‘History. 
Democritus. 
Hippocra- 
tes, 380 
A.C. 
186 
The Pythagorean school sent forth many mathema- 
ticians ; of these Archytas claims attention, because of 
his solution of the problem of finding two mean pro- 
portionals; also on account of his heing one of the first 
that employed the geometrical analysis, which he had 
learnt from Plato, and by means of which he made 
many discoveries. He is said to have applied geome- 
try to mechanics, for which he was blamed by Plato ; 
but probably it was rather for applying, on the con- 
trary, mechanics to geometry, as he employed motion 
in geometrical resolutions and constructions. 
Democritus of Abdera studied geometry, and was a 
profound mathematician. From the titles of his works, 
it has been conjectured that he was one of the prin- 
cipal promoters of the elementary doctrine respecting 
the contact of circles and spheres, and concerning ir- 
rational numbers and solids. He treated besides of 
some of the principles of optics and perspective. _ 
Hippocrates was originally a merchant, but having 
no turn for commerce, his affairs went into disorder ; 
to repair them, he came to Athens, and was one day 
led by curiosity to visit the schools of philosophy: 
There he heard of geometry for the first time; and as 
probably there is a natural adaptation of certain minds 
to particular studies, he was instantly captivated with 
the subject, and became one of the best geometers of 
his time. He discovered the quadrature of a space 
bounded by half the circumference of one circle, and 
one fourth the circumference of another, their convexi- 
ties being turned the same way. This figure, called a 
lune, he shewed to be equal to a right angled triangle 
having its sides about the right angle equal, and the 
remaining side equal to the common chord of the two 
ares ; and thus he was the first that proved a curvilineal 
to be equal to a rectilineal space. But although a kind 
of quadrature, it cannot be compared as a discovery 
with the quadrature of the parabola found afterwards 
by Archimedes: the former is merely a geometrical 
trick, which leads to nothing further; but the latter 
was an important step in the progress of the science. 
Hippocrates attempted the quadrature of the circle, but 
if ns mode of reasoning has been correctly handed 
down to us, he committed a blunder: this is the oldest 
pasa in geometry upon record. On the other 
and it must be mentioned to his credit, that he first 
proved the duplication of the cube to depend on the 
finding of two mean proportionals between two given 
lines: (See Introduction to Conic Sections.) He 
was also the first that composed Elements of Geometry, 
which, however, have been lost, and are only to be re- 
" gretted, because we might have learnt from them the 
Bryson and 
Antiphon. 
state of the science at that period. It has been said 
that, notwithstanding his want of success in commerce, 
he retained something of the mercantile spirit: he ac- 
cepted money for teaching geometry, and for this he 
was expelled the school of the Pythagoreans. This of- 
fence we think might have been forgiven, in consi- 
deration of his misfortunes. 
Two geometers, Bryson and Antiphon, appear to 
have lived about the time of Hippocrates, and a little 
before Aristotle. _ These are only known by some ani- 
madversions of this last philosopher on their attempts 
to square the circle. It appears that before this’ time, 
geometers knew that the area of a circle was equal to a 
triangle whose base was equal to the circumference, 
and perpendicular equal to the radius. 
_ Having briefly traced the progress of geometry du- 
ring the two first ages after its introduction into Greece, 
we — now to the origin of the Platonic school; 
GEOMETRY. 
which may be considered as an gera in the history of 
the science. Its celebrated founder had been tke dis. . 
ciple of a philosopher (Socrates) who set little value on Flatonie 
geometry ; but Plato entertained a very different opi- founded 
nion of its utility. After the example of Thales and about 40 
Pythagoras, he travelled into Egypt, to study under A.C. — 
the priests. He also went into Italy, to consult the fa- 
mous Pythagoreans, Philolaus, Timaus of Locris, and 
Archytas, and to Cyrene to hear the mathematician 
Theodorus. On his return to Greece, he made mz 
matics, and especially etry, the basis of his ins 
structions. He put an inscription over his school, fora 
bidding any one to enter, that did not understand geo- ' 
metry ; and when questioned concerning the probable 
employment of the Deity, he answered, that he geome- 
trized continually, meaning no doubt that he governed 
the universe by geometrical laws. | 
It does not appear that Plato composed any work Inventic 
himself on mathematics, but he is reputed to have in- of geom 
vented the Geometrical Analysis: (See Anatysts,) — 
The theory of the Conic Sections originated in this 
school; some have even supposed that Plato himself 
invented it, but there does not seem to be any sufficient 
ground for this opinion. See Conic Sections. 
A third discovery due to the Platonic school was that Of geor 
of the geometrical loci ; when the conditions which de- trical loci. 
termine the position of a point are such as to admit of 
its being any where in a line of a particular kind, but 
do not admit of its being out of that line, then the line 
is called the locus of the point: Thus, if one end of a 
straight line of a given length be at a given point, the 
locus of the other end will be the circumference of a 
given circle: Again, if the base of a triangle of a given 
area be given in position and magnitude, the locus of 
its vertex will be a given straight line, which will be 
parallel to the base ; also, if the base of a triangle be 
given in position and gt ca and its vertical angle 
be given in magnitude, the Jocus of its vertex will be 
the circumference of a given circle: all this is evident 
from the elements of geometry. Geometrical loci, con- 
sidered merely as speculative truths, are interesting; 
but their chief value arises from their utility in the re- 
solution of problems, of which, in general, they suggest 
the most elegant solutions. See Locus. 
The celebrated problem concerning the duplication puplica 
of the cube, acquired its celebrity about the time of of the eu 
Plato. Its origin, however, was earlier; for it ap] q 
that Hippocrates had reduced it to the determination 
of two mean proportionals between two given lines ; 
but it had not then excited much attention amon 
meters. We have already given its history in the in- 
troduction to Conic Sections. Plato himself gave a 
solution, and it was also resolved by Archytas, Eu- 
doxus, Eratosthenes, and Menechmus. The solutions 
of eleven of the ancient geometers, are preserved in 
Eutocius’ commentary on Archimedes, de Sph. et Cyl. 
It is probable that the érisection of an angle, a pro= mo 
blem of the same difficulty as the duplication of the of an 
cube, was likewise considered in the Platonic school, 
There is no absolute testimony of its being so ancient ; 
but, according to the natural progress of the human 
mind, it must have occurred as soon as aS« 
sumed the form of a science; for the transition from 
the bisection of an arc to its division into three, or any 
number of equal parts, or into parts which have a given 
ratio to one another, is easy. The quadratrix, a curve 
almost as old as the time of Plato, appears to have been 
invented with a view to the solution of the problem in 
its most general form. One difficulty in the problems 
