Five Equations That Changed the World: The Power and Poetry of Mathematics by Guillen, Dr. Michael

Five Equations That Changed the World emphasizes how mathematics (specifically, the five equations) has profoundly shaped our history in the last few centuries. Dr. Guillen tells how these five seemingly abstract and far-from-our-daily-lives equations have had very tangible and revolutionary impacts on our lives and changed the way we make sense of our universe. He does it in such an engaging and relatable way, which is why this book will always be one of my favorites. He tells and reminds us how the universe speaks to us in the poetic language of mathematics.

Below are my key takeaways from the book on each equation.

1. The Law of Universal Gravitation (Isaac Newton):

What keeps the Moon in its orbit? Why do apples fall straight down to the ground from the trees? Isaac Newton's law of universal gravitation not only did answer these questions but also united the Earth "underneath" the Moon with the Heavens "above" the Moon (Aristotelian and Ptolemaic view). He also brought the permanent divorce of science and religion.

What does it say?: Every object in the universe is attracted to every other object. Newton's law of universal gravitation describes this. The strength of this attraction depends on how heavy the objects are and the distance between them. The heavier the objects are, the stronger the attraction. The bigger the distance in between, the weaker the attraction.

So what?: This discovery bridged the divide between the celestial and terrestrial, powerfully showing that the same physical laws governing an apple falling from a tree also govern the orbits of the planets. (Before Newton and this discovery, Aristotle believed that the universe was divided into two regions: the central region, which included the Earth and its atmosphere, and the celestial region beyond the moon.)

The implications were monumental. It shook the theological and scientific beliefs of the time at their cores. It set the stage for modern physics and laid the foundation for space exploration. Newton, with this equation, brought about the permanent divorce of science and religion.

How did he discover it?: While the story of the apple falling from the tree is famous, Newton's path to his gravitational law was more complex. Combining observations with rigorous mathematics and experimentation, Newton sought to explain the motions of celestial bodies. He integrated ideas from his own work on calculus, which he developed concurrently with Leibniz, to arrive at his law of gravitation.

Imagine yourself swinging a stone tied to a rope in a circle. The stone stays in its path because the rope pulls it in. If the rope breaks, the stone flies away due to the centrifugal force. Now, visualize the Moon orbiting Earth, just like the stone you were swinging in a circle. What's keeping it from flying off into space? Newton believed it was an invisible force, just like that rope you had, pulling the Moon towards Earth. He realized that this same invisible force was what made an apple fall from a tree. Dug deeper, he also found that this force's strength depended on the objects' size (or mass) and how far apart they were. Bigger objects have a stronger pull, and the closer they are, the stronger the attraction. By fusing this together, he introduced the Law of Universal Gravitation, which says every object in the universe pulls on every other object based on its mass and the distance between them.

2. Hydrodynamics of Fluid Flow (Daniel Bernoulli):

Life on Earth is all about fluids. While our feet are mostly on the ground, the water we drink and the air we breathe keep us alive. Without water, we would be gone in days. Without air, we would be gone in a few minutes. Before Daniel Bernoulli, we understood a lot about solids but not about fluids and how they behaved. (Except for Archimedes' discoveries on hydrostatics, the study of captive fluids, which did not have anything to say about flowing water (hydrodynamic). That remained mostly as a mystery) Thanks to his unquenchable curiosity and relentless efforts, we began to demystify the fluid (liquids and gases) physics that we deeply rely on for our lives.

What does it say?: Fluids do not move randomly. While different, they also show patterns and follow physical rules for their behaviors. Bernoulli's principle sheds light on this by explaining how the speed of a fluid relates to the pressure it generates. The faster the fluid flows, the smaller its pressure. The slower the fluid flows, the bigger its pressure.

So what?: Bernoulli's principle has played a substantial role in many aspects of our lives. One of the most fascinating ones is how airplanes fly. Aircraft wings are designed based on this principle. The shape of the wing (different curvatures of the wing's top and below sides) makes air travel faster over the top than below. This creates lower pressure above the wing and higher pressure below the wing. Lift off..!

How did he discover?: Prior to Bernoulli, calculating the pressure in static fluids was an easy task (simply dividing the weight of the fluid by the area of its supported surface, similar to the pressure of solids). However, for moving fluids, it was not easy to measure or define the pressure of something whose weight and shape (therefore, the area of its supported surface) were constantly changing.

Coming from a family of renowned mathematicians and driven by his curious observations of nature, Bernoulli came up with a very simple yet profoundly genius idea for the time...

He punctured a hole in the wall of a pipe and inserted a glass tube into the hole. Then, he ran water through the pipe as usual and observed. He observed (with elation) that as the water flowed past the opening, a small column of water rose up in the glass tube and stopped at a certain height. This height was a measure of the flowing water's pressure. He was able to measure it successfully. (As an after-fact, how simple it is, isn't it? Yet not so easy to come up with in the first place)

The smaller the speed, the bigger the flowing fluid's pressure. The bigger the speed, the smaller the flowing fluid's pressure.

Pressure + Vis Viva = Constant

(Vis Viva means "living force" in Latin, it is basically the kinetic energy.)

From which, Bernoulli's principle came to be written:

P + ρ × ½ v^2 = CONSTANT

3. The Electromagnetic Field (Michael Faraday):

The electromagnetic force is all around us, literally everywhere. The light bulbs in our homes, the smartphones we use every day, and the dynamo that I was so excited to install on my bike when I was a kid, all use electromagnetism. However, before Michael Faraday, we did not understand how electricity and magnetism were connected. We thought electricity, magnetism, and gravitation were the three separate, important forces that governed the universe. Faraday’s unparalleled curiosity and tenacity in experimentation compensated for his lack of formal education and helped him discover that electricity and magnetism were related. His discovery has had a huge impact on our world and the way we live. His excellence in experiments and his true passion for science rose him from the lowest echelon of English society to the head of the Royal Institution.

What does it say?: Simply put, when there is a change in magnetic field, it produces electricity. Similarly, when electric current flows, it creates a magnetic field. This relation between electricity and magnetism gave birth to the concept of the electromagnetic field.

So what?: A lot…The implications of Faraday's discoveries are monumental and are virtually in every aspect of our modern lives. Telecommunications, wireless technologies, televisions, radios, medical equipment (such as MRI machines), and even the batteries powering electric cars are direct results of Faraday’s groundbreaking work and discovery.

How did he discover?: Faraday's journey was strongly founded in meticulous experimentation (including top-notch observation, note-taking, and thorough analysis). One of his most iconic experiments involved a coil of wire, a magnet, and a galvanometer (a device to measure electric current). When he moved a magnet in and out of the coil of wire, the needle of the galvanometer that was connected to the wire twitched. This indicated the flow of current in the wire, although there was no battery connected to the wire. This was groundbreaking as just a magnet and a coil of wire produced electricity. Faraday deduced that the movement of the magnet created an electric field, which induced a current in the coil.

Faraday continued his experiments and established the principles of electromagnetic induction. His work beautifully proved that electricity and magnetism were not separate forces but two aspects of the same fundamental phenomenon, the electromagnetic field.

Later, a young Scottish scientist, James Clerk Maxwell, translated Faraday's groundbreaking discovery into a mathematical equation, ∇ × E =–∂ B/ ∂ t. Through their efforts, we know can comprehend and leverage the invisible world of forces around us, which enabled a new era of technological innovation and progress.

4. The Second Law of Thermodynamics (Rudolf Clausius):

It’s no good crying over spilt milk, because all the forces of the universe were bent on spilling it.—SOMERSET MAUGHAM

Can you describe Life and Death not emotionally but quantitatively? That's what Rudolf Clausius' discovery of the second law of thermodynamics does. We will come to that but first a little background.

In the universe, there are only two types of processes: those that can be undone and those that cannot. The actions that can be undone are called revocable processes. The actions that cannot be undone are called irrevocable processes.

As steam engines started powering the Industrial Revolution and transforming the world, the need to understand the principles behind their operation more fundamentally started growing. While the conservation of energy (the First Law of Thermodynamics) was already known, it did not explain why certain processes were irreversible. This is where Rudolf Clausius, the German physicist, stepped in and gave us one of the most fundamental laws in physics.

What does it say?: In simple terms, Clausius (the Second Law of Thermodynamics) says that an isolated system's total entropy (disorder) can only increase over time. Energy tends to disperse and spread out if it is not restricted. This means the sum of the entropies of the participating bodies always increases.

So what?: This law defines the direction in which thermodynamic processes can occur. For instance, This law tells us why heat flows from hot to cold and not the other way around. It also introduces the concept of entropy (a deliberate choice of word by Clausius to resemble the word "energy") as a measure of disorder. On a cosmic scale, it gives us a peak of the universe's ultimate fate: a state of maximum entropy where all processes cease, which is termed the "heat death" of the universe. On a technological level and in everyday life, refrigerators and air conditioners are designed and operated in accordance to this law.

How did he discover?: Clausius formulated the Second Law of Thermodynamics after closely observing and thinking about the operating principles of the steam engine. He was trying to determine the efficiency limits of the steam engine and why not all the heat energy could be transformed into work. To explain this, he introduced the concept of entropy and said that the systems naturally evolve towards a state of greater disorder (entropy).

Clausius' discovery was not just a scientific observation to understand the steam engine's operating principle. He uncovered a universal truth about the nature of our universe and its tendency to disorder. This provided a mathematical framework for the arrow of time, which could be considered as the quantitative narrative for the life cycle of energy in our universe, in a sense, explaining life and death not emotionally but quantitatively.

5. The Theory of Relativity (Albert Einstein):

What would happen if you were traveling at the speed of sound? Imagine you are running away from your mom at the speed of sound, and she is chasing you while yelling at you (Yes, I was a naughty kid!). When you look back, you would see your mom coming at you, but you would not hear her words of love (!) because the soundwaves coming from her would not reach you. Einstein used a similar thought experiment, but he tried to travel along the light, at the speed of light and imagined what would happen!

What does it say?: Einstein's theory of relativity, which is expressed in the famous equation, E=mc^2, tells us that energy and mass are different forms of the same thing. They are interchangeable and convertible to each other. "c" in this equation is the speed of light, which is ~300,000,000 m/s. It is one million times faster than sound. The fact that the speed of light is a massive constant in this equation implies that even a fraction of mass can be converted into a huge amount of energy. (As in the Sun or other stars or atomic bombs)

So what?: Relativity theory's implications are vast, spanning from understanding the glowing stars to nuclear energy, both as a power source and, unfortunately, as a weapon. It also gave us a new perspective to look thru and make sense of our universe. Similar to what Newton did centuries ago, Einstein, with this new perspective, showed us that time and space were relative, not absolute, which was the way Newton saw the universe.

How did he discover it?: Einstein was a huge fan of thought experiments and they played a pivotal role in his discovery of initially the special theory of relativity and then the general theory of relativity. Just like how we started this section by imagining moving at the speed of soundwaves, Einstein took it a step forward and imagined riding alongside a beam of light! He wondered how the universe would look if he moved at the speed of light. He thought, he would observe the light as a static electromagnetic field, which would violate Maxwell's assertion that the speed of light in vacuum is always the same for all observers, regardless of their relative motion. This was irreconcilable with Newtonian physics, which advocated for absolute space and time. Therefore, Einstein abandoned the traditional absolute concepts of space and time and proposed that space and time would be relative to the observer's motion, hence the concepts of "time dilation" and "length contraction".

In summary, Five Equations That Changed the World beautifully presents some of the universe's laws in a very relatable and easy-to-read way, as they were discovered by some of the most brilliant minds in history. All these five equations/laws/principles show us how certain patterns of the cosmos govern our lives and existence. Next time you look around and try to make sense of the universe, you might try remembering these equations where relevant and observe the universe thru the lens of these equations and discoveries.

Kaan Demiryurek

11 November 2023