Standing on the Shoulders of Giants: The Case of Archimedes
by CJ Uberroth
It is said that during the Roman siege of Syracuse (212-214 BC), the Romans were stalemated by the Syracusans’ complex methods of defense. Marcus Claudius Marcellus, a renowned Roman General, found himself at a loss during the siege. Apparently, the mathematician, Archimedes, had used his knowledge to invent a claw capable of toppling siege weapons and sinking ships, as well as a mirror designed to focus light and heat on the sails of advancing ships, in order to set them ablaze. The defense was so potent that upon the final push into Syracuse, Marcus ordered that Archimedes be spared so that he might be honored for his genius. Though this order was ignored and Archimedes was killed in his home, it is said that Marcus arranged special funeral for his adversary in honor of the battle. With Archimedes’ death, we would wait centuries for another equally brilliant mind to emerge.
There are any number of achievements in modern mathematics that demonstrate the depth and scope of Archimedes’ intelligence, but three iconic moments in mathematical and scientific history exemplify it.
The trial of Galileo by the Catholic Church for his assertion that the earth revolves around the sun along with the other planets is well-known, but what is much less known or appreciated is the fact neither Galileo’s nor even Copernicus’s advancement of the theory of heliocentrism was the first time that this picture of the universe had been proposed.
Archimedes wrote a letter commonly known as The Sand Reckoner in which he set out to debunk the “sand-problem.” The idea was the notion that the sands of the world were either infinite or uncountable due to the lack of a number system large enough to count them. While The Sand Reckoner is considered one of Archimedes’ most accessible works, it is not the beauty of his proof that is relevant in this case. Archimedes’ method of proof was exhaustive: he showed that if you imagine a quantity of sand greater than that of the earth and prove that it is countable and finite, then this would imply that the sands of the world were countable and finite. In his attempt to demonstrate this, Archimedes applies the concept of heliocentrism:
Now you are aware that ‘universe’ is the name given by astronomers to the sphere whose center is the center of the earth and whose radius is equal to the straight line between the center of the sun and the center of the earth. This is the common account (τὰ γραφόμενα), as you have heard from astronomers. But Aristarchus of Samos brought out a book consisting of some hypotheses, in which the premises lead to the result that the universe is many times greater than that now so called. His hypotheses are that the fixed stars and the sun remain unmoved, that the earth revolves about the sun in the circumference of a circle, the sun lying in the middle of the orbit, and that the sphere of the fixed stars, situated about the same center as the sun, is so great that the circle in which he supposes the earth to revolve bears such a proportion to the distance of the fixed stars as the center of the sphere bears to its surface.
As it turns out, then, Galileo and Archimedes had more in common than simply dying in their homes. Indeed, one can only wonder how vigorously Archimedes must have been turning in his grave, when the Inquisition sentenced Galileo, or how human cultures around the world could have held fast to false ideas for so long.
In the 17th and 18th centuries, mathematics really began to take off, with the work of Gauss, Euler, and L’Hopital, and was the period which saw the development of infinitesimal calculus. Some may know the story of Leibniz and Newton’s battle to create calculus as we know it today, and they are credited with developing the Fundamental Theorem of Calculus, which connects differential calculus and integral calculus. Essentially, what they did was give prove of and provide a notation for the idea that you can imagine rectangles below a curve that estimates the value of the space between said curve and the x-axis. If you then shrink these rectangles to an infinitely small width, you can obtain the exact area of the space below the curve.
This theorem and its subsequent use have led to countless advancements in mathematics and science. And yet, again, Leibniz and Newton were not the first to do it, but rather, Archimedes, who had made these comparisons of figures in his letters On the Sphere and the Cylinder and On the Measurement of a Circle. Most notably, he used inscribed polygons within circles to demonstrate formulas for areas of conic sections and then “rotated” them into three dimensional figures to extrapolate the formulas for volumes. These are all things we teach in modern calculus, but the history of these formulas and their creation largely goes unnoticed.
Finally, it is important to mention Leonhard Euler. From his development of a modern notation for algebra and calculus to the discovery of his namesake value, e, Euler really informed just about every facet of the mathematics of his day. Given what we have already seen, however, I’m sure that readers will not be surprised to discover that like Galileo, Leibniz, and Newton, Euler was standing on the shoulder of a much older mathematical giant. For whether it is his creation of common notations for summations and pi, his analysis of arithmetic, geometric, and recursive series, or his incredible work in physics and mechanics with Bernoulli, what we find is that Euler’s achievements arose on top of Archimedes’ work.
Archimedes demonstrated the constancy of the ratio between a circle’s circumference and area (pi) and also developed his own arguments for using arithmetic to solve for the value of a series (a series is a long chain of adding values, created by some mathematical rule — an example is adding the first one-hundred integers). Indeed, Archimedes even dabbled in mechanics. In a letter to Eratosthenes, Archimedes laid out proofs involving centers of gravity and the change in these centers under different circumstances, and advancement that we really would not fully grasp until the modern scientific and industrial revolutions. What is particularly amazing is the fact that this letter had no real practical purpose behind it. Archimedes simply wanted to share his knowledge with a respected colleague, who had shown interest in the relationship between mathematics and the principles of mechanics.
The truth is that all of Archimedes’ letters have this quality, their ultimate aim being to create a commonwealth of knowledge that ultimately would know no borders, in time or space. It would seem that he never kept anything for himself or for his people, and in that sense, he imagined the international, apolitical community of mathematicians and scientists that exists today and has proven to be resilient even in the most politically contentious of times. The only shame is that the long history of human contentiousness, not to mention an increasing and appalling level of historical ignorance, has contributed to Archimedes going largely un-credited for the intellectual civilization to which, in many ways, he gave life.