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number, the product of equals. More legitimate is their application asceticism of the Jewish sect of the Essenes seems, as Zeller conof number to geometry, according to which“ was identified | tends, to be due to a strong infusion of Neopythagorean elements. with the point, “two" with the line, “three” with the surface, At a still later period Neopythagoreanism set up Pythagoras and and "four" with the cube. In the history of music the Pytha- | Apollonius of Tyana not only as ideals of the philosophic life but gorcan school is also of considerable importance from the develop- also as prophets and wonder-workers in immediate communication ment which the theory of the octave owes to its members; according with another world, and in the details of their “lives” it is easy to some accounts the discovery of the harmonic system is due to to read the desire to emulate the narrative of the Gospels. The Pythagoras himself.

Life of Apollonius by Philostratus, which is for the most part an As already mentioned, the movements of the heavenly bodies historical romance, belongs to the 3u Christian century. formed for the Pythagoreans an illustration on a grand scale of the Zeller's discussion of Pythagoreanism, in his Philosophie d. Griechen, book i., truth of their theory. Their cosmological system is also interest

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 ing on account of peculiarities which mark it out from the current and the probable origin of the Essenes. The numerous monographs dealing conceptions of antiquity and bring it curiously near to the modern with special parts of the subject are there examined aud sisted. (A. SE.) theory. Conceiving the universe, like many early thinkers, as a sphere, they placed in the heart of it the central fire, to which they give the name of Hestia, the hearth or altar of the universe, tlie

Pythagorean Geometry. citadel or throne of Zeus. Around this move the ten heavenly boilies—farthest off the heaven of the fixed stars, then the five

As the introduction of geometry into Greece is by planets known to antiquity, then the sun, the moon, the carth,

common consent attributed to Thales, so all are agreed that and lastly the counter-earth (avtixowr), which revolves between to Pythagoras is due the honour of having raised mathiethe earth and the central fire and thus completes the sacred decade. matics to the rank of a science. We know that the early Revolving along with the earth, the last-mentioned body is always Pythagoreans published nothing, and that, moreover, they interposed as a shield between us and the direct rays of the central fire. Our light and leat come to us indirectly by way of reflexion referred all their discoveries back to their master. (See from the sun.

When the earth is on the same side of the central PHILOLAUS.) Hence it is not possible to separate his work fire as the sun, we have olay ; when it is on the other side, vight. from that of his early disciples, and we must therefore treat This attribution of the changes of day and night to the earth's own motion led up directly to the true thicory, as soon as the machinery

the geometry of the early Pythagorean school as a whole. of the central fire and the counter-carth was dispensed with. The We know that Pythagoras made numbers the basis of his counter-carth became the western hemisphere, and the earth ro- philosophical system, as well physical as metaphysical, volved on its own axis instead of round an imaginary centre. But, and that he united the study of geometry with that of as appears from the above, the Pythagorean astronomy is also arithmetic. remarkable as having attributed a planetary motion to the earth instead of making our globe the centre of the universe. Long after

The following statements have been handed down to warls, when the church condemned the theory of Copernicus, the (a) Aristotle (1let., i. 5, 985) says “the Pythagoreans indictment that lay against it was its heathen and “Pythagorean ” | first applied themselves to mathematics, a science which character.

they improved; and, penetrated with it, they fancied that The doctrine which the memory of mankind associates most closely with Pythagoras's name is that of the transmigration of the principles of mathematics were the principles of all souls - Mereul'sychosis (9.v.). Though evidently of great import- things.” (V) Eudemus informs us that “Pythagoras changed ance for Pythagoras himself

, it does not stand in any very obvious geometry into the form of a liberal science, regarding its the idea from the orphic Systeries." The bodily life of the sotar, principles in a purely abstract manner, and investigated accoriling to this doctrine, is an imprisonment suflered for sins

its theorems from the immaterial and intellectual point of comunitted in a former state of existence. At death the soul reaps

view (Ams kui Voepôs).”] (c) Diogenes Laertius (viii. wshant it has sown in the present life. The reward of the best is to 11) relates that "it was Pythagoras who carried geometry enter the cosmos, or the higher and purer regions of the universe, to perfection, after Mæris 2 had first found out the prinwhile the direst crimes receive their punishment in Tartarus. But the general lot is to live afresh in a series of human or animal forms, ciples of the elements of that science, as Anticlides tells us the nature of the bodily prison being determined in each case by in the second book of his History of Alexander; and the the deeds done in the life juist endel. This is the same doctrine part of the science to which Pythagoras applied himself of retribution and purificatory wandering which meets us in Plato's above all others was arithmetic.” (d) According to Arismythical descriptions of a future life. They are borrowed by him in their substance from the Pythagoreans or from a common source

toxenus, the musician, Pythagoras seems to have esteemed in the Mysteries. In accordance with this religious view of life as

arithmetic above everything, and to have advanced it by a stage of probation were the ethical precepts of the school, inculeat- diverting it from the service of commerce and by likening ing reverence towarıls the gods and to parents, justice, gentleness, | all things to numbers.3 (e) Diogenes Laertius (viii. 13) temperance, purity of life, prayer, regular self-examination, and the observance of various ritual requirements.

reports on the same authority that Pythagoras was the Connecting its ethics in this way with religion and the idea of a

first person who introduced measures and weights among future life, the Pythagorean societies had in them from the be- | the Greeks. (1) He discovered the numerical relations of ginning a germ of asceticism and contemplative mysticism which the musical scale (Diog. Laert., viii. 11). (9) Proclus + it was left for a later age fully to develop The Pythagorean life

is destined to survive the peculiar doctrines of the Pythagorean says that “the word ' mathematics' originated with the philosophy and to be grafted on later philosophie icleas.

The Pythagoreans." (1) We learn also from the same authorasceticism which characterized it appears in the 4th century B.C. itys that the Pythagoreans made a fourfold division of in close connexion with the Orphic Mysteries; and the “L'ytha- mathematical science, attributing one of its parts to the goreans" of that time are frequently the butts of the New Athenian

“how many” (T) Tórov) and the other to the "how much ” struck terper roots ; in Alexandria and elsewhere schools of men (mylixov); and they assigned to each of these parts a amose calling themselves Pythagoreans, but more accurately dis

twofold division. They said that discrete quantity or the tinguished by modern criticism as leopythagoreans, seeing that “how many” is either absolute or relative, and that contheir philosophical doctrines are evidently derived in varying protinued quantity or the “how much” is either stable or in Bez said that they develop the mystic siile of the Platonie doctrine: motion. Hence they laid down that arithmetic contemand only so far as this is connected with the similar speculations plates that discrete quantity which subsists hy itself, but of Pythagoras can they claim to be followers of the latter

. Hence music that which is related to another; and that geometry men like Plutarch, who personally prefer to call themselves considers continued quantity so far as it is immorable, Platonists, may also be considered as within the scope of this Pythagorean riviral. The link that really connects these Veo. pythagoreans with the Samian philosopher and distinguishes them

i Proclus Diadochus, In primum Euclidis Elemeutorum librum from the other schools of their time is their ascetic ideal of life and Commentarii, ed. Friedlein, p. 65. their preoccupation with religion. In religious speculation they ? Væris was a king of Egypt who, Herodotus tells us, lived 900 years pared the way for the Neoplatonie conception of God as immeasur- before his visit to that country, ably transceniling the world ; and in their thirst for prophecies, 3 Aristor., Fragm., ap. Stob., Eclog. Phys., i. 2, orules, and signs they gave espression to the prevalent longing Procl., op. cit., p. 15. for a supernatural rerelation of the divine nature and will. The

Op. cit., p. 35.

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but that astronomy (obalpukń) contemplates continued goreans and was called by them “health” (úyreía).? (11) quantity so far as it is of a self-motive nature. (1) Diogenes The discovery of the law of the three squares (Euclid I. Laertius (viii. 25) states, on the authority of Favorinus, 47), commonly called the “theorem of Pythagoras," is that Pythagoras “employed definitions in the mathema- attributed to him by many authorities, of whom the oldest tical subjects to which he applied himself."

is Vitruvius. (12) One of the methods of finding rightThe following notices of the geometrical work of Pytha- angled triangles whose sides can be expressed in numbers goras and the early Pythagoreans are also preserved. (1) | (Pythagorean triangles)—that setting out from the odd The Pythagoreans define a point as "unity having position' numbers—is referred to Pythagoras by Heron of Alex(Procl., op. cit., p. 95). (2) They considered a point as andria and Proclus.9 (13) The discovery of irrational analogous to the monad, a line to the duad, a superficies quantities is ascribed to Pythagoras by Eudemus (Procl., to the triad, and a body to the tetrad (ib., p. 97). (3) op. cit., p. 65). (14) The three proportions-arithmetical, They showed that the plane around a point is completely geometrical, and harmonical-were known to Pythagoras. I filled by six equilateral triangles, four squares, or three (15) Iamblichus 11 says, "Formerly, in the time of Pytharegular hexagons (ib., p. 305). (4) Eudemus ascribes to goras and the mathematicians under him, there were three them the discovery of the theorem that the interior angles means only—the arithmetical, the geometrical, and the of a triangle are equal to two right angles, and gives their third in order which was known by the name sub-contrary proof, which was substantially the same as that in Euclid I. (úmevavría), but which Archytas and Hippasus designated

(ib., p. 379). (5) Proclus informs us in his comment- the harmonical, since it appeared to include the ratios ary on Euclid I. 44 that Eudemus says that the problems concerning harmony and melody.” (16) The so-called concerning the application of areas --- where the term

-where the term most perfect or musical proportion, e.g., 6:8::9:12, "application” is not to be taken in its restricted sense which comprehends in it all the former ratios, according (Tapaßolý), in which it is used in this proposition, but to Iamblichus, 12 is said to be an invention of the Babyalso in its wider signification, embracing umepBody and lonians and to have been first brought into Greece by čldecyıs, in which it is used in Book VI. Props. 28, 29 Pythagoras. (17) Arithmetical progressions were treated —are old, and inventions of the Pythagoreans 2 (ib., p. by the Pythagoreans, and it appears from a passage in 419). (6) This is confirmed by Plutarch, who says, after Lucian that Pythagoras himself had considered the special Apollodorus, that Pythagoras sacrificed an ox on finding case of triangular numbers : Pythagoras asks some one, the geometrical diagram, either the one relating to the “How do you count?” he replies, “One, two, three, four.” hypotenuse, namely, that the square on it is equal to the Pythagoras, interrupting, says, "Do you see? what you sum of the squares on the sides, or that relating to the take to be four, that is ten and a perfect triangle and our problem concerning the application of an area. 4 (7) oath.”13 (18) The odd numbers were called by the PythaPlutarch also ascribes to Pythagoras the solution of the goreans "gnomons," 14 and were regarded as generating, inproblem, To construct a figure equal to one and similar to another given figure. (8) Eudemus states that Pythagoras

? Lucian, Pro Lapsu in Salut., s. 5; also schol. on Aristoph., Nub.,

611. discovered the construction of the regular solids (Procl., laniblichus (De Vit. Pyth., c. 33, ss. 237 and 238). This figure is

That the Pythagoreans used such synıbols we learn from op. cit., p. 65). (9) Hippasus, the Pythagorean, is said referred to Pythagoras himself, and in the Middle Ages was called to have perished in the sea on account of his impiety, Pythagoræ figura ; even so late as Paracelsus it was regarded by him inasmuch as he boasted that he first divulged the know

as a symbol of health. It is said to have obtained its special name ledge of the sphere with the twelve pentagons (the in

from the letters v, Y; 1, 0 (=el), a having been written at its prominent

vertices. scribed ordinate dodecahedron): Hippasus assumed the 8 De Arch., ix., Præf., 5, 6, 7. Amongst other authorities are glory of the discovery to himself, whereas everything be- Diogenes Laertius (viii

. 11), Proclus (op. cit., p. 426), and Plutarch longed to Him-“for thus they designate Pythagoras, and (ut sup., 6). Plutarch, however

, attributes to the Egyptians the knowdo not call him by name. (10) The triple interwoven

ledge of this theorem in the particular case where the sides are 3, 4,

and 5 (De Is. et Osir., c. 56). triangle or pentagram-star-shaped regular pentagon- 9 Heron Alex., Geom. et Stereom. Rel., ed. F. Hultsch, pp. 56, was used as a symbol or sign of recognition by the Pytha- 146 ; Procl., op. cit., p. 428. The method of Pythagoras is as

follows :-hé took an odd number as the lesser side; then, having

squared this number and diminished the square by unity, he took ? We learn, however, from a fragment of Geminus, which has been half the remainder as the greater side, and by adding unity to this handed down by Eutocius in his commentary on the Conics of Apol.

number he obtained the hypotenuse, e-9., 3, 4, 5; 5, 12, 13. lonius (Apoll., Conica, ed. Halleius, p. 9), that the ancient geometers

10 Nicom, Ger., Introd. Ar., c. xxii. observed two right angles in each species of triangle, in the equilateral 11 In Nicomachi Arithmeticam, ed. S. Tennulius, p. 141. first, then in the isosceles, and lastly in the scalene, whereas later 12 Op. cit., p. 168. As an example of this proportion Nicomachus writers proved the theorem generally thus—"The three internal angles and, after him, Iamblichus give the numbers 6, 8, 9, 12, the harmonical of every triangle are equal to two right angles.'

and arithmetical means between two numbers forming a geometric 2 The words of Proclus are interesting. According to Eudemus

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proportion with the numbers themselves the inventions respecting the application, eccess, and (lefect of areas are

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a+. 2 ancient, and are due to the Pythagorcans. Moderns, borrowing these blichus further relates (1.c.) that many Pythagoreans made use of this names, transferred them to the so-called conic lines , the parabola, proportion, as Aristæus of Crotona, Timæus of Locri

, Philolaus and the hyperbola, the ellipse, as the older school, in their nomenclature Archytas of Tare

and many others, and after them Plato in his concerning the description of areas in plano on a finite right line, re- T'imæus (see Nicom., Inst. Arithm., ed. Ast, p. 153, and Animadgarded the terms thus :-An area is said to be applied (mapaßámleiv) versiones, pp. 327-329; and Iambl., op. cit., p. 172 sq.). to a given right line when an area equal in content to some given one 13 Blwv apãous, 4, vol. i. p. 317, ed. C. Jacobitz. is described thereon ; but when the base of the area is greater than 14 Tvúpwv means that by which anything is known, or“criterion”; its the given line, then the area is said to be in excess (üttepßáll elv); but oldest concrete signification seems to be the carpenter's square (norma) when the base is less, so that some part of the given line lies without by which a right angle is known. Hence it came to denote a perthe described area, then the area is said to be in defect (elleleLv). pendicular, of which, indeed, it was the archaic name (Proclus, op.cit., Euclid uses in this way in his sixth book the terms eccess and defect. p. 283). Gnomon is also an instrument for measuring altitudes, by

The term application (Trapaßaller), which we owe to the Pythia- means of which the meridian can be found ; it denotes, further, the goreans, has this signification.

index or style of a sun-dial, the shadow of which points out the hours. 3 Non posse suaviter vivi sec. Epicurum, c. xi.

In geometry it means the square or rectangle about the diagonal of a 4 Eťte apóßimua Tepi Toû xwplov tỉns Trapaßolîs. Some authors, square or rectangle, together with the two complements, on account rendering the last five words "concerning the area of the parabola, of the resemblance of the figure to a carpenter's square; and then, have ascribed to Pythagoras the quadrature of the parabola, which was more generally, the similar figure with regard to any parallelogram, one of the great discoveries of Archimedes.

as defined by Euclid II. Def. 2. Again, in a still more general 5 Symp. viii., Quæst. 2, c. 4.

signification, it means the figure which, being added to any figure, 6 Iamblichus, De Vit. Pyth., c. 18, s. 88.

preserves the original form. See Heron, Definitiones (59). When

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asmuch as by the addition of successive gnomons—consist- | The original square is thus dissected into the four triangles as ing each of an odd number of unit squares—to the original before and the ligure within, which is the square on the hypotenuse. square unit or monad the square form was preserved. (19) the sides of the right-angled triangle

.

This square, therefore, must be equal to the sum of the squares on In like manner, if the simplest oblong (érepojenkes), consist- It is well known that the Pythagoreans were much occupied ing of two unit squares or monads in juxtaposition, be taken with the construction of regular polygons and solids, which in and four unit squares be placed about it after the manner their cosmology played an essential part as the fundamental forms

of the elements of the universe. We can trace the origin of these of a gnomon, and then in manner six, eight ... unit

mathematical speculations in the theorem (3) that “the plane squares be placed in succession, the oblong form will be around a point is completely filled by six equilateral triangles, preserved. (20) Another of his doctrines was, that of all four squares, or three regular hexagons.” Plato also makes the solid figures the sphere was the most beautiful, and of all Pythagorcan Timæus explain--"Each straight-lined figure consists

of triangles, but all triangles can be dissected into rectangular plane figures the circle.? (21) According to Iamblichus

ones which are either isosceles or scalene. Among the latter the the Pythagoreans are said to have found the quadrature of

most beautiful is that out of the doubling of which an equilateral the circle.

arises, or in which the square of the greater perpendicular is three On examining the purely geometrical work of Pythagoras and times that of the smaller, or in which the smaller perpendicular his early disciples, as given in the preceding extracts, we observe is half the hypotenuse. But two or four right-angled isosceles that it is much concerned with the geometry of areas, and we are triangles, properly put together, form the square ; two or six of indeed struck with its Egyptian character. This appears in the the most beautiful scalene right-angled triangles form the equitheorem (3) concerning the filling up a plane with regular figures - lateral triangle; and out of these two figures arise the solids which for floors or walls covered with tiles of various colours were common correspond with the four elements of the real world, the tetrain Egypt; in the construction of the regular solids (8), for some hedron, octahedron, icosahedron, and the cube”* (Timæus, 53, of them are found in Egyptian architecture ; in the problems con- 54, 55). The construction of the regular solids is distinctly cerning the application of areas (5); and lastly, in the theorem of ascribed to Pythagoras himself by Eudemlis (8). Or these five Pythagoras (11), coupled with his rule for the construction of right- solids three—the tetrahedron, the cube, and the octahedron—were angled triangles in numbers (12). We learn from Plutarch that known to the Egyptians and are to be found in their architecture. the Egyptians were acquainted with the geometrical fact that a Let us now examine what is required for the construction of the triangle whose sides contain three, four, and five parts is right other two solids-- the icosahedron and the dodecahedron. In the angled, and that the square of the greatest side is equal to the formation of the tetrahedron three, and in that of the octahedron squares of the sides containing the right angle. It is probable too four, equal equilateral triangles had been placed with a common that this theorein was known to them in the simple case where the verter and adjacent sides coincident; and it was known that is six right-angled triangle is isosceles, inasmuch as it would be at once such triangles were placed round a common vertex with their suggested by the contemplation of a floor covered with square tiles adjacent sides coincident, they would lic a plane, and that, -the square on the diagonal and the sum of the squares on the therefore, no solid could be formed in that manner from them. It sides contain each four of the right-angled triangles into which

remaineil, then, to try whether five such equilateral triangles could ono of the squares is divided by its diagonal. It is easy now to be placed at a common vertex in like manner; on trial it would see how the problem to construct a square which shall be equal to be found that they could be so placed, and that their bases would the sum of two squares could, in some cases, be solved numerically: form a regular pentagon. The existence of a regular pentagon From the observation of a chequered board it would be perceived

would thus become known. It was also known from the formation that the element in the successive formation of squares is the

of the cube that three squares could be placed in a similar way gnomon or carpenter's square. Each gnomon consists of an odıl with a common vertex; and that, further, if three equal and number of squares, and the successive gnomons correspond to the

regular hexagons were placed round a point as common vertex successive odd numbers, and include, therefore, all odd squares,

with adjacent sides coincident, they would form a plane. It reSuppose, now, two squares are given, one consisting of sixteen and mained in this case too only to try whether three equal regular the other of nine unit squares, and that it is proposed to form from pentagons could be placed with a common vertex and in a similar them another square. It is evident that the square consisting of

way; this on trial would be found possible and would lead to the nine unit squares can take the form of the fourth gnomon, which, construction of the regular dodecahedron, which was the regular being placed round the former square, will generate a new square

solid last arrived at. containing twenty-five unit squares. Similarly it may have been

We sec that the construction of the regular pentagon is required observed that the twelfth gnomon, consisting of twenty-five unit

for the formation of each of these two regular solids, and that, squares, could be transformed into a square each of whose sides therefore, it must have been a discovery of Pythagoras. If we contains five units, and thus it may have been seen conversely that

examine now what knowledge of geometry was required for the the latter square, by taking the gnomonic or generating form with solution of this problem, we shall see that it depends on Euclid IV. respect to the squaro ou twelve units as base, would produce the

10, which is reduced to Euclid II. 11, which problem is reduced to square of thirteen units, and so on. This methol required only to

the following: To produce a given straight line so that the rectbe generalized in order to enablo Pythagoras to arrive at his rule angle under the whole line thus produced and the produced part for finding right-angleil triangles whose sides can be expressed shall be equal to the square on the given line, or, in the language in numbers, which, wo are toll, sets out from the odd numbers, of the ancients, To apply to a given straight line a rectangle which The nth square together with the uth gnomon forms the (n + 1)th shall be equal to a given area-in this case the square on the given square; if the nth gnomon contains ma unit squares, m being an

line-and which shall be excessive by a square. Now it is to bo m? - 1

observed that the problem is solved in this manner by Euclid (VI. oll number, we have 21+1=m?, .:1= which gives the 30, 1st methol), and that we know on the authority of Eudemus rule of lythagoras.

that the problems concerning the application of areas and their The general proof of Euclid I. 47 is attributed to Pythagoras,

crcess and defect are old, and inventions of the Pythagoreans (5). but we have the express statement of l'roclus (op. cit., p. 426) that

Hence the statements of lambliehus concerning Hippasus (9) — this theorein was not proved in the first instance as it is in the

that he divulged the sphere with the twelve pentagons--and of Elements. The following simple and natural way of arriving at

Lucian and the scholiast on Aristophanes (10)--that the pentathe theorem is suggested by Bretschneider after Camerer. 3 A

gram was used as a symbol of recognition amongst the Pythagoreans square can be dissected into the sum of two squares and two equal

become of greater importance. rretangles, as in Euclid II. 4; these two rectangles can, by draw:

Further, the discovery of irrational magnitudes is ascribed to ing their diagonals, be decomposed into four equal right-angleil

Pythagoras by Eudemus (13), and this discovery has been ever triangles, the sum of the sides of each being equal to the side of

regarded as one of the greatest of antiquity. It is commonly the square ; agun, these four right-angled triangles can be placed

assumed that Pythagoras was led to this theory from the consideraso that a vérter of each shall be in one of the corners of the square

tion of the isosceles right-angled triangle. It seems to the present in such a way that a greater and less siile are in continuation.

writer, however, more probable that the discovery of incommen

surable magnitudes was rather owing to the problem: To cut a gomons are added successively in this manner to a square mona, line in extreme and mean ratio. From the solution of this problem the first gnomon may be regarded as that consisting of three square it folloirs at once that, if on the greater segment of a line so cut monals, and is indeed the constituent of a simple Greek fret; the a part be taken equal to the less, the greater segment, regarded second of five square monads, &c. ; hence we have the gnomonic as a new line, will be cut in a similar manner; and this process oombers

can be continued without end. On the other hand, if a similar Diog, Laert, De rit, Pyth., viii, 19.

method be adopted in the case of any two lines which can be reSimplicius In Aristotelis Physicorum libros quattuor priores Com- presented numerically, the process would end. llence would arise suan, ed. H. Diels, p. 60. See Bretsch., Die Geom. ior Euklides, p. 82; Camerer, Euclidis

+ The dodecahedron was assigned to the fifth element, quinta pars, bem, rol i p. 444, and the references given thera.

æther, or, as some think, to the universe. (See PHILOLACS.)

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the distinction between commensurable and incommensurable same surface the sphere is the greatest. We must also deny to
quantities. A reference to Euclid X. 2 will show that the method Pythagoras and his school a knowledge of the conic sections, and
above is the one used to prove that two magnitudes are incommen- in particular of the quadrature of the parabola, attributed to him
surable; and in Euclid* x. 3 it will be seen that the greatest | by some authors; and we have noticed the misconception which
common measure of two commensurable magnitudes is found by gave rise to this erroneous inference.
this process of continued subtraction. It seems probable that Let us now see what conclusions can be drawn from
Pythagoras, to whom is attributed one of the rules for representing the foregoing examination of the mathematical work of
the sides of right-angled triangles in numbers, tried to find the
sides of an isosceles right-angled triangle numerically, and that,

Pythagoras and his school, and thus form an estimate of
failing in the attempt, he suspected that the hypotenuse and a the state of geometry about 480 B.C. First, as to matter.
side had no common measure. He may have demonstrated the It forms the bulk of the first two books of Euclid, and
incommensurability of the side of a square and its diagonal. The includes a sketch of the doctrine of proportion—which
nature of the old proof—which consisted of a reductio ad absurd-
um, showing that, if the diagonal be commensurable with the was probably limited to commensurable magnitudes,
side, it would follow that the same number would be odd and together with some of the contents of the sixth book. It
eveni-makes it more probable, however, that this was accom- contains too the discovery of the irrational (ädoyov) and
plished by his successors. The existence of the irrational as
well as that of the regular dodecahedron appears to have been

the construction of the regular solids, the latter requiring regarded by the school as one of their chief discoveries, and to

the description of certain regular polygons—the foundahave been preserved as a secret; it is remarkable, too, that a story tion, in fact, of the fourth book of Euclid. Secondly, as similar to that told by Iamblichus of Hippasus is narrated of tlie to form. The Pythagoreans first severed geometry from person who first published the idea of the irrational, namely, that he suffered shipwreck, &c.?

the needs of practical life, and treated it as a liberal Eudemus ascribes the problems concerning the application of science, giving definitions and introducing the manner of figures to the Pythagoreans. The simplest cases of the problems, proof which has ever since been in use. Further, they Euclid VI. 28, 29—those, namely, in which the given parallelogram distinguished between discrete and continuous quantities, is a square-correspond to the problem : To cut a given straight and regarded geometry as a branch of mathematics

, of segments shall be equal to a given rectilineal figure. The solution

which they made the fourfold division that lasted to the of this problem-in which the solution of a quadratic equation is

Middle Ages--the quadrivium (fourfold way to knowledge) implicitly contained-depends on the problem, Euclid II. 14, and of Boetius and the scholastic philosophy. And it may be the theorems, Euclid Il. 5 and 6, together with the theorem of observed that the name of a mathematics," as well as Pythagoras. It is probable

that the finding of a mean proportional that of " philosophy," is ascribed to them. Thirdly, as to between two given lines, or the construction of a square which shall be equal to a given rectangle, is due to Pythagoras himself.

method. One chief characteristic of the mathematical The solution of the more general problem, Euclid VI. 25, is also work of Pythagoras was the combination of arithmetic attributed to him by Plutarch (7). The solution of this problem with geometry. The notions of an equation and a propordepends on that of the particular case and on the application of areas ; it requires, moreover, a knowledge of the theorems: Similar

tion—which are common to both, and contain the first rectilineal figures are to each other as the squares on their homo germ of algebra-were introduced among the Greeks by logous sides (Euclid VI. 20); and, If three lines are in geometrical Thales. These notions, especially the latter, were elaboproportion, the first is to the third as the square on the first is rated by Pythagoras and his school, so that they reached to the square on the second. Now Hippocrates of Chios, about 440 B.C., who was instructed in geometry by the Pythagoreans,

the rank of a true scientific method in their theory of possessed this knowledge. We are justified, therefore, in ascrib- proportion.* To Pythagoras, then, is due the honour of ing the solution of the general problem, if not (with Plutarch) to having supplied a method which is common to all branches Pythagoras, at least to his early successors.

of mathematics, and in this respect he is fully comparable The theorem that similar polygons are to each other in the duplicate ratio of their homologous sides involves a first sketeh, at

to Descartes, to whom we owe the decisive combination of least

, of the doctrine of proportion and the similarity of figures: algebra with geometry.
That we owe the foundation and development of the doctrine of See C. A. Bretschneider, Die Geometrie u. die Gcometer vor Era
proportion to Pythagoras and his school is confirmed by the testi- klides (Leipsic, 1870); H. Hankel, Zur Geschichte der Mathematik
mony of Nicomachus (14) and Iamblichus (15 and 16). From these (Leipsic, 1974); F. Hoefer, Histoire des Mathématiques (Paris,
passages it appears that the early Pythagoreans were acquainted, 1874); G. J. Allman, “Greek Geometry from Thales to Euclid,"
not only with the arithmetical and geometrical means between in IIermathena, Nos. v., vii., and x.°(Dublin, 1877, 1881, and
two magnitudes, but also with their harmonical mean, which was 1884); M. Cantor, Vorlesungen über Geschichte der Mathematik
then called " subcontrary." The Pythagoreans were much occupicu (Leipsic, 1880). The recently published Short History of Grech
with the representation of numbers by geometrical figures. These Mathematics by James Gow (Cambridge, 1884) will be found a

,
convenient compilation.

(G. J. A.)
the summation of the natural numbers, the odd numbers, and the PYTHEAS of Massilia was a celebrated Greek navi-
even numbers, all of which are capable of geometrical representa- gator and geographer, to whom the Greeks appear to have
tion. See the passage in Lucian (17) and the rule for finding been indebted for the earliest information they possessed,
Pythagorean triangles (12) and the observations thereon supra.
On the other hand, there is no evidence to support the statement

of at all a definite character, concerning the western regions of Montucla that Pythagoras laid the foundation of the doctrine of of Europe, and especially the British Islands. The period isoperimetry, by proving that of all figures having the same peri: at which he lived cannot be accurately determined ; but it meter the circle is the greatest, and that of all solids having the

is certain that he wrote, not only before Eratosthenes, who 1 For this proof, see Euclid X. 117; see also Aristot., Analyt. Pr., relied much upon his authority, but before Dicæarchus, i. c. 23 and c. 44.

who was a pupil of Aristotle, and died about 285 B.C. 2 Knoche, Untersuchungen über die neu aufgefundenen Scholien

Hence he may probably be regarded as about contemdes Proklus Diadochus zu Euclid's Elementen, pp. 20 and 23, Herford,

porary with Alexander the Great. His work is now 1865.

3 It is agreed on all hands that these two theories were treated at wholly lost, and appears to have been consulted in the
length by Pythagoras and his school. It is almost certain, however, original by comparatively few ancient writers, most of the
that the theorems arrived at were proved for commensurable magni-
tudes only, and were assumed to hold good for all. The Pythagoreans Proportion was not regarded by the ancients merely as a branch
themselves seem to have been aware that their proofs were not rigor- of arithmetic. We learn from Proclus that “Eratosthenes looked on
ous, and were open to serious objection ; in this we may have the proportion as the bond of mathematics” (op. cit., p. 43). We are
explanation of the secrecy which was attached by them to the idea of

also told in an anonymous scholium on the Elements of Euclid, which the incommensurable and to the pentagram which involved, and indeed Knoche attributes to Proclus, that the fifth book, which treats of prorepresented, that idea. Now it is remarkable that the doctrine of portion, is common to geometry, arithmetic, music, and, in a word, proportion is twice treated in the Elements of Euclid—first, in a general to all mathematical science. And Kepler, who lived near enough to manner, so as to include incommensurables, in Book V., which tradition the ancients to reflect the spirit of their methods, says that one part of ascribes to Eudoxus, and then arithmetically in Book VII., which, as geometry is concerned with the comparison of figures and quantities, Hankel has supposed, contains the treatment of the subject by the whence proportion arises, He also adds that arithmetic and geometry older Pythagoreans.

afford mutual aid to each other, and that they cannot be separated.

X

4

statements cited from it being confined to detached points, the great Phænician emporium, and probably the farthest which may easily have been derived at second or even point familiar to the Greeks, round Spain and Gaul to third hand. We are hence left almost wholly in the dark the British Islands, and that he followed the eastern coast as to the form and character of the work itself, but the of Britain for a considerable distance to the north, obtainvarious titles under which it is cited by later writers point ing information as to its farther extension in that direction rather to a geographical treatise, in which he had embodied | which led him greatly to exaggerate its size. At the same the results of his observations, than to a continuous narra- time he heard vaguely of the existence of a large island to tive of his voyage like that of a modern navigator. the north of it-probably derived from the fact of the

Some modern writers have supposed Pytheas to have groups of the Orkneys and Shetlands being really found been sent out at the public expense, in command of an in that position—to which he gave the name of Thule. expedition organized by the republic of Massilia ; but No ancient writer (except a late astronomer, who merely there is no ancient authority for this, and the statement refers to it in a passing notice and obviously at second of Polybius, who had unquestionably seen the original hand) asserts that Pytheas had himself visited Thule ; his work, is express, that he had undertaken the voyage in account of the Sluggish Sea beyond it was, as stated by a private capacity and with limited means. All that we | Polybius himself in the passage already cited, derived know concerning the voyage of Pytheas (apart from such inerely from hearsay. detached notices as those already referred to) is contained But the most important statement made by Pytheas in in a brief passage of Polybius, cited by Strabo, in which regard to this unknown land of Thule, and which has given he tells us that Pytheas, according to his own statement, rise to most controversy in modern times, was that connected had not only visited Britain but had personally explored with the astronomical phenomena affecting the duration a large part of it, and stated its circumference at more than of day and night in these remote arctic regions. Un10,000 stadia (1000 geographical miles). To this he fortunately the reports transmitted to us at second hand added the account of Thule (wbich he placed six days' in our existing authorities differ so widely that it is almost voyage to the north of Britain) and the adjoining regions, impossible to determine what Pytheas himself really stated. in which there was no longer any distinction between the It is, however, probable that the version given in one air and earth and sea, but a kind of mixture of all three, passage by Pliny (I.N., iv, 16, 101) correctly represents forming a substance resembling the gelatinous molluse his authority. According to this le reported as a fact known as the Pulmo marinus, which rendered all naviya- that at the summer solstice the days were twenty-four tion and progress in any other mode alike impossible. hours in length, and conversely at the winter solstice the This substance he had himself seen, but the other state-nights were of equal duration. Of course this would be ments he derived from hearsay. Returning from thence strictly true had Thule really been situated under the arctic he visited the whole of the coasts of Europe bordering circle, which Pytheas evidently considered it to be, and on the ocean as far as the Tanais (Polyb. ap. Strab., ii. his skill as an astronomer would thus lead him to accept p. 101). This last sentence has led some modern writers readily as a fact what he knew (as a voyager proceeded

suppose that he made two different voyages; but this onwards towards the north) must be true at some point. is highly improbable, and the expressions of Polybius But this statement certainly affords no evidence that he certainly imply that his explorations in both directions, had himself actually visited the mysterious land to which first towards the north and afterwards towards the east, it refers. (See THULE.) formed part of one and the same voyage.

Still more difficult is it to determine the extent and The circumstance that the countries visited, and to a character of Pytheas's explorations towards the east. The certain extent explored, by Pytheas were not only pre- statement of Polybius that he proceeded along the whole viously unknown to the Greeks-except perhaps by vague of the northern coasts of Europe as far as the Tanais is hearsay accounts received through the Phænicians-but evidently based upon the supposition that this would be a were not visited by any subsequent authority during a simple and direct course along the coast of Germany and period of more than two centuries led some of the later Scythia, Polybius himself, in common with theother Greek Greek geographers altogether to disregard his statements, geographers till a much later period, being wholly ignorant and even to treat the whole story of his voyage as a fiction. of the vast projection of the Cimbric peninsula, and the Eratosthenes, indeed, who wrote about a century after his long circumnavigation that it involved, -of all which no time, was disposed to attach great value to his authority, trace is found in the extant notices of Pytheas. Notwiththough doubting some of his statements; but Polybius, standing this, some modern writers have supposed him to about half a century later, involved the whole in one have entered the Baltic and penetrated as far as the mouth sweeping condemnation, treating the work of Pytheas as of the Vistula, which he erroneously supposed to be the A mere tissue of fables, like that of Euhemerus concerning Tanais. The only foundation for this highly improbable asPunchwa; and even Strabo, in whose time the western sumption is to be found in the fact that in a passage cited by regions of Europe were comparatively well known, adopted Pliny (11.2., xxxvii. 2, 35) Pytheas is represented as stating to a great extent the same view with Polybius.

that amber was brought from an island called Abalus, In modern times a more critical examination has arrived distant a day's voyage from the land of the Guitones, a at a more favourable judgment, and, though Gossellin in German nation who dwelt on an estuary of the ocean called his Recherches sur la Géographie des Anciens (vol. iv. pp. Mentonomus, 6000 stadia in extent. It was a production 168-180) and Sir G. C. Lewis in his Iistory of Ancient thrown up by the waves of the sea, and was used by the Astronomy (pp. 466-481) revived the sceptical view, the inhabitants to burn instead of wood. It is not improbable tendency of modern critics has been rather to exaggerate that the "estuary” here mentioned really refers to the than to depreciate the value of what was really added by Baltic, the existence of which as a separate sea was unPytheas to geographical knowledge. The fact is that our known to all ancient geographers; but the obscure manner information concerning him is so imperfect, and the scanty in which it is indicated, as well as the inaccuracy of the notioes preserved to us from his work at once so meagre

statements concerning the place from whence the amber and discordant

, that it is very ditficult to arrive at anything was actually derived, both point to the sort of hearsay like a sound conclusion. It may, however, be considered accounts which Pytheas might readily have picked up on is fairly established that Pytheas really made a royage the shores of the German Ocean, without proceeding farther round the western coasts of Europe, proceeding from Gädes, than the mouth of the Elbe, which is supposed by Ukert

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