attributed to Thales what other geometrical knowledge he must have had is a peculiarly fascinating inquiry. It has been already suggested that he knew, in some form, the theorem Eucl. VI. 4. To this Dr Allman adds also two other inferences. If, he argues, Thales knew that the angle in a semicircle is a right angle, he must have known also that 'the interior angles of a triangle are equal to two right angles' (Euclid 1. 32, pt. 2). He infers this, not from the fact that Euclid uses the proposition I. 32, in the proof of III. 31, pt. 1,' but in another way. Thales knew that the angle in a semicircle is a right angle: if he had then joined the apex of the triangle containing that right angle with the centre of the circle, he would have obtained two isosceles triangles, in which, as he also knew, the angles at the base are equal. Hence, he could not have failed to see that the interior angles of a right-angled triangle were equal to two right angles, and since any triangle may be divided into two right angled triangles, the same proposition is true of every triangle. It is justifiable, no doubt, to ascribe so much intelligence to Thales, but it is another matter to attribute to him a particular piece of knowledge and a particular method of proof on the same plan, Thales might be held to have known the first six books of Euclid. It will be remembered that Geminus, in the extract quoted above, attributes to "the ancients" (oi Taλaioi) the knowledge of the proposition that the interior angles of a triangle are equal to two right angles. may be conceded that he alludes here to Thales among others, but it is also to be borne in mind that he says that this proposition was separately proved for the different classes of triangles. Hence Dr Allman suggests, as an alternative, that the theorem was arrived at from inspection of Egyptian floors paved with tiles of the form of equilateral triangles, or squares, or hexagons2.

1 There would be two objections at least to such an inference, viz. that Euclid 1. 32 contains two propositions, of which only the first, which is not the prop. in question, is used in III. 31: and also that Euclid 1. 32 is said by Proclus (p. 379) to have been proved almost as it stands by the Py

It

thagoreans. Cantor, however (p. 120), is inclined to attribute to Thales Euclid's proof (or something very like it) of III. 31.

2 Proclus, p. 305, attributes to the Pythagoreans the theorem that only three regular polygons, the equilateral triangle, the square and the hexagon,

If, for instance, Thales observed that six equilateral triangles could be placed round a common vertex, he would also notice that six equal angles make up four right angles, and therefore the angles of each equilateral triangle are equal to two rightangles. Hankel (pp. 95, 96) suggests a similar theory, which is adopted also by Cantor (pp. 120-121), with the addition that the scalene triangle was divided into two right-angled triangles, each of which was considered as half a rectangle. It seems needless to dwell further on this proposition.

86. Dr Allman, however, makes a second inference of a far bolder character. He converts the theorem that the angle in a semicircle is a right angle into a theorem that, if on a given straight line as base, there be described any number of triangles each having a right angle at the vertex, then the locus of their vertices is the circumference of a circle described on the given base as diameter, and attributes to Thales, therefore, the conception of geometrical loci. If Thales proved the first theorem empirically, by constructing a great number of rightangled triangles on the same base, no doubt the notion of a locus may have occurred to him: but what becomes then of that deductive, that essentially Greek character which Thales is always said to have imparted to Egyptian geometry?1 There will not be left a single theorem, attributed to Thales, which he is not likely to have discovered by inspection or inductively. He may, no doubt, have arrived at any theorem in two ways, at first inductively or by inspection, and later also by a formal deductive process, but there is no available evidence on this ✓matter. If he used deduction only for this particular theorem, he would probably not have conceived a locus. If he used induction only, he might have conceived a locus, but there would have been no great merit in the conception.

Of speculation in this style there is no end, and there is hardly a single Greek geometer who is not the subject of it. A

can be placed about a point so as to fill a space, but Dr Allman (p. 169 note) supposes, no doubt rightly, that the Egyptians habitually used these figures for tiles.

1 The Eudemian summary expressly says that Thales "attacked some questions in their general form" (xa0oλικώτερον).

GREEK GEOMETRY TO EUCLID.

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mathematician, writing for mathematicians, is perhaps entitled, and may even be required, to fill up with his own opinions the gaps in his evidence. But his theories, however ingenious, are necessarily of such a kind that even a non-mathematical reader can see that they are, for the most part, imaginary, and a mathematician will think he can make better for himself. A history, like this, of which the utility will no doubt vary as the brevity, had best omit long and inconclusive discussions. Suffice it then to say, of Thales, that he certainly introduced geometry to the Greeks, that he probably improved upon Egyptian geometry by teaching more particularly of lines than of areas, and by giving deductive instead of inductive proofs, and that at any rate he formed a school which derived from him its subjects and methods of inquiry, its belief in the stability of natural laws, its tradition of the beauty and utility of the intellectual life1.

87. The Eudemian summary names, immediately after Thales, Mamercus, the brother of the poet Stesichorus, as one of the founders of Greek geometry. Nothing more is known of this person, and his name itself is exceedingly doubtful. Stesichorus lived in Sicily, and died about 560 B.C. nevertheless may have been a pupil of Thales, for it is difficult Mamercus to imagine how he could have learnt any geometry in Sicily at that time. However this may be, Thales undoubtedly had some pupils (e.g. Mandryatus of Priene) whom the Eudemian summary does not mention. Thales, Anaximander of Miletus, became very famous. He Another pupil of was born about 611 B.C., and died about 545 B.C.3. He also, like Thales, devoted himself mainly to physical speculations and to astronomy. It has been already mentioned that he first introduced the gnomon and the polos or sundial into Greece1.

1 Thales apparently composed some astronomical treatise in verse, but the authorities on his writings are conflicting. See Bretschneider § 39, pp. 54, 55.

2 Apuleius, Florida, iv. n. 18, ed. Hildebr. p. 88, ed. Delphin. p. 817. Bretschneider, pp. 53, 56.

G. G. M.

3 Diog. Laert. II. c. 1.

4 The gnomon was an upright staff placed in the centre of three concentric circles, so that at the summer solstice its shadow at noon just reached the inner circle, at the equinoxes the middle, at the winter solstice the outer. Afterwards in places, of which the meridian

10

Simplicius also relates (in Ar. de Coelo, ed. Brandis, p. 497 a), on the authority of Eudemus, that Anaximander ascertained the relative sizes and distances of the planets: and Diogenes states that he first constructed terrestrial and celestial globes1. These facts favour a presumption that Anaximander also was greatly interested in geometry, and Suidas, in particular, attributes to him a work entitled ὑποτύπωσις τῆς γεωμετριάς, which would seem to mean ‘a collection of figures illustrative of geometry. Pliny (H. N. II. c. 76) as was mentioned above (p. 67 n.), attributes the introduction of the gnomon to the younger philosopher, Anaximenes, who lived B. C. 570-499, and there may be some confusion between him and Anaximander. Nothing is known of any geometrical work by Anaximenes, and the same might be said of the more famous Anaxagoras of Clazomenae, (B. C. 500-428) were it not that the Eudemian summary expressly mentions him as a geometer; that Plutarch (de exilio, c. 17), relates that when in prison he wrote a treatise on quadrature of the circle, and that Vitruvius (vii. praef.), ascribes to him a work on perspective.

88. We may add finally to the Ionic school, with which he seems to have had most affinity, Enopides of Chios, a contemporary perhaps of Anaxagoras, or according to the Eudemian summary, a little later. Of him Diodorus, as quoted above (p. 131), relates that he studied in Egypt. He was certainly devoted chiefly to astronomy; and Elian (Var. Hist. x. 7), says that he invented a "great year" of 59 years, that is, a period at the end of which, according to his observations, the lunar and solar years would exactly coincide. He was however interested in geometry, and Proclus attributes to him the was known, the circles were omitted and three spots, marked on the meridian line, were substituted. The polos can hardly have been similar to our sundials, but was probably a staff placed in the centre of six concentric circles, such that every two hours the shadow of the staff passed from one circle to the next. Bretschneider, p. 60. Cantor, p. 92.

1 His fellow-townsman, Hecataeus,

4

made about the same time the first тар.

2 Anaxagoras lived, in his later years, with Pericles at Athens.

3 Censorinus c. 18, says that a "great year" of this length was attributed also to Philolaus, the Pythagorean. See the note to Elian in Gronovius' ed. II. p. 655.

4 Ed. Friedlein, pp. 283 and 333. Eudemus is cited in the latter passage.

solution of two problems, 'To draw a straight line perpendicular to a given straight line of unlimited length, from a given point without it' (Euclid I. 12.) and 'At a given point in a given straight line to make a rectilineal angle equal to a given rectilineal angle' (Euclid I. 23). On the first of these, Proclus' note is curious and worth quoting. He says, "Enopides first invented this problem, thinking it useful for astronomy. He calls the perpendicular (κá@ETOS) in the antique manner a 'gnomon,' because the gnomon is at right angles (πρòs ỏρlás) to the horizon, and the line drawn is at right angles to the given line, differing in plane only (T σxéσe), but not in principle (κατὰ τὸ ὑποκείμενον).”

It is plain enough from these scanty facts and from their scantiness, that the Ionic school did not, in nearly two hundred years, do anything like what might have been expected for the advancement of geometry. It introduced the study, kept it alive, and by working at astronomy, opened up a vast field of research, to which geometry soon became essential. The progress of geometry itself, however, was due mainly to the Pythagoreans in Italy.

(c.) The Pythagoreans.

89. Pythagoras, the son of Mnesarchus, was born in Samos, probably about 580 B. C. The date of his birth, however, and the other facts of his biography are the subject of disputes, which, owing to the nature of the evidence, can never be satisfactorily settled. The following summary statement perhaps excludes most of the very doubtful matter. Pythagoras was at first the pupil of Pherecydes of Syros1, but afterwards visited Thales, and was by him incited to study in Egypt, particularly at Memphis or Diospolis (Thebes). In pursuance of this

1 Pherecydes is said (Suidas, s. v. Pliny H. N. VII. 56) to have been the first writer of prose. He is also said

to have introduced the doctrine of metempsychosis, which Pythagoras a

dopted. See Ritter and Preller, Hist. Philos. c. II. § 92.

2 Iamblichus (Vita Pyth. c. 2) is the authority for this statement, which is not intrinsically improbable.