uggling for the sovereignty of the republic. Among these was the
young Sextus Pompeius, whose large fleet made him for a time master
of Sicily and of the sea; and he was said to have been admitted by the
Queen of Egypt as a lover. But he was able to be of but little use
to her in return for her favours, as his fleet was soon defeated by
Octavianus.

Caesar had left behind him, in the neighbourhood of Alexandria, a large
body of Roman troops, in the pay and nominally under the orders of
Cleopatra, but in reality to keep Egypt in obedience. There they lived
as if above all Egyptian law or Roman discipline, indulging in the vices
of that luxurious capital. When some of them in a riot, in the year 45
B.C., killed two sons of Bibulus the consul, Cleopatra was either afraid
or unable to punish the murderers; the most she could do was to get
them sent in chains into Syria to the grieving father, who with true
greatness of mind sent them back to the Egyptian legions, saying that it
was for the senate to punish them, not for him.

While Ptolemy her second husband was a boy and could claim no share
of the government, he was allowed to live with all the outward show of
royalty, but as soon as he reached the age of fifteen, in B.C. 44, at
which he might call himself her equal and would soon be her master,
Cleopatra had him put to death. She had then reigned four years with
her elder brother and four years with her younger brother, and from that
time forward she reigned alone, calling her child by Caesar her colleague
on the throne.

At a time when vice and luxury claimed the thoughts of all who were not
busy in the civil wars, we cannot hope to find the fruits of genius in
Alexandria; but the mathematics are plants of a hardy growth, and are
not choked so easily as poetry and history. Sosigenes was then the
first astronomer in Egypt, and Julius Caesar was guided by his advice
in setting right the Roman Calendar. He was a careful and painstaking
mathematician, and, after fixing the length of the year at three hundred
and sixty-five days and a quarter, he three times changed the beginning
of the year, in his doubts as to the day on which the equinox fell; for
the astronomer could then only make two observations in a year with a
view to learn the time of the equinox, by seeing when the sun shone
in the plane of the equator. Photinus the mathematician wrote both
on arithmetic and geometry, and was usually thought the author of a
mat