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Portrait of a scholar (Archimedes?). Domenico Fetti, 1620.
Early life and education
Archimedes was born into a family connected to the ruling elite of Syracuse. In his work ‘The Sand-Reckoner’, Archimedes gives his father’s name as Phidias, an astronomer about whom nothing else is known, it is assumed this is what likely sparked Archimedes’ early interest in the sciences. Little is known about Archimedes’ early years, but it is believed he travelled to Alexandria in Egypt, the intellectual hub of the Mediterranean world, to study under the successors of Euclid. This connection with Alexandria placed Archimedes within the vibrant mathematical community of his time.
Upon returning to Syracuse, Archimedes dedicated himself to a life of inquiry and discovery. His surviving written work indicates a keen desire to explore both the abstract principles of mathematics and their practical applications. He had a deep love for pure mathematics but was equally committed to solving real-world problems.
Mathematical contribution
Archimedes’ work in mathematics established him as a towering figure in the discipline. One of his most celebrated achievements is the approximation of pi (π). Archimedes developed a method of calculating the value of π by inscribing and circumscribing polygons around a circle, finding their perimeters and using this data to estimate π with remarkable precision. His method, known as the method of exhaustion, is considered an early form of integral calculus and remained the most accurate approximation of π until modern times.
Another significant contribution is Archimedes’ principle of the lever. In his treatise ‘On the Equilibrium of Planes’, Archimedes laid down the laws governing levers and balance. His famous statement, “Give me a place to stand, and a lever long enough, and I will move the Earth, (World),” captures the essence of his discovery that a small force, applied correctly using a lever, can move large objects. This principle revolutionized mechanics and influenced engineering practices for centuries.
Archimedes also made strides in understanding the concept of centroids in geometry. In his work, ‘On the Quadrature of the Parabola’, he calculated the area of a parabolic segment and demonstrated that the centroid of a parabola lies along its axis. His ability to combine abstract reasoning with physical insight was unparalleled in the ancient world.
Hydrostatics and the principle of buoyancy
Archimedes is perhaps best known for his work in hydrostatics, particularly his discovery of the principle of buoyancy. This principle states that any object immersed in a fluid experiences an upward force equal to the weight of the fluid displaced by the object. The story of Archimedes’ Eureka, (Greek: εὕρηκα, I have found), moment is legendary, even if not exactly what occurred.
King Hiero II of Syracuse asked Archimedes to determine whether a crown made for him was pure gold or adulterated with silver. While bathing, Archimedes realized that the water displaced by his body in the bath provided a way to measure the volume of the crown and thus its density. Leaping from his bath in excitement, Archimedes is said to have run through the streets shouting Eureka!, I have found it!
Archimedes’ principle of buoyancy became a cornerstone of fluid mechanics, underpinning theories of flotation and sinking. This discovery had practical applications in shipbuilding, as it allowed engineers to calculate whether ships would float based on their weight and the water they displaced.
Inventions and engineering feats
Archimedes’ inventive genius was not limited to theoretical work; he also designed practical machines that were ahead of their time. One of his most famous inventions is the Archimedean screw, a device used to raise water. The screw consists of a helical surface inside a cylinder, and when turned, it would lift water from a lower to a higher elevation. Originally designed for irrigation in Egypt, the Archimedean screw is still used today in some parts of the world for pumping water.
In the realm of warfare, Archimedes applied his knowledge of mechanics to develop powerful machines that helped defend Syracuse during the Second Punic War. His military devices were so effective that they became the stuff of legend. Among these were massive catapults capable of hurling projectiles at enemy ships, and large cranes or “claws” that could grab ships, lift them out of the water, and then smash them back down, sinking them. Another invention, known as the Archimedes’ heat ray, allegedly used mirrors to focus sunlight onto Roman ships, setting them on fire. While the historical accuracy of the heat ray remains debated, it highlights the blend of scientific theory and military necessity in Archimedes’ work.
The method of mechanical theorems
One of Archimedes’ lesser-known yet most significant contributions to mathematics is the method of mechanical theorems, detailed in his work of the same name. In this treatise, Archimedes explored the use of mechanical reasoning to discover mathematical theorems. By imagining geometric figures as physical bodies with weight, he was able to “balance” them and deduce geometric relationships. This approach was revolutionary because it introduced an early form of integral calculus long before Newton and Leibniz developed the field in the 17th century.
Archimedes’ spiral—a curve traced by a point moving uniformly away from a central point while the point revolves around that point—was another breakthrough. He used this spiral to square the circle, showing that the area under one turn of the spiral was equal to a quarter of a circle’s area. His work on spirals remains relevant in modern mathematics, particularly in polar coordinate systems.
The Sand Reckoner, challenging the size of the universe
Archimedes’ curiosity extended beyond the Earth and into the cosmos. In his treatise ‘The Sand Reckoner’, he set out to determine whether it was possible to calculate the number of grains of sand that could fill the known universe. While this might seem like a trivial exercise, Archimedes used it as a way to propose a new system for expressing large numbers, challenging the conventional Greek numeral system. His estimate pushed the boundaries of ancient cosmology and provided a method for handling astronomical magnitudes, illustrating his unique ability to connect mathematics with natural philosophy.
Archimedes’ death and legacy
The legacy of Archimedes is indelibly tied to his tragic death during the Siege of Syracuse in 212 BCE. The Romans, led by General Marcus Claudius Marcellus, laid siege to the city, and Archimedes played a critical role in designing machines that delayed the Roman advance. According to Plutarch, Archimedes was so engrossed in his work during the siege that he was oblivious to the Roman army entering the city. A Roman soldier found Archimedes working on a mathematical problem, allegedly Archimedes responded, “Do not disturb my circles,” referring to the geometric figures he was drawing in the sand, the soldier, either misunderstood or ignored the orders, killing Archimedes on the spot, despite orders from Marcellus that Archimedes was to be spared.
Archimedes’ contributions to science and engineering left an enduring legacy. His surviving works were studied, copied, and transmitted through the centuries, influencing Islamic scholars in the medieval period and later the European Renaissance. His method of approximation and mechanical theorems paved the way for the development of modern calculus, while his inventions demonstrated the power of applied science. Archimedes’ influence can be seen in the works of later astronomers and mathematicians such as Galileo Galilei, Isaac Newton, and even physicists mathematicians and engineers of today.
Rediscovery of Archimedes’ works
Many of Archimedes’ original writings were lost over time, but his ideas and concepts survived through Latin and Arabic translations. One of the most exciting discoveries related to Archimedes’ work came in 1906 when the Archimedes Palimpsest was uncovered. This medieval manuscript contained previously unknown works by Archimedes, including The Method of Mechanical Theorems. The palimpsest was produced when monks in the 13th century scraped off the original Archimedean text and reused the parchment for religious writings. Modern technology has allowed scholars to recover the erased text, providing new insight into Archimedes’ mathematical genius.
Archimedes stands as a beacon of human ingenuity and intellectual curiosity. His profound insight into mathematics, physics, mechanics, and applied science such as hydrostatics continues to resonate today, demonstrating how a single individual’s quest for knowledge can shape the course of human history.
From his invention of the Archimedean screw to his principles of levers and buoyancy, Archimedes’ achievements spanned both theoretical brilliance and practical applications. His life and work exemplify the synergy between scientific inquiry and the real-world problems that science seeks to solve, making him one of the most remarkable figures of the ancient world.
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Notes:
Surviving writing of Archimedes:
· Measurement of a Circle
· The Sand Reckoner
· On the Equilibrium of Planes
· Quadrature of the Parabola
· On the Sphere and Cylinder
· On Spirals
· On Conoids and Spheroids
· On Floating Bodies
· Ostomachion, (Loculus of Archimedes or Archimedes’ Box)
· The cattle problem
· The Method of Mechanical Theorems
· Book of Lemmas or Liber Assumptorum is a treatise with 15 propositions on the nature of circles
· Archimedes’ Palimpsest described above includes treatises:-
· On the Equilibrium of Planes
· On Spirals
· Measurement of a Circle
· On the Sphere and Cylinder
· On Floating Bodies
· The Method of Mechanical Theorems
· Stomach-ion
· Speeches by the 4th century BC politician Hypereides
· A Commentary on Aristotle’s Categories
· Associated other works
Euclid
Euclid (Εὐκλείδης) was a Greek mathematician, often referred to as the “father of geometry.” He lived around 300 BCE, during the reign of Ptolemy I in Alexandria. His most famous work, Elements, is a comprehensive compilation of the mathematical knowledge of his time and became the foundational text for geometry for centuries.
Elements consist of 13 books covering geometry, number theory, and mathematical logic. Euclid also contributed to optics, astronomy, and the study of conic sections. His work laid the groundwork for deductive reasoning in mathematics, a method still used today.