instance, that with a segment of a circle the angle in it is given, a corresponding porism is to find the ratio of the angle to a right angle. But though porisms occur in the Elements, they were used chiefly in higher geometry and Pappus says that Euclid's Porismata formed part of the collection Τόπος ἀναλυόμevos, like the Data. He proceeds then1 to discuss the nature of porisms, which he first defines, like Proclus, as intermediate between a problem and theorem, subsequently as "a proposition for the purpose of finding the thing proposed," afterwards again (but this, he asserts, is only a partial definition) as “that which is inferior by hypothesis to a local theorem” (τὸ λεῖπον ὑποθέσει τοπικοῦ θεωρήματος)" of which οἱ τόποι are the commonest examples. He then describes with some fulness two types of porisms contained in Euclid's book, but gives 28 more types with horrible brevity, e. g. in the first book, 'This line is given in position,' in the third book, 'The sum of these two straight lines has a given ratio to a straight line drawn from this point to a given points. No figures are appended. The whole work contained, in three books, 171 propositions, to which Pappus supplies 38 lemmas. Upon these statements of Pappus, which Halley and Prof. de Morgan found unintelligible, Simson framed a definition of a porism as "a proposition in which it is to be proved that one or several things is or are given which (like any one of an infinite number of things not given but having the same relation to the things which are given) has or have a certain property, described in the proposition." Chasles, who approves of this

1 VII. p. 648. 18 sqq.

2 The translation in the text is from Chasles. It seems, on authority, to be right. Heiberg explains it as "a local theorem with incomplete hypothesis." Whatever it may mean, it clearly is only intended to describe a special class of porisms, used by writers later than Euclid who, without attempting to find the thing proposed, merely declared that it was possible to do so (e.g. Archimedes, De Spir. propp. 5-9, cited by Heiberg, pp. 68, 69). Pappus then adds that oi Tóπo belonged to this class of porisms but, owing to

their number, were collected in a separate work (κεχωρισμένον τῶν πορισμάτων ἤθροισται).

3 See Nos. v. and xx. The whole list is given in Hultsch, pp. 654 sqq. Heiberg, pp. 73-77. The Greek of xx, is ὅτι λόγος συναμφοτέρου πρός τινα ἀπὸ τοῦδε ἕως δοθέντος. Halley, Simson and Heiberg interpret this dark saying as above: Chasles and Hultsch translate "the sum of these two rectangles has a given ratio to the segment lying between this point and a given point."

De Porismatibus, p. 347, quoted by Chasles, Le Livre de Porismes, p. 27.

definition, then proceeds to show the similarity between porisms and the propositions called Tóπou, for a Tóπos "is a proposition in which it is declared that certain points subject to the same known law are on a line of which the nature is enunciated and of which it remains to find the magnitude and the position. Example: two points being given, as also a ratio, the locus of a point, the distances of which from the two given points are in the given ratio, is the circumference of a circle given both in magnitude and in position." Hence, also, a connexion exists between the two meanings of 'porisma,' for every porism may be put as the corollary of a local theorem and the close connexion between the porism and the datum is equally obvious. Further, Chasles suggests a new definition of porism, which shall combine all the older definitions. Porisms, according to him, are incomplete theorems, "expressing certain relations between things variable according to a common law relations indicated in the enunciation of the porism but requiring to be completed by the determination of the magnitude and the position of certain things which are the consequence of the hypothesis and which would be determined in the enunciation of a theorem properly so-called." In order to exhibit the similarity of porisms with the most usual propositions of modern geometry, Chasles gives the following example (among others): If in the diameter of a circle there be taken two points which divide it harmonically, the ratio of the distances between these two points and any point on the circumference will be constant." Substitute here "given" for "constant" and this proposition is a porism. Find the ratio and include it in the enunciation, and you have a complete theorem.

Upon the preliminary discourse of Chasles, from which these remarks are taken, Heiberg (pp. 56-79) has many criticisms, supported by much learning, to offer, but his observations are

Playfair (in Trans. of R. S. of Edinburgh, 1792), improving on Simson, suggested a def. of a porism as "a proposition affirming the possibility of finding such conditions as will render a certain problem indeterminate, or capable of innumerable solutions."

Chasles, pp. 31, 32, objects to this. 1 Chasles, Porismes, pp. 33–36. 2 Ibid. pp. 36-38.

3 Ibid. pp. 42, 43. The porisms cited by Diophantus (supra, p. 121) are closely similar to data.

relevant mainly to the form of the enunciation of a porism and its relations, by virtue of its enunciation and hypothesis, to the TÓTTOS and the local theorem'. The passage of Pappus, on which Chasles and Heiberg, and every other would-be restorer of Euclid's work must necessarily rely, is so obscure and is suspected of so many interpolations and mutilations, that I could not, save at inconvenient length, give the details of the controversy, which, after all, is of no practical importance. I have therefore preferred to accept Chasles's theories, which are founded on adequate learning and are followed by a restoration of Euclid's Porisms with which, at present, no serious fault has been found3.

One of the types of porisms which Pappus describes at any length, is as follows: "If from two given points, two straight lines be drawn, which cut one another on a straight line given in position, and one of which intercepts on a straight line, given in position, a segment extending to a given point on it, the other will intercept on another straight line a segment which has a given ratio." This type was treated in one or more propositions early in the First Book, and this statement, together with the 38 lemmas of Pappus, gave Chasles his clue. The Porisms of the First Book, in his view, deal with propositions suggested by a hypothesis in which we suppose two straight lines to turn about two fixed points, to cut one another on a straight line given in position, and to make on two other fixed straight lines (or on one only) two segments which have to one another a certain constant relation. In the Second Book, the segments are, as a rule, formed on one line only. In the Third Book, the two fixed points are on the circumference of a circle and the two revolving straight lines cut one another on this circumference. “Almost, if not quite, all the relations of segments in the first two Books are 1 E.g. according to Heiberg, a porism proper has nothing whatever to do with a corollary. A TÓTOS was, as Simson defined it, a proposition 'to find a locus,' and therefore TÓTоι were a kind of porisms. The propositions, which Chasles calls 'local problems' and distinguishes from 'loci' and 'local theorems,' are really identical with 'loci' and are porisms, etc.

2 See Hultsch's edition. Heiberg accepts the whole of the text.

3 Heiberg himself has very few criticisms to make, even on the enunciations, which, he admits, are generally of the true porismatic form. The one obvious error in Chasles' book is that his restored Porism xv. (p. 119) is identical with the 8th Lemma of Pappus, which is only ancillary to a porism.

such as express that two variable points on two straight lines, or on one only, form two homographic divisions." It should be added that Chasles has had the good fortune to produce 201 porisms, or 30 more than Euclid himself composed'. The original porisms were used, as their place in the Tóπоs avaλvóμevos indicates, in the analysis, or in the synthesis, of a problem which was solved analytically. No doubt, a porism of the form 'it is possible to find' would be used in analysis, like the Data; a porism of the form 'to find' would be used in the synthesis.

125. The immediate successors of Euclid, as heads of the Alexandrian mathematical school, seem to have been Conon of Samos, who added "Berenice's hair" to the constellations, and Dositheus of Colonus. Perhaps also a certain Zeuxippus and Nicoteles of Cyrene were at Alexandria during this period. But nothing is known of these persons, save that Conon, Dositheus and Zeuxippus corresponded with Archimedes, who had a high opinion of their abilities (especially of Conon's3) and that Apollonius acknowledges some obligation to discoveries in conic sections by Conon and Nicoteles*.

But Archimedes, the greatest mathematician of antiquity, lived not at Alexandria but at Syracuse. He is said by Tzetzes3

1 A summary of the more interesting portion of Chasles' book is given in Taylor's Ancient and Modern Conics, pp. LII-LIV. Chasles himself says, p. 14, "Si ce livre de Porismes nous fût parvenu, il eût donné lieu depuis longtemps à la conception et au développement des théories élémentaires du rapport anharmonique, des divisions homographiques et de l'involution."

2 Catullus LXVI. 7, 8, translating Callimachus. Delambre (1. p. 215) suggests that Callimachus invented the name of the constellation himself and attributed it to Conon. The Berenice in question was wife of Ptolemy III. (Euergetes). Ptolemy, the astronomer, cites some observations of Conon.

3 See the prefaces to Sph. et Cyl. and Arenarius, ed. Torelli, pp. 63, 64, 319.

4 Conica, Pref. to Bk. Iv. Halley's ed. pp. 217, 218. A very important astronomer, Aristarchus of Samos, belongs to this interval. His extant work on the Sizes and Distances of the Sun and Moon is printed in the 3rd Vol. of Wallis's works. His proofs of course are geometrical (e.g. Prop. 2 is "If a greater sphere illuminate a less, more than half the latter is illuminated") but add nothing to geometry.

5 Chiliad. II. 35, 105. Proclus, p. 68, cites Eratosthenes as witnessing that he was a contemporary of Archimedes. The chief authority on the life of Archimedes is Plutarch, Vita Marcelli, cc. 14-19. A biography, which was used by Eutocius, was written by one Heracleides who perhaps was the friend whom Archimedes mentions pp. 217, 318 (Torelli).

(an authority as late as the 12th century) to have died at the age of 75, and, as it is well attested that he was killed in the sack of Syracuse B. C. 212, he was probably born about 287 B. C. Diodorus1 says that he visited Egypt and it is certain that he was a friend of Conon and Eratosthenes, who lived in Alexandria. His writings also show a most thorough acquaintance with all the work previously done in mathematics, and it may therefore be inferred that he was a disciple of the Alexandrian school. He returned, however, to Syracuse and lived there on intimate terms with King Hieron and his son Gelon, to whom possibly he was related by blood'. He made himself useful to his patrons by his extraordinary ingenuity of mechanical invention, a gift by which he himself set little store. He is said, by various contrivances, to have inflicted much loss on the Romans during the siege by Marcellus, but the city was ultimately taken and Archimedes perished in the indiscriminate slaughter. Marcellus wished to preserve his life but he was slain by accident. The story is that he was contemplating a geometrical figure drawn on the ground when a Roman soldier entered. Archimedes bade him stand off and not spoil the diagram, but the soldier, insulted at this behaviour, fell upon him and killed him. Marcellus raised in his honour a tomb bearing the figure of a sphere inscribed in a cylinder. Cicero had the honour of restoring this during his quaestorship in Sicily B. C. 75o.

1 Diod. v. 37.

2 Plutarch, Marcell. 14.

3 Ibid. 17, πᾶσαν ὅλως τέχνην χρείας ἐφαπτομένην ἀγεννῆ καὶ βάναυσον ἡγηoάuevos, "thinking that every kind of art, which was connected with daily needs, was ignoble and vulgar.”

4 Cic. Verr. IV. 131, Livy xxv. 31, Plut. Marc. 19, Pliny, Hist. Nat. vII. 125.

5 This tale is told in many slightly different forms. Plutarch loc. cit. Valerius Maximus VIII. 7, 7, Tzetzes II. 35. 135, Zonaras Ix. 5.

6 Cic. Tusc. Disp. v. 64, 65. The

authorities for Archimedes' life are collected and generally quoted in Torelli's Preface, pp. 11 and 12, and Heiberg's Quaestiones Archimedeae, Copenhagen, 1879, pp. 1-9. This little monograph deals chiefly with the text, but contains much very minute information on the arithmetic of Archimedes. Heiberg has since edited the text (Leipzig, 1880), but I have quoted always from Torelli, whose edition I happen to have. The errors and misprints which Heiberg points out in Torelli, are not such as to seriously affect his value for the present purpose.