Pythagoras: His Life and Teaching, a Compendium of Classical Sources (25 page)

G
EOMETRY

P
ythagoras (says Iamblichus) is reported to have been much addicted to Geometry, for amongst the Egyptians, of whom he learned it, there are many geometrical problems.
⁶³²
The most learned Egyptians were continually, for many ages of gods and men, required to measure their whole country by reason of the overflowing and decrease of the Nile—whence it is called Geometry.
⁶³³

Some there are who ascribe all theorems concerning Lines jointly to the Egyptians and the Chaldeans; and all these, they say, Pythagoras took, and augmenting the science, explained them accurately to his disciples. Proclus affirms that Pythagoras first advanced the geometrical part of learning into a liberal science, considering the principles more sublimely (than Thales, Ameristus, and Hippias, his predecessors in this study) and thoroughly investigated the theorems immaterially and intellectually.
⁶³⁴
Timaeus says that he first perfected geometry, the elements whereof (as Anticlides affirms), were invented by Moeris.
⁶³⁵
Aristoxenus says that Pythagoras first introduced measures and weights amongst the Grecians.
⁶³⁶

CHAPTER 1

O
F A
P
OINT
, L
INE
, S
UPERFICIES AND
S
OLID

P
ythagoras asserted a Point to be correspondent in proportion to a unit; a Line to two; a Superficies to three: a Solid to four.
⁶³⁷

The Pythagoreans define a Point as a Monad having Position.
⁶³⁸

A Line, being the second, and constituted by the First Motion from indivisible nature, they called Duad.
⁶³⁹

A Superficies they compared to the number three, for that is the first of all causes which are found in figures: for a circle, which is the principle of all round figures, occultly comprises a triad in center, space, and circumference.
⁶⁴⁰
But a triangle, which is the first of all rectilinear figures, is manifestly included in a ternary, and receives its form according to that number.

Hence the Pythagoreans affirm, that the triangle is simply the Principle of Generation, and of the formation of things generable. Whereupon Timaeus says that all proportions, natural as well as of the constitution of elements, are triangular; because they are distant by a threefold interval and are collective of things every way divisible. Triangles are variously permutable and are replenished with material infinity, and represent the natural conjunctions of bodies dissolved. As triangles are comprehended by three right lines, they also have angles which collect the multitude of lines, and give the additional property of an angle and conjunction to them.
⁶⁴¹

With reason therefore did Philolaus dedicate the angle of a triangle to four gods: Saturn, Pluto, Mars, and Bacchus—comprehending in these the whole quadripartite ornament of elements coming down from heaven or from the four quarters of the Zodiac. For Saturn constitutes an essence wholly humid and frigid; Mars wholly fiery; Pluto comprises all terrestrial life; Bacchus predominates over humid and hot generation, of which wine is a sign being humid and hot. All these differ in their operations upon second bodies, but are united to one another, for which reason Philolaus collected their union according to one angle.

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

P
ROPOSITIONS

O
f 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
(
a
²
+
b
²
=
c
²
)

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.