Placing well-known names in “top-ten” lists by profession or mentioning them in comparison to one another as to how “great” they were is not the right way to ascribe merit when it comes to mathematicians. While “tops” lists provide an objective way to classify sport legends in terms of scores, records, and memorable matches, they are not quite adequate for saying how “great” a mathematician is, in terms of the number of his or her discoveries. For what really counts in mathematics is fruitfulness, fertileness, and inspiring work. The main criterion for qualifying a discovery as remarkable is the degree to which that discovery or individual influences subsequent discoveries or theories.
Archimedes is such a figure in the history of mathematics in that both his work and he as a man inspired scientists and mathematicians, and he should inspire us all. Indeed, it is not accidental at all that his “Eureka!” became and has remained over millennia the paradigmatic cry of victory of the arduous, rational self-search for a solution to a theoretical issue. Perhaps you yourself shouted it after struggling to solve a difficult math problem.
Archimedes of Syracuse lived in the third century B.C. in Greece and is known as a Greek mathematician, physicist, engineering inventor, and astronomer. Such distinct qualifications are specific only for modern and contemporary time, as in that ancient time there were not such disciplinary distinctions — mathematics and natural sciences were all under the same umbrella, usually called ‘philosophy’ or ‘natural philosophy’. Differentiation of these disciplines came in the 16th-17th century, the age of birth of modern science.
Archimedes’ theoretical achievements decisively influenced contemporary mathematics and physics. His thinking focused equally on both geometry and its physical applications, like all sorts of useful mechanical devices, astronomy, and nontraditional measurements.
As a mathematician, he discovered the method of exhaustion for computing the area of (or under) a curved shape. This method stemmed from the idea that a curved shape is the image of an infinite aggregation and accumulation of straight segments, and implicitly, its area consists of an infinite aggregation of regular shapes such as triangles and other polygons. Although such an idea can be expressed rigorously only in terms of present-day mathematical analysis with the notion of limit being essential, the tools of Euclidean geometry sufficed for Archimedes at that time to make his point and effectively provide precise measurements of such areas.
In brief, the method of exhaustion assumes inscribing inside the area to be measured a sequence of polygons (properly chosen) whose areas converge to the area of that shape (or approximate the area, in the terms of that period). The difference between the area of the initial shape and that of the n-th polygon will become arbitrarily small as n increases. Then, the possible values of the area in question are successively eliminated with the lower areas in that sequence, by using the reductio ad absurdum logical principle.
An easy example of using the method of exhaustion is what is known as Archimedes’ quadrature of the parabola. He proved that the area determined by a parabola and a straight line intersecting it is the infinite sum .
The idea was to dissect that parabolic segment into an infinite number of triangles (a procedure known as triangulation), as shown in Figure 1. Each of these triangles is inscribed in its own parabolic segment in the same way that the initial triangle is inscribed in the large segment. Assuming the area of the initial triangle is 1, Archimedes proved the area of the parabolic segment to be 4/3.
What is remarkable is that Archimedes computed the infinite sum above (which is the limit of a geometrical series with the common ratio ¼, in modern terms), also by geometrical means. He considered a unit square tiled with four squares of side length ½, then each of these smaller squares tiled with four squares of side length ¼ , and so on, as in Figure 2. Then he showed that the areas of all the squares lining up with the diagonal of the initial square cover 1/3 of the area of the initial square. You could try to reconstruct this proof as an exercise; it is not hard at all.
The method that Archimedes used for finding the area of curved shapes is simultaneously simple and complex — complex because he had to deal with infinity and argue for such concept. The concept of infinity was present in Euclid’s fundamental concepts of his geometry (just a few decades before), where segments, lines, and shapes are sets containing an infinite number of points; yet this infinity reflected “numerosity” and continuity, a “large” and “dense” infinity. However, the infinity that Archimedes found and used in his sequences of shapes was of another kind: it was a “sequential” infinity, “smaller than” the geometrical infinity. It was “as large as” the infinity of natural numbers; however, it had the peculiarity (at that time) of accumulating itself near a certain number and as such, generating numbers arbitrarily close to zero.
These concepts, although having a geometrical origin in Archimedes’ work, inspired two fundamental concepts for modern integral and differential calculus (pioneered by Leibniz, in the 17th century), namely the infinitesimals and the idea of limit.
Infinitesimals are positive quantities so small that there is no mathematical way to measure them. They designate a concept that over the history of mathematics gave serious headaches to mathematicians who struggled to provide axiomatic constructions for them in order to embed them as numbers among the reals. Such constructions were necessary since the naturals, rationals, and reals benefited from rigorous systematic constructions that defined them as numbers. One of the main features of mathematics as a discipline is that any of its fundamental concepts must have its own mathematics, including sets and numbers. Detected with the ancient work of Archimedes, infinitesimals came to be quite controversial in modern and contemporary mathematics. Several number systems have been axiomatically designed to embed infinitesimals; however, not all mathematicians accepted them as real numbers.
Still related to curved shapes, it is worth mentioning that Archimedes provided one of the first approximations for , by placing it in the interval . He did that by inscribing and circumscribing a circle with two similar 96-sided regular polygons.
There are plenty of contributions of Archimedes in various fields of mathematics besides geometry. One of my math professors commented that a mathematician becomes really great when his name comes to be “adjectivized”. Well, there are many notions that are called after Archimedes: Archimedean absolute value, Archimedean circle, Archimedean field, Archimedean group, Archimedean spiral, to name just notions from mathematics.
However, Archimedes is best known for his principle in the physics of water. The legendary tale about his discovering this principle with his “Eureka!” while taking a bath is well known (though some historians don’t take it as true, arguing that the story appeared nowhere in the known written works of Archimedes). The tale says that in searching for a way to determine whether the crown of King Hiero II of Syracuse was made of pure gold or of a metal surreptitiously alloyed with silver, without damaging the crown, he discovered the principle of buoyancy, which states that every object immersed in a liquid is buoyed up by a force equal to the weight of the liquid displaced by the object. Applied to his particular crown problem, this principle allowed Archimedes to find the volume of the crown and then its density, by dividing mass by volume. If this density were found to be lower than the usual density of gold, then less dense metals had been added. Archimedes’ principle is nowadays a canon law of physics fundamental for hydrostatics and fluid mechanics.
If the crown tale is true, it is just another example of Archimedes’ inclination toward practical application of his knowledge. He was not content simply with his theoretical achievements, but wanted the applications of those achievements to benefit the community. His mechanical inventions — the screw for raising water, block-and-tackle pulley systems, the odometer (measuring distances), the planetarium — stand as beautiful evidences of his caring about people’s daily life.
Archimedes also involved himself in the military actions defending the city of Syracuse against enemy attacks, not as a soldier, but as a brilliant inventor of weapons. He designed a claw consisting of a long arm from which a large metal grappling hook was suspended. When the claw was dropped onto a rival ship, the arm would swing upwards, lifting the ship out of the water. It is known as Archimedes’ claw.
Yet, the most ingenious weapon seems to be Archimedes’ heat ray device. The device was a system of mirrors acting together as a parabolic reflector of the sunlight. The heated rays converged toward the focus and set fire to the enemy’s ships. Several recent experiments have reproduced Archimedes’ heat ray device and proved it to be functional and effective.
We know that many brilliant mathematicians are so dedicated to their theoretical work and creation that they tend to remain “suspended” in the abstract realm of their pursuits. There, they try to exploit their minds to reach as high an intellectual plane as possible, and it is often difficult for them to get back “on the ground” to make their knowledge work for practical applicative purposes. The ancient ages provided us with a marvelous counterexample of that tendency in the person of Archimedes, who dedicated his time and knowledge to serve his community, from providing people with effective tools and annihilating enemy ships, to leaving them mathematical and physical principles they will use fundamentally in their future science over millennia.
Catalin Barboianu is mathematician and philosopher of science. He is the founder of PhilScience.