number, the product of equals. More legitimate is their application of number to geometry, according to which “one" was identified with the point, “two" with the line, “three" with the surface, and “four" with the cube. In the history of music the Pythagorean school is also of considerable importance from the development which the theory of the octave owes to its members; according to some accounts the discovery of the harmonic system is due to Pythagoras himself. As already mentioned, the movements of the heavenly bodies formed for i. Pythagoreans an illustration on a grand scale of the truth of their theory. Their cosmological system is also interesting on account of peculiarities which mark it out from the current conceptions of antiquity and bring it curiously near to the modern theory. Conceiving the universe, like many early thinkers, as a sphere, they placed in the heart of it the central fire, to which they gave the name of Hestia, the hearth or altar of the universe, the citadel or throne of Zeus. Around this move the ten heavenly bodies—farthest off the heaven of the fixed stars, then the five planets known to antiquity, then the sun, the moon, the earth, and lastly the counter-earth (dwrix0av), which revolves between the earth and the central fire and thus completes the sacred decade. Revolving along with the earth, the last-mentioned body is always interposed as a shield between us and the direct rays of the central fire. Our light and heat come to us indirectly by way of reflexion from the sun. When the earth is on the same side of the central fire as the sun, we have day; when it is on the other side, night. This attribution of the changes of day and night to the earth's own motion led up directly to the true theory, as soon as the machinery of the o fire and the counter-earth was dispensed with. The counter-earth became the western hemisphere, and the earth revolved on its own axis instead of round an imaginary centre. Put, as appears from the above, the Pythagorean astronomy is also remarkable as having attributed a planetary motion to the earth instead of making our globe the centre of the universe. Long afterwards, when the ... condemned the theory of Copernicus, the indictment that lay against it was its heathen and “I’ythagorean " character. The doctrine which the memory of mankind associates most closely with Pythagoras's name is that of the transmigration of souls —Merewisychosis (q.v.). Though evidently of great importance for Pythagoras himself, it does not stand in any very obvious connexion with his philosophy proper. He seems to have adopted the idea from the Orphic Mysteries. The bodily life of the soul, according to this doctrine, is an imprisonment suffered for sins committel in a former state of existence. At death the soul reaps what it has sown in the present life. The reward of the best is to enter the cosmos, or the W. and purer regions of the universe, while the lirest crimes receive their punishment in Tartarus. 13ut the general lot is to live afresh in a series of human or animal forms, the nature of the bodily prison being determined in each case by the loods done in the life just ended. This is the same doctrine of retribution and purificatory wandering which meets us in Plato's mythical descriptions of a future life. They are borrowed by him in their substance from the Pythagoreans or from a common source in the Mysteries. In accordance with this religious view of life as a stage of probation were the ethical precepts of the shool, insulating reverence towards the gods and to parents, justice, gentleness, temporance, purity of life, prayer, regular self-examination, and the observance of various ritual requirements. tonnecting its ethics in this way with religion and the idea of a future lif., the Pythagorean societies had in them from the boginning a gorm of asceticism and contemplative mysticism which it was left for a later age fully to develop. The l’ythagorean life w is destinel to survive the peculiar doctrines of the l’ythagoon philosophy and to be grafted on later philosophio ideas. The i-..tion which characterized it appears in the oth century 1:..". in one connexion with the Orphic Mysteries; and the “I’ythagoreans" of that time are frequently the butts of the New Athenian tomoly. In the Alexandrian period the Pythagorean tradition -: nek deeper roots: in ol. and elsewhere schools of men cro- alling themselves Pythagoreans, but more accurately lison suished by modern criticism as Neopythagoreans, seeing that "h ir philosophical doctrines are evidently delived in varying proo from Plato, Aristotle, and the Stoics. In general it may - aid that they develop the mystic side of the Platonic dotrine : an only so far as this is connected with the similar speculations of Pythigoras can they claim to be followers of the latter. Hono mon like Plutarch, who personally prefer to all themselves Platanists, may also be considered as within the scope of this Pythagorean revival. The link that really connects these Neoythagoreans with the Samian philosopher and distinguish's thon, }. the other schools of their time is their ascetic ideal of life and the ur preo eupation with religion. In religious speculation they pavel the way for the Neoplatonic conception of God as immeasurably transceniling the world; and in their thirst for ..". oracles, and signs they gave expression to the prevalent, longing for a supernatural revelation of the divine nature and will. The asceticism of the Jewish sect of the Essenes seems, as Zeller contends, to be due to a strong infusion of Neopythagorean elements. At a still later period Neopythagoreanism set up Pythagoras and Apollonius of Tyana not only as ideals of the philosophic life but also as prophets and wonder-workers in immediate communication with another world, and in the details of their “lives” it is easy to read the desire to emulate the narrative of the Gospels. The Life of Apollonius by Philostratus, which is for the most part an historical romance, belongs to the 3d Christian century. Zeller's discussion of Pythagoreanism, in his Philosophie d. Griechen, book i., is very full ; he also deals at considerable length in the last volume of the work with the Neopythagoreans, considered as the precursors of Neoplatonism and the probable origin of the Essenes. The numerous monographs dealing with special parts of the subject are there examined and sisted. (A. S.E.) Pythagorean Geometry. As the introduction of geometry into Greece is by common consent attributed to Thales, so all are agreed that to Pythagoras is due the honour of having raised mathematics to the rank of a science. We know that the early Pythagoreans published nothing, and that, moreover, they referred all their discoveries back to their master. (See PHILoLAUs.) Hence it is not possible to separate his work from that of his early disciples, and we must therefore treat the geometry of the early Pythagorean school as a whole. We know that Pythagoras made numbers the basis of his philosophical system, as well physical as metaphysical, and that he united the study of geometry with that of arithmetic. The following statements have been handed down to us. (a) Aristotle (Jset., i. 5, 985) says “the Pythagoreans first applied themselves to mathematics, a science which they improved; and, penetrated with it, they fancied that the principles of mathematics were the principles of all things.” (b) Eudemus informs us that “Pythagoras changed geometry into the form of a liberal science, regarding its principles in a purely abstract manner, and investigated its theorems from the immaterial and intellectual point of view (di’Asos kui vospios).”” (c) Diogenes Laertius (viii. 11) relates that “it was Pythagoras who carried geometry to perfection, after Mooris - had first found out the principles of the elements of that science, as Anticlides tells us in the second book of his //istory of .1/orander ; and the part of the science to which Pythagoras applied himself above all others was arithmetic.” (7) According to Aristoxenus, the musician, Pythagoras seems to have esteemed arithmetic above everything, and to have advanced it by diverting it from the service of commerce and by likening all things to numbers.” (, ) I)iogenes Laertins (viii. 13) reports on the same authority that Pythagoras was the first person who introduced measures and weights among the Greeks. (f) He discovered the numerical relations of the musical scale (l)ing. Laurt., viii. 1 l). (of) I'rolus' says that “the word mathematics' originated with the Pythagoreans.” (h) We learn also from the same authority " that the Pythagoreans made a fourfold division of mathematical science, attributing one of its parts to the “how many" (Ti, Zoror) and the other to the “how much” (r. 7] Aikor); and they assign, d to each of these parts a twofold division. They said that discrete quantity or the “how many ” is either als, oute or relative, and that continued quantity or the “how much is either stable or in motion. Hence they laid down that arithmetic contomplates that discrete quantity which subsists ly itself. lut music that which is relat, d to another; and that on try considers continued quantity so far as it is immoval!", but that astronomy () orbatpukij) 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 mathematical subjects to which he applied himself.” The following notices of the geometrical work of Pythagoras 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" (ib., p. 379). (5) Proclus informs us in his commentary 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 (Tapas}oAs), in which it is used in this proposition, but also in its wider signification, embracing Teps30Ås, and &\Aeolus, in which it is used in Book VI. Props. 28, 29 —are old, and inventions of the Pythagoreans” (ib., p. 419). (6) This is confirmed by Plutarch,” 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.” (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. (S) 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 knowledge of the sphere with the twelve pentagons (the inscribed ordinate dodecahedron): Hippasus assumed the glory of the discovery to himself, whereas everything belonged to Him—“for thus they designate Pythagoras, and do not call him by name.” (10) The triple interwoven triangle or pentagram—star-shaped regular pentagon— was used as a symbol or sign of recognition by the Pytha * We learn, however, from a fragment of Geminus, which has been handed down by Eutocius in his commentary on the Comics of Apollonius (Apoll., Comica, 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.” * The words of Proclus are interesting. “According to Eudemus the inventions respecting the application, ercess, 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 arcas in plano on a finite right line, regarded the terms thus:–An area is said to be applied (TapašáN\cup) 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 eccess (trepSá\\ew); 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 (é\\ettreuv). Euclid uses in this way in his sixth book the terms eccess and defect. . . . The term application (Tapagá\\euv), which we owe to the Pythagoreans, has this signification.” * Von posse suaviter vivi sec. Epicurum, c. xi. * Eire Tpó3\mua Tepl ros, Xoptov Tijs trapagoNos. 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. * Symp. viii., Quaest. 2, c. 4. " Iamblichus, De Wit. Pyth., c. 18, s. 88. goreans and was called by them “health” (äyteia).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.” (12) One of the methods of finding rightangled 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.” (13) The discovery of irrational quantities is ascribed to Pythagoras by Eudemus (Procl., op. cit., p. 65). (14) The three proportions—arithmetical, geometrical, and harmonical—were known to Pythagoras.” (15) Iamblichus" says, “Formerly, in the time of Pythagoras 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 (Tevavría), 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 Iamblichus,” is said to be an invention of the Babylonians 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? what you take to be four, that is ten and a perfect triangle and our oath.” (18) The odd numbers were called by the Pythagoreans “gnomons,” and were regarded as generating, in * Lucian, Pro Lapsu in Salut., s. 5; also schol. on Aristoph., Yub., 611. That the Pythagoreans used such symbols we learn from Iamblichus (1)e I’lt. Pyth., c. 33, ss. 237 and 238). This figure is referred to Pythagoras himself, and in the Middle Ages was called Pythagora 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, t, 0 (=et), a having been written at its prominent vertices. * De Arch., ix., Praef., 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 knowledge of this theorem in the particular case where the sides are 3, 4, and 5 (I)e Is. et Osir., c. 56). * Heron Alex., Geom. et Stercom. 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. * Nicom. Ger., Introd. Ar., c. xxii. * In Vicomachi Arithmeticam, ed. S. Tennulius, p. 141. * Op. cit., p. 168. As an example of this proportion Nicomachus and, after him, Ianblichus give the numbers 6, 8, 9, 12, the harmonical and arithmetical means between two numbers forming a geometric proportion with the numbers themselves (a. #: .";”. •) . Iamblichus further relates (l.c.) that many Pythagoreans made use of this proportion, as Aristasus of Crotona, Timaeus of Locri, Philolaus and Archytas of Tarentum, and many others, and after them Plato in his Timaeus (see Nicom., Inst. Arithm., ed. Ast, p. 153, and Animadversiones, pp. 327-329; and Iambl., op. cit., p. 172 sq.). * Bowy Tpaats, 4, vol. i. p. 317, ed. C. Jacobitz. * I'viouw 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 perpendicular, 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 - asmuch as by the addition of successive gnomons—consisting each of an odd number of unit squares—to the original square unit or monad the square form was preserved. (19) In like manner, if the simplest oblong (śrepôpokes), consisting of two unit squares or monads injuxtaposition, be taken and four unit squares be placed about it after the manner of a gnomon, and then in like manner six, eight . . . unit squares be placed in succession, the oblong form will be preserved. (20) Another of his doctrines was, that of all solid figures the sphere was the most beautiful, and of all plane figures the circle." (21) According to Iamblichus the Pythagoreans are said to have found the quadrature of the circle.” On examining the purely geometrical work of Pythagoras and his early disciples, as given in the preceding extracts, we observe that it is much concerned with the geometry of areas, and we are indeed struck with its Egyptian character. This .. in the theorem (3) concerning the filling up a plane with regular figures— for floors or walls covered with tiles of various colours were common in Egypt; in the construction of the regular solids (8), for some of them are found in Egyptian architecture; in the problems concerning the application of areas (5); and lastly, in the theorem of Pyt ras (11), coupled with his rule for the construction of rightangled triangles in numbers (12). We learn from Plutarch that the o: were acquainted with the geometrical fact that a triangle whose sides contain three, four, and five parts is rightangled, and that the square of the greatest side is equal to the squares of the sides containing the right angle. It is probable too that this theorem was known to them in the simple case where the right-angled triangle is isosceles, inasmuch as it would be at once suggested by the contemplation of a floor covered with square tiles —the square on the diagonal and the sum of the squares on the sides contain each four of the right-angled triangles into which one of the squares is divided by its diagonal. It is easy now to see how the problem to construct a square which shall be equal to the sum of two squares could, in some cases, be solved numerically. From the observation of a chequered board it would be perceived that the element in the successive formation of squares is the gnomon or carpenter's square. Each gnomon consists of an odd number of squares, and the successive gnomons o to the successive odd numbers, and include, therefore, all odd squares. Suppose, now, two squares are given, one consisting of sixteen and the other of nine unit squares, and that it is proposed to form from them another square. It is evident that the square consisting of nine unit squares can take the form of the fourth gnomon, which, being placed round the former square, will generate a new o containing twenty-five unit squares. Similarly it may have been observed that the twelfth gnomon, consisting of twenty-five unit slunres, could be transformed into a square each of whose sides contains five units, and thus it may have been seen conversely that the latter square, by taking the gnomonic or generating form with respect to the square on twelve units as base, would produce the square of thirteen units, and so on. This method required only to be generalized in order to enable Pythagoras to arrive at his rule for finding right-angled triangles whose sides can be expressed in numbers, joi. we are told, sets out from the odd numbers. The nth square together with the nth gnomon forms the (n + 1}th square; if the nth gnomon contains mo unit squares, on being an 2 rule of Pythagoras. The general proof of Fuelil I. 47 is attributed to Pythagoras, but we have the express statement of Proclus (op. cit., P. 426) that this theorem was not proved in the first instance as it is in the Elements. The following simple and natural way of arriving at the theorem is suggested by Bretschneider after Camerer.” A stuare can be dissected into the sum of two squares and two equal octangles, as in Euclid II. 4; these two rectangles can, by drawing their diagonals, be decomposed into four equal right-angled triangles, the sum of the sides of each being equal to the side of the sluare; again, these four right-angled triangles can be placed so that a vertex of each shall be in one of the corners of the square in such a way that a greater and less side are in continuation. l, which gives the The original square is thus dissected into the four triangles as before and the figure within, which is the square on the hypotenuse. This square, therefore, must be equal to the sum of the squares on the sides of the right-angled triangle. It is well known that the Pythagoreans were much occupied with the construction of regular . and solids, which in their cosmology played an essential part as the fundamental forms of the elements of the universe. . We can trace the origin of these mathematical speculations in the theorem (3) that “the plane around a point is completely filled by six equilateral triangles, four squares, or three regular hexagons.” Plato also makes the Pythagorean Timaeus explain—“Each straight-lined figure consists of triangles, but all triangles can be dissected into rectangular ones which are either isosceles or scalene. Among the latter the most beautiful is that out of the doubling of which an equilateral arises, or in which the square of the greater perpendicular is three times that of the smaller, or in which the smaller perpendicular is half the hypotenuse. But two or four .." isosceles triangles, properly put together, form the square; two or six of the most beautiful scalene right-angled triangles form the equilateral triangle; and out of these two figures arise the solids which correspond with the four elements of the real world, the tetrahedron, octahedron, icosahedron, and the cube" " (Timaeus, 53, 54, 55). The construction of the regular solids is distinctly ascribed to Pythagoras himself by Eudemus (8). Of these five solids three—the tetrahedron, the cube, and the octahedron—were known to the Egyptians and are to be found in their architecture. Let us now examine what is required for the construction of the other two solids the icosahedron and the dodecahedron. In the formation of the tetrahedron three, and in that of the octahedron four, equal equilateral triangles had been placed with a common vertex and adjacent sides coincident; and it was known that is six such triangles were placed round a common vertex with their adjacent sides coincident, they would lie in a plane, and that, therefore, no solid could be formed in that manner from them. It remained, then, to try whether five such equilateral triangles could be placed at a common vertex in like manner; on trial it would be found that they could be so placed, and that their bases would form a regular pentagon. The existence of a regular pentagon would thus become known. It was also known from the formation of the cube that three squares could be placed in a similar way with a common vertex; and that, further, if three equal and regular hexagons were placed round a point as common vertex with adjacent sides coincident, they would form a plane. It remained in this case too only to try whether three equal regular Pentagons could be placed with a common vertex and in a similar way ; this on trial would be found possible and would lead to the construction of the regular dodecahedron, which was the regular solid last arrived at. We see that the construction of the regular pentagon is required for the formation of each of these two regular solids, and that, therefore, it must have been a discovery of Pythagoras. If we examine now what knowledge of geometry was required for the solution of this problem, we shall see that it depends on Euclid IV. 10, which is .. to Euclid II. 11, which problem is reduced to the following: To produce a given straight line so that the rectangle under the whole line thus produ'rol and the produced part shall be equal to the square on the given line, or, in the language of the ancients, To o to a given straight line a rectangle which shall be equal to a given area – in this case the square on the given line—and which shall be crossiro by a square. Now it is to be observed that the problem is solved in this manner by Euclid (VI. 30, 1st method and that we know on the authority of Eulemus that the problems oncerning the application of areas and their cross and dos et are old, and inventions of the Pythagoreans (5). Hence the statements of Iambliehus concerning Hippasus 9 – that he divulged the sphere with the twelve pentagons—and of Lucian and the scholiast on Aristophanes 10–that the pontagram was used as a symbol of recognition amongst the Pythagoreans —become of greater importance. Further, the dis overy of irrational magnitudes is ascribed to Pythagoras by Eudemus loo, and this list overy has been ever regarded as one of the greatest of antiquity. It is commonly assumed that Pythagoras was led to this theory from the consideration of the isosceles right-angled triangle. It seems to the present writer, however, more probable that the dis overy of in or surable magnitudes was rather owing to the poll in line in extreme and mean ratio. From the solution of this pioiom it follows at once that, if on the greater segment of a line so out a loart be taken colual to the loss, the greater segment, regard-l as a new line, will be out in a similar manner; and this to o can be continued without end. On the other hand, if a sin.iiar method be adopted in the case of any two lines whi. h. van be represented numerically, the process would end. Hence wouil arise * The dodecahedron was assigned to the fifth element, quinto 1-1 is, same surface the sphere is the greatest. We must also deny to Pythagoras and his school a knowledge of the conic sections, and in particular of the quadrature of the parabola, attributed to him by some authors; and we have noticed the misconception which gave rise to this erroneous inference. Let us now see what conclusions can be drawn from the foregoing examination of the mathematical work of Pythagoras and his school, and thus form an estimate of the state of geometry about 480 B.C. First, as to matter. It forms the bulk of the first two books of Euclid, and includes a sketch of the doctrine of proportion—which was probably limited to commensurable magnitudes— together with some of the contents of the sixth book. It contains too the discovery of the irrational (&Aoyov) and the construction of the regular solids, the latter requiring the description of certain regular polygons—the foundation, in fact, of the fourth book of Euclid. Secondly, as to form. The Pythagoreans first severed geometry from the needs of practical life, and treated it as a liberal science, giving definitions and introducing the manner of proof which has ever since been in use. Further, they distinguished between discrete and continuous quantities, and regarded geometry as a branch of mathematics, of which they made the fourfold division that lasted to the Middle Ages—the quadrivium (fourfold way to knowledge) of Boetius and the scholastic philosophy. And it may be observed that the name of “mathematics,” as well as that of “philosophy,” is ascribed to them. Thirdly, as to method. One chief characteristic of the mathematical work of Pythagoras was the combination of arithmetic with geometry. The notions of an equation and a proportion—which are common to both, and contain the first germ of algebra—were introduced among the Greeks by Thales. These motions, especially the latter, were elaborated by Pythagoras and his school, so that they reached the rank of a true scientific method in their theory of proportion.* To Pythagoras, then, is due the honour of having supplied a method which is common to all branches of mathematics, and in this respect he is fully comparable to Descartes, to whom we owe the decisive combination of algebra with geometry. See C. A. Bretschneider, Die Geometric u. die Geometer vor EuŽlides (Leipsic, 1870); II. Hankel, Zur Geschichte der Mathematik (Leipsic, 1874); F. Hoefer, Histoire des Mathématiques (Paris, 1874); G. J. Allman, “Greek Geometry from Thales to Euclid,” in Hermathcna, Nos. v., vii., and x. (Dublin, 1877, 1881, and 1884); M. Cantor, Porlesungen über Geschichte der Mathematik (Leipsic, 1880). The recently published Short History of Greek Mathematics by James Gow (Cambridge, 1884) will be found a convenient compilation. (G. J. A. PYTHEAS of Massilia was a celebrated Greek navigator and geographer, to whom the Greeks appear to have been indebted for the earliest information they possessed, of at all a definite character, concerning the western regions of Europe, and especially the British Islands. The period at which he lived cannot be accurately determined; but it is certain that he wrote, not only before Eratosthenes, who relied much upon his authority, but before Dicaearchus, who was a pupil of Aristotle, and died about 285 B.C. IIence he may probably be regarded as about contemporary with Alexander the Great. His work is now wholly lost, and appears to have been consulted in the original by comparatively few ancient writers, most of the the distinction between commensurable and incommensurable quantities. A reference to Euclid X. 2 will show that the method above is the one used to prove that two magnitudes are incommensurable; and in Euclid X. 3 it will be seen that the greatest common measure of two commensurable magnitudes is found by this process of continued subtraction. . It seems probable that Pythagoras, to whom is attributed one of the rules for representing the sides of right-angled triangles in numbers, tried to find the sides of an isosceles right-angled triangle numerically, and that, failing in the attempt, he suspected that the hypotenuse and a side had no common measure. He may have demonstrated the incommensurability of the side of a square and its diagonal. The nature of the old proof–which consisted of a reductio ad absurdwm, showing that, if the diagonal be commensurable with the side, it would follow that the same number would be odd and even 1–makes it more probable, however, that this was accomplished by his successors. The existence of the irrational as well as that of the regular dodecahedron appears to have been regarded by the school as one of their chief discoveries, and to have been preserved as a secret; it is remarkable, too, that a story similar to that told by Iamblichus of Hippasus is narrated of the person who first published the idea of the irrational, namely, that he suffered shipwreck, &c.” Eudemus ascribes the problems concerning the application of figures to the Pythagoreans. The simplest cases of the problems, ouclid VI. 28, 29—those, namely, in which the given parallelogram is a square—correspond to the problem : To cut a given straight line internally or externally so that the rectangle under the segments shall be equal to a given rectilineal figure. The solution of this problem—in which the solution of a quádratic equation is implicitly contained—depends on the problem, Euclid II. 14, and the theorems, Euclid II. 5 and 6, together with the theorem of Pythagoras. It is probable that the finding of a mean proportional between two given lines, or the construction of a square which shall be equal to a given rectangle, is due to Pythagoras himself. The solution of the more general problem, Euclid VI. 25, is also attributed to him by Plutarch (7). The solution of this problem depends on that of the particular case and on the application of areas; it requires, moreover, a knowledge of the theorems: Similar rectilineal figures are to each other as the squares on their homologous sides (Euclid VI. 20); and, If three lines are in geometrical proportion, the first is to the third as the square on the first is to the square on the second. Now Hippocrates of Chios, about 440 B.C., who was instructed in geometry by the Pythagoreans, Yossessed this knowledge. We are justified, therefore, in ascribing the solution of the general problem, if not (with Plutarch) to Pythagoras, at least to his early successors. The theorem that similar polygons are to each other in the duplicate ratio of their homologous sides involves a first sketch, at least, of the doctrine of proportion and the similarity of figures.” That we owe the foundation and development of the doctrine of proportion to Pythagoras and his school is confirmed by the testimony of Nicomachus (14) and Iamblichus (15 and 16). From these passages it appears that the early Pythagoreans were acquainted, not only with the arithmetical and geometrical means between two magnitudes, but also with their harmonical mean, which was then called “subcontrary.” The Pythagoreans were much occupied with the representation of numbers by geometrical figures. These speculations originated with Pythagoras, who was o with the summation of the natural numbers, the odd numbers, and the even numbers, all of which are capable of geometrical representation. See the passage in Lucian (17) and the rule for finding Pythagorean triangles (12) and the observations thereon suprat. On the other hand, there is no evidence to support the statement of Montucla that Pythagoras laid the foundation of the doctrine of isoperimetry, by proving that of all figures having the same perimeter the circle is the greatest, and that of all solids having the * For this proof, see Euclid X. 117; see also Aristot., Analyt. Pr., i. c. 23 and c. 44. * Knoche, Untersuchungen üller die new aussefundenen Scholien des Proklus Diadochus zu Euclid's Elementen, pl.). 20 and 23, Herford, 1865. * It is agreed on all hands that these two theories were treated at length by Pythagoras and his school. It is almost certain, however, that the theorems arrived at were proved for commensurable magnitudes only, and were assumed to hold good for all. The Pythagoreans themselves seem to have been aware that their proofs were not rigorous, and were open to serious objection; in this we may have the explanation of the secrecy which was attached by them to the idea of the incommensurable and to the pentagram which involved, and indeed represented, that idea. Now it is remarkable that the doctrine of proportion is twice treated in the Elements of Euclid—first, in a general manner, so as to include incommensurables, in Book V., which tradition ascribes to Eudoxus, and then arithmetically in Book VII., which, as Hankel has supposed, contains the treatment of the subject by the older Pythagoreans. 4 Proportion was not regarded by the ancients merely as a branch of arithmetic. We learn from Proclus that “Eratosthenes looked on proportion as the bond of mathematics” (op. cit., p. 43). We are also told in an anonymous scholium on the Elements of Euclid, which Knoche attributes to Proclus, that the fifth book, which treats of proportion, is common to geometry, arithmetic, music, and, in a word, to all mathematical science. And Kepler, who lived near enough to the ancients to reflect the spirit of their methods, says that one part of geometry is concerned with the comparison of figures and quantities, whence proportion arises. He also adds that arithmetic and geometry afford mutual aid to each other, and that they cannot be separated. statements cited from it being confined to detached points, which may easily have been derived at second or even third hand. We are hence left almost wholly in the dark as to the form and character of the work itself, but the various titles under which it is cited by later writers point rather to a geographical treatise, in which he had embodied the results of his observations, than to a continuous narrative of his voyage like that of a modern navigator. Some modern writers have supposed Pytheas to have been sent out at the public expense, in command of an expedition organized by the republic of Massilia; but there is no ancient authority for this, and the statement of Polybius, who had unquestionably seen the original work, is express, that he had undertaken the voyage in a private capacity and with limited means. All that we know concerning the voyage of Pytheas (apart from such detached notices as those already referred to) is contained in a brief passage of Polybius, cited by Strabo, in which he tells us that Pytheas, according to his own statement, had not only visited Britain but had personally explored a large part of it, and stated its circumference at more than 40,000 stadia (4000 geographical miles). To this he added the account of Thule (which he placed six days' voyage to the north of Britain) and the adjoining regions, in which there was no longer any distinction between the air and earth and sea, but a kind of mixture of all three, forming a substance resembling the gelatinous mollusc known as the Pulmo marinus, which rendered all navigation and progress in any other mode alike impossible. This substance he had himself seen, but the other statements he derived from hearsay. Iteturning from thence he visited the whole of the coasts of Europe bordering on the ocean as far as the Tanais (Polyb. ap. Strab., ii. p. 104). This last sentence has led some modern writers to suppose that he made two different voyages; but this is highly improbable, and the expressions of Polybius certainly imply that his explorations in both directions, first towards the north and afterwards towards the east, formed part of one and the same voyage. The circumstance that the countries visited, and to a certain extent explored, by Pytheas were not only previously unknown to the Greeks—except perhaps by vague hearsay accounts received through the Phoenicians—but were not visited by any subsequent authority during a period of more than two centuries led some of the later Greek geographers altogether to disregard his statements, and even to treat the whole story of his voyage as a fiction. Eratosthenes, indeed, who wrote about a century after his time, was disposed to attach great value to his authority, though doubting some of his statements; but l'olybius, about half a century later, involved the whole in one sweeping condemnation, treating the work of Pytheas as a mere tissue of fables, like that of Euhemerus concerning Panchara; and even Strabo, in whose time the western regions of Europe were comparatively well known, adopted to a great extent the same view with Polybius. In modern times a more critical examination has arrived at a more favourable judgment, and, though Gossellin in his Recher'hes sur la Géofro.hie dos Anciens (vol. iv. pp. 168-180) and Sir G. C. Lewis in his History of Ancient ...tstronomy (pp. 466-481) revived the sceptical view, the tendency of modern critics has been rather to exaggerate than to depreciate the value of what was really added by Pytheas to geographical knowledge. The fact, is that our information concerning him is so imperfect, and the scanty notices preserved to us from his work at once so meagre and discordant, that it is very difficult to arrive at anything like a sound conclusion. as fairly established that Pytheas really made a voyage round the western coasts of Europe, proceeding from Gades, It may, however, be considered the shores of the German Ocean, without proceeding farther the great Phoenician emporium, and probably the farthest point familiar to the Greeks, round Spain and Gaul to the British Islands, and that he followed the eastern coast of Britain for a considerable distance to the north, obtaining information as to its farther extension in that direction which led him greatly to exaggerate its size. At the same time he heard vaguely of the existence of a large island to the north of it—probably derived from the fact of the groups of the Orkneys and Shetlands being really found in that position—to which he gave the name of Thule. No ancient writer (except a late astronomer, who merely refers to it in a passing notice and obviously at second hand) asserts that Pytheas had himself visited Thule; his account of the Sluggish Sea beyond it was, as stated by Polybius himself in the passage already cited, derived merely from hearsay. But the most important statement made by Pytheas in regard to this unknown land of Thule, and which has given rise to most controversy in modern times, was that connected with the astronomical phenomena affecting the duration of day and night in these remote arctic regions. Unfortunately the reports transmitted to us at second hand in our existing authorities differ so widely that it is almost impossible to determine what Pytheas himself really stated. It is, however, probable that the version given in one passage by Pliny (//..Y., iv. 16, 104) correctly represents his authority. According to this he reported as a fact that at the summer solstice the days were twenty-four hours in length, and conversely at the winter solstice the nights were of equal duration. Of course this would be strictly true had Thule really been situated under the arctic circle, which Pytheas evidently considered it to be, and his skill as an astronomer would thus lead him to accept readily as a fact what he knew (as a voyager proceeded onwards towards the north) must be true at some point. But this statement certainly affords no evidence that he had himself actually visited the mysterious land to which it refers. (See Thri.P.) Still more difficult is it to determine the extent and character of Pytheas's explorations towards the cast. The statement of Polybius that he proceeded along the whole of the northern coasts of Europe as far as the Tanais is evidently based upon the supposition that this would be a simple and direct course along the coast of Germany and Scythia, --Polybius himself, in common with the other Greek geographers till a much later period, being wholly ignorant of the vast projection of the Cimbric peninsula, and the long circumnavigation that it involved.---of all which no trace is found in the extant notices of Pytheas. Notwithstanding this, some modern writers have supposed him to have entered the Baltic and penetrated as far as the mouth of the Vistula, which he erroneously supposed to be the Tanais. The only foundation for this highly improbable assumption is to be found in the fact that in a passage cited by Pliny (//..V., xxxvii. 2, 35) I'ytheas is represented as stating that amber was brought from an island called Abalus, distant a day's voyage from the land of the Guttones, a (;erman nation who dwelt on an estuary of the ocean called Mentonomus, 6000 stadia in extent. It was a production thrown up by the waves of the sea, and was used by the inhabitants to burn instead of wood. It is not improbable that the “estuary " here mentioned really refers to the Baltic, the existence of which as a separate sea was unknown to all ancient geographers; but the obscure manner in which it is indicated, as well as the inaccuracy of the statements concerning the place from whence the amber was actually derived, both point to the sort of hearsay accounts which Pytheas might readily have picked up on than the mouth of the Elbe, which is supposed by I kert |