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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 “one was identifiel 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 philosophié life but gorean school is also of considerable importance from the develop- also as prophets and wonder-workers in iminediate communication inent which the thcory 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 harnionic 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 alrearly mentioned, the movements of the heavenly boilies historical romance, belongs to the 3d Christian century. formed for the Pythagoreans an illustration on a grand scale of the Zeller's discussion of Pythagoreanism, in his Philosophie dl. 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 ing on account of peculiarities which mark it out from the current

work with the Neopythagoreans, considered as the precursors of Neoplatonism

and the probable origin of the Essenes. The numerous monographis dealing conceptions of antiquity and bring it curiously near to the modern with special parts of the subject are there examined and sisted. (A. SE.) theory. Conceiving the universe, like many carly thinkers, as a sphere, they placed in the heart of it the central fire, to which they have the name of Hestia, the hearth or altar of the universe, thie

Pythagorean Geometry. citailel or throne of Zeus. Around this move the ten heavenly bolics-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-carth (avtixowv), which revolves between to Pythagoras is due the honour of having raised mathethe earth and the central fire anıl thus completes the sacred decade. matics the rank of a science. We know that the early Revolving along with the carth, the last-mentioned body is always Pythagoreans published nothing, and that, moreover, they fire. Our light and heat 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 | PiilolAUS.) Hence it is not possible to separate his work firo as the sun, we have day; when it is on the other side, pight. from that of his early disciples, and we must therefore treat This attribution of the changes of day and night to the eartlı's own motion led up directly to the true theory, as soon as the machinery

the geometry of the early Pythagorean school as a whole. of the central fire and the counter-earth was dispensed with. The We know that Pythagoras made numbers the basis of his countor-carth became the western hemisphere, and the carth re. philosophical system, as well physical as metaphysical, volved on its own axis insteal of round an imaginary centre. But, and that he united the study of geometry with that of ay appears from the above, the Pythagorean astronomy is also arithmetic. remarkable as having attributed a planetary motion to ihe carth 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 (Jet., i. 5, 985) says "the l’ythagoreans inlictment that lay against it was its heathen and “Pythagorean” | first applied themselves to mathematics, a science which character.

The doctrino which the memory of mankind associates most they improved ; and, penetrated with it, they fancied that losely with Pythagoras's name is that of the transmigration of the principles of mathematics were the principles of all souls - JeremissCHONES (7.v.). Though evidently of great import- things.” (?) Endemus informs us that"Pythagoras changed ance for Pythagoras himself, it loes not stand in any very obvious geometry into the form of a liberal science, regarding its connexion with his philosophy proper. Ile seems to have adopted principles in a purely abstract manner, and investigated the idea froin thu Orphic Mysteries. The bodily life of the soul, its theorems from the immaterial and intellectual point of 3cording to this doctrine, is an imprisonment suffered for sins committeil in a former state of existence. At death the soul reaps

view (úilos Kuì l'oepows).”] (s) Diogenes Laertius (viii. what it has sown in the present life. The rewarıl of the best is to 11) relates that "it was l’ythagoras who carried geometry enter the cosmos, or the higher and purer regions of the universe,

to perfection, after Merishad first found out the prinwhile the direst crimes receive their punishment in Tartarus. But trgoneral lot is to livo nfresh in a series of human or animal forms, ciples of the elements of that science, as Anticlides tells us th muture of the bolily prison being determined in each case by in the second book of his History of 21leermelor; and the the boils done in the life just endel. This is the same dortrin. 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." (el) According to Arisuythical descriptions of a future life. They are borrowed by him in their substance from the Pythagoreans or from a common source

toxenus, the musician, l’ythagoras 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 stages of probation were the ethical preempts of the school, inculcat-diverting it from the service of commerce and by likening ing riverence towards the gols and to parents, justice, gentleness, all things to numbers." () Diogenes Laertins (viii. 13) tulapuraus, purity of life, prayer, regular self-examination, and reports on the same authority that lythagoras was the the observance of various ritual requirements.

Connerting its othies in this way with religion and the iilea of a first person who introdureal measures and weights among four lif, the Pythagorean societies had in them from the be- the (reeks. (1) le discovered the numerical relations of winning a gorm of as ticism and contemplative mysticism which the musical scale (Dios. Lacrt., viii. 11). (:) I'roclus ! I w.24 left for a later ngo fully to develop. The lythagorean lite

says that "the word 'mathematics' originated with the Wilestinel to survive the paculiar doctrines of the Pythagorean priophy and to be grafteid on later philosophie ideas." The Pythagoreans." (h) We learn also from the same authoriwi. 1-11 which vhir..terizel it appears in theith cintury city that the Pythagoreans maile a fourfoll division of 111119 comcion with the Orphic Mysteries ; and the Pytha mathematical science, attributing one of its parts to the Sun" of that time are frequently the boutes of the Vows Athmian low many" (7) "1") and the other to the "how much“ Tants. ukol*per ruots; in Alexandria and olsu len slowls of men (7animor); anıl the samemu to each of these parts a malling themselves Pythagoreans, but more accurately dis twofoll division. They said that dirrete quantity of the

in all by malim criticism as Scopithagoreans, sering that “how many is either alisht or relatives and that con•!- ir philophucal lextrines arevidently derived in vargius pro tinned quantity or the "ww.mh" is either stable or in 1 vuil that they develop the mystic side of the Platonic vortrine motion. Hence they livid down that vrithmetic contemand only s far as this is conncited with the similar spekulations i plates that discrete quantity which will,~;:ta lix it-ri, lilit or lithiuris an they claim to lw followers of the latter. Hence music that which is related to another; and that comery men like Plutarch, who pærsonally prefer to call them considers contime quantity 10 for its it is imurile!! l'angists, may also be considered as within the scope of this Pythagorean niriral. The link that really connects these Veopithograns with the Samian philosopher and distinguishes them i l'ro. 111 Dillotis, 1../

1. from the other whools of thrir time is their asetic ideal of life and (mwilini, Mh. Frivile'll, ) their procupation with religion. In religious speculation they - Miris was a hill of For vir. 115:11: Hamarin moral the way for the Neoplatonii conception of cool as immeasur Before lus visit to that o'ntry. hly transeniling the world; and in their thirst for prophecies, :? Ir-,., F1 1. S..!.., E., P., 2.1, orales, and signs thru pare expression to the prevalent longing + I'r.l. ........ 15. fup a supernatural revelation of the divine nature and will. The I it', : 3,3.

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but that astronomy (ý opalpıkń) contemplates continued goreans and was called by them “health” (üyleía).? (11) quantity so far as it is of a self-motive nature. (i) 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.8 (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.

andria and Proclus. (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. 16 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. (iteravtíu), but which Archytas and Hippasus designated 32 1 (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 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 (Tupaßoln), 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 inepßoly and lonians and to have been first brought into Greece by člderyes, in which it is used in Book VI. Props. 28, 29 Pythagoras. (17) Arithmetical progressions were treated

are old, and inventions of the Pythagoreans? (ib., 1. 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. (7) oath. (18) The odd numbers were called by the PythaPlutarch • also ascribes to Pythagoras the solution of the goreans “ynomons,” 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 Sulut., s. 5; also schol. on Aristoph., Vub.,

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

That the Pythagoreans used such symbols 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 jipura; 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

from the letters ', , , 0 ( = el), a having been written at its prominent ledge of the sphere with the twelve pentagons (the in

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), Íroclus (op. cit., p. 426), and Plutarch longed to Him“for thus they designate Pythagoras, and (vt sup., 6): Plutarch, however, attributes to the Egyptians the knowdo not call him by name. .”6 (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 :- he took an odd number as the lesser side ; then, having

squared this number and diminished the square by unity, he took 1 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 ou the Conics of A pol. number he obtained the hypotenuse, c.9., 3, 4, 5; 5, 12, 13. lonius (Apoll., C'onicil, el. Halleius, p. 9), that the ancient geometers

10 Vicom. Ger., Introd. or., c. xxii. observed two right angles in each species of triangle, in the equilateral 11 In Vicomachi 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

2ab a + 6 the inventions respecting the application, ercess, and (lefect of areas are

proportion with the numbers themselves (a : ancient, and are due to the Pythagoreans. Moderns, borrowing these blichus further relates (?.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 Tarentum, and many others, and after then Plato in his concerning the (lescription of arcas in pluno on a finite right line, re T'imæus (see Nicom., Inst. Arithm., ed. Ast, p. 153, and Animadgardeil the terms thus :-An area is said to be applied (Tapaßállav) 247siones, 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 Blwy apaois, 4, vol. i. p. 317, ed. C. Jacobitz. is described thereon ; but when the base of tho area is greater than 14 I'vouwv means that by which anything is known, or“criterion”; its the given line, then the area is said to be in ercess (ütepß&Mev); 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 «lescribed area, then the area is said to be in clefect (Alelt el). pendicular, of which, indeel, 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 (TapaßálXelv), which we owe to the Pytha 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 Von posse suaviter rivi sec. Epicurum, c. xi.

In geometry it means the square or rectangle about the diagonal of a 4 Είτε πρόβλημα περί του χωρίου της παραβολής. 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 Symp. viii., Quæst. 2, c. 4.

signification, it means the figure which, being added to any figure, 6 Tamblichus, De l'it. 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 figure within, which is the square on the hypotenuse. square unit or monad the square form was preserved. (19)

This square, therefore, must be equal to the sum of the squares on

the sides of the right-angled triangle. In like manner, if the simplest oblong (étepóunkes), 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 a gnomon, and then in like manner six, eight . . . unit

of the elements of the universe. We can trace the origin of these squares be placed in succession, the oblong form will be mathematical speculations in the theorem (3) that the plane

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 Pythagorean Timæus explain—“Each straight-lined figure consists plane figures the circle. (21) According to Iambilichus

of triangles, but all triangles can be dissected into rectangular

ones which are cither 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 ; an 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 heilron, 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 Eudemus (8). Of these fivo 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 dovlecahedron. 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, cqual equilateral triangles had been placed with a common that this theorem was known to them in the simple case where the vertex and adjacent siles 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 coinciclent, they would lie in 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 one of the squares is divided by its diagonal. It is easy now to be placed at a coinmon 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 woulu the sum of two squares could, in some cases, be solved numerically: form a regular pentagon. The existence of a regular pent:igon From the observation of a chequered board it would be perceived woull 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 oleh 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 nuinbers, 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 maineal 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 place 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 generato a new square solidl last arrived at. containing twenty-five unit squares. Similarly it may have been We see that the construction of the regular pentagon is required observed that the twelfth gnomon, consisting of twenty-five unit for the formation of cach of these two regular soliids, and that, snares, 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 cxamine now what knowledge of geometry was required for the the latter square, by taking the groinonic or generating form with solution of this problem, we shall see that it depends on Euclid IV. respect to the square ou twelve units as base, would produce the 10, which is reduced to Eucli II

. 11, which problem is reduced to suare of thirteen units, and so on. This methol required only to the following: To produce a given straight line so that the rerthe generalized in order to enable Pythagoras to arrive at his rule angle under the whole line thus prolume and the produced part for finding right-angle triangles whose sides can be expressed shall be equal to the square on the given line, or, in the language in numbers, which, we are toli, 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 nth gnomon forms the (92 +1}th shall be equal ton given arca -- 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 cressiry by a square. Now it is to be m?-1

observed that the problem is solved in this manner by Euclid (II. orel number, we have 21+1=:1?,.,n= which gives the 30, 1st method, and that we know on the authority of Eulemus rule of Pythagoras.

that the problems concerning the application of areas and their The genem proof of Euclil I. ti is attributed to Pythagoras,

crves and dirt are olil, and intentions of the l’ythagorcans 15). but we have the express statement of l'rolus (op. cit., p. 4.26) that

Hence the statements of lamblielius concerning lippasus 9.this theorein was not proved in the first instance as it is in the

that he divulgad the sphere with the twrlve juntayons-and of Elronde. The following simple anil natural way of arriving at

Lucian and the scholiast on Aristophanes , 10—that the puntathe 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 –lecome of greater importance. metangles, as in Euclid II. 4; these two rertangles can, by draw: Further, the discority of irrational magnitudes is ascribed to ing their diagonals, be decomposed into four equal right-angle Pythagoras by Enimus , 13, anul this oli rovery las been ever triangles, the sum of the siles of each being cqual to the side of riguroland as one of the croatest of antiquity. It is commonly the square : aguin, these four right-angled triangles can be placeol a smerl that lythagoras was del to this tlieory from the con-ider. so that a rerter of cach shall be in one of the corners of the square

tion of the isosceles right-angled triangle. It sells to the prosent in such a way that a greater and less side are in continuation. writer, however, more probable that the discovery of it:1.10

surable maguitule was rather owing to the mobilem : To si a guomons are aided successively in this manner to a square monad, line in catrome and mean ratio. From the wintion of this poem the first gnomon may be regarded as that consisting of three sure it follows at one that, if on the A Teatrs smelt of a line wint monals, and is indeed the constituent of a simple Greek fret; the a purt love caken cul to the less the greater sent, toarelo-1 Arcond of five square monads, &c.; hence we have the gnomonic as a new lime, will leiut in a similar mandhari and this fotos por aumbers

can be continued without end. On the oiher hand, if a similar Ding. Laert, De l'it. Pyth., viii. 19.

method be a lopted in the case of any two lines whiih can lw peus Simplicius, in Aristotelis Paysicorum libros quattuor priores l'um presented numerically, the preniess monil enl. Hence Boll arirentaria, ed. HL Diels, p. 60.

See Bretsch, Die Geom. tor Eullides, p. 82; Camerer, Eudidis 4 The darle ahe iron was 2-sigels) the f.fth element, quintal 3-7 ", Elem, vol. L p. 444, and the references given there.

Imer, or, as some think, to the universe. See PHILLACS.)

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
lythagoras, to whom is attributer 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 Pythagoras and his school

, and thus form an estimate of
sides of an isosceles right-angled triangle numerically, and that,
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 liave 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 absurdum, 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 even makes it more probable, however, that this was accom contains too the discovery of the irrational (aloyov) and plished by his successors. The existence of the irrational as

the construction of the regular solids, the latter requiring well as that of the regular dodecahedron appears to have been regardled by the school as one of their chief (liscoveries, 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 Ianıblichus of Hippasus is narrated of the to form. The Pythagoreans first severed geometry from person who first published the idea of the irrational, namely, that the needs of practical life, and treated it as a liberal he suffered shipwreck, &c.2

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 line internally or externally so that the rectangle under the 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—lepends on the problem, Euclid II. 11, and of Boetius and the scholastic philosophy. And it may be the theorems, Euclid Il. 5 and 6, together with the theorem of Pythagoras.

observed that the name of "mathematics," as well as 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 Vl. 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

tion—which are common to both, and contain the first areas ; it requires, moreover, a knowledge of the theoreins: Similar rectilineal figures are to each other as the squares on their homo

germ of algebra—were introduced among the Greeks by logous sidles (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 410 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 l’ythagoras, at least to his early successors.

of mathematies, 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 sketch, at

to Descartes, to whom we owe the decisive combination of least, of the doctrine of proportion and the similarity of figures. 3

algebra with geometry. That we owe the foundation and development of the doctrine of

See ('. A. Bretschneider, Die Geometrie u. die Gcometer vor Euproportion to Pythagoras and his school is confirmed by the testi- klides (Leipsie, 1870); II. IIankel, Zur Geschichte der Vathematik mony of Nicomachus (14) and Iamblichus (15 and 16). From these (Leipsic, 1871); F. Hoefer, Histoire des Jathématiques (Paris, passages it appears that the carly Pythagoreans were acquainted, 1876); G. J. Allman, “Greek Geometry from Thales to Euclid,' not only with the arithmetical and geometrical means between in Iermathena, Nos. v., vii., and x. (Dublin, 1877, 1881, and two magnitudes, but also with their harmonical mean, which was

1881); M. Cantor, Porlesungen über Geschichte der Mathematik then called “subcontrary.” The Pythagoreans were much occupied (Leipsic, 1880). The recently published Short History of Greck with the representation of numbers by geometrical figures. These Muthematics by James Gow (Cambridge, 1884) will be found a speculations originated with Pythagoras, who was acquainted with

convenient compilation.

(G. J. A.) the summation of the natural numbers, the odd numbers, and the PITHEAS of Massilia was a celebrated Greek navieven 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, Pythayorean triangles (12) and the observations thereon supru. 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 lythagoras 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 For this proof, see Euclid X. 117; see also Aristot., Inalyt. 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 IIence he may probably be regarded as about contemdles Proklus Diadochus zu Euclid's Elementen, Pp. 20 and 23, Herford,

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

3 It is agreeil 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 cominensurable magni-
tudes only, and were assumed to holil good for all. The l’ythagoreans 4 Proportion was not regarded by the ancients merely as a branch
themselves seem to have been aware that their proofs were uot 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 iden of also toll in an anonymous scholium on the Elements of Euclid, which
the incommensurable and to the pentagram which involved, and indeel Knoche attributes to Proclus, that the fifth book, which treats of pro-
represented, that idea. Now it is remarkable that the doctrine of portion, is common to geometry, arithmetic, music, and, in a word,
proportion is tucice 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
olde: Pythagoreans.

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

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PY THE AS

143 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 Slugyish Sea beyond it was, as stated by a private capacity and with limited means. All that we Polybius himself in the passaye already cited, derived know concerning the voyage of Pytheas (apart from such merely from hearsay. (letached 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 controversyin 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. In10,000 stadia (1000 geographical miles). To this he fortunately the reports transmitted to us at second hand added the account of Thule (which 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, passaye hy Pliny (11..1., iv. 16, 10.4) correctly represents forming a substance resembling the gelatinous molluschis anthority. According to this he 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. 108). This last sentence has led some modern writers readily as a fact what he knew (as a voyager proceeded to 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 cast. The certain extent explored, by Pytheas were not only pre- statement of Polybius that he proceeded along the whole viously unknown to the (reeks -except perhaps by vayne of the northern coasts of Europe as far as the Tanais is hearsay accounts received through the Phænicians—but evilently 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 Creek Creek 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 ('imbrie 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 anthority; trace is found in the extant notices of Pytheas. Votwiththough doubting some of his statements; but Polybius, standing this, some moulern 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 wweeping condemnation, treating the work of Pytheas as of the Vistula, which he crroneously supposed to be the . mere tissue of fables, like that of Euhemerus concerning Tanais. The only foundation for this highly improbable asPanchæa ; and even Strabo, in whose time the western sumption is to be found in the fact that in apa-aye cited by regions of Europe were comparatively well known, adopted I'liny (11.1., xxxvii. 2., 33.5) lytheais represente notating 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 Guttunes, a at a more favourable judgment, and, though Gossellin in German nation who dwelt on an estuary of the occan called his Recherches sur la Ciérophie des anciens (vol. iv. pp. Mentonomus, 6000 stailia in extent. It was a pirouluetion 168-180) and Sir G. C. Lewis in his History of Ancient thrown up ly the wives of the sea, and was 11see liy the Astronomy (pp. 466-481) revived the sceptical view, the inhabitants to burn instead of wil. It is not improbable tendency of modern critics has been rather to exaggerate that the "estuary" here mentioned really refers to tlie than to depreciate the value of what was really added by Baltic, the existence of which as a separate sea was unytheas to geographical knowledge. The fact is that our known to all ancient groumphers; but the obscure manner information concerning him is so imperfect, and the scanty in which it is indicateil, as well as the inaccuracy of the noticev preserved to us from his work at once so meagre statements concerning the place from whence the amber unul dixordant, that it is very difficult to arrive at anything was artually deriverel, luth gwint 10 the sort of hearsay like a sound conclusion. It may, however, be considered accounts which Pytheas miglit readily have picked up on this fairly established that Pytheas really made a voyage' the shores of the German (rean, without princeealing farther roand the western coasts of Europe, proceeding from Gates, than the month of the Elle, which is suure-l loy Ikert

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