Archimedes

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*Archimedes Thoughtful by Fetti*

born c. 290–280 bce, Syracuse, Sicily
[now in Italy]

died 212/211 bce, Syracuse

the
most famous mathematician and inventor
of ancient Greece. Archimedes is
especially important for his discovery
of the relation between the surface and
volume of a sphere and its
circumscribing cyclinder. He is known
for his formulation of a hydrostatic
principle (known as Archimedes’
principle) and a device for raising
water, still used in developing
countries, known as the Archimedes
screw.

**His
life**

Archimedes probably spent some time in
Egypt early in his career, but he
resided for most of his life in
Syracuse, the principal Greek city-state
in Sicily, where he was on intimate
terms with its king, Hieron II.
Archimedes published his works in the
form of correspondence with the
principal mathematicians of his time,
including the Alexandrian scholars Conon
of Samos and Eratosthenes of Cyrene. He
played an important role in the defense
of Syracuse against the siege laid by
the Romans in 213 bce by constructing
war machines so effective that they long
delayed the capture of the city. When
Syracuse eventually fell to the Roman
general Marcus Claudius Marcellus in the
autumn of 212 or spring of 211 bce,
Archimedes was killed in the sack of the
city.

Far
more details survive about the life of
Archimedes than about any other ancient
scientist, but they are largely
anecdotal, reflecting the impression
that his mechanical genius made on the
popular imagination. Thus, he is
credited with inventing the Archimedes
screw, and he is supposed to have made
two “spheres” that Marcellus took back
to Rome—one a star globe and the other a
device (the details of which are
uncertain) for mechanically representing
the motions of the Sun, the Moon, and
the planets. The story that he
determined the proportion of gold and
silver in a wreath made for Hieron by
weighing it in water is probably true,
but the version that has him leaping
from the bath in which he supposedly got
the idea and running naked through the
streets shouting “Heurēka!” (“I have
found it!”) is popular embellishment.
Equally apocryphal are the stories that
he used a huge array of mirrors to burn
the Roman ships besieging Syracuse; that
he said, “Give me a place to stand and I
will move the Earth”; and that a Roman
soldier killed him because he refused to
leave his mathematical diagrams—although
all are popular reflections of his real
interest in catoptrics (the branch of
optics dealing with the reflection of
light from mirrors, plane or curved),
mechanics, and pure mathematics.

According to Plutarch (c. 46–119 ce),
Archimedes had so low an opinion of the
kind of practical invention at which he
excelled and to which he owed his
contemporary fame that he left no
written work on such subjects. While it
is true that—apart from a dubious
reference to a treatise, “On
Sphere-Making”—all of his known works
were of a theoretical character, his
interest in mechanics nevertheless
deeply influenced his mathematical
thinking. Not only did he write works on
theoretical mechanics and hydrostatics,
but his treatise Method Concerning
Mechanical Theorems shows that he used
mechanical reasoning as a heuristic
device for the discovery of new
mathematical theorems.

**His works**

There are nine extant treatises by
Archimedes in Greek. The principal
results in On the Sphere and Cylinder
(in two books) are that the surface area
of any sphere of radius r is four times
that of its greatest circle (in modern
notation, S = 4πr2) and that the volume
of a sphere is two-thirds that of the
cylinder in which it is inscribed
(leading immediately to the formula for
the volume, V = 4/3πr3). Archimedes was
proud enough of the latter discovery to
leave instructions for his tomb to be
marked with a sphere inscribed in a
cylinder. Marcus Tullius Cicero (106–43
bce) found the tomb, overgrown with
vegetation, a century and a half after
Archimedes’ death.

Measurement of the Circle is a fragment
of a longer work in which π (pi), the
ratio of the circumference to the
diameter of a circle, is shown to lie
between the limits of 3 10/71 and 3 1/7.
Archimedes’ approach to determining π,
which consists of inscribing and
circumscribing regular polygons with a
large number of sides (see the
animation), was followed by everyone
until the development of infinite series
expansions in India during the 15th
century and in Europe during the 17th
century. This work also contains
accurate approximations (expressed as
ratios of integers) to the square roots
of 3 and several large numbers.

On
Conoids and Spheroids deals with
determining the volumes of the segments
of solids formed by the revolution of a
conic section (circle, ellipse,
parabola, or hyperbola) about its axis.
In modern terms, these are problems of
integration. (See calculus.) On Spirals
develops many properties of tangents to,
and areas associated with, the spiral of
Archimedes—i.e., the locus of a point
moving with uniform speed along a
straight line that itself is rotating
with uniform speed about a fixed point.
It was one of only a few curves beyond
the straight line and the conic sections
known in antiquity.

On the
Equilibrium of Planes (or Centres of
Gravity of Planes; in two books) is
mainly concerned with establishing the
centres of gravity of various
rectilinear plane figures and segments
of the parabola and the paraboloid. The
first book purports to establish the
“law of the lever” (magnitudes balance
at distances from the fulcrum in inverse
ratio to their weights), and it is
mainly on the basis of this treatise
that Archimedes has been called the
founder of theoretical mechanics. Much
of this book, however, is undoubtedly
not authentic, consisting as it does of
inept later additions or reworkings, and
it seems likely that the basic principle
of the law of the lever and—possibly—the
concept of the centre of gravity were
established on a mathematical basis by
scholars earlier than Archimedes. His
contribution was rather to extend these
concepts to conic sections.

Quadrature of the Parabola demonstrates,
first by “mechanical” means (as in
Method, discussed below) and then by
conventional geometric methods, that the
area of any segment of a parabola is 4/3
of the area of the triangle having the
same base and height as that segment.
This is, again, a problem in
integration.

The
Sand-Reckoner is a small treatise that
is a jeu d’esprit written for the
layman—it is addressed to Gelon, son of
Hieron—that nevertheless contains some
profoundly original mathematics. Its
object is to remedy the inadequacies of
the Greek numerical notation system by
showing how to express a huge number—the
number of grains of sand that it would
take to fill the whole of the universe.
What Archimedes does, in effect, is to
create a place-value system of notation,
with a base of 100,000,000. (This was
apparently a completely original idea,
since he had no knowledge of the
contemporary Babylonian place-value
system with base 60.) The work is also
of interest because it gives the most
detailed surviving description of the
heliocentric system of Aristarchus of
Samos (c. 310–230 bce) and because it
contains an account of an ingenious
procedure that Archimedes used to
determine the Sun’s apparent diameter by
observation with an instrument.

Method
Concerning Mechanical Theorems describes
a process of discovery in mathematics.
It is the sole surviving work from
antiquity, and one of the few from any
period, that deals with this topic. In
it Archimedes recounts how he used a
“mechanical” method to arrive at some of
his key discoveries, including the area
of a parabolic segment and the surface
area and volume of a sphere. The
technique consists of dividing each of
two figures into an infinite but equal
number of infinitesimally thin strips,
then “weighing” each corresponding pair
of these strips against each other on a
notional balance to obtain the ratio of
the two original figures. Archimedes
emphasizes that, though useful as a
heuristic method, this procedure does
not constitute a rigorous proof.

On
Floating Bodies (in two books) survives
only partly in Greek, the rest in
medieval Latin translation from the
Greek. It is the first known work on
hydrostatics, of which Archimedes is
recognized as the founder. Its purpose
is to determine the positions that
various solids will assume when floating
in a fluid, according to their form and
the variation in their specific
gravities. In the first book various
general principles are established,
notably what has come to be known as
Archimedes’ principle: a solid denser
than a fluid will, when immersed in that
fluid, be lighter by the weight of the
fluid it displaces. The second book is a
mathematical tour de force unmatched in
antiquity and rarely equaled since. In
it Archimedes determines the different
positions of stability that a right
paraboloid of revolution assumes when
floating in a fluid of greater specific
gravity, according to geometric and
hydrostatic variations.

Archimedes is known, from references of
later authors, to have written a number
of other works that have not survived.
Of particular interest are treatises on
catoptrics, in which he discussed, among
other things, the phenomenon of
refraction; on the 13 semiregular
(Archimedean) polyhedra (those bodies
bounded by regular polygons, not
necessarily all of the same type, that
can be inscribed in a sphere); and the
“Cattle Problem” (preserved in a Greek
epigram), which poses a problem in
indeterminate analysis, with eight
unknowns. In addition to these, there
survive several works in Arabic
translation ascribed to Archimedes that
cannot have been composed by him in
their present form, although they may
contain “Archimedean” elements. These
include a work on inscribing the regular
heptagon in a circle; a collection of
lemmas (propositions assumed to be true
that are used to prove a theorem) and a
book, On Touching Circles, both having
to do with elementary plane geometry;
and the Stomachion (parts of which also
survive in Greek), dealing with a square
divided into 14 pieces for a game or
puzzle.

Archimedes’ mathematical proofs and
presentation exhibit great boldness and
originality of thought on the one hand
and extreme rigour on the other, meeting
the highest standards of contemporary
geometry. While the Method shows that he
arrived at the formulas for the surface
area and volume of a sphere by
“mechanical” reasoning involving
infinitesimals, in his actual proofs of
the results in Sphere and Cylinder he
uses only the rigorous methods of
successive finite approximation that had
been invented by Eudoxus of Cnidus in
the 4th century bce. These methods, of
which Archimedes was a master, are the
standard procedure in all his works on
higher geometry that deal with proving
results about areas and volumes. Their
mathematical rigour stands in strong
contrast to the “proofs” of the first
practitioners of integral calculus in
the 17th century, when infinitesimals
were reintroduced into mathematics. Yet
Archimedes’ results are no less
impressive than theirs. The same freedom
from conventional ways of thinking is
apparent in the arithmetical field in
Sand-Reckoner, which shows a deep
understanding of the nature of the
numerical system.

In
antiquity Archimedes was also known as
an outstanding astronomer: his
observations of solstices were used by
Hipparchus (flourished c. 140 bce), the
foremost ancient astronomer. Very little
is known of this side of Archimedes’
activity, although Sand-Reckoner reveals
his keen astronomical interest and
practical observational ability. There
has, however, been handed down a set of
numbers attributed to him giving the
distances of the various heavenly bodies
from the Earth, which has been shown to
be based not on observed astronomical
data but on a “Pythagorean” theory
associating the spatial intervals
between the planets with musical
intervals. Surprising though it is to
find these metaphysical speculations in
the work of a practicing astronomer,
there is good reason to believe that
their attribution to Archimedes is
correct.

**His influence**

Given the magnitude and originality of
Archimedes’ achievement, the influence
of his mathematics in antiquity was
rather small. Those of his results that
could be simply expressed—such as the
formulas for the surface area and volume
of a sphere—became mathematical
commonplaces, and one of the bounds he
established for π, 22/7, was adopted as
the usual approximation to it in
antiquity and the Middle Ages.
Nevertheless, his mathematical work was
not continued or developed, as far as is
known, in any important way in ancient
times, despite his hope expressed in
Method that its publication would enable
others to make new discoveries. However,
when some of his treatises were
translated into Arabic in the late 8th
or 9th century, several mathematicians
of medieval Islam were inspired to equal
or improve on his achievements. This
holds particularly in the determination
of the volumes of solids of revolution,
but his influence is also evident in the
determination of centres of gravity and
in geometric construction problems.
Thus, several meritorious works by
medieval Islamic mathematicians were
inspired by their study of Archimedes.

The
greatest impact of Archimedes’ work on
later mathematicians came in the 16th
and 17th centuries with the printing of
texts derived from the Greek, and
eventually of the Greek text itself, the
Editio Princeps, in Basel in 1544. The
Latin translation of many of Archimedes’
works by Federico Commandino in 1558
contributed greatly to the spread of
knowledge of them, which was reflected
in the work of the foremost
mathematicians and physicists of the
time, including Johannes Kepler
(1571–1630) and Galileo Galilei
(1564–1642). David Rivault’s edition and
Latin translation (1615) of the complete
works, including the ancient
commentaries, was enormously influential
in the work of some of the best
mathematicians of the 17th century,
notably René Descartes (1596–1650) and
Pierre de Fermat (1601–1665). Without
the background of the rediscovered
ancient mathematicians, among whom
Archimedes was paramount, the
development of mathematics in Europe in
the century between 1550 and 1650 is
inconceivable. It is unfortunate that
Method remained unknown to both Arabic
and Renaissance mathematicians (it was
only rediscovered in the late 19th
century), for they might have fulfilled
Archimedes’ hope that the work would
prove useful in the discovery of
theorems.

*
Gerald J. Toomer*