Page:Encyclopædia Britannica, Ninth Edition, v. 20.djvu/152

140 PYTHAGORAS but that astronomy (17 (r^xupiKTj) contemplates continued quantity so far as it is of a self -motive nature, (i) Diogenes Laertius (viii. 25) states, on the authority of Favorinus, that Pythagoras "employed definitions in the mathema- tical subjects to which he applied himself." The following notices of the geometrical work of Pytha- goras and the early Pythagoreans are also preserved. (1) The Pythagoreans define a point as "unity having position" (Procl., op. cit., p. 95). (2) They considered a point as analogous to the monad, a line to the duad, a superficies to the triad, and a body to the tetrad (ib., p. 97). (3) They showed that the plane around a point is completely filled by six equilateral triangles, four squares, or three regular hexagons (ib., p. 305). (4) Eudemus ascribes to them the discovery of the theorem that the interior angles of a triangle are equal to two right angles, and gives their proof, which was substantially the same as that in Euclid I. 32 l (ib., p. 379). (5) Proclus informs us in his comment- ary on Euclid I. 44 that Eudemus says that the problems concerning the application of areas where the term "application" is not to be taken in its restricted sense (Tra.pa.(3ofy, in which it is used in this proposition, but also in its wider signification, embracing virfp/BoX^ and lAAei^is, in which it is used in Book VI. Props. 28, 29 are old, and inventions of the Pythagoreans 2 (ib., p. 419). (6) This is confirmed by Plutarch, 3 who says, after Apollodorus, that Pythagoras sacrificed an ox on finding the geometrical diagram, either the one relating to the hypotenuse, namely, that the square on it is equal to the sum of the squares on the sides, or that relating to the problem concerning the application of an area. 4 (7) Plutarch 5 also ascribes to Pythagoras the solution of the problem, To construct a figure equal to one and similar to another given figure. (8) Eudemus states that Pythagoras discovered the construction of the regular solids (Procl., op. cit., p. 65). (9) Hippasus, the Pythagorean, is said to have perished in the sea on account of his impiety, inasmuch as he boasted that he first divulged the know- ledge of the sphere with the twelve pentagons (the in- scribed ordinate dodecahedron): Hippasus assumed the glory of the discovery to himself, whereas everything be- longed to Him " for thus they designate Pythagoras, and do not call him by name." 6 (10) The triple interwoven triangle or pentagram star -shaped regular pentagon was used as a symbol or sign of recognition by the Pytha- 1 We learn, however, from a fragment of Geininus, which has been handed down by Eutocius in his commentary on the Conies of Apol- lonius (Apoll., C'onica, ed. Halleius, p. 9), that the ancient geometers observed two right angles in each species of triangle, in the equilateral first, then in the isosceles, and lastly in the scalene, whereas later writers proved the theorem generally thus " The three internal angles of every triangle are equal to two right angles." 2 The words of Proclus are interesting. " According to Eudemus the inventions respecting the application, excess, and defect of areas are ancient, and are due to the Pythagoreans. Moderns, borrowing these names, transferred them to the so-called conic lines, the parabola, the hyperbola, the ellipse, as the older school, in their nomenclature concerning the description of areas in piano on a finite right line, re- garded the terms thus : An area is said to be applied (irapa^deLv) to a given right line when an area equal in content to some given one is described thereon ; but when the base of the area is greater than the given line, then the area is said to be in excess (vwfpj3<ieiv) ; but when the base is less, so that some part of the given line lies without the described area, then the area is said to be in defect (AXehre(i'). Euclid uses in this way in his sixth book the terms excess and defect. . . . The term application (irapa.pdeit>), which we owe to the Pytha- goreans, has this signification." 3 J.VOTI posse suaviter vivi sec. JEpicurum, c. xi. 4 Efre Trp6^rifjM wepl rou x u pt v T ^ s Tapa/3o?)j. Some authors, rendering the last five words "concerning the area of the parabola," have ascribed to Pythagoras the quadrature of the parabola, which was one of the great discoveries of Archimedes. 5 Symp. viii., Quaest. 2, c. 4. 6 lamblichus, De Vit. Pyth., c. 18, s. 88. goreans and was called by them "health" (vyieia). 7 (11) The discovery of the law of the three squares (Euclid I. 47), commonly called the "theorem of Pythagoras," is attributed to him by many authorities, of whom the oldest is Vitruvius. 8 (12) One of the methods of finding right- angled triangles whose sides can be expressed in numbers (Pythagorean triangles) that setting out from the odd numbers is referred to Pythagoras by Heron of Alex- andria and Proclus. 9 (13) The discovery of irrational quantities is ascribed to Pythagoras by Eudemus (Procl., op. cit., p. 65). (14) The three proportions arithmetical, geometrical, and harmonica I were known to Pythagoras. 10 (15) lamblichus 11 says, "Formerly, in the time of Pytha- goras and the mathematicians under him, there were three means only the arithmetical, the geometrical, and the third in order which was known by the name sub-contrary (vTTfvavTia), but which Archytas and Hippasus designated the harmonical, since it appeared to include the ratios concerning harmony and melody." (16) The so-called most perfect or musical proportion, e.g., 6 : 8 : : 9 : 12, which comprehends in it all the former ratios, according to lamblichus, 12 is said to be an invention of the Baby- lonians and to have been first brought into Greece by Pythagoras. (17) Arithmetical progressions were treated by the Pythagoreans, and it appears from a passage in Lucian that Pythagoras himself had considered the special case of triangular numbers : Pythagoras asks some one, " How do you count ? " he replies, " One, two, three, four." Pythagoras, interrupting, says, " Do you see 1 what you take to be four, that is ten and a perfect triangle and our oath." 13 (18) The odd numbers were called by the Pytha- goreans "gnomons," 14 and were regarded as generating, in- 7 Lucian, Pro Lapsu in Salut., s. 5 ; also schol. on Aristoph., Nub., 611. That the Pythagoreans used such symbols we learn from lamblichus (De Vit. Pyth., c. 33, ss. 237 and 238). This figure is referred to Pythagoras himself, and in the Middle Ages was called Pythagorse figura ; even so late as Paracelsus it was regarded by him as a symbol of health. It is said to have obtained its special name from the letters v, y, i, 6 ( = )> a having been written at its prominent vertices. 8 De Arch., ix., Pnef., 5, 6, 7. Amongst other authorities are Diogenes Laertius (viii. 11), Proclus (op. cit., p. 426), and Plutarch (ut sup., 6). Plutarch, however, attributes to the Egyptians the know- ledge of this theorem in the particular case where the sides are 3, 4, and 5 (De Is. et Osir. , c. 56). 9 Heron Alex., Geom. et Stereom. Rel., ed. F. Hultsch, pp. 56, 146; Procl., op. cit., p. 428. The method of Pythagoras is as follows : he took an odd number as the lesser side ; then, having squared this number and diminished the square by unity, he took half the remainder as the greater side, and by adding unity to this number he obtained the hypotenuse, e.g., 3, 4, 5 ; 5, 12, 13. 10 Nicom. Ger., Introd. Ar., c. xxii. 11 In Nicomachi Arithmeticam, ed. S. Tennulius, p. 141. 12 Op. cit., p. 168. As an example of this proportion Nicomachus and, after him, lamblichus give the numbers 6, 8, 9, 12, the harmonical and arithmetical means between two numbers forming a geometric proportion with the numbers themselves ( a : r : : -jr : b . lam- blichus further relates (I.e.) that many Pythagoreans made use of this proportion, as Aristaeus of Crotona, Timaeus of Locri, Philolaus and Archytas of Tarentnm, and many others, and after them Plato in his Timeeus (see Nicom., Inst. Arithm., ed. Ast, p. 153, and Animad- versiones, pp. 327-329 ; and Iambi., op. cit., p. 172 sq.). 13 Tttwv irpcicris, 4, vol. i. p. 317, ed. C. Jacobitz. 14 TvufjLuv means that by which anything is known, or "criterion "; its oldest concrete signification seems to be the carpenter's square (norma) by which a right angle is known. Hence it came to denote a per- pendicular, of which, indeed, it was the archaic name (Proclus, op. cit., p. 283). Gnomon is also an instrument for measuring altitudes, by means of which the meridian can be found ; it denotes, further, the index or style of a sun-dial, the shadow of which points out the hours. In geometry it means the square or rectangle about the diagonal of a square or rectangle, together with the two complements, on account of the resemblance of the figure to a carpenter's square ; and then, more generally, the similar figure with regard to any parallelogram, as defined by Euclid II. Def. 2. Again, in a still more general signification, it means the figure which, being added to any figure, preserves the original form. See Heron, Definitiones (59). When