# 
1802.]} af Sketch of the Hiftory of Pure Mathematics. 
‘the problem ; and who, by doubling the 
fides,’ conftruéted an altar, not double, 
but oétuple. Meanwhile the plague con- 
tinued to rage, for the god abfolutely re- 
quired an altar exaétly double.’” A new 
deputation was fent to him, and was an- 
. {wered, that the thing required was not 
performed. It was then’ fulpécted, that 
this duplication was a myfterious, thing ; 
and affiftance was fought fromthe moft 
famous geometricians, who were them- 
felves much puzzled witlrit. a 
20. This ftory appears to have been a 
fable contrived by fome mathematician, 
who wifhed to give importance to the pro- 
blem of two mean proportionals; by find- 
ing which, Hippocrates of Chio doubled 
the cube. 
21. Whatever was the origin of this 
problem of the duplication of the cube, 
Plato gave a commodious, practical folu- 
tion of it. Menechmus propofed two 
Jearned folutions of it, which deferve 
praife, as being the firft known applica- 
tion of the geometric Joci,.and of the 
conic feétions. 
_ 22. It is probable, that the trifection 
of an angle alfo, exercifed the induftry of 
the Platonic geometricians. Being a pro- 
blem of the fame order with that of the 
duplication of the cube, it-requires like 
it, other refources than thofe of plain Ge- 
ometry. ‘The ancients difcovered feveral 
folutions of it, fome of them remarkable 
for their elegance and fimplicity. 
23. All the refined methods fketched 
by the difciples of Plato, received fuch 
augmentations from the hands of his fuc- 
cefiors, as to furnifh the materials of fe- 
_veral confiderable works. 
24. What the {chool of Plato was for 
' Geometry in particular, that of Alexan- 
dria became for the mathematics in gene- 
ral. Among the learned men whom the 
patronage of the Ptolemies firft attracted 
to Alexandria, we find Exclid, the geome- 
trician. It is believed, that he ftudied at 
Athens, under ‘the fcholars of Plato. 
‘But we are ignorant of his country, and 
the events of his life, It is only known, 
that he was gentle, modeft, and favour- 
ably received thofe who cultivated the ac- 
curate fciences. He was unacquainted 
with the art of difguifing the truth; for 
when king Ptolemy afked him, if Geome- 
try m;ght not be acquired in a way lefs 
troubjefome than that commonly ufed? 
Euclid anfwered drily, ‘‘No, Prince, 
there is no royal road to Geometry.”” 
25. Euclid has been immortalized by 
the Elements which bear his name. He 
collected into that work the geometrical 
/ 
109 
propofitions, which had been difeovered 
before his time, and which, till then, were 
fcattered and unconneéted. ‘He added ta 
them a great number of others, and con- 
catenated the whole into a fyftem, juftly 
admired by the lovers of geometrical 'ri- 
gour. I know not whether he really de- 
ferves the blame thrown upon him, for 
having employed too many definitions, for 
want of order, ahd for having been too 
ferupulous in demonftrating truths which 
are felf-evident. But he has found zealous 
defenders in the Englith geometricians*. 
26. The Elements, when they came 
from the hand of Euclid, contained but 
‘thirteen books}, namely, ten on Geome- 
try, and three on Arithmetic. 
“37, About halfacentury after Euclid, 
‘appeared: Archimedest, the greatelt geo- 
metrician- of-antiquity.. He opened new 
profpetts 
— * Note by the Tranflator Out author might 
have ufed the epithet able, as well as xealous. 
See particularly Dr. Barrow’s Mathematical 
“Letures,; Dr. Keill’s preface to his tranflae 
tion of Commandini’s Euclid, Dr. Saunder- 
fon’s Iluftration of the sth Book, in his 
Algebra; and Dr, Simfon, of Glafgow, in 
the notes annexed to his Euclid. It may be 
remarked, that this lait learned author, al- 
ways fuppofing the elements to have come 
faultlefs and immaculate from the pen of 
Euclid, contrives to lay all the blame of the 
Inaccuracies he has occafion to obferve, on 
his commentators and tranfcribers ; thus ex- 
alting his venerable author into a fort of 
mathematical Pope. Something of the fame 
{pirit is obfervable in Saunderfon and BaYrow. 
** The 14th and sth books now found in - 
the Elements of Euclid, were the work of 
Hypficles of Alexandria, who lived about 
100 years before Chrift. 
f Archimedes was botn/at Syracufe, about 
287 years before the Chriftian era, having 
been related to king Hiero. Endved with a 
fuperior genius, he enlarged the bounds of 
all the branches of the mathematics, efpe- 
cially of Geometry. A queftion propofed to 
him by Hiero, gave occafion to his hydroftati- 
cal difcoveries. That prince, having fent to 
‘a goldfmith a certain quantity wf gold, to be 
made into a crown, the artift was fufpeted of 
having retained part of the gold, and of fub- 
ftituting an equal weight of filver, As 
Hiero withed not to deface fo exquifite a 
piece of workmanthip, Archimedes was con« 
fulted on the means proper for difcovering the 
extent of the fraud; and he determined it, 
‘by a principle, the difcovery of which filled 
him with tranfport, namely, that/all bodies 
immerfed in a fluid, lofe a part of their 
weight equal to the. weight. of the fluid 
which they difplace, Archimedes applied to 
the 
