GEOMETRY. 
of doubling a cube, and trisecting isecting an angle, must have 
arisen i the tate f resolving them by 
straight lines and circles alone ; and of this the ancient 
metrical analysis gave no certain indication. The 
modern analysis teaches how to resolve every such pro- 
blem, and also shews by what lines it may be effected, 
These discoveries must be senbuted to Hae Flatanie 
achool in general ; for it is impossible to say with whom 
each pa nt Some of advanced years frequented. the 
school, as friends of its celebrated head, or out,of respect 
for his doctrines ; and others, chiefly young ns, as 
disciples and pupils. Of the first class were Laodamus, 
Archytas, an ewtetus. Laodamus was one of the 
first to whom Plato communicated his method of ana- 
lysis, before he made it public; and he is said by Pro- 
us to have profited greatly by this instrument of dis- 
covery. Archytas was a Pythagorean. of extensive 
knowledge in geometry and mechanics. He had a great 
friendship for Plato, and frequently visited him at 
Athens ; but in one of his voyages he perished by ship- 
wreck. Theetetus was a heh citizen. of Athens, anda 
friend and fellow-student of Plato under Socrates, and 
Theodorus of Cyrene, the geometer. He appears to 
have cultivated and extended the theory of the regular 
solids, 
The progress that geometry had then made, from the 
time of Hi pocrates of Chic, required, that the ele- 
ments of the science should be new. modelled. This 
was done by Leon, a scholar of Neoclis, or Neoclide, a 
piilosaphes who had studied under Plato. To Leon 
been ascribed also the invention of that part of the 
solution of a problem called its determination, which 
treats of the limits, or the cases in which it is possible. 
Eudoxus of Cnidus was one of the most celebrated of 
the friends and contemporaries of Plato. He generali- 
zed many theorems, quulthereby greatly advanced geo- 
metry. It is believed that he cultivated the theory of 
the conic sections; and its invention has been attribu- 
ted to him. . He resolved the problem of the duplica- 
tion of the cube; and itis to be regretted that. Euto- 
eius, who despised his solution, has not thought fit. to 
record it with the others, in his Commentary on Archi- 
medes. Diogenes Laertius has attributed to him the 
invention of curve lines in general; from which we 
may infer, that other curves than the conic sections 
were known in the school of Plato. Archimedes says, 
in the beginning of his treatise on the sphere and cy- 
linder, that Eudoxus found the measure of the pyramid 
and cone, and that he had treated of solids ; and others 
in have supposed, that he was the author. of the 
a of proportion as given in the fifth book. of Eu- 
clid’s Elements. 
Passing over various geometers who are said to have 
distinguished themselves, but of whom hardly any 
thing more than the names are now known, we shall 
only mention Menzchmus, and his brother Dinostratus. 
The former extended the theory of conic sections, in- 
s,. somuch that Eratosthenes seems to have given him the- 
honour of their discovery, calling them the curves of 
Menechmus. His two solutions of the problem of two. 
mean Bp tbe, are a proof of his geometrical skill, 
Several discoveries have been given to Dinostratus; but 
he is chiefly known by a property which he discovered 
of the quadratriz, a curve, supposed to have, been. ins 
vented by Hippias of Elis. 
After the death of Plato, his.school’ was divided into- 
two, which, upon some points, held opposite sentiments,. 
but agreed in regarding a knowledge of the mathema-~ 
tics as absolutely necessary to such.as would study phi« 
187 
losophy. Thus the geometrical theories which had been 
cultivated with so much ardour in his lifetime, still con- 
tinued to make progress. Xenocrates, the successor of 
Plato after Speusippus, wrote on geometry and arith. 
metic. The prifcipal geometers were all bred in the 
Platonic schoo], and among these probably we ought 
to reckon Aristeeus, who is now. little known, because 
his works are lost: we learn, however, from Pappus, 
that he was ms of the ancients who had made the most 
ess in their sublime geometry. He composed a 
Saitioe on solid Joci, in Gra hocks. mods ariother teenie 
sections, also in five books, which last contained the 
eatest part of what was afterwards given by Apol- 
onius in the first four books of his work. Pappus 
placed this work after the conics. of Apollonius, in 
the order of study which he prescribed to his son: 
This shews that it was a profound theory, and supposed 
the doctrine of conics to be previously known. He is 
reputed to have been the friend and preceptor of Euclid. 
The progress of geometry among the Peripatetics was 
not so brilliant as it had been in the school of Plato, 
but the science was by no means neglected, The suc- 
cessor of Aristotle composed sevetel works relating to 
mathematics, and particularly a complete history of 
these sciences down to his own time: there were four 
books on the history of geometry, six on that. of astro- 
nomy, and one on that of arithmetic. What a treasure 
this would have been, had. we now possessed it! 
The next remarkable epoch in the history of geome- 
try, after the time of Plato, was the establishment of the 
ool of Alexandria, by Ptolem 
History. 
Xenocratess 
Aristeus. 
School of 
Alexandria: 
Lagus, about 300 founded 
years before the Christian era. This event was highly 3° -- © 
propitious to learning in general, and particularly to 
every branch of mathematics then known ; for the whole 
was then cultivated with as much attention as had been 
bestowed on geometry alone in the Platonic school. 
It was here that the celebrated geometer Euclid flou- Buclid 
rished under the first of the Ptolemies : his native place flourished: 
is not certainly known, but he appears to have studied 
at Athens, under the disciples of Plato, before he sets 
tled at Alexandria. Pappus, in the introduction to the 
seventh book of his Collections, gives him an excellent 
character, describing him as gentle, modest, and be-. 
nign towards all, and: more especially such as. cultis 
vated and improved the mathematics. Thezg is an 
anecdote recorded of Euclid; which seems ¢o,shew he 
was not much of a courtier: Ptolemy. Philadelphus 
having asked him whether there was. any easier way 
of studying geometry than that commonly taught ; his. 
reply was, “ there is no royal road to geometry.” This 
celebrated man composed treatises on various branches 
of the ancient mathematics, but he is best known: by 
his Elements, a work on etry and arithmetic, in 
thirteen books, under which he has collected all the éle« 
mentary truths of geometry, which had been found be+ 
fore his time. The selection and arrangement. have 
been made with such judgment, that, after a period of ’ 
2000 years, and notwithstanding the: great additions 
made to mathematical science, it is still generally allowed 
to be the best. elementary work on geometry, extant. 
Numberless treatises. have been written since the revi- 
val of learning, some with a view to improve, and’ 
others to supplant the work of the Greek geometer; but 
in this country, at least, they have been generally ne- 
lected and forgotten, and Euclid maintains, his place 
in our schools, 
280 A, €.. 
Of, Euclid’s Elements, the first four books treat. of pyctid’s 
the properties of plane figures; the fifth contains the Elements. 
theory of proportion; and the: sixth its application to 
