not be taken on either base. Suddenly he passes thence to problems (or porisms) to find given points on a sphere, e.g. the point which is nearest to a given plane or the points in which a given straight line will cut the sphere. Then he shews how to inscribe seven similar regular hexagons in a circle, one having the same centre as the circle, the other six standing each on one side of the first. This problem serves for the construction of cogwheels and extracts from the Baρoûλos and the Mechanics of Heron, added perhaps by a later hand, conclude the collection.

156. To the development of Greek geometry the Collection of Pappus can hardly be deemed really important. It is evidence, indeed, that the geometrical school of Alexandria was still flourishing after 600 years and it shews what subjects were studied there. But among his contemporaries Pappus is like the peak of Teneriffe in the Atlantic. He looks back, from a distance of 500 years, to find his peer in Apollonius. In the long interval, only two or three writers, Zenodorus and Serenus and Menelaus, had produced in pure geometry a little work of the best order, and there are none such to follow. The Collection of Pappus is not cited by any of his successors1, and none of them attempted to make the slightest use of the proofs and aperçus in which the book abounds. It becomes interesting only in the history of mathematics during the 17th and 18th centuries, when there were again geometers capable of using it and others who independently struck out and pursued lines of investigation which were more or less clearly anticipated by Pappus. To give here an elaborate account of Pappus would be to create a false impression. His work is only the last convulsive effort of Greek geometry which was now nearly dead and was never effectually revived. It is not so with Ptolemy or Diophantus. The trigonometry of the former is the foundation of a new study which was handed on to other nations indeed but which has thenceforth a continuous history of progress. Diophantus also represents the outbreak of a movement which probably was not Greek in its origin, and

1 Hultsch's Preface to Vol. III. p. 3. Eutocius however, (in Torelli p. 139) referring to the μηχανικαί εἰσαγωγαί οἱ

Pappus, cites the proposition VIII. 11 of the Collectio. (This is also in Bk. III. pp. 64-69 of Hultsch).

which the Greek genius long resisted, but which was especially adapted to the tastes of the people who, after the extinction of Greek schools, received their heritage and kept their memory green. But no Indian or Arab ever studied Pappus or cared in the least for his style or his matter. When geometry came once more up to his level, the invention of analytical methods gave it a sudden push which sent it far beyond him and he was out of date at the very moment when he seemed to be taking a new lease of life.

A few lines only will be sufficient to call attention to some passages of Pappus in which modern geometers still take an antiquarian interest'. These occur mostly in Book VII. Here (p. 682) occurs the theorem, afterwards re-discovered or stolen by Guldin (1577-1643), that the volume of a solid of revolution is equal to the product of the area of the revolving figure and the length of the path of its centre of gravity. Here also (p. 1013) Pappus first found the focus of a parabola and suggested the use of the directrix. Here in the lemmas to the Sectio Determinata the theory of points in involution is propounded: and among those to the De Tactionibus the problem is solved, to draw through three points lying in the same straight line, three straight lines which shall form a triangle inscribed in a given circle. Here also (p. 678) occurs the problem "given several straight lines, to find the locus of a point such that the perpendiculars, or more generally straight lines at given angles, drawn from the point to the given lines shall satisfy the condition that the product of certain of them shall be in a given ratio to the

1 Some of these have been mentioned before à propos of the books to which the lemmas of Pappus refer. A summary of a kind more satisfying to the modern geometer will be found in Chasles Aperçu pp. 28-44. Cantor pp. 382-386 cites generally the same propositions as Chasles, but adds some remarks on hints of algebraical symbolism in Pappus. Taylor (Anc. and Mod. Conics. pp. lii-liv) gives little more than the lemmas to Euclid's porisms from Book VII.

2 On this problem (no. 117) Chasles has the following remarks. "The props. 105, 107, 108 are particular cases of it. One of the points is there supposed to be at infinity. The problem, generalised by placing the points anywhere, has become celebrated by its difficulty, by the fame of the geometers who solved it and especially by the solution, as general and simple as possible, given by a boy of 16, Ottaiano of Naples." Aperçu, pp. 44, 328.

product of the rest "1. Descartes and Newton brought this into celebrity as the "problem of Pappus." But though the seventh Book, which contains the lemmas to the TÓTоs avaλvóμevos is by far the most important, there is matter in the other books of a very surprising character. The 4th Book, which deals with curves, contains a great number of brilliant propositions, especially on the quadratrix and the Archimedean spiral. Pappus supplements the latter by producing (p. 261 sqq.), a spiral on a sphere, in which a great circle revolves uniformly about a diameter, while a point on the circle moves uniformly along its circumference. He then finds the area of the surface so determined, "a complanation which claims the more lively admiration, if we remember that, though the whole spherical superficies was known since Archimedes' time, to measure portions of it, such as spherical triangles, was then and for long afterwards an unsolved problem"". The 8th Book (p. 1034 sqq.) contains a proposition to the effect that the centre of gravity of a triangle is that of another triangle of which the vertices lie on the sides of the first and divide them all in the same ratio. All these, and many more of equal difficulty, seem to be new and of Pappus' own invention. It ought not, however, to be forgotten that in at least three cases, which have been noticed above in their proper places, Pappus seems to have assumed credit to which he is not entitled. In Book III. he gives as his own a solution of the trisection-problem with a conchoid, which can hardly be other than the solution which Proclus ascribes to Nicomedes: in Book IV. he gives 14 propositions of Zenodorus without so much as naming that author: and in Book VIII. he solves the problem 'to move a given weight with a given power' in a manner which differs only accidentally from Heron's. It is probable that many

It is in this problem that Pappus objects to having more than 4 straight lines, on the ground that a geometry of more than three dimensions was absurd.

2 Cantor p. 384.

3 Pappus supposes points, starting simultaneously from the three vertices, to move along the sides with velocities

proportionate to the length of the sides. 4 In Heron the weight is 1000 talents, the power 5, and he solves the problem by a series of cogwheels, the diameters of each pair being in the ratio 5:1. Pappus takes the weight 160, power 4 and the diameters 2:1. See Pappus VIII. prop. 10 (p. 1061 sqq.) and Vincent's Heron cited supra, p. 278 n.

works of ancient geometers were, in Pappus' time, becoming rare. Pappus himself, for instance, does not seem to have seen Euclid's Conics and Eutocius and Proclus (much later) had certainly not seen many books which they knew by name'. It was therefore possible to appropriate many proofs without much chance of detection and it may be that Pappus used this opportunity.

157. It was suggested at the beginning of this chapter, that possibly the Jews had something to do with the revival of the arithmetical investigations which culminate about this time in the Algebra of Diophantus. It is possible also that the decay of Greek geometry was due to the gradual advance of peoples who have never, at any time, cared much for this branch of mathematics, though they have a surprising natural talent for the other. At any rate, nearly all the leading writers of the Neo-Platonic and Neo-Pythagorean schools were not Greeks. Philo was a Jew: Nicomachus was an Arabian: Ammonius the founder of Neo-Platonism was an Egyptian: so was Plotinus Porphyrius came from Tyre: the name of Anatolius, wherever he was born, means literally 'Oriental': Iamblichus was a native of Chalcis, in Colesyria. These are the philosophers who, in the first four centuries of our era, commanded the largest influence and not one of them was a geometer. Nevertheless, the world is wide and the geometrical school at Alexandria was still largely attended, though it produced no brilliant professors after Pappus. Perhaps Patricius, the author of two rules now inserted in Heron's works (Geom. 104* and Stereom. I. 22) belonged to this time, but there are two persons of this name, one a Lydian of about A.D. 374, the other somewhat later, a Lycian and the father of Proclus. Theon of Alexandria was certainly making astronomical observations in A.D. 365 and 372, and he as certainly held classes (ovvovolaı) for which he prepared his edition of Euclid. We have seen also that the preface to Euclid's Optics consists of notes from Theon's lectures. He also wrote a commentary on the Almagest, (ed. Halma. 1821) most of which is extant and which is perhaps in

1 Heiberg, Litterargesch. Euklid. p. 89.

great part founded on the similar work of Pappus'. This also contains many little historical notices which have been extracted above in their proper places, and the commentary to Book I. of Ptolemy is especially valuable for its specimens of Greek arithmetic. Theon's daughter Hypatia (ob. A.D. 415), seems to have been a better mathematician than her father. The story of her life and her tragical death are familiar to English readers through Kingsley's novel. None of her works are extant, but Suidas (sub voce) says she wrote “ὑπόμνημα εἰς Διοφάντην τὸν ἀστρονομικὸν κανόνα εἰς τὰ κωνικὰ ̓Απολλωνίου ὑπόμνημα”. This may mean three works, viz.: notes to Diophantus, the astronomical canon and notes to Apollonius' conics, or (altering Διοφάντην to Διοφάντου) may refer to two only, notes to the astronomical canon of Diophantus and notes to the conics. Hypatia was the last of the Alexandrian professors who attained any fame. The Neo-Platonic school in Athens, under Syrianus, now began to attract more attention, and in the interests of Platonism the historical study of geometry was for a time revived. Proclus the successor (diádoxos) of Syrianus at the Athenian school (A.D. 410-485), studied in Alexandria and there acquired that general acquaintance with Greek geometry which enabled him to write his commentary on Euclid's Elements. His notes on the first Book are still extant2, and contain a very large proportion of all the most valuable information we possess on the history of Greek geometry. But Proclus himself is a wordy and obscure writer and his best things are taken from Geminus and Eudemus. Proclus' pupil Marinus of Neapolis (i.e. Flavia Neapolis, the ancient Sychem in Palestine) wrote the life of his master and is the author of the preface to Euclid's Data. He also was at the head of the Athenian school. Isidorus succeeded him and was the teacher of Damascius of Damascus, who appended the 15th Book to

1 The MSS. have a fragment of Pappus's commentary at the beginning of Theon's to Book v. and in Theon's to Book 1. occurs a tractate on calculation with sexagesimal fractions which is, in some мss, attributed to

Pappus or Diophantus.

2 Some of the extant scholia to the other books are thought to be by Proclus. See Knoche's essay, cited above p. 74n.