William Smith’s Archimedes: Principles of Discovery explores the life and achievements of one of the greatest mathematicians, scientists and engineers of the ancient world. Rather than presenting Archimedes simply as a historical figure, the book examines the principles behind his discoveries and the way he approached difficult problems. Curiosity, observation, mathematical reasoning and persistence were central to his work, and these qualities allowed him to transform practical questions into discoveries that would influence science for centuries.

Archimedes lived more than two thousand years ago, yet many of his ideas remain fundamental to modern mathematics, physics and engineering. His work on buoyancy, levers, centers of gravity, geometry, π, mechanical devices and astronomy demonstrates an extraordinary ability to connect abstract mathematical ideas with the physical world.

Archimedes and the World of Syracuse

Archimedes was born around 287 BC in Syracuse, on the island of Sicily. He lived during the Hellenistic period, when Greek mathematics and science were developing rapidly. Alexandria had become a major intellectual center, and Archimedes is traditionally believed to have studied there before returning to Syracuse.

Syracuse was an important and wealthy Greek city, ruled by King Hiero II. Archimedes was connected to the royal court, giving him opportunities to work on both theoretical mathematics and practical engineering problems.

One of the most remarkable aspects of his career was his ability to move between theory and practice. A problem involving a machine could lead him toward a mathematical principle, while a mathematical problem could suggest a practical engineering solution. This combination became one of the defining characteristics of his work. Archimedes and Syracuse are featured in Indiana Jones and the Dial of Destiny, a movie we recommend.

The Eureka Story and the Principle of Buoyancy

The most famous story about Archimedes concerns a golden crown belonging to King Hiero. The king supposedly suspected that the crown contained silver mixed with gold but wanted to determine whether it was genuine without destroying it.

According to the traditional story, Archimedes discovered the solution while taking a bath. He noticed that his body displaced water and realized that the volume of an irregular object could be determined from the amount of water it displaced. He supposedly ran through Syracuse shouting “Eureka!”—“I have found it!”

The details of this story come from later sources, so historians cannot establish every part of the episode. However, Archimedes’ work on hydrostatics and floating bodies is unquestionably genuine.

Archimedes’ principle states that an object immersed in a fluid experiences an upward force equal to the weight of the fluid it displaces. This explains why objects float or sink and provides a mathematical foundation for understanding buoyancy.

The importance of this discovery is enormous. Archimedes transformed observations about water and floating objects into a systematic scientific theory. His work became a foundation of hydrostatics and remains essential to modern engineering, shipbuilding and physics.

The Lever and the Foundations of Mechanics

Archimedes also made fundamental contributions to mechanics through his study of levers and equilibrium.

A lever allows a relatively small force to move a much larger weight when the force is applied at an appropriate distance from the fulcrum. Archimedes did more than recognize this practical effect: he developed a mathematical explanation for it.

His studies of equilibrium and centers of gravity established relationships between weight, distance and balance. These ideas helped transform mechanics from practical craftsmanship into a mathematical science.

The famous quotation traditionally attributed to Archimedes—“Give me a place to stand, and I will move the Earth”—expresses the principle of mechanical advantage. Although the exact historical wording is uncertain, it captures the idea behind his work.

The significance of the lever is therefore much greater than the simple machine itself. Archimedes demonstrated that the behavior of machines could be predicted through mathematics.

The Archimedes Screw

The Archimedes screw is one of the inventions most strongly associated with his name. It consists of a spiral surface arranged around a shaft inside a tube or cylinder. As the screw rotates, water is transported upward.

The device is traditionally associated with Archimedes’ work in Egypt, where water management and irrigation were major practical concerns. It may have been based on earlier technologies, but Archimedes is credited with developing or improving the mechanism.

The screw demonstrates how rotational movement can be converted into the controlled movement of water. Variations of the same basic principle are still used in modern pumping systems.

It is an excellent example of Archimedes’ ability to turn mathematical and mechanical principles into practical technology.

Archimedes and Mathematics

Although his inventions have made him famous, mathematics was at the heart of Archimedes’ work. He developed sophisticated methods for calculating areas and volumes and investigated geometric shapes including spheres, cylinders, parabolas and spirals.

One of his most important techniques was the method of exhaustion. Archimedes could approximate a complicated figure using increasingly precise simpler figures. By narrowing the difference between the approximation and the desired quantity, he could establish remarkably accurate mathematical results.

This approach anticipated important ideas later associated with calculus.

Archimedes did not invent modern calculus—Newton and Leibniz developed the formal system almost two thousand years later—but his methods demonstrated that areas and volumes could be calculated through processes resembling integration.

His surviving work shows how advanced Greek mathematics had become and how close Archimedes came to ideas that would not be formalized until the scientific revolution.

Archimedes and π

Archimedes also made one of antiquity’s most important contributions to the calculation of π.

He used polygons placed inside and outside a circle to establish increasingly precise upper and lower limits for the value of π. By increasing the number of polygon sides, he could make the approximation progressively more accurate.

The importance of his achievement is not simply the numerical value he obtained. His method demonstrated a powerful general mathematical strategy: when an exact quantity is difficult to calculate directly, establish boundaries around it and progressively narrow those boundaries.

This method of approximation remains fundamental to mathematics.

The Sphere and the Cylinder

Archimedes was particularly proud of his discovery concerning the relationship between a sphere and its enclosing cylinder.

He demonstrated that the volume of a sphere is two-thirds that of a cylinder with the same diameter and a height equal to that diameter. A corresponding relationship exists between their surface areas.

According to tradition, Archimedes considered this one of his greatest discoveries and wanted a sphere and cylinder represented on his tomb.

The story illustrates the importance Archimedes placed on pure mathematical discovery. Not all of his work was intended to produce a practical machine. For him, understanding the mathematical structure of the universe was an achievement in itself.

The Parabola and Infinite Processes

Archimedes also calculated the area of a segment of a parabola using geometric methods involving an infinite series.

He divided the area into smaller and smaller geometric components and demonstrated a precise relationship between them. Once again, this resembles concepts that would later become central to calculus.

What makes this achievement remarkable is that Archimedes had none of the modern mathematical notation for limits, derivatives or integrals. He developed rigorous geometric arguments that achieved many of the same conceptual goals.

His work therefore represents an important bridge between ancient geometry and later mathematical analysis.

The Sand-Reckoner

In The Sand-Reckoner, Archimedes tackled a completely different mathematical problem: how to represent enormous numbers.

He imagined calculating the number of grains of sand that could theoretically fill the universe. The purpose was not really to count sand but to demonstrate that mathematics could describe numbers vastly larger than those normally used.

Archimedes developed a numerical system capable of representing enormous quantities. The work also contains astronomical discussions and refers to the heliocentric model associated with Aristarchus of Samos.

The Sand-Reckoner demonstrates Archimedes’ willingness to combine mathematics, astronomy and imagination to investigate questions about the scale of the universe.

Astronomy and Mechanical Models

Archimedes was also interested in astronomy and is traditionally associated with a mechanical model representing the movements of the Sun, Moon and planets.

Ancient writers described a device capable of reproducing celestial motions. Although the precise construction of this machine is uncertain, the accounts demonstrate the ancient fascination with using mechanical systems to represent the heavens.

For Archimedes, mathematics was not isolated from nature. The movements of celestial bodies could be described through mathematical and mechanical relationships.

This approach anticipated a central idea of later science: that nature can be understood through mathematical models.

Archimedes as an Engineer

Archimedes’ reputation as an engineer was strengthened during the Roman siege of Syracuse in the Second Punic War.

Ancient accounts describe defensive machines attributed to him, including powerful catapults and mechanisms capable of attacking ships at different distances. The famous “Claw of Archimedes” was supposedly a large crane-like mechanism capable of grabbing attacking vessels and lifting or overturning them.

Archimedes’ role in Syracuse’s defense illustrates how his theoretical knowledge could be applied to practical engineering problems.

The Mechanical Method of Discovery

Perhaps the most fascinating insight into Archimedes’ thinking comes from The Method of Mechanical Theorems.

Archimedes used mechanical reasoning to help discover mathematical relationships. He could imagine geometric figures divided into extremely thin sections and treat those sections as weights arranged around a balance.

This allowed him to discover relationships between areas and volumes before proving them rigorously through geometry.

The distinction is important. Mechanical reasoning helped Archimedes discover what might be true, while mathematical proof established that it was true.

This approach is remarkably similar to modern scientific practice, where experiments, models and intuition can suggest an answer that must then be tested or demonstrated rigorously.

Archimedes’ Lasting Legacy

Archimedes’ greatest achievement may be the way he united mathematics, physics and engineering.

The lever was both a machine and a mathematical principle. The Archimedes screw demonstrated the practical application of geometry and mechanics. Hydrostatics transformed physical observation into mathematical law. His study of areas and volumes anticipated calculus, while his work on π demonstrated sophisticated approximation techniques.

His influence continued long after his death. Renaissance mathematicians and scientists rediscovered and studied his works, and his ideas contributed to the development of modern mechanics and mathematical physics.

What makes Archimedes particularly remarkable is that he achieved so much without the tools available to modern scientists. He had no modern algebraic notation, calculus, computers or advanced measuring instruments. Instead, he relied on observation, logical reasoning, geometric imagination and mathematical proof.

Archimedes: Principles of Discovery

William Smith’s Archimedes: Principles of Discovery provides an opportunity to look beyond the familiar image of Archimedes running from his bath shouting “Eureka.” His real legacy is far greater.

Archimedes established fundamental principles of buoyancy and mechanics, developed mathematical theories of levers and centers of gravity, improved mechanical devices such as the water screw, calculated π with remarkable precision, investigated the volumes and areas of complex geometric shapes, studied parabolas and infinite series, explored enormous numbers and contributed to mathematical astronomy.

More importantly, he demonstrated a powerful method of thinking. He approached difficult problems by breaking them down into fundamental principles, using observation and intuition to find possible solutions, and then applying rigorous mathematics to prove them.

That is why Archimedes remains one of the most important figures in the history of science. His inventions and discoveries changed mathematics, physics and engineering, but his greatest contribution may have been showing that the physical world could be understood through mathematical reasoning.

More than two thousand years later, the principles behind his work remain part of modern science. From ships floating in water to machines that lift heavy objects, from mathematical calculations of π to the study of areas and volumes, the influence of Archimedes can still be seen everywhere.

His enduring message is remarkably simple: seemingly impossible problems can often be solved when we find the right principle.