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Category Archives: Mathematicians

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Euclid: The Father of Geometry

by December 12, 2018

By Jocelyn Hitchcock, Contributing Writer, Classical Wisdom
We all know the varsity team: Einstein, Newton, Pythagoras, Descartes. These names are drilled into our heads all through grade school math and history classes, and possibly accompanied by an under-the-breath curse from a disgruntled calculus or physics student. However, another mathematician should receive our attention: Euclid of Alexandria.
It is difficult to underscore enough the importance and significance of Euclid and his impact on mathematics for the subsequent 2000 years. Nonetheless, we will try.
Portrait of the Mathematician Euclid

Euclid of Alexandria

Euclid’s Early Life
The documentation of Euclid’s life is scant, at best. While we have a great deal of his work in extant, facts about Euclid as a person come down to us mainly through little snippets by Proclus and Apollonius (among others). Proclus, a Greek philosopher living in the 5th century CE, writes retrospectively and so his information must be taken with a grain of salt. He writes that Euclid taught in Alexandria, Egypt during the time of Ptolemy I Soter (4th-3rd century BCE). This places him as younger than Plato, but slightly older than Archimedes. Euclid was likely born around 300 BCE and resided in Alexandria, Egypt for most, if not all, of his life. Some historians think that he might have studied for a bit at Plato’s Academy in Athens, but this is conjecture based on his style of teaching. Past this, nothing is known of Euclid.
Map of Alexandria, Egypt, birthplace of Euclid

Map Showing Alexandria, Egypt, birthplace of Euclid

Euclid’s Career
Often called the “Father of Geometry,” Euclid was a teacher of mathematics, cultivating a school of pupils not unlike the style of the Academy. Proclus writes that Ptolemy once asked Euclid if there was a “shortened way to study geometry than the Elements, to which Euclid replied that there was no royal road to geometry.” This would suggest that not only was Euclid noteworthy among mathematicians and scientists in Alexandria, but was prominent enough to have an audience with the ruler of Egypt. As with details of his early life, we don’t know specifics regarding his career, save for his extant works and the fact that he was a prominent teacher in Alexandria.
Euclid’s Works and Achievements
Euclid likely wouldn’t have reached his level of renown if his works didn’t survive to such an incredible extent. His main work, The Elements, is a proto-textbook of 13 sections pulling together definitions, theories, and constructions of mathematics at the time. He covers geometry, number theory, and incommensurate lines- all subjects that have proved to be invaluable over the development of mathematics.
Book cover of Euclid's work

The cover of Euclid’s Elements

The Elements consisted of five general axioms and five geometrical postulates. Euclid provided the basic model for mathematical argument that follows logical deductions from initial assumptions. For those of us (including myself) who are not so mathematically inclined to understand the nitty-gritty details of Euclid’s Elements, Sir Thomas Heath sums them up in his 1908 publication The Elements of Euclid:
The 5 Axioms of Euclid:

1. Things which are equal to the same thing are also equal to one another
2. If equals are added to equals, the whole (sums) are equal
3. If equals are subtracted from equals, the remainders (differences) are equal
4. Things that coincide with one another are equal to one another
5. The whole is greater than the part

The 5 Geometric Postulates:

1. It is possible to draw a straight line from any point to any point
2. It is possible to extend a finite straight line continuously in a straight line
3. It is possible to create a circle with any center and distance
4. All right angles are equal to one another
5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the straight lines, if produced indefinitely, will meet on that side on which the angles are less than two right angles.

Other important contributions from Euclid is his proof of Pythagoras‘ Theorem, providing us with formulas that calculate the volumes of solids like cones, pyramids, and cylinders, as well as identifying the first four ‘perfect numbers,’ among a dozen or so other theories and proofs.
Euclid's Proposition 1.47

The most famous proposition from The Elements, Proposition 1.47. Also known as the Pythagorean Theorem.

In addition to the Elements, five other works of Euclid have come down to us and have been able to be interpreted: Data, dealing with the nature and implications of “given” information in geometry; On Divisions of Figures, dealing with the division of geometrical figures into two or more equal parts or into parts in given ratios; Catoptrics, dealing with the theory of mirrors and the images formed in plane and spherical concave mirrors; Phaenomena, a treatise on spherical astronomy; and Optics the earliest surviving Greek treatise on perspective.
Euclid’s Death and Legacy
Portrait of Euclid

Euclid of Alexandria

We can assume that Euclid died in the mid-3rd century BCE in Alexandria, but that is all we know. However, he left behind a legacy that has survived almost two and a half millennia. His work on geometry and theory is still in use today, governing even advanced models of dimension mathematics. He is considered one of the greatest mathematicians to have ever lived, and a European Space Agency’s spacecraft was even named in his honor, the Euclid Spacecraft.

Archimedes The Super Villain

by July 28, 2014

An ancient mathematician from the coastal city of Syracuse, Archimedes is largely considered one of the most prolific and most brilliant scientific minds of antiquity.
ArchimedesHis work focused on, but was not limited to, applying the concept of infinitesimals and the method of exhaustion to prove a number of geometrical theorems.

Notably, he was able to find the area of a circle, the surface area of a sphere, the area under a parabola, he accurately approximated the value of pi, and anticipated the creation of the study of calculus. He was, undoubtedly, a genius.

He might have also been a super villain. I don’t know. Maybe. I’m just saying.

That’s right dear reader. While Archimedes is largely remembered as a mathematician and a scientist, he was also a very crafty inventor. And it was reported that he amassed an array of some of the most unique and terrifying weapons of the ancient world. The Roman army might have had a superior army and an impressive navy, but the city of Syracuse had a “death ray” for crying out loud.
But now I’m getting ahead of myself. What were some of these ancient weapons of mass destruction? What exactly did Archimedes invent that would have put any Bond villain to shame?
Well, for starters, there were…

1. Long Range Catapults

All of the weapons that are mentioned in this article were said to have found use during the siege of General MarcellusSyracuse in 214 BCE.
It was the height of the Second Punic War and it was feared by the Roman Republic that the Kingdom of Syracuse might ally with their enemy, the Carthaginian Empire.

To prevent such a disaster, the Roman General Marcus Claudius Marcellus was ordered to lay siege to the city of Syracuse, breach its walls, and place the civilization firmly under Roman control. It was a task that might have been significantly easier if it were not for the deadly weapons of Archimedes.
When the Roman offensive was underway many of the invading ships fell victim to the city’s artillery. Archimedes had developed new and improved catapults with longer range and a heaver payload. The catapults were so effective that they were said to have caused massive Roman casualties even while the ships were still at a great distance from the city walls.

“But Archimedes had constructed artillery which could cover a whole variety of ranges, so that while the attacking ships were still at a distance he scored so many hits with his catapults and stone-throwers that he was able to cause them severe damage and harass their approach.” -Polybius (Universal Histories)

As if that were not bad enough, Archimedes also had prepared…
2. The Scorpions
For the ships who managed to somehow avoid the deadly artillery, there was now the task of scaling the city walls, which were “by no means easy to climb”. For those soldiers who did attempt to scale the battlements, they would be forced to contend with Archimedes’ next array of death machines; a series of projectile weapons known as “the scorpions”.

It was said that across the city wall, there were a series of holes that had been drilled through. These loopholes within the walls were said to have the breadth of a palms width. Behind these peepholes and within the city walls were stationed a number of archers with rows of the so-called “scorpions”.

A smaller catapult, or possibly a very large crossbow, the weapon discharged iron darts at the invading mariners. Deadly and impossible to counterattack, the projectile weapon would be the woe of General Marcellus. In the words of Polybius, the scorpion “put many mariners out of action”.
And if the ships still managed to out maneuver the long range artillery and the deadly “scorpions”, they still had to contend with…

3. The Claw of Archimedes
Sometimes referred to as “The Iron Hand” or “The Sinker of Ships”, this weapon was said to have been a massive grappling hook. It was said to have been dropped from the top of the city fortifications onto an enemy ship. From there, the claw would be lifted back up, bringing the vessel and the entire crew with it. The ship would be dashed against the rocks or Archimedes' Clawsimply capsized. The crew, weighted down by heavy armor, might have sunk beneath the waves and drowned.
The destruction that was unleaded by Archimedes claw was described by Plutarch when he writes…

“A ship was frequently lifted up to a great height in the air (a dreadful thing to behold), and was rolled to and fro, and kept swinging, until the mariners were all thrown out, when at length it was dashed against the rocks, or let fall.” -Plutarch (Parallel Lives:Marcellus)

I know what you are thinking. There is no way, right? Well, as it just so happens there is so a way. While this seems like a weapon that would have been impossible for ancient engineers to construct, the plausibility of the weapon was tested in 2005 by The Discovery Chanel in their special Super Weapons of the Ancient World.
Long story short, it is very plausible that such a weapon could have been constructed and then effectively used in wartime, especially when the designer was the brilliant Archimedes. Historians believe that the claw might have operated something like this.
Okay, so even if a ship could get past the artillery, dodge the deadly scorpions, and then not fall victim to the claw of Archimedes, they would still have to contend with an even deadlier weapon.
4. Archimedes’ Death Ray
Okay, truth be told, it probably wasn’t called a “death ray” in 218 BCE. However, it amounts to the same thing.
It is reported that by using a number of highly polished bronze or copper shields, the soldiers of Syracuse were able to redirect and focus the light of the sun onto enemy ships. Supposedly, the ships would burst into flames in a matter of seconds.

heat ray
“At last in an incredible manner he burned up the whole Roman fleet. For by tilting a kind of mirror toward the sun he concentrated the sun’s beam upon it; and owing to the thickness and smoothness of the mirror he ignited the air from this beam and kindled a great flame, the whole of which he directed upon the ships that lay at anchor in the path of the fire, until he consumed them all.” -Dio Cassius (Roman History)

Okay, you may have put your doubts aside to accept the claw of Archimedes, but surely the creation of a deadly heat ray in 200 BCE is pure myth, right?
The existence of such a weapon has been a matter of some debate over the centuries. Several scientists have attempted to recreate the machine with varying success.

In 1973 a Greek scientist by the name of Ioannis Sakkas carried out his own experiment in Skaramagas naval base outside of Athens. By using seventy copper coated mirrors, Sakkas redirected the suns rays toward a mock plywood ship some 200 feet away. After only a few seconds, the model boat burst into flames, proving that such a heat ray was, in fact, plausible. It might have looked something like this.
These weapons were said to have been so effective that the Roman army was eventually forced to call of their general offensive of the city. Having no real way to contend with the weapons of Archimedes, General Marcellus decided to take the city by means of siege.
By cutting off all trade routes in and out of the city, the Romans forced the city of Syracuse to consume all of their resources until they were eventually defeated by starvation.
The invading army stormed the city and took it with ease. It was said that General Marcellus had ordered that Archimedes be spared. However, such an order would not be followed.

Archimedes painting
The story goes that a Roman soldier came upon Archimedes in his home where the scientist was busying himself with his work. Flustered that somebody had interrupted him, Archimedes ordered the soldier to leave. The Roman either did not recognize Archimedes or realized that he was the man responsible for hundreds of Roman deaths. Either way, the story ends the same. Archimedes, now in his late eighties, was slain in his workshop by the invaders.

While all of Archimedes’ weapons are impressive, it is the heat ray more than any other that captures the imagination and makes one wonder whether Archimedes might have spent his evenings sitting in a war room somewhere, stroking a cat and secretly plotting world domination.
And if you think I am going too far with this whole Archimedes-could-have-been-a-Bond-villain thing, then I’m just going to point out that the James Bond franchise liked the idea of a heat ray so much that they actually created a movie where the main antagonist had, you guessed it, a heat ray. The film was “Die Another Day”. Go watch it and tell me that I’m wrong.
Despite my very humorous observations, there is absolutely no evidence that Archimedes ever had any intention to use his weapons for any other means than to protect his home of Syracuse. An inventor, a mathematician, a brilliant scientist, Archimedes is many things. However, I suppose we will have to conclude that he probably was not a super villain.

Pythagoras and the Revolution of Mathematics

by June 5, 2013

pythagoras bust


While Pythagoras has become known as one of the first revolutionaries in the field of philosophy and mathematics, surprisingly little is known about him, as he kept no writings of his own. All that we have learned about Pythagoras and his teachings has come to us indirectly through the writings of others.

What we do know about Pythagoras is that he was born in about 570 BCE. He visited Egypt as a young man and learned extensively about mathematical relations, proportions, and a rudimentary form of geometry. It is believed he studied within the Milesian school that had been founded by Thales a generation before. This school was devoted to uncovering the nature of the universe through objective, non mystical means; and while Pythagoras would not outright disagree with the ideas of Thales, he did take a different approach to exploring metaphysics. To Pythagoras, mathematics was the key. It was the form and order that existed in all things and through it the universe had come into being.

While in Egypt, Pythagoras had learned much about geometry, and he was taught that a triangle with ratios of 3:4:5 would always have  a right angle. While this was very useful in fields such as architecture, Pythagoras would go one step further. He uncovered the underlying equation that constituted this observation. Pythagoras was the original intellectual who discovered that the square of a triangle’s hypotenuse equals the sum of the squares of the other two sides.

The Father of Mathematics

This idea is more commonly known by the equation: A²+B²=C². That’s right, Pythagoras was the creator of everybody’s favorite equation from high school math class. This discovery was profound, revolutionary in fact. Never before had a thinker discovered this truth that has been shown to be universally true. This concept was so extraordinary that Pythagoras interpreted it as being divine.
While modern thinkers usually take the line of thought that objective, scientific thinking is contradictory to religion, Pythagoras took quite the opposite position. His foray into mathematics lead him to believe that numbers and mathematical relations held a particular divinity that could explain the universe. To Pythagoras, all the universe was constructed of mathematical relations, and so the cosmos themselves could be explained by the study of numbers.
Pythagoras hypothesized that before the universe there existed infinity with no limits or structure. God would then have created a limit upon all things so that infinity would be able to take on an actual size. In this way a measurable unity was created from which all things formed. The Pythagoreans held so much faith in numbers that they actually started applying characteristics to them. They believed that even numbers were “good” and odd numbers were “evil”. While Pythagoras might have gotten a bit carried away on that point, there is still much support to his claim that the natural world is governed by mathematical relations.

Pythagoras in “The School of Athens”

Pythagoras studied harmonics and learnt that much of music was governed by consistent mathematical ratios. It was said Pythagoras was given inspiration while passing a blacksmiths shop. One blacksmith had an anvil half the size of the other, and the sounds that were produced when the hammers struck were exactly an octave apart. Pythagoras experimented with musical strings and realised that there was a constant ratio between the sizes of strings that determined if they were harmonious when plucked together. This idea would become known as the harmonic series and would become very important in the fields of harmonics and even music.

Pythagoras gained quite a following during his time on earth. The Pythagoreans, as they would be known as, set up a structured community in Croton, southern Italy. This community of 300 people lived by a structured set of rules and standards. Pythagoras was the community leader and taught his ideas of mathematics and divine structure to all who would listen. This community was said to have verged on a cult, as people living together, worshiping numbers certainly would appear to be sect-like. In fact, it was told that this group of Pythagoreans were the subject of much hostility and were eventually run out of town. Pythagoras and his followers were eventually forced to settle in Metapontum, also in southern Italy.

“Number is the ruler of forms and ideas” -Pythagoras

Pythagoras remains one of the most influential thinkers of ancient times. His belief that the cosmos were governed by mathematical forms would be built upon by later astronomers, including the famous Galileo. Additionally, the idea that the natural world was built upon predictable ratios was used by English chemist John Newlands, who discovered that when chemical elements were ordered by atomic weight, those with similar properties occur at every eighth element. This would lead to our understanding of the periodic law of chemical elements that is still used today.
Pythagoras believed in reason above all else. He came to believe that all the universe was governed by mathematical laws. To him, the very structure of the cosmos, existence and even God was possible only through numbers. He was a man whose influence would be felt for centuries to come. He remains one of the fathers of mathematics and one of the most important of the ancient philosophers.