185 
GEOMETRY. 
Gromertry is that branch of mathematics which treats 
of the ies of extension and figure. The name is 
derived from yiwwsrgi«, the science of land measuring. 
_ There is a certain degree of geometrical knowledge 
which naturally arises out of the wants of man, in every 
state of society. It is impossible to build houses and 
temples, or to apportion territory, without employing 
some of the principles of geometry. Hence we cannot 
c to find a period of society, or a country in which 
it was altogether unknown. 
Ancient writers have generally supposed. that it was 
first cultivated in Egypt; and, according to some, it de- 
rived its origin from the necessity of determining every 
year the just share of land that belonged to each pro- 
prietor, after the waters of the Nile, which annually 
overflowed the country, had returned into their ordina- 
channel, It may however be remarked, that the 
dbliteration of the lemmas. by the inundation, is 
quite a conjecture, and not a very probable one. 
discoveries are attributed to him, concerning the circle, 
and the comparison. of triangles, In particular, he 
first found that all angles in a semicircle are eet 
angles; a. discovery which is said to have excited in 
his mind that lively emotion, which is perhaps only 
felt by poets and geometers: he foresaw the import- 
ant consequences to which this proposition led, and 
he expressed his gratitude to the muses by a sacrifice, 
This, however, is but a small part of what geometry 
owes him; and it is much to be regretted that the loss 
of the ancient history of the science should have left 
us in uncertainty as to the full extent of the obliga- 
tion. 
It is probable that the 
of Thales were acquaint 
of Ameristus and Anaximander only have reached our 
History. 
eater number of the disciples Early geo- 
with geometry ; but the names meters. 
-Ameristus 
times, The first is said to have been a skilful geometer; and Anaxi- 
the other composed a kind of elementary treatise or in- mander. 
troduction to 
metry, the earliest on record. Thales Born 610 
was succeeded in his school by Anaximander, who is “ © 
said to have invented the sphere, the gnomon, geogra- 
phical charts, and sun-dials; he was succeeded by Anaxi- 
menes ; and this philosopher again was succeeded by Anaxime- 
his scholar Anaxagoras, who, being cast into prison on ac- nes. Born 
Some ters, among whom is Herodotus, fix the 
igin of geometry at the time when Sesostris intersect- 
ed Egypt by numerous canals, and divided the country 
among the inhabitants. Sir Isaac Newton has adopted 
this 
inion in his araye Ye and has supposed that 
hot 
this division was made by h the minister of Sesos- count of his opinions relating to astronomy, employed 554 A. C. 
tris, who, according to him, was the same as Osiris; himself in attempting to square the circle. This is the ey 
and this conjecture is supported by some ancient autho- earliest effort on record, to resolve the most celebrated non ‘ae a 
rities. Aristotle has however attributed the invention problem in geometry. 
tothe Egyptian priests, who, living secluded from the 
world, had leisure for study. Thus, various opinions 
have been entertained respecting the origin of geome- 
try, but all have agreed in fixing it in E ype 
The celebrated philosopher, Thales of Miletus, trans- 
lanted the sciences, and particularly mathematics, from 
gypt into Greece. He was born about 640 years be- 
fore Christ, and being unable to gratify his ardent de- 
sire for knowledge at home, he travelled into Egypt, at 
an advanced period of life, where he conversed with 
the priests, the only depositories of learning in that 
country. Diogenes Laertius relates, that he measured 
the height of the pyramids, or rather the obelisks, by 
means of their shadow ; and Plutarch says, that the 
king Amasis was astonished at this instance of sagacity 
in the Greek philosopher; which is a proof that the 
tians had made but little progress in the science. 
It is also stated” by Proclus, that Thales employed the 
principles of geometry to determine the distance of 
vessels remote from the shore. On his return to Greece, 
his celebrity for learning drew the attention of his 
countrymen: he soon had disciples, and hence the 
‘foundation of the Ionian school, so called from Ionia 
his native country. 
There were some slight traces of what may be called 
natural geometry in Greece, before the time of Thales : 
Thus, Euphorbus of Phrygia is said to have discovered 
some of the properties of a triangle ; the square and the 
level have been ascribed to Theodoras of Samos; and the 
compasses to the nephew of Dedalus, But these can 
only be considered as a kind of instinctive geometry ; 
the origin of the true geometry among the Greeks must 
be fixed to the period of the return of Thales. It was 
he that laid the foundation of the science, and inspired 
his countrymen with a taste for its study ; and various 
VOL, X. PART I. 
Pythagoras. was one of the earliest and most success- 
ful cultivators of geometry. He was born about 580 
years before the Christian era; he studied under Thales, 
and by his advice travelled into Egypt, Here he is said 
to have consulted the columns of Sothis, on which that 
celebrated person had engraven the principles of geo- 
metry, and which were deposited in subterranean va- 
ses. A learned curiosity induced him to travel also 
into India ; and it is far from being improbable, that he 
was more indebted for his knowledge to the Brahmins, 
on the banks of the Ganges, than to the priests of 
Pythagoras, 
Born 580 
A. C. 
Egypt. On his return,. finding his native country a pythaso- 
prey to tyranny, he settled in Italy, and there founded dion sical 
one of the most celebrated schools of antiquity. . He is founded 
said to have discovered that in any right angled triangle, *out 550 
the square on the side opposite the right angle, is equal 
to the twowsquares on the sides containing it; and, on 
this account, to have sacrificed one hundred oxen, to 
express his gratitude to the muses. This, however, was 
incompatible with his moral principles, which led him 
to abhor the shedding of blood on any account what« 
ever ; and besides, the moderate fortune of a philoso. 
pher would not admit of such an expensive proof of his 
piety. The application which the Pythagoreans made 
of geometry gave birth to several new theories, such as 
the incommensurability of certain lines, for example, 
the side of a square, and its diagonal, also the doctrine 
of the regular solids, which, although of little use in it- 
self, must have led to the discovery of many proposi- 
tions in geometry. Diogenes Laertius has attributed to 
Pythagoras the merit of having discoyered,. that of all 
figures having the same boundary, the circle among 
alee figures, and the sphere among solid figures, are 
the most capacious: if this was so, he is the first on 
-record that has treated of isoperimetrical problems. 
2A ; 
