But if the differences of triangles conduce to generation, we must justly acknowledge the triangle to be the principle and author of the constitution of sublunary things. For the right angle gives them essence, and determines the measure of its being; and the proportion of a rectangle triangle causes the essence of generable elements; the obtuse angle gives them all distance, the proportion of an obtuse-angled triangle augments material forms in magnitude, and in all kinds of mutation; the acute angle makes their nature divisible, the proportion of an acute-angled triangle prepares them to receive divisions into infinite; and, simply, the triangular proportion constitutes the essence of material bodies, distant and every way divisible. Thus much for triangles.
Of quadrangular figures, the Pythagoreans hold that the square chiefly represents the Divine Essence, for by it they principally signify pure and immaculate order; for rectitude imitates inflexibility, equality firm power; for motion proceedeth from inequality, rest from equality.642 The gods therefore—who are authors in all things of firm consistence, pure incontaminate order, and inevitable power—are not improperly represented by the figure of a square.
Moreover Philolaus, by another apprehension, calls the angle of a square the angle of a Rhea, Ceres, and Vesta. Seeing that the Square constitutes the Earth and is the nearest element to it (as Timaeus teaches), but that the Earth itself receives genital seeds and prolific power from all these gods, he not unaptly compares the angle of a square to all these life-communicating deities. For some call the Earth and Ceres herself, Vesta; and Rhea is said wholly to participate of her, and that in her is all generative causes. Whence Philolaus says the angle of a square, by a certain terrestrial power, comprehends one union of these divine kinds.
The Greek understanding of geometry can be observed in many surviving remants of the ancient world, including coinage. This silver stater issued on the island of Aegina c.480-457 B.C. pairs a sea turtle with a square incuse punch divided into five sections.
Photo courtesy of Numismatica Ars Classica
CHAPTER 2
PROPOSITIONS
Of the many Geometrical theorems invented by Pythagoras and his followers, these are particularly known as such.
Only these three Polygons fill up the whole space about a point: the equilateral Triangle, the Square, and the Hexagon equilateral and equiangle.643 The equilateral triangle must be taken six times, for six two-thirds make four right angles; the hexagon must be taken thrice, for every six angular angle is equal to one right angle, and one third; the square four times, for every angle of a square is right. Therefore six equilateral triangles joined at the angles, complete four right angles, as do also three hexagons and four squares. But of all other polygons whatsoever, joined together at the angles, some exceed four right angles, others fall short. This Proclus calls a Celebrious Theorem of the Pythagoreans.
Every triangle has the internal angles equal to two right angles.644 This theorem, Eudemus the Peripatetic ascribes to the Pythagoreans. For their manner of demonstration see Proclus.
In rectangle triangles, the square which is made of the side opposite the right angle [the hypotenuse], is equal to the squares which are made of the sides containing the right angle.645
This theorem Pythagoras found out; and by it he showed how to make a gnomon or square (which the carpenters cannot do without much difficulty and uncertainty), not mechanically, but according to rule. For if we take three rulers: one of them being three feet long, the second four feet, the third five feet, and put these three so together that they touch one another at the ends in a triangle, they make a perfect square. Now if to each of these rulers be ascribed a square, that which consisted of three feet will have nine; that which of four will have sixteen; that which of five will have twenty-five. So that how many feet the areas of the two lesser squares of three and four make, so many will the square of five make.646
Illustration of the Pythagorean Theorem (a2+b2=c2)
Apollodorus the Logician,647 and others relate that upon the invention of this Theorem, Pythagoras sacrificed a Hecatomb to the Muses,648 in confirmation whereof they alleged this epigram,
That noble Scheme Pythagoras devis'd,
For which a Hecatomb he sacrific'd.
Plutarch says, it was only one ox 649 and even that is questioned by Cicero as inconsistent with his doctrine, which forbade bloody sacrifices.650 The more accurate therefore relate (says Porphyry), that he sacrificed an ox made of flower; or, as Gregory Nazianzen says, of clay.651
But Plutarch doubts whether it was for the invention of the forementioned proposition that Pythagoras sacrificed an ox, or for the problem concerning the area of a Parabola.652 Indeed, the application of spaces or figures to lines is (as his follower Eudemus affirms), an invention of the Pythagorean Muse: Parabola, Hyperbola, Ellipsis.653 From them, the later writers taking these names, transferred them to conical lines, calling one parabola, another hyperbola, another, ellipsis. Whereas those ancient divine persons, the Pythagoreans, signified by those names the description of places applied to a determinate right line. For when a right line being proposed, the space given is wholly adequate to the right line, then, they say the space is applied . But when you make the length of the space greater than that of the right line,654 then they say it exceeds . But when less, so as the space being described there is some part of the right line beyond it, then it falls short . In this sense Euclid uses parabola, Liber I, prop. forty-four, and hyperbola and ellipsis, in the sixth book.
CHAPTER 3
HOW HE DETERMINED THE STATURE OF HERCULES
Plutarch, in his treatise discussing how great difference there is in the souls and bodies of men as to ingenuity and strength, relates that Pythagoras reasoned curiously and subtly in finding out and collecting the extraordinary stature and length of the body of Hercules.655 For it being manifest that Hercules measured with his feet the running course of Olympian Jupiter at Pisa, and that he made it 600 feet long, and that all the other running courses in Greece instituted afterward by others were 600 feet long, yet shorter than this; Pythagoras easily understood the measure of Hercules's foot. He determined that it was proportionably as much longer than that of other men as the Olympic course he established was longer than all others. And having comprehended the size of Hercules's foot, he considered what length of body did suit with that measure, according to the natural proportion of all the members one to another. He concluded that Hercules was so much taller in body than others, by how much the Olympic course was longer than the rest which were made after the same number of feet.
ASTRONOMY
N either did the Pythagoreans superficially consider the speculation of celestial things, in which Pythagoras was also exquisite, as appears by these few remains.656
Although only a demi-god, Hercules (Heracles to the Greeks) was as familiar as any of the Olympian deities. He is shown holding a drinking vessel (a rhyton) and his club on this electrum stater struck in about 380 B.C. at the city of Cyzicus on the southern shore of the Propontis.
Photo courtesy of Numismatica Ars Classica
CHAPTER 1
THE SYSTEM OF THE SPHERES
The word , Heaven, is taken three ways: first, for the Sphere of Fixed Stars; second, for all that is between the Sphere of Fixed Stars and the Moon; lastly, for the whole world, both Heaven and Earth.657
The anonymous writer of the life of Pythagoras affirms that Pythagoras said there are twelve orders in Heaven. The first and outmost is the fixed Sphere; next to this is the star of Saturn; and then the other six planets, Jupiter, Mars, Venus, Mercury, Sun, and Moon; next these, the Sphere of Fire, then that of Air, then of Water, last of all the Earth.658
But they who seem more strictly to follow the mind of Pythagoras and his disciples, aver that they held the celestial Spheres to be ten—whereof nine only are visible to us (the fixed Sphere, the seven planets, and our Earth). The tenth is Antichthon, an Earth above, or opposite to ours.659 This Antichthon they added to make up the numbe
r of the moving bodies.660 For they considered that the affections and proportions of music consist in numbers; that all other things appear to be assimilated to numbers; that numbers are the first of all nature; and that the elements of numbers are the elements of all beings. They therefore asserted that all Heaven is harmony and number, and that the affections and parts of Heaven are correspondent to number. And collecting these, they adapted them to the composition of the whole; wherein if anything were wanting they supplied it, that the whole might be alike compacted. Thusly, because the Decad seems to be perfect and to comprehend the whole nature of numbers, they asserted the celestial spheres to be ten. Now there being nine only visible to us, hereupon they conceived the tenth to be Antichthon, an Earth opposite to ours.
As concerning the order and system of these, the Pythagoreans held, that in the middle of the world is Fire.661 Or, as Stobaeus says,662 in the midst of the four elements is the fiery globe of unity which they term Vesta and Monad.663 Simplicius says that they who understand this thing more intimately state that this fire is the procreative, nutritive, and excitative power which is in the midst of the Earth. But Simplicius himself seems not to have apprehended the right meaning of the Pythagoreans—who by this fire, or fiery globe of unity, meant nothing else but the Sun seated in the midst of the universe, immoveable, about which the other parts of the world are moved. This opinion Pythagoras seems to have derived from the Egyptians, who hieroglyphically represented the Sun by a beetle. They chose this symbol because the beetle, having formed a ball of cow dung, and lying upon its back, rolls it about from claw to claw; so the other parts of the world are moved and rolled by and about the Sun.664
By this immovable fire in the midst of the Universe, they understood not (as Simplicius conceives) that the Earth is manifest. Forasmuch as they further held that the Earth is not immovable,665 nor seated in the midst of the globe, but suspended, as being one of the stars carried about the fire which is in the middle; and that thereby it makes day and night.666 The reason why the Earth ought not to have the middle place is because the most excellent body ought to have the most excellent place. Fire is more excellent than Earth, and the center more excellent than all places without it; therefore they conceived that not the Earth, but the Fire is placed in the midst.667 Moreover, because that which is the most excellent of the universe, ought principally to be preserved, and the middle is such, therefore they term the Fire ,† the custody of Jupiter.
The same they held of the Antichthon also, that like our Earth it is suspended, as being one of the stars carried about the Fire, and thereby makes day and night.668 By this Antichthon, Clemens says they understood Heaven. Simplicius says the Moon, as being a kind of aetherial Earth—as well for that it eclipses the light of the Sun which is proper to the Earth, as for that it is the bound of celestials, as the Earth of sublunaries. But the contrary is manifest, as well from the completing of the number ten (in respect whereof this Antichthon was imagined). For they held it is not visible to us by reason that following the motion of this Earth, it is always opposite to, or beneath us, and the bigness of our Earth hinders us from seeing it.669 And Aristotle affirms there were some who conceived the Antichthon to be the cause why there are more eclipses of the Moon than of the Sun, which may likewise happen by reason of many other bodies invisible to us.
Laertius, who, says Philosaus, was the first who conceived the Earth to have a circular motion, seems to mean no more than that he first committed this opinion of Pythagoras to writing and first made it public.670 For Eusebius expressly affirms that he committed to writing the dissertations of Pythagoras. His opinion, as delivered by Plutarch and Stobaeus, is exactly the same: for he placed fire in the midst, which he called the genius of the universe, and the mansion of Jupiter, and the mother of gods, and altar, and ward, and measure of nature. He conceived that the ten celestial bodies move about it—Heaven, the Sphere of Fixed Stars, the five planets, the Sun, the Moon, the Earth, and lastly the Antichthon.
From the same fountain seems Aristarchus the Samian to have derived this hypothesis, though some ascribe the invention thereof to him. For he supposed that the Sun and planets move not, but that the Earth moves round about the Sun which is seated in the middle.671 Plutarch adds that Plato in his old age repented for that he had placed the Earth in the midst of the universe, and not in its proper place.672
This opinion was of late revived by Nicolaus Copernicus, who considering how inconvenient and troublesome it is to understand and maintain the motions of the Heavens and immobility of the Earth, explained it with admirable ingenuity after the mind of the Pythagoreans. According to whose hypothesis, the Sun, as we said, is settled in the midst of the world, immovable. The Sphere of Fixed Stars in the extremity or outside of the world is immovable also. Between these are disposed the planets, and amongst them the Earth as one of them. The Earth moves both about the Sun, and about his proper axis. Its diurnal motion by one revolution, makes a night and a day; its annual motion about the Sun, by one revolution makes a year. So as by reason of its diurnal motion to the east, the Sun and other stars seem to move to the west; and by reason of its annual motion through the Zodiac, the Earth itself is in one sign, and the Sun seems to be in the sign opposite to it. Between the Sun and the Earth they place Mercury and Venus. Between the Earth and the Fixed Stars are Mars, Jupiter, and Saturn. The Moon, being next the Earth, is continually moved within the great orb between Venus and Mars, round about the Earth as its center. Its revolution about the Earth is completed in a month; about the Sun (together with the Earth) in a year.
CHAPTER 2
THE MOTIONS OF THE PLANETS
As concerning the course and revolution of the planets, they affirm the great year to be the revolution of Saturn. For the rest of the planets complete their periods in a shorter time; but Saturn in no less then thirty years. Jupiter in twelve years; Mars in two; the Sun (speaking according to the phenomenon) in one; Mercury and Venus as the Sun (or to speak more exactly, Mercury in three months, Venus in eight); the Moon as being next the Earth, soonest, in a month.673
According to this inequality appears the motion of the planets to our sight, by reason that the eye is out of the center of the orb. But in the whole course of Astronomy (says Geminus) are supposed the motions of the Sun, Moon, and five planets, equal and circular; contrary to the diurnal revolution of the world. The Pythagoreans, first applying themselves to these disquisitions, supposed circular and equal motions of the Sun, the Moon, and five planets. For they admitted not such irregularity in eternal and divine bodies, that sometimes they should move swifter, sometimes slower, and sometimes stand still (as the stationary points in the planets). Neither in any sober, well-tempered person could we admit such irregularity of pace. Indeed, the necessities of life often cause men to move faster or slower; but in the incorruptible nature of the stars, there cannot be alleged any cause to swiftness and slowness. Wherefore the Pythagoreans proposed this question, how the phenomena's might be salved by circular and equal motions.
That Pythagoras himself observed these irregularities and the ways to assuage them, appears from Iamblichus, who says he communicated a revelative right knowledge of all manner of motion of the spheres and stars, [“their oppositions, their eclipses, inequalities, eccentricities and epicycles”]. ‘ [“oppositions”] is the anticipation of any planet, either in respect to some other planet or to the Fixed Stars. ‘ [“eclipses”] is the falling later of any planet, either in respect to some other planet, or to the Fixed Stars. ‘, Inequality,† is when the same planet moves slower and faster according to its distance from the Sun in the Pythagorean hypothesis (or from the Earth in the Ptolemaic), slower in its aphelion, faster in its perihelion.
The two ways of solving these phenomena are by eccentrics or by epicycles. For a homocentric with an epicycle (as Eudoxus first demonstrated), is equipollent to an eccentric. Eccentricity is when the center of their equal motion is distant from the center of their apparent motion. Both these Iamb
lichus ascribes to Pythagoras,674 from whom perhaps they were communicated to Eudoxus, to whose invention others ascribe them.675
CHAPTER 3
THE INTERVALS AND HARMONY OF THE SPHERES
Pythagoras (says Censorinus) asserted that this whole world is made according to musical proportion; and that the seven planets between Heaven and the Earth, which govern the nativities of mortals, have a harmonious motion. And they have Intervals correspondent to musical Diastemes, and render various sounds according to their several heights so consonant that they make most sweet melody. But to us these sounds are inaudible by reason of the greatness of the noise, which the narrow passage of our ears is not capable to receive.676
For as Eratosthenes determined that the largest circumference of the Earth is 252,000 stadia; so Pythagoras declared how many stadia there are between the Earth and every star. In this measure of the world, we are to understand the Italic stadium, which consists of 625 feet. (For there are others of a different length, such as the Olympic of 600 feet, and the Pythic of 500.) From the Earth therefore to the Moon, Pythagoras conceived to be about 12,600 stadia. And that distance, according to musical proportion, is a tone. From the Moon to Mercury (who is called [“twinkling”]) half as much, as it were a hemitone. From thence to Phosphorus, which is the star Venus, almost as much, that is, another hemitone. From thence to the Sun, twice as much, as it were a tone and a half. Thus the Sun is distant from the Earth three tones and a half, which is called diapente; from the Moon, two and a half, which is diatessaron. From the Sun to Mars, who is called [“Fire”], there is the same interval as from Earth to the Moon, which makes a tone; from thence to Jupiter, who is called [“radiant”], half as much, which makes a hemitone. From there to the Supreme Heaven where the signs are is a hemitone also. So that the diasteme from the Supreme Heaven to the Sun is diatessaron, that is, two tones and a half: from the same Heaven to the top of the Earth six tones, a diapason concord. Moreover he referred to other stars many things which the masters of music treat of, and showed that all this world is Enharmonic. Thus Censorinus. But Pliny, delivering this opinion of Pythagoras, reckons seven tones from the Earth to the Supreme Heaven; for whereas Censorinus accounts for a hemitone from Saturn to the Zodiac, Pliny makes it sesquiduple.677
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