% supplied. ‘The whole subject is, however, so much 
his: own, that what is te the spiral of Conon, is 
ustially called the spiral of Archimedes. He has also 
ited Of the Equilibrium Planes, or of their centres 
: avity, in two books; and next’Of the Quadrature of 
Me Porchota This is the first complete quadrature of 
a curve that was ever found. He here shews, that 
the area of any ent of a parabola cut off by a 
chord, is two-thirds of the circumscribing parallelo- 
, and this he proves by two different methods. 
is Arenarivs was written to evince the possibility 
of expressing, numbers, the grains of sand that 
might fill the whole space of the universe. Here 
he introduces a of a geometrical Sa 
sion, that has since been made cae a eras ; the 
th of logarithms ; but it wou going too far to 
‘sips that Archimedes had made any approach to 
that valuable invention. This tract is. valuable, not on 
account of the subject on which he treats, but because 
of the information it contains ecting the ancient 
astronomy, and the application which it gives of the 
Greek arithmetic. In addition to the works we have 
enumerated, there is a treatise on bodies which are car- 
ried on a fluid, in two books, and a book of Lemmas, 
which is a collection of theorems and problems, curious 
in themselves, and useful to the geometrical analysis. 
’ These are all the writings of Archimedes now extant, 
but many have been lost. , 
The writings of Archimedes are the most precious 
relict of the ancient geometry: they shew to what an 
‘extent such a genius as his could carry its method of 
demonstration ; but they likewise prove, that there 
were certain limits beyond which it Lacinha inapplica- 
ble, on account of the unwieldiness of the machinery. 
In general, the’ pro, of discovery is slow; but 
Archimedes took up the subject where men of ordin 
ities were ata stand, and, by the vigour of his 
mind, anticipated the labour of ages: he was undoubt- 
edly the Newton of antiquity. 
tocius has written a commentary on a part of the 
works of Archimedes, viz. on the books De Sphera et 
lindro, de dimensione circuli et de aquiponderantibus. 
th the year 1543, Nicolas Tartalea translated from 
' Greek into Latin, and published at Venice, the treatises 
1. De Centris Gravium, &c. 2. Quadratura Parabole. 
8. De Insidentibus aque, liber primus ; and, in'1555, 
the two books De Insidentibus* aque appeared at 
Venice. In 1543, an edition of the works of Archi- 
medes was published at Basle, with the Latin trans- 
lation of Seki of Cremona, and revised by Regimon= 
tanus. In this, the two books De Insidentibus in 
Fluido, and the Lemmata were wanting, but it con- 
tained the commentary of Eutocius in Greek and 
Latin. Other editions of his works, or parts of them, 
have been given by Commandinus, Renault, Greaves 
and Foster, Borelli, Barrow, Maurolicus, Wallis, some 
with commentaries ; but these are in a manner super- 
seded by the Oxford edition of Torelli in Greek and 
Latin, printed in 1792, and the French translation of 
Pp in 4to and 8vo, the latter printed in 1808. 
For farther information respecting this geometer, see 
ARCHIMEDES, | me 
__ Eratosthenes flourished in the Alexandrian school, 
about the time of Archimedes: his extensive acquire- 
ments in all branches of knowledge induced the third’ 
_ Ptolemy to make him his librarian. As a geometer, he 
might rank with Aristeus, Euclid, and Apollonius. 
come down to us in Eutocius’ Commentary on. Archi- 
medes; and we find it recorded in Pappus, that he 
GEOMETRY. 
His construction of’ the duplication of the: cube, has - 
189 
wrote two books on a branch of the Y pits bepe analy- _ History. 
sis, which were entitled De Locis ad medielates ; these 
appear to have been conic sections. There is an arith- 
metical invention attributed to him, by which the prime , 
numbers may be determined. Its. nature has been de- 
scribed in our article ArttuMetic, page $72. It may 
be presumed that Eratosthenes composed many works; 
one is said to have been on the conic sections, and 
others on astronomy, but these are now completely lost. 
About the time that Archimedes finishe 
another geometer of the highest order appeared. This 240 A.C. 
was Apollonius of Perga, a town in Pamphylia. He 
was born towards the middle of the third century, be~ 
fore the Christian era, and he flourished principally un- 
der Ptolemy Philopater, or towards the el of that 
century. He studied inthe Alexandrian school under 
the successors of Euclid ; and so highly esteemed were 
his discoveries, that he acquired the name of the Great 
Geometer. It is mortifying to reflect, that sometimes 
consummate abilities are aio ed with great moral de- 
fects ; Apollonius had a mind of the highest order, yet 
he was vain, jealous of merit in others, and always dis« 
sed to detraction. He was, however, one of the most 
inventive and profound writers that has treated of the 
mathematics, and it was in a great measure from his 
‘works that the true spirit of the ancient geometry was 
to be learnt. In the introduction to our article Conie 
‘Sections, we have had occasion to Yaar of his trea- 
tise on that subject ; which contributed principally to his 
celebrity. The most material of his other works were 
‘the following treatises: 1. On the Section of a Ratio; 
2. On the Section of a Space; 3. On Determinate 
Section ; 4. On Tangencies ; 5. On Inclinations; 6. On 
Plane Loci: The nature and contents of each of these 
has been icularly described in our article on ANa= 
LYSIS. e have understood that Peyrard, the learned 
French editor of the works of Euclid and Archimedes, 
had it in contemplation to give French translations of 
the writings of Apollonius, as well as the other ancient 
eters, as far as they have been preserved ; but we 
ear that the state of France is not likely to be soon fa- 
vourable to the execution of his views. 
The names of several mathematicians of antiquity 
contemporary with Archimedes and Apollonius, ive 
come down to us. Apollonius has addressed the three 
first books of his conics to Eudemus of Pergamus, and 
speaks of him as a good judge in these matters, but he 
being dead before the fourth book’ was finished, Apol« 
lonius addressed it to Attalus. He says, in his first ad- 
dress to Eudemus, that Naucrates had instigated him 
to study the conics ; and in that which precedes the se- 
cond book, he requests Eudemus to communicate it to 
Philonides of Ephesus, 
It appears that there was a geometer named Trasi- 
deus, who c mded with Conon of Samos on the 
conic sections, and another Nicoteles the Cyrenean, 
who animadverted on some mistakes committed by Co- 
non. Here, then, are five or six geometers besides 
Apollonius, who all cultivated the theory of. conics. 
The regret which Archimedes expressed for the loss of 
Conon, gives us reason to think highly of him; but 
this is almost the only ground upon which we can form 
an idea of his skill as a geometer. 
* Dositheus was also a friend of Archimedes, who ad- 
dressed to him several of his works. It is probable 
that Nicomedes, the inventor of the conchoid, lived Nicome- 
about the period at which we are now arrived. This des. 
curve, and: the application he made of it to the finding 
of two mean proportionals, are the only vestiges that 
now remain of his labours, 
his career, Apollonius, 
