190 
History. As we descend towards. the commencement of the 
—_— Christian era, we find a numerous list of mathemati- 
cians, most of whom are chiefly known as cultivators 
of astronomy, and some as writers on geometry. In 
this number were Geminus of Rhodes, who composed 
a work called Enarrationes Geometricew, which consist« 
ed of six books; Philo, who gave a solution of the 
problem of two mean proportionals ; Possidonius, who 
was a geometer, an astronomer, a mechanician, and a 
geographer. Dionysiodorus, who resolved a difficult 
problem of Archimedes, namely to divide a hemi- 
‘sphere in a given ratio by a plane parallel to its base ; 
Theodosius, and Theodosius, the author of an excellent treatise on 
504.C. Spherics, in three books, which has been preserved, and 
which constitutes 2 part of the precious remains of the 
ancient geometry. 
The astronomer and geometer Menelaus of Alexan- 
dria, lived in the second century of the Christian era: 
he composed a treatise on Trigonometry, in six books; 
and another on Spherics, in three books, “which is 
still extant. He appears also to have treated of the 
geometry of curve lines. 
Possido« 
nius. 
Menelaus. 
Ptolemy. The astronomer Ptolemy must be reckoned among 
PEN * 4 the geometers of the second century. His work on 
UD. ie 
Optics, which however is now lost, * is supposed 
to have contained some beautiful applications of geo- 
metry. . 
There were several geometers who flourished in the 
period of the three or four first centuries of the Chris- 
tian era; but the exact time of, each is not certainly 
known; as Serenus of Antissensis, who wrote on cy- 
linders and cones ; Hypsicles of Alexandria, who wrote 
two books on regular.solids ; Perseus Citticus, the in- 
ventor of certain lines called x yep which were 
curves made by the section of a plane and a solid, form- 
ed by the revolution of an arc of a circle about a given 
axis. Philo of Thyaneus, who appears to have treated 
of certain curves, which were also considered by Mene- 
Jaus, but whose nature is not now known. Pappus 
also mentions Demetrius of Alexandria, as the author 
of a work which treated of curves, and hence it has 
been conjectured that the ancients had gone farther 
into this,subject than has been generally supposed. 
We are now come to the period when learning be-~ 
gan to decline, so that instead of brilliant discoveries, 
and original treatises, we have only commentaries and 
annotations on the works of former times, a presage of 
the approach of ignorance and barbarism. Of this na- 
Pappus, ture were the works of Pappus, and Theon of Alex. 
A, D. 330.. andria, two mathematicians who lived towards the end 
of the fourth century. The former of these, however, 
ranks in a higher class, on account of the genius dis. 
played in his writings. Geometry is particularly obli- 
ged to him for his Mathematical Collections, which 
originally consisted of eight books ; but of these, the 
first and half of the second are now lost. He seems to 
have intended to collect, into one body, several scatter- 
ed discoveries, and to illustrate and complete, in man 
places, the writings of the most celebrated mathemati- 
cians, in particular Apollonius, Archimedes, Euclid, 
and Theodosius ; for this purpose he has given a mul« 
titude of lemmas, and curious theorems, which they had 
supposed known ;. and he has also described the diffe- 
rent attempts which had been made to resolve the 
more difficult problems, as the duplication of the cube, 
and the trisection of an angle. The preface to his se- 
147, 
Serenus 
Hypsicles, 
* A Latin translation of the Optics of Ptolemy has lately been discovered’ in the Royal Library at: Paris. M. Le Chevalier 
Delambre, who mentioned to the Editor this curious fact, ‘has given an analysis of the work in the Connoissances des Tems for 1816, 
See our article Ortics. En, 
GEOMETRY. 
venth book is inestimably precious, for it has preserved 
from oblivion many analytical works on ¢ try, of 
which we should otherwise-have been entirely ignorant, P 
The abridgment which he has given of these is all that 
remains of the greater number ; yet it has served to give 
a continuity to the history of geometry, and to inspi 
modern mathematicians with a high opinion of the 
theories of the ancients. In fact, such of their geome- 
trical writings as have descended to our times are meres 
ly elementary ; their more recondite works have either 
been entirely lost, or are only known by the account 
which Pappus has given of them. The books that ree 
main of Pappus have suffered much from the injuriesof __ 
time ; there are many inaccuracies, and some passages ; 
so mutilated as to be hardly intelligible. The original 
Greek, except some extracts, has never been che 
The only translation that has been given, at is by 
Commandinus, was published at Pesara in 1588, and 
again, with little variation, in 1660, at Bol Com- % 
mandinus appears to have had access to only one ma- i 
nuscript, which wanted the first two books, and through- 
out was very faulty. There are, however, several. ma- \ 
nuscripts of Pappus in the libraries; the university of i 
Oxford possesses two; one has half of the second book, : 
and this part was published, with a Latin translation, in 
1688, by Dr Wallis, It treats of arithmetic, so that 
probably the first two books treated of this subject. 
e university has already conferred. a great-favour on 
geometrical science, by the elegant editions it has given 
of Euclid, Apollonius, and Peni ead ras and it is to’be 
wished that the obligation were increased by an edition 
of Pappus. Our limits will not admit. of a detailed 
statement of the contents of this valuable work ; some 
account of it may be seen in Dr Hutton’s Mathematical 
Dictionary, and also in Dr Traill’s Life of Simson. 
Theon, the associate of Pappus in the Alexandrian ‘1 
school, wrote Scholia, or Notes on Euclid, which —_ 
mandinus has given in one of his Latin editions of that 
author. He is supposed, however, to have ly vie 
tiated the text; and Dr Simson, the learned editor of 
Euclid, has bestowed great labour in freeing it from 
what he supposed to be Theon’s interpolations. 
Theon had a beautiful and accomplished daughter, 
named Hypatia, who cultivated geometry ; and so learn= 1 
ed was she in the science, that she was judged worthy Died 
to succeed her father in te ks tg a Be She 
wrote commentaries on. onius and Diophantus, 
which are now lost. It is infinitely to be lamented that 
so exalted a being should have had so tragical a fate, 
This woman, the ornament of her sex and of human na- 
ture, fell a sacrifice to the blind fury of a fanatical mob, 
about the beginning ot the fifth century. t 
The philosopher Proclus, the chief of the Platonics 
at Athens, transferred thither, in some degree, the seat 
of the mathematical sciences, towards the middle of the 
fifth century ; although he did not extend .geometry, 
yet he held it in esteem, _ His very. prolix commentary 
on the first book of Euclid, has made us acquainted. 
with many traits in the history of the ancient geometry, 
and excited a regret that he did not extend it to the res 
maining books. Proclus was succeeded. in his school 
by Marinus of Neapolis, who formed with Isidore of Mari 
Miletus and Eutocius of Ascalon, a kind of succession, pro 
which brings the history of ry downto the reign 
of Justinian. Marinus wrote a ce to Euclid’s book 
of Data, which Dr Simson has rejected in his edition 
