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), Mathematical Exposition]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.
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
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, z2 = 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 z2 = xy) and, on the same diagram, the other conic section with equation y2 = 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.
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last modified 9/4/02
Copyright (c) 2000. Daniel E. Otero