Maths, Mechanics & Eureka

When we study the history of science, specifically physics, we find that a good idea simultaneously existed in various places. This suggests it may be better not to overemphasize a person for the origin of an idea. If we focus on the context, utility, and exchange of ideas, we get a broader picture of scientific ideas.
This approach creates a spatio-temporal network, an interesting way to view the historical evolution of ideas across the humanities, space, and time. People, ideas, and technologies collectively progress the frontiers of science in various places at different times. In that sense, science, including physics, is a global human endeavor.
This is evident when we look into the history of mechanics from ancient times until now. Mechanics is a fundamental sub-discipline of physics and has a strong connection to mathematics and engineering. It has evolved with logical reasoning and understanding of natural phenomena.

Engineering structures, which humanity has long been interested in, have played a significant role. Mechanics has been the playground of philosophers, scientists, and engineers. The questions it raised led to new thinking and new technologies.
Mechanics offers a way to understand and engineer the universe. In this episode, we explore its links to mathematics, engineering, and key thinkers.

The importance of counting.

Counting has played a critical role in human life for a long time. In fact, we use fingers as a counting device, and this has been a very powerful tool for a significant period. So much so that it has been utilized to keep a tally of small numbers, which can be counted and analyzed in everyday life. Interestingly, as the numbers became larger, one had to externalize the counting process, and in ancient times, people had very creative methods to count objects. One of the fascinating aspects of counting was to use bones. Yes, bones were used as platforms on which marks were created, and these marks were utilized as the counts in a tally.

One of the pieces of evidence for such behavior was found in the Congo Basin in a place called Ishango. The bones that were found are dated around 9000 BCE to 6500 BCE (although a big debate is going on regarding the dates, with some putting it beyond 20,000 BCE), on which marks have been identified that look like counts registered on the platform. Indeed, it is quite fascinating to see how people used various devices to enumerate objects.
If you ask any child what is 1 plus 1, they will be able to say that it is 2. This may sound trivial, but the concept of addition itself was not in the historical context. To consider two numbers and add them together needs a certain degree of abstraction, which has been part of the evolution of mathematics since ancient times. A variety of counting methods have been devised in different civilizations, which have added a kind of flavour to the history of mathematics.

Importantly, language has played a critical role in facilitating a vocabulary for counting. Depending upon the syntax and the order of letters, numbers have been represented in a variety of ways across space and time of human history.

In this context, let me emphasize two important aspects of numbers and their representation. The first aspect is related to the positional notation. What is it? Let me give you an example. If you take numbers 24 and 42, the position of the number 4 is different. In 24, the number 4 is in the unit’s place, and in the number 42, it is in the tens place. So, the position of the number determines its value, and this is an important concept that civilizations have thought about and utilized in their counting systems.
The second aspect is the concept of zero. Let me give an example. If you consider number 007 and compare it to 700, obviously, depending on the location of the zero, the value of the number drastically changes. But what is intriguing is that various civilizations used a variety of symbols, such as dots, circles, and empty spaces, to represent nothingness.
What is not trivial is the recognition of zero as a number by itself. This needs a leap of thought because it must be an abstraction of a concept where the nothingness has to be associated with a number, and hence the association to zero. It was Brahmagupta around the year 628 CE in his Brahma Sputa Siddhanta that we first encountered the concept of zero as a number. This is indeed one of the great achievements because, without zero as a number, one cannot build mathematical concepts. It is both fundamental and profound. It has further played a critical role in laying the foundation of mathematics as we know it today.

02 Geometry

Now, let’s look at the connection between geometry and physics. Across various civilizations, the size and shapes of objects were curiosities, and understanding them was an important necessity for everyday life. Given that objects in the natural and artificial world come in various sizes and shapes, it was necessary to understand them for further utilization.

In ancient Greece, Thales of Miletus was one of the earliest to use a mathematical way of thinking and to formulate a framework to understand nature through logical analysis and not based on faith or myths. This thinking further percolated to all the subsequent philosophers, and that included a person named Pythagoras. Pythagoras’ life and times are not as well documented as those of other Greek philosophers, but by some estimates, he was supposed to have lived around 570 BCE to 495 BCE.

During that time, the Greeks had colonized various parts of Europe, and this included some parts of Italy. Pythagoras remained in that colonized part of Greece, where he had established a school, which was also interestingly a mythical cult. His school hardly shared any information with the outside world, and this is probably one of the reasons why there is very little known about Pythagoras’ life and times.

In fact, none of his writing has survived to date, and most of the information that we get is from indirect sources. However, the attribution of some scholars to Pythagoras’ work needs attention. Interestingly, Pythagoras did some experiments and tried to understand the production of sound.

He made an interesting connection to the strings and the pleasant sound that they produce. He hypothesized that there is a rational number of steps in strings that led to the pleasant sound. This thinking was further extrapolated to rational numbers, and that became an interesting connection. Pythagoras has also been attributed to have thought about astronomical objects.

Earth being spherical is one of the concepts that he had thought about and played a role in rationalizing the distances of objects such as the Sun, Moon, and planets. This kind of methodical thinking further influenced many Greek schools of thought, and this included the famous Plato’s Academy. Plato himself was a renowned philosopher, but he had a very strong inclination towards mathematics.

He also came up with the five solids and the four elements, which played a critical role in his interpreting of the natural world based on them. But it is in 300 BCE that we see an epoch in geometry in the form of Euclid’s Elements. Euclid of Alexandria was one of the great mathematicians whose work is still of significant relevance today.

Euclid, like many of his predecessors, had a life immersed in the ancient university system—in his case, the University of Alexandria. Again, not much is known about Euclid’s life and times, except for the fact that he wrote 13 volumes of his magnificent book titled Elements.

This book has turned out to be the foundation of mathematics and has played a critical role in creating a new worldview both for natural scientists and abstract mathematicians. Most of what we know today about Euclid is thanks to a Greek commentator, Theon of Alexandria, who lived roughly 700 years after Euclid. He played a critical role in interpreting and highlighting the works of Euclid, and going forward in time, Arabs took a keen interest in Euclid’s geometry and incorporated it into their education and research.

Euclid’s work on geometry is a masterpiece, which has 13 books in a series and contains 465 theorems. Each of them contains foundational knowledge about geometrical entities, including lines, angles, shapes, and solid geometries that past people had discussed. It is a tribute to his knowledge that Euclid’s Elements is still in print, and this shows how much the impact of Euclid has been over the centuries.

Importantly, the geometrical way of thinking has deeply influenced physics, along with the principle of counting and geometry. Physics, armed with mathematics, became an important way of looking at natural life in ancient times. This way of thinking further influenced another remarkable thinker named Archimedes of Syracuse.

If you want to think about a remarkable person who has deeply contributed to science and mathematics from ancient times, there is nobody better than Archimedes. Born in 287 BCE, Archimedes had a remarkable life because the number of things that are associated with him related to science, mathematics, and technology is probably unsurpassed compared to anybody else across the ages. Generally, when we talk about Archimedes, we associate him with the famous Eureka, where he probably ran naked in the excitement of discovering a specific concept related to buoyancy.

Of course, this might be altogether a myth, but the science that Archimedes did was indeed real and outstanding. He contributed to various areas in science, including mechanics, hydrodynamics, optics, engineering, and mathematics in both the pure and applied forms. We know about his achievements thanks to nine ancient Greek treatises, which give us a glimpse of his work.

An important aspect related to mechanics is the fact that in the ancient age, one can divide the contributions in terms of statics and dynamics. The dynamics aspect was mainly related to thinking driven by Aristotle and his school, which has turned out to be kind of incorrect from the modern viewpoint. But when it comes to statics, Archimedes had a very important role to play, and many of the discoveries he made have turned out to be correct and highly useful.

Related to statics, he wrote many interesting treatises, one of them being On the Equilibrium of Planes. In this book, he talks about the concept of the lever and utilizes the concept of the center of gravity. It is in this treatise where the concept of the center of gravity is used to understand various geometries, and he discusses the center of gravity of different geometrical objects.

Another important book related to Archimedes is On Floating Bodies. In this book, he discusses buoyancy and gives an important hypothesis to understand bodies immersed in a fluid. His discoveries were very critical in naval architecture.

Archimedes was not only an outstanding scientist but also an excellent engineer. He designed a water pump in which a hollow cylinder had a rotating helical shaft, which could pump water efficiently. This is usually called the Archimedes pump, and it has been used even to date.

Archimedes also contributed to the development of mathematics. He wrote On the Sphere and Cylinder and The Method, used for mechanical analysis.

One has to wonder whether he was one single person or many. Steven Strogatz’s book related to calculus, titled Infinite Powers: How Calculus Reveals the Secrets of the Universe, has a beautiful description of Archimedes’ contribution to understanding curves, including the circle and the determination of pi in an ingenious way. The logical process Archimedes used was unsurpassed for his time, and his contributions to science are among the most important from the ancient age.

There are also interesting stories related to his work, and one of them is called the Archimedes Palimpsest. A palimpsest is a technique in which one writes something, erases it, and rewrites on the same surface. In 1906, a Danish professor, Johan Heiberg, visited Constantinople to examine documents related to prayers, dated from the 13th century. To his surprise, it turned out that there was an underlying document beneath that prayer text, which was Archimedes’ writing.

It is truly outstanding that someone could discover such an important document after such a long time, and that’s another reason why one should do archival work—because you never know what kind of jewels one can discover. Archimedes contributed to various aspects of science, mathematics, and technology, but it is also vital to appreciate that he used a logical way of thinking. Such thinking had a deep influence on people who followed him, and even today, the process of his analysis stands up to scholarly scrutiny. It’s critical for us to realize that such people play a key role in spreading important ideas in science, in physics, and, in this case, mechanics.

Archimedes will surely be remembered as one of the greatest human beings who propelled human scientific thought. The legacy of Archimedes has been kept alive by introducing his figure on the Fields Medal, a major prize in mathematics. It’s considered the Nobel Prize equivalent in mathematics.

On the medal, there is an engraving with the quote: “Rise above oneself and grasp the world.” It is a great quotation to not only engrave on a medal but also to follow in letter and spirit.

With the same spirit to rise and grasp the world, we will explore the physics of mechanics going forward.

References:

GoogleTalksArchive, dir. 2012. The Archimedes Palimpsest. https://www.youtube.com/watch?v=Xe9uQVGkz9k.

Heath, T. L. 1897. The Works Of Archimedes. Cambridge University Press. http://archive.org/details/worksofarchimede029517mbp.

“Ishango Bone.” 2025. In Wikipedia. https://en.wikipedia.org/w/index.php?title=Ishango_bone&oldid=1280156982.

“Mathematics in India – Bhāvanā.” n.d. Accessed April 20, 2025. https://bhavana.org.in/mathematics-in-india-6/.

Padmanabhan, Thanu, and Vasanthi Padmanabhan. 2019. The Dawn of Science: Glimpses from History for the Curious Mind. Springer.

Stein, Sherman. 1999. Archimedes: What Did He Do Beside Cry Eureka?

Strogatz, Steven. 2019. Infinite Powers: How Calculus Reveals the Secrets of the Universe. Boston New York: Mariner Books.

Wu, Shiyue, and Francesco Perono Cacciafoco. 2024. “Understanding through the Numbers: Number Systems, Their Evolution, and Their Perception among Kula People from Alor Island, Southeastern Indonesia.” Humans 4 (1): 34–49. https://doi.org/10.3390/humans4010003.

Rene Dugas. 1955. A History Of Mechanics. http://archive.org/details/ahistoryofmechanics_201907.