollonius tried to prove Axiom 1, and the equal incorrectness of assuming what really requires proof, “as Euclid did in the fourth postulate [equality of right angles] and in the fifth postulate [the parallel-postulate][95]”;
on Postulates 1, 2, 3, which Geminus makes depend on the idea of a straight line being described by the motion of a point[96];
(in the note on Postulate 5) on the inadmissibility in geometry of an argument which is merely plausible, and the danger in this particular case owing to the existence of lines which do converge ad infinitum and yet never meet[97];
(in the note on I. 1) on the subject-matter of geometry, theorems, problems and διορισμοί (conditions of possibility) for problems[98];
(in the note on I. 5) on a generalisation of I. 5 by Geminus through the substitution for the rectilineal base of “one uniform line (curve),” by means of which he proved that the only “uniform lines”
(alike in all their parts) are a straight line, a circle, and a cylindrical helix[99];
(in the note on I. 10) on the question whether a line is made up of indivisible parts (ἀμερῆ), as affecting the problem of bisecting a given straight line[100];
(in the note on I. 35) on topical, or locus-theorems[101], where the illustration of the equal parallelograms described between a hyperbola and its asymptotes may also be due to Geminus[102].
Other passages which may fairly be attributed to Geminus, though his name is not mentioned, are the following:
in the prologue, where there is the same allusion as in the passage (8) above to a remark of Aristotle that it is equally absurd to expect scientific proofs from a rhetorician and to accept mere plausibilities from a geometer[103];
a passage in the prologue about the subject-matter, methods, and bases of geometry, the latter including axioms and postulates[104];
another on the definition and nature of elements[105];
a remark on the Stoic use of the term axiom for every simple statement (ἀπόφανσις ἁπλῆ)[106];
another discussion on theorems and problems[107], in the middle of which however there are some sentences by Proclus himself[108].
another passage, in connexion with Def. 3, on lines including or not including a figure (with which cf. part of the passage (4) above)[109];
a classification of different sorts of angles according as they are contained by simple or mixed lines (or curves)[110];
a similar classification of figures[111], and of plane figures[112];
Posidonius’ definition of a figure[113];
a classification of triangles into seven kinds[114];
a note distinguishing lines (or curves) producible indefinitely or not so producible, whether forming a figure or not forming a figure (like the “single-turn spiral”)[115];
passages distinguishing different sorts of problems[116], different sorts of theorems[117], and two sorts of converses (complete and partial)[118];
the definition of the term “porism” as used in the title of Euclid’s Porisms, as distinct from the other meaning of “corollary”[119];
a note on the Epicurean objection to I. 20 as being obvious even to an ass[120];
a passage on the properties of parallels, with allusions to
Apollonius’ Conics, and the curves invented by Nicomedes, Hippias and Perseus[121];
a passage on the parallel-postulate regarded as the converse of I. 17[122].
Of the authors to whom Proclus was indebted in a less degree the most important is Apollonius of Perga. Two passages allude to his Conics[123], one to a work on irrationals[124], and two to a treatise On the cochlias (apparently the cylindrical helix) by Apollonius[125]. But more important for our purpose are six references to Apollonius in connexion with elementary geometry.
He appears as the author of an attempt to explain the idea of a line (possessing length but no breadth) by reference to daily experience, e.g. when we tell someone to measure, merely, the length of a road or of a wall[126]; and doubtless the similar passage showing how we may in like manner get a notion of a surface (without depth) is his also[127].
He gave a new general definition of an angle[128].
He tried to prove certain axioms[129], and Proclus gives his attempt to prove Axiom 1, word for word[130].
Proclus further quotes:
Apollonius’ solution of the problem in Eucl. I. 10, avoiding Euclid’s use of I. 9[131],
his solution of the problem in I. 11, differing only slightly from Euclid’s[132], and
his solution of the problem in I. 23[133].
Heiberg[134] conjectures that Apollonius departed from Euclid’s method in these propositions because he objected to solving problems of a more general, by means of problems of a more particular, character. Proclus however considers all three solutions inferior to Euclid’s; and his remarks on Apollonius’ handling of these elementary matters generally suggest that he was nettled by criticisms of Euclid in the work containing the things which he quotes from Apollonius, just as we conclude that Pappus was offended by the remarks of Apollonius about Euclid’s incomplete treatment of the “three- and four-line locus[135].” If this was the case, Proclus can hardly have got his information about these things at second-hand; and there seems to be no reason to doubt that he had the actual work of Apollonius before him. This work may have been the treatise mentioned by Marinus in the words “Apollonius in his general treatise” (Ἀπολλώνιος ἐν τῇ καθόλου πραγματείᾳ)[136]. If the notice in the Fihrist[137] stating, on the authority of Thābit b. Qurra, that Apollonius wrote a tract on the parallel-postulate be correct, it may have been included in the same work. We may conclude generally that, in it, Apollonius tried to remodel the beginnings of geometry, reducing the number of axioms, appealing, in his definitions of lines, surfaces etc., more to experience than to abstract reason, and substituting for certain proofs others of a more general character.
The probabilities are that, in quoting from the tract of Ptolemy in which he tried to prove the parallel-postulate, Proclus had the actual work before him. For, after an allusion to it as “a certain book[138]” he gives two long extracts[139], and at the beginning of the second indicates the title of the tract, “in the (book) about the meeting of straight lines produced from (angles) less than two right angles,” as he has very rarely done in other cases.
Certain things from Posidonius are evidently quoted at second-hand, the authority being Geminus (e.g. the definitions of figure and parallels); but besides these we have quotations from a separate work which he wrote to controvert Zeno of Sidon, an Epicurean who had sought to destroy the whole of geometry[140]. We are told that Zeno had argued that, even if we admit the fundamental principles (ἀρχαί) of geometry, the deductions from them cannot be proved without the admission of something else as well, which has not been included in the said principles[141]. On I. 1 Proclus gives at some length the arguments of Zeno and the reply of Posidonius as regards this proposition[142]. In this case Zeno’s “something else” which he considers to be assumed is the fact that two straight lines cannot have a common segment, and then, as regards the “proof” of it by means of the bisection of a circle by its diameter, he objects that it has been assumed that two circumferences (arcs) of circles cannot have a common part. Lastly, he makes up, for the purpose of attacking it, another supposed “proof” of the fact that two straight lines cannot have a common part. Proclus appears, more than once, to be quoting the actual words of Zeno and Posidonius; in particular, two expressions used by Posidonius about “the acrid Epicurean” (τὸν δριμὺν Ἐπικούρειον)[143] and his “misrepresentations” (Ποσειδώνιός φησι τὸν Ζήνωνα συκοφαντεῖν)[144]. It is not necessary to suppose that Proclus had the original work of Zeno before him, because Zeno’s arguments may easily have been got from Posidonius’ reply; but he would appear to have quoted direct from the latter at all events.
The work of Carpus mechanicus (a treatise on astronomy) quoted from by Proclus[145] must have been accessible to him at first-hand, because a portion of the extract from it about the relation of theorems and problems[146] is reproduced word for word. Moreover, if he were not using the book itself, Proclus would hardly be in a position to question whether the introduction of the subject of theorems and problems was opportune in the place where it was found (εἰ μὲν κατὰ καιρὸν ἢ μή, παρείσθω πρὸς τὸ παρόν)[147].
It is of course evident that Proclus had before him the original works of Plato, Aristotle, Archimedes and Plotinus, as well as the Συμμικτά of Porphyry and the works of his master Syrianus (ὁ ἡμέτερος καθηγεμών)[148], from whom he quotes in his note on the definition of an angle. Tannery also points out that he must have had before him a group of works representing the Pythagorean tradition on its mystic, as distinct from its mathematical, side, from Philolaus downwards, and comprising the more or less apocryphal ἱερὸς λόγος of Pythagoras, the Oracles (λόγια), and Orphic verses[149].
Besides quotations from writers whom we can identify with more or less certainty, there are many other passages which are doubtless quoted from other commentators whose names we do not know. A list of such passages is given by van Pesch[150], and there is no need to cite them here.
Van Pesch also gives at the end of his work[151] a convenient list of the books which, as the result of his investigation, he deems to have been accessible to and directly used by Proclus. The list is worth giving here, on the same ground of convenience. It is as follows:
Lastly we come to the question what passages, if any, in the commentary of Proclus represent his own contributions to the subject. As we have seen, the onus probandi must be held to rest upon him who shall maintain that a particular note is original on the part of Proclus. Hence it is not enough that it should be impossible to point to another writer as the probable source of a note; we must have a positive reason for attributing it to Proclus. The criterion must therefore be found either (1) in the general terms in which Proclus points out the deficiencies in previous commentaries and indicates the respects in which his own will differ from them, or (2) in specific expressions used by him in introducing particular notes which may indicate that he is giving his own views. Besides indicating that he paid more attention than his predecessors to questions requiring deeper study (τὸ πραγματειῶδες) and “pursued clear distinctions” (τὸ εὐδιαίρετον μεταδιώκοντας)[152]—by which he appears to imply that his predecessors had confused the different departments of their commentaries, viz. lemmas, cases, and objections (ἐνστάσεις)[153]—Proclus complains that the earlier commentators had failed to indicate the ultimate grounds or causes of propositions[154]. Although it is from Geminus that he borrowed a passage maintaining that it is one of the proper functions of geometry to inquire into causes (τὴν αἰτίαν καὶ τὸ διὰ τί)[155], yet it is not likely that Geminus dealt with Euclid’s propositions one by one; and consequently, when we find Proclus, on I. 8, 16, 17, 18, 32, and 47[156], endeavouring to explain causes, we have good reason to suppose that the explanations are his own.
Again, his remarks on certain things which he quotes from Pappus can scarcely be due to anyone else, since Pappus is the latest of the commentators whose works he appears to have used. Under this head come
his objections to certain new axioms introduced by Pappus[157],
his conjecture as to how Pappus came to think of his alternative proof of I. 5[158],
an addition to Pappus’ remarks about the curvilineal angle which is equal to a right angle without being one[159].
The defence of Geminus against Carpus, who combated his view of theorems and problems, is also probably due to Proclus[160], as well as an observation on I. 38 to the effect that I. 35—38 are really comprehended in VI. 1 as particular cases[161].
Lastly, we can have no hesitation in attributing to Proclus himself (1) the criticism of Ptolemy’s attempt to prove the parallel-postulate[162], and (2) the other attempted proof given in the same note[163] (on I. 29) and assuming as an axiom that “if from one point two straight lines forming an angle be produced ad infinitum the distance between them when so produced ad infinitum exceeds any finite magnitude (i.e. length),” an assumption which purports to be the equivalent of a statement in Aristotle[164]. It is introduced by words in which the writer appears to claim originality for his proof: “To him who desires to see this proved (κατασκευαζόμενον) let it be said by us (λεγέσθω παρ’ ἡμῶν)” etc.[165] Moreover, Philoponus, in a note on Aristotle’s Anal. post. I. 10, says that “the geometer (Euclid) assumes this as an axiom, but it wants a great deal of proof, insomuch that both Ptolemy and Proclus wrote a whole book upon it[166].”