Part IX.

le-
(r) Muf. lib. rag* 95-
for moving particular kpaffions, there^ is a ^ markable fragment of YOumon the Muficiarij cited . by'Cr) Anjlides.
SECT. 3.
Geometry.
{a) Cap, 29 p. 144.
J)Ythagoras (faith {a) Jdmblichus') is re¬
ported to have been much addiQed to Geome try •, for^ amangji the Egyptians C of whom he learned it 3 there ^re many Geometrical Problems ^ ibemoft learned of them having been continually^ jor many . ages of gods and men:, necejjitated tomea- f lire their lohole country^ by reajon of the overflow¬ ing and decreafe Nilus^ whence it is called Geo¬ metry. (b) Some there arc who afcribe all Theo¬ rems conceininghines.,jointly to the Egyptians Chaldeans i and all thefe^ they fay^ Pythago¬ ras took., and augmenting the Science.,explainedthem (c) Ill Euclid, Dijcipfes. {c) Troclus affirms
(b) Ibid.
lib.
(d) Liert.
(c) Laert.
that he jirji advanced theGeometrical part of Learn ing into a Liberal Science., confldering the Princi¬ ples more fublimely ( than Thales, Amerifius, and Hippicis.hX^ predeceflbrs in this ftudy) andperfcru- tating thcThcorems immaterially and intellehlually: (d) Tim£ifs faith. That he flrji perfeSed Geome-. try the Elements whereof, (as Anticlides affirms ) were invented by Moeris. (e) Arilfoxenus, that he firjl introduced Meajures and Weights amongji the Greecians.
CHAP. I.
Of a Point, Line, Superficies and Solid.
(a) Prhd.m (dfVyLthagoras Point to be correfpon-
Euchd. lib. 2. JL dent in proportion ’ to an unite ; a Line, 1 ■ 3 Superficies, to 3 *, a Solid, to 4. The
define a point, z Monad having po-
def. I. fition.
(0 iW.lib.i. firjAlrne being the Second, and conftituted def. 2. by the firft Motion, from indivifible nature, they called Duad.
(d) A fuperficies they compared to the Num-
(d) Pml.lib.2,
def. 5.
(e) Frocl. in End. lib. 2. def. 24.
ber 3. for that is the firft of' all caufes which are found in figures : for a Circle, which is the Principle of all round figures, occultly compri- feth a Triad in center, fpace, and circumference. But a Triangle, which is the firft of all re£tiline figures, is manifeftly included in a Ternary, and receiveth its form according to that number. (e) Ucnce xhe Pythagoreans that the Tri¬ angle is fimply the Principle of generation, and of the formation of things generable*, whereupon Tunxm faith, that all proportions, as well natu¬ ral, as of the conftitution of Elements, are Tri¬ angular, becaufe they are diftant by a threefold interval, and are colleOiive of things every way divifible, and varioufly permutable, and are re- plenifhed with Materia, infinity, and reprefent the natural conjunctions of bodies diflblved, as Triangles which are comprehended by three right Lines ; bur they have Angles which collect the multitude of Lines, and give an adventitious Angle and Conjunction to them. With reafbn therefore did Philolaus dedicate the Angle of a 1 riangle to four Gods, Saturn, Pluto, Mars, Bac-
chm, comprehending in thefe the whole quadri- rartite Ornament of Hem ents coming down from -leaven, or from the four quarters of theZodiack. orSaturn conftituteth . an eflence wholly humid and frigid : Mars wholly fiery, Pluto comprifeth all Terreftrial life, Bacchus predominates over hu¬ mid and hot generation, of which Wine is afign, being humid and hot. All thefe differ in their operations upon fecond bodies, but are united to one another, for which realon Philolaus collected their Union according to one Angle. But if the differences of Triangles conduce to generation, we muftjuftly acknowledge the Triangle to be the Principle and Author of the conftitution of fub- lunary things, for the right Angle gives them effence, and determines the meafure of its being i and the proportion of a reUangle triangle cau- feth the effence of generable Elements *, the ob- tufe Angle giveth them all diftance, the propor¬ tion of an obtufe angled triangle augmenteth material forms in magnitude, and in all kinds of mutation^ the acute Angle maketh their nature divifible, the proportion of an acute-angled Tri¬ angle prepares them to receive divifions into infi¬ nite ; and fimply, the Triangular proportion con¬ ftituteth the effence of Material bodies, diftant and every way divifible: Thus much for Triangles.
if) Of quadrangular ^%mts,x\iQpythagoreans (f) hold that the fquare chiefly reprefenteth the di- vine effence, for by it they principally fignifie pure and immaculate order •, for reUitude imitateth in¬ flexibility, equality firm power; forMotionpro- ceedeth from inequality, reft from equality. The Gods therefore, who are Authors in all things of firm Gonfiftence, and pure incontaminate order, . and inevitable power ; are not improperly repre- fented by the figure of a Square. Moreover, Phi¬ lolaus by another apprehenfion calleth the Angle of a Square, the Angle of a Rhea, Ceres, andFir- Jia-, for feeing that the Square conftituteth the Earth, and is the neareft Element to it, as Timeerse teacheth, but the earth it felf receiveth Genital feeds and Prolifick power from all thefe gods ; he not unaptly compareth the Angle of a Square to all thefe life-communicating Deities. Forlbme call the Earth and Ceres her felf Vefla ; and Rhea is faid wholly to participate of her, and that in her are all generative caufes. Whence Philolaus faith, the Angle of a Square by a certain terre¬ ftrial power, comprehends one union of thefe divine kinds.
Trod, lib. 2* 34*
SECT. II.
Propofltions.
the many Geometrical Theorems inven-
thefe
ted by Pythagoras, and his followers, are particularly known as fuch.
{a) Only thefe three Polygones fill up the whole fpace about a point. The xqui lateral Triangle, and ^e^i. the Square, and the hexagone equilateral and^om. equiangle. The Gtquilateral Triangle muft be taken fix times, for fix two thirds make four right Angles ; the Hexagone muft be taken thrice, for every fex angular Angle is equal to one right Angle, and one third ; the fquare four times, for every Angle of a Square is right. Thereibre fix equilateral Triangles joined as the Angles,
frod. ill . Jib. g. ;
> 20. I
com-:
391