A Dictionary of Greek and Roman biography and mythology. By various writers. Ed. by William Smith. Illustrated by numerous engravings on wood.

272 ARCHIMEDES.
any chord is equal to two-thirds of the parallelogram of which one side is the chord in question,
and the opposite side a tangent to the parabola.
This was the first real example of the quadrature
of a curvilinear space; that is, of the discovery of
a rectilinear figure equal to an area not bounded
entirely by straight lines.
A treatise on the Sphere and Cylinder, in which
various propositions relative to the surfaces and
volumes of the sphere, cylinder, and cone, were
demonstrated for the first time. Many of them
are now familiarly known; for example, those
which establish the ratio (|) between the volumes,
and also between the surfaces, of the sphere and
circumscribing cylinder; and the ratio (1) between
the area of a great circle and the surface of the
sphere. They are easily demonstrable by the
modern analytical methods; but the original discovery and geometrical proof of them required the
genius of Archimedes. Moreover, the legitimacy
of the modern applications of analysis to questions
concerning curved lines and surfaces, can only be
proved by a kind of geometrical reasoning, of
which Archimedes gave the first example. (See
Lacroix, Dilf. Cal. vol. i. pp. 63 and 431; and
compare De Morgan, Dif. Cal. p. 32.)
The book on the Dimension of the Circle consists
of three propositions. 1st. Every circle is equal
to a right-angled triangle of which the sides containing the right angle are equal respectively to its
radius and circumference. 2nd. The ratio of the
area of the circle to the square of its diameter is
nearly that of 11 to 14. 3rd. The circumference
of the circle is greater than three times its diameter
by a quantity greater than 0 of the diameter but
less than of the same. The last two propositions are established by comparing the circumference of the circle with the perimeters of the
inscribed and circumscribed polygons of 96 sides.
The treatise on Spirals contains demonstrations
of the principal properties of the curve, now known
as the Spiral of Archimedes, which is generated by
the uniform motion of a point along a straight line
revolving uniformly in one plane about one of its
extremities. It appears from the introductory
epistle to Dositheus that Archimedes had not been
able to put these theorems in a satisfactory form
without long-continued and repeated trials; and
that Conon, to whom he had sent them as probleins along with various others, had died without
accomplishing their solution.
The book on Conoids and Spheroids relates
chiefly to the volumes cut off by planes from the
solids so called; those namely which are generated
by the rotation of the Conic Sections about their
principal axes. Like the work last described, it
was the result of laborious, and at first unsuccessful, attempts. (See the introduction.)
The Arenarius (d VapnF'irqs) is a short tract
addressed to Gelo, the eldest son of Hiero, in
which Archimedes proves, that it is possible to
assign a number greater than that of the grains of
sand which would fill the sphere of the fixed stars.
This singular investigation was suggested by an
opinion which some persons had expressed, that
the sands on the shores of Sicily were either infinite, or at least would exceed any numbers which
could be assigned for them; and the success with
which the difficulties caused by the awkward and
imperfect notation of the ancient Greek arithmetic
are eluded by a device identical in principle with

ARCHIMEDES.
the modern -method of logarithms, affords one of
the most striking instances of the great mathematician's genius. Having briefly discussed the
opinions of Aristarchus upon the constitution and
extent of the Universe [ARIsTARCHUS], and
described his own method of determining the apparent diameter of the sun, and the magnitude of
the pupil of the eye, he is led to assume that the
diameter of the sphere of the fixed stars may be
taken as not exceeding 100 million of millions of
stadia; and that a sphere, one deIcruAos in diameter, cannot contain more than 640 millions of
grains of sand; then, taking the stadium, in round
numbers, as not greater than 10,000 58drcrvAot, he
shews that the number of grains in question could
not be so great as 1000 myriads multiplied by the
eighth term of a geometrical progression of which
the first term was unity and the common ratio a
myriad of myriads; a number which in our notation would be expressed by unity with 63 ciphers
annexed.
The two books On Floating Bodies (HIepi rv
'Oxeový'vwv) contain demonstrations of the laws
which determine the position of bodies immersed
in water; and particularly of segments of spheres
and parabolic conoids. They are extant only in
the Latin version of Commandine, with the exception of a fragment lepl V rv "TSart a q)r/raTpifvwv in Ang. Mai's Collection, vol. i. p. 427.
The treatise entitled Lemmata is a collection ol
15 propositions in plane geometry. It is derived
from an Arabic MS. and its genuineness has beer
doubted. (See Torelli's preface.)
Eutocius of Ascalon, about A. D. 600, wrote
commentary on the Treatises on the Sphere an(
Cylinder, on the Dimension of the Circle, and or
Centres of Gravity. All the works above men
tioned, together with this Commentary, were founm
on the taking of Constantinople, and brought firs
into Italy and then into Germany. They wer
printed at Basle in 1544, in Greek and Latin, bHervagius. Of the subsequent editions by far th
best is that of Torelli, "Archim. quae supers
omnia, cum Eutocii Ascalonitae commentarii:
Ex recens. Joseph. Torelli, Veronensis," Oxoi
1792. It was founded upon the Basle editioi
except in the case of the Arenarius, the text (
which is taken from that of Dr. Wallis, who pul
lished this treatise and the Dimensio Circuli, wit
a translation and notes, at Oxford, in 1679. (The
are reprinted in vol. iii. of his works.)
The Arenarius, having been little meddled wil
by the ancient commentators, retains the DoI
dialect, in which Archimedes, like his countryms
Theocritus, wrote. (See Wallis, Op. vol. iii. p
537, 545. Tzetzes says, E'AEye sd ical Swptoau
c(wvPr ZvpaKovefa, Hr /C3, Kal xapriwut 'l dv 'y
IcivIscw rdayv.) A French translation of t
works of Archimedes, with notes, was publish,
by F. Peyrard, Paris, 1808, 2 vols. 8vo., and
English translation of the Arenarius by G. And(
son, London, 1784.
(G. M. Mazuchelli, Notizie istoriche e critii
intorno alla vita, alle invenzioni, ed agli scritti
Archimede, Brescia, 1737, 4to.; C. M. Brande
Dissertatio sistens Archimedis vitam, ejusque
Mathesin merita, Gryphiswald.1789,4to.; Mirte
in Ersch und Gruber, Allgemeine Encycclopjii
art. Archimedes; Quarterly Review, vol. iii. f
Peyrard's Archimedes; Rigaud, The Arenarius
Archimedes, Oxford, 1837, printed for the Ashr