In the third century BCE, Rome
was involved in a series of military conflicts (the Punic Wars) with the
Greek city-state of Carthage,
situated across
the Mediterranean Sea on the African coast. Caught in the middle of
these conflicts was Syracuse, another city-state on the coast of Sicily,
which was claimed by both sides. Initially allied with Carthage against
Rome at the outset of the First
Punic War in 263 BCE, Syracuse soon switched allegiance. The King of
Syracuse, Hiero
II, managed to keep war at bay by honoring this treaty with Rome, but
the situation became precarious in the later years of the century as the
Carthaginian general Hannibal
was gaining the upper hand in Spain and Italy against poorly managed Roman
armies.

Archimedes (287 - 212 BCE), son of Phidias, an astronomer, was thought to have been a kinsman of Hiero. In his youth, Archimedes ventured to Alexandria in Egypt to avail himself of the best education to be found in the Greek world. There he would have been able to study the texts at the great Library of Alexandria, where Euclid had worked, and he made friendships with other philosopher-mathematicians, most notably Conon of Samos with whom he corresponded for many years. Archimedes eventually returned to Syracuse, where he earned fame as an "engineering consultant" to the king, inventing many clever devices for the military defense of the city: catapults, grappling hooks, and improvements to the architecture of the city walls.

Hiero died in 215 and was succeeded by his young grandson Hieronymus, who switched allegiance to Carthage just as the Second Punic War began. The Romans soon dispatched their navy to take back Syracuse by force and managed to have the 15-year-old Hieronymus assassinated. The accounts of the subsequent siege of Syracuse by the Roman general Marcellus in 213 in the military histories of Plutarch and Livy tell a fascinating story of the success enjoyed by Archimedes in the defense of the city, which managed to hold off the attacking Romans for many months. The Roman army finally entered the city when its defenses were down during a festival held to honor the goddess Artemis. In the ensuing plunder, Archimedes was killed. The story of his death is also the subject of some legendary histories.

Esteemed by many as the greatest
mathematician of the Greek era, spanning roughly the thousand years 500
BCE to AD 500, Archimedes produced many deep and far-reaching results in
geometry, especially in the service of mechanics and hydrostatics. He is
the author of 9 books that survive to the present, including the two from
which we read in this course, and a handful that are lost; another book,
the *Method*, was thought to have been lost until a copy was discovered
in 1906 by J. L. Heiberg in Constantinople (modern-day Istanbul). The fascinating
story of the reappearance of this manuscript in 1998 can be read at a wonderfully
creative Archimedes
website prepared by the Walters Museum in Baltimore.

Back in the fifth century, BC,
Hippocrates
of Chios, in addition to working on the problem of the quadrature of
the circle, also considered another problem that would vex geometers for
centuries, the problem of
**the duplication of the cube**. As
the legend goes,

when the god announced to the Delians [inhabitants of Delos] by oracle that to get rid of a plague they must construct an altar double of the existing one, their craftsmen fell into great perplexity in trying to find how a solid could be made double of another solid, and they went to ask Plato about it. He told them that the god had given this oracle, not because he wanted an altar of double the size, but becasue he wished, in setting this task before them, to reproach the Greeks for their neglect of mathematics and their contempt for geometry. [from Theon of Smyrna (ca. 100AD),Assuming that the "altar" was cubical with side of unit length, doubling the altar meant doubling the volume, from 1 to 2. But the length of the side of the doubled cube is then equal to the cube root of 2: we are required to find a line segment of length equal to the cube root of 2. You will recall that for the Greeks, this meant to construct the appropriate segment with straightedge and compass. The concerted efforts of many geometers yielded nothing toward resolving this puzzle, and like the problem of the quadrature of the circle, the duplication of the cube stood as a challenge for centuries.Mathematical Exposition]

One early advance in this endeavor
was made by Hippocrates. Hippocrates knew that the determination
of a square root of *x* corresponds to finding the **geometric mean**,
or **mean proportional**, between 1 and *x*:

*b* is the mean proportional
between *a* and *c* if it satisfes the proportion *a*
: *b* = *b* : *c*

Thus, if *y* is the number
that satisfies the proportion 1 : *y* = *y* : *x*,
then cross multiplication shows that *y* is also the square
root of *x*. Hippocrates then reasoned that the duplication
problem, and indeed the construction of any cube root, was equivalent to
finding *two* mean proportionals between 1 and *x*, for if *y*
and *z* satisfy the proportion

1 : *y* = *y* : *z*
= *z* : *x*

then

thereby solving the problem. Of course, this simply replaced one hard problem with another equally difficult one!

The next major advance came from
Menaechmus
(ca. 350BC), one of Eudoxus' students. The problem of the two mean
proportionals can be solved by *simultaneously* finding two square
roots

but this is made intractable because
of the fact that neither *y* nor *z* are given values, so each
of these two relationships carries *two* unknowns rather than one.
Menaechmus then tackles the problem by solving every square root problem
at once! To find the square root *z* of the variable quantity
*xy*,
for instance, he lays out circles with diameters *x* +
*y* for
many values of *y*. The altitudes to those circles from the
point on the diameter that separates *x* from *y* are the corresponding
values of their geometric mean *z*(see the diagram below). Connecting
all the points which are a distance *y* from the vertical axis that
separates *x* from
*y* along the diameter, and a distance *z*
from the diameter, generates a **locus** of points that forms a curve.

We see in this diagram a type of
coordinate system in which the horizontal and vertical lines represent
a *z*-axis and *y*-axis respectively. In this coordinate
system the points on the curve satisfy the relation given by the mean proportionals,
*z*^{2} = *xy*. This is easily recognized today
as the equation of a parabola. Menaechmus, however, interprets this
diagram differently. For him, this is a picture of a three-dimensional
object: the system of circles are viewed as cross sections of a cone, viewed
from above, the smaller ones lying atop the larger ones. The apex
of the cone lies above the common intersection point of all the cross-sectional
circles at the left of the diagram. The vertical *z*-axis is
the edge-on view of a plane that cuts down through the cone, and the horizontal
*y*-axis, together with the rectangles and the curve, have been swung
up out of this cutting plane into the viewing plane. A perspective
drawing of this appears below.

The curve is therefore identified
as a cross section of a right-angled cone (right-angled because one side
of the cone is at right angles with its base), or simply a **conic section**.
(Menaechmus would not have used the term 'parabola', for this word was
coined centuries later by Apollonius.)

While it is possible to use a straightedge
and compass to determine any number of points on this conic section, as
is done above, the entire curve is not constructible. Nonetheless,
Menaechmus conceived of drawing this curve (with equation *z*^{2}
= *xy*) and, on the same diagram, the other conic section with equation
*y*^{2} = *z*.

Similar uses were found for the
other conic sections that appear when an acute-angled or obtuse-angled
cone are sliced by a vertical plane. Thus were born the conic sections.

Another geometric problem of
note is one that arises for every curve. Given a point
*P* on
the curve we can draw a line through *P* to cut the curve in a second
point *Q*. Any such line is called a **secant** to the curve
through *P* and *Q*(from the Latin *secare*, to cut).
But often there can also be drawn a line through *P* which cuts the
curve in *no* other point, a line that is seen to just 'touch' the
curve there; such a line is said to be **tangent** to the curve at *P*(from
the Latin *tangere*, to touch). The earliest form of the tangent
problem is seen in *Elements*
iii.16 cor., where Euclid finds that the tangent to a circle at a given
point is simply the line perpendicular to the diameter at that point.

It soon became
standard operating procedure, whenever a curve was introduced for study,
for geometers to ask the question of how one determined the tangent to
the curve at a given point. Since the conic sections were among the
first curves that geometers considered, the tangent problem for the conic
sections was naturally an important problem. We find that it was
solved by the time of Euclid, since it is known to have appeared in a lost
work of his (called *Conics*, strangely enough). In the text
that follows, we will see that Archimedes quotes some of these results
without demonstration. Later, we will see their proofs in a work
of Apollonius.

Read the text

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last modified 9/4/02

Copyright (c) 2000. Daniel E. Otero