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Beware the Ides of March

by March 15, 2018

“Beware the Ides of March.”
You may here that phrase today because the 15th of March is referred to as the ‘Ides of March’ and marks the anniversary of the assassination of Julius Ceasar in 44 BC.
Bust of Caesar

Bust of Caesar

Gaius Julius Caesar was a Roman general, Consul, statesman, and notable author of Latin prose. He was both a conquering hero and a dictator. He played an essential role in the history of Ancient Rome, acting out pivotal parts in events that led to the demise of the Republic and the rise of the Empire.
Caesar started off as an accomplished military man, fighting for the glory of Rome. He was able to extend Roman territory to the English channel and the Rhine in his conquest of Gaul, completed by 51 BC.
He became the first Roman general to invade Britain.
His achievements awarded him the position of an unmatched military prowess, but also threatened to eclipse the role of Pompey, the military and political leader of the late Roman Republic. Pompey, who had previously held an alliance with Caesar and Crassus, had realigned himself with the senate after Crassus’ death in 53 BC.
When the Gallic wars had finished, the Senate ordered Caesar to lay down his weapons and commanded him to return to Rome. Caesar, however, refused. In 49 BC he crossed the Rubicon with a legion. This was the moment that marked his defiance; he had left his province and illegally entered Roman territory, bearing arms. A civil war ensued, but Caesar emerged as the unrivaled leader of Rome.
Caesar assumed control of the government and then proceeded to install a program of social and governmental reforms, such as the creation of the Julian Calendar. He centralised the bureaucracy of the Republic and eventually was proclaimed “dictator in perpetuity”.
Caesar, however, was not popular with everyone – especially the politicos he had ignored.
On March 15th, 44 BC, the Ides of March, Caesar was stabbed to death at a senate meeting.
According to Plutarch, Caesar had been told that this would come to pass. A seer had warned Caesar that harm would come to him, no later than the Ides on March. Then on that fateful day, Caesar passed the prophesier on his walk to the Theatre of Pompey, the place were he would be murdered. He quipped, “The ides of March have come”, thinking that the morbid prophecy had not been fulfilled.
To this the seer replied, “Aye, Caesar; but not gone.”
Beware the Ides of March
It is thought that as many as 60 conspirators were involved in the assassination, led by Brutus and Cassius. This scene, as dramatized by William Shakespeare has given us the famous lines, “Beware the Ides of March” and “Et tu, Brute?”
Whether the date was, in fact, the 15th of March is up to debate, as the Roman calendar was structured differently from our modern calendars. For one thing, they only had 10 months. Additionally, the Romans did not number the days of the month sequentially from the first to the last. They actually counted back from three fixed points with in the month and varied depending on the length of the month. These included the Nones (5th or 7th), the Ides (13th or 15th), and the Kalends (1st of the following month).
The Ides occurred on the 15th for March, May, July and October, and were supposed to be determined by the full moon. (This reflects the lunar origins of the Roman calendar).

It’s Pi Time!

by March 14, 2018

The history of pie is both storied and interesting in and of itself. Did you know, for instance, that the Ancient Greeks are thought to have originated the pie pastry, as can be seen in Aristophanes’ plays (5th century BC), where there are mentions of sweetmeats including small pastries filled with fruit? Neither did I until today.
But that’s not what we are talking about, delicious as it sounds.
Instead, we are delving into a different pi, though it also shares some of its origins in Ancient Greece.
Yes, dear reader, we are investigating the captivating phenomena of a mathematical constant, the algorithm of which was devised by the brilliant (and perhaps evil) Archimedes around 250 BC.
While math may not peak your interest initially, the history and discovery of this unique irrational number should. And if nothing else, it might take you back to your old school days.
Think of it as a mathematical madeleine moment.
But first, let’s go over what exactly Pi is.
This curious number (of which the nerdist of memory masters like to repeat) was originally defined as the ratio of a circle’s circumference to its diameter. In other words, pi equals the circumference divided by the diameter (π = c/d). Conversely, the circumference is equal to pi times the diameter (c = πd). No matter how large or small a circle is, pi will always work out to be the same number.
Following me still? Well, it’s going to get tricker for a moment.
π is also an irrational number, and no, not like your mother in law. This means it can never be written as a fraction of two whole numbers, and it does not have a terminating or repeating decimal expansion. Essentially, there is no exact value, seeing as the number does not end. The decimal expansion of π goes on forever, never showing any repeating pattern. I’ll say it again for emphasis – it never repeats and it goes on forever.
I’ll just let the magnitude of that sink in for a moment.
Pi gif

Animation of the act of unrolling circumference of a circle having diameter 2, illustrating the ratio π.

Now, since π is irrational, all we can ever hope to do is get better and better decimal approximations… which is where the magnificent, albeit ultra nerdy, competition of reciting known numbers comes in.
Why Pi? Why not cake?
Good question, dear reader – I can tell you are hungry…. For knowledge. (waaa waaa)
Pi is called Pi because the 16th letter of the Greek alphabet, π (Pi), was employed as an abbreviation of the Greek word for periphery (περιφέρεια) which means circumference.
That was easy enough! Now, onto the history of this minxy little number.
Pi actually goes much further back that Ancient Greece. In fact, several ancient civilizations came up with fairly accurate values for π, including the Egyptians and Babylonians, both within one percent of the true value.
In Babylon, a clay tablet dated 1900–1600 BC has a geometrical statement that, by implication, treats π as 25/8 = 3.125. In Egypt, the Rhind Papyrus, dated around 1650 BC but copied from a document dated to 1850 BC, has a formula for the area of a circle that treats π as (16/9)2 ≈ 3.1605.
There is even a biblical verse where it appears pi was approximated:
And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it about. — I Kings 7:23 (King James Version)
Then, around 250 BC the Greek mathematician Archimedes created the first ever algorithm for calculating it. His system was so popular, it dominated the math scene for over 1,000 years, and as a result, π is sometimes referred to as “Archimedes’ constant”.
Around 150 AD, Greek-Roman scientist Ptolemy, in his Almagest, gave a value for π of 3.1416, which he may have obtained from Archimedes or from Apollonius of Perga.

Engraving of a crowned Ptolemy being guided by the muse Astronomy, from Margarita Philosophica by Gregor Reisch, 1508.

Later on in 480 AD, Chinese mathematicians used geometrical techniques to approximate to seven digits and Indian mathematics made it to about five in the 5th century AD. The historically first exact formula for π, based on infinite series, was not available until a millennium later, when in the 14th century the Madhava–Leibniz series was discovered once again in India.
Today we use algorithms based on the idea of infinite series from calculus, and our ever-faster computers allow us to find trillions of digits of π.
Now, while all this history is no doubt fascinating, these pages are dedicated to the Greeks and Romans! So let’s go back to Archimedes and his revolutionary technique.
First off, if you don’t know much about Archimedes, you should…and not only because he was likely a evil super-mind.

Archimedes Thoughtful by Fetti (1620)

Born in Syracuse on the island of Sicily in 287 BC, Archimedes was a Greek mathematician, scientist, mechanical engineer and inventor who is considered one of the greatest mathematicians of the ancient world. Among his many epithets, he is considered the father of simple machines, as he introduced the concept of the lever, the compound pulley, as well as inventions ranging from water clocks to the famous Archimedes screw.
He also designed devices to be used in warfare such as the catapult, the iron hand, and the death ray. In fact, Archimedes was one of the world’s first mathematical physicists whose inventions were actually applied to the physical world. (And there are fantastic re-enactments of these cruel creations. As seen here and here).

So it should be no surprise that such a mind was capable of figuring out the humble circle!

But how did it do it?

Archimedes used the Pythagorean Theorem to find the areas of two polygons. He then approximated the area of a circle based on the area of a regular polygon inscribed within the circle and the area of a regular polygon within which the circle was circumscribed. The polygons, as Archimedes mapped them, gave the upper and lower bounds for the area of a circle, and he approximated that pi is between 3 1/7 and 3 10/71.
Wait…. what? Let’s go over that again, but with pictures.
Basically, Archimedes observed that polygons drawn inside and outside a circle would have perimeters somewhat close to the circumference of the circle.
As described in Jorg Arndt and Cristoph Haenel’s book Pi Unleashed, Archimedes started with hexagons:
We start with a circle of diameter equal to one, so that, by definition, its circumference will equal π. Using some basic geometry and trigonometry, Archimedes observed that the length of each of the sides of the inscribed blue hexagon would be 1/2, and the lengths of the sides of the circumscribed red hexagon would be 1/√3.
The perimeter of the inscribed blue hexagon has to be smaller than the circumference of the circle, since the hexagon fits entirely inside the circle. The six sides of the hexagon all have length 1/2, so this perimeter is 6 × 1/2 = 3.
Similarly, the circumference of the circle has to be less than the perimeter of the circumscribed red hexagon, and this perimeter is 6 × 1/√3, which is about 3.46.

This gives us the inequalities 3 <π < 3.46, already moving us closer to 3.14. Archimedes, through some further clever geometry, figured out how to estimate the perimeters for polygons with twice as many sides. He went from a 6-sided polygon, to a 12-sided polygon, to a 24-sided polygon, to a 48-sided polygon, and ended up with a 96-sided polygon. This final estimate gave a range for π between 3.1408 and 3.1428, which is accurate to two places.

If I haven’t lost you, this was a revolutionary method, one that differed from the earlier approximations in a fundamental way.
Previously, the number was calculated in a very approximate way, usually by simply comparing the area or perimeter of a certain polygon with a circle.
Archimedes’ technique was new as it was an iterative process, one where you can get a more accurate approximation by repeating the process, using the previous calculation of pi to obtain a new, more correct number.
Now, some consider the celebration of Pi and its forever, never repeating decimals as an overblown party. But, we here at Classical Wisdom beg to differ. Any day that makes people stop and appreciate math and its marvels is certainly worthwhile… and I think we all know the late, great Stephen Hawking would agree, may he rest in peace.

Thucydides: Scientific Historian and Political Realist

by March 8, 2018

By Ben Potter
424 BC: Seven raging years after the start of the Peloponnesian War. Seven years of Greek on Greek, sword on sword, blood on blood. Seven years which have brought pain and pride to Spartans and Athenians alike.
Now is the time for a great man to come to the fore, to turn the screw, to be a hero.
The venue? The city of Amphipolis (literally ‘around the city’) up in the Thracian heartland. The players? The Spartan commander Brasidas and the newly elected Athenian strategos (general), Thucydides.
Ancient Siege

Ancient Siege

Brasidas has assaulted Amphipolis and is attempting to negotiate with its people. Thucydides marches his troops on the city knowing he has the upper-hand. After all, this is his territory.
Though an Athenian, the wealthy and aristocratic Thucydides owns land and gold mines in this area of Northern Greece. He can thus exert considerable influence over the local populace.
So he and his men arrive prepared for battle, ready to whet their double-edged xiphos swords on the briny blood of ignoble Spartans.
But…. disaster strikes. The perfidious Brasidas has talked the Amphipolitans round with terms of moderation.
The city is lost; Thucydides is disgraced.
His voted honour of strategos is stripped from him, as are all his rights of citizenship. He is cast out to wander a lonely and forlorn figure, branded forever with the stigma of exile.
However, it’s hard to keep a good man down….
Harder too if a man is independently wealthy, well-educated and related, not only to Miltiades and Cimon, but also to Thracian royalty.
He was also blessed with a robust constitution. Despite falling ill, Thucydides survived the great Athenian plague (430 – 426 BC). He commented on this truly catastrophic event in his ‘History of the Peloponnesian War’:

“As the disaster passed all bounds, men, not knowing what was to become of them, became equally contemptuous of the gods’ property and the gods’ dues.”

Athenian Plague

The Plague of Athens, Michiel Sweerts, c. 1652-1654

Throughout the war (431- 404 BC), Thucydides had been making copious notes and recording important speeches in order to write his History.
Now, unfettered from partisanship, exile gave him the freedom to travel extensively and unmolested. Not only because he was no longer occupied as a solider, but because he was not viewed as an enemy by any state. Instead he became that strangest of creatures; an ex-patriot expatriate.
The father of ‘scientific history’ and ‘political realism’ originally embarked upon his writing project as he had been able to augur the magnitude of the war from its outset.
Thucydides was well aware, much like a muddied, bloodied and bewildered soldier at Ypres would have been, that he was living through a time of unusually powerful danger and destruction.
Thucydides considered the war to be an event, not a time-frame and tackled it thus.
He considered the history of the war unique unto itself; hermetically sealed away from entertaining trivialities like art, literature and society.
The great man himself said:
“To hear this history rehearsed, for that there be inserted in it no fables, shall be perhaps not delightful. But he that desires to look into the truth of things done, and which (according to the condition of humanity) may be done again, or at least their like, shall find enough herein to make him think it profitable. And it is compiled rather for an everlasting possession than to be rehearsed for a prize.”
Bust of Thucydides

The historian, Thucydides

Thucydides, though with the noblest of intentions, would be considered by modern standards something of a hack. He was well-intentioned and capable, but never enlightens with the scholarly cut and thrust of Polybius.
Regardless of his bias for the politician Pericles (or against Cleon) there is one overriding problem with the text. Buckley put it succinctly: “The thorniest problem in using Thucydides as a reliable historical source concerns the authenticity of his speeches”.
Thucydides himself reinforces this:
“It has been difficult for me to remember the exact words that were spoken in the speeches that I myself heard, and for those who brought me reports of other speeches. Therefore it has been my method to record speeches which I thought were the most appropriate for each speaker to give in each situation, while keeping as close as possible to the general sense of what was actually said”.
However, quarter must be given as, at the time of Thucydides, historiography is in its infancy, still being fired in the crucible of time.
As Terry Buckley concludes: “Thucydides is by far the best of our literary sources and where there is a direct conflict in the evidence supplied by him and by other historians, his version is to be preferred”.
His usefulness and legacy outstrip those of all his contemporaries. If, for no other reason, than the fact that his work increased accountability – it let leaders know that their blunders wouldn’t be lost to the ages, but read, reread and analysed, potentially to their detriment.
Regarding the outcome of the Peloponnesian War, Thucydides claims it pivoted on the fate of the perverted genius Alcibiades who was also forced into exile: “they personally objected to his private habits; and so speedily shipwrecked the state”.
Thus the destiny of the war may have been decided by hot-headed voters exiling their brightest and their best as a consequence of short-term outrage.
Thucydides, despite his supplies of gold was no mere member of the idle-rich, but a dynamic, wealthy warrior and thinker, who the Athenians were also foolish enough to turn away from their society.
It is because of Thucydides’ tenacity and foresight that his wisdom, and the folly of the Athenian demos, live on.

The Lost Poetess

by March 1, 2018

Classical Wisdom‘s First ever Webinar, The Lost Poetess, is available for a limited time. Click below to register and watch for free.
The Lost Poetess
by Anya Leonard, Co-Founder of Classical Wisdom
Who was the Lost Poetess?
Considered equal to Homer and praised by Plato (who didn’t even like poetry!), this ancient poet has almost completely disappeared. We’ll look at who she is, what happened to her work, and whether she was really worth the ancient hype.
The Webinar The Lost Poetess was presented by Classical Wisdom in conjunction with Ancient Origins.

The Theory of Recollection: Immortal Soul Required

by March 1, 2018

By Ben Potter
The Phaedo takes places in 399 BC at the scene of the final days of Socrates’ life. The dialogue is primarily an argument for the immortality of the soul that Socrates is trying to convince his grief-ridden colleagues, and maybe indeed himself, of in order to prove that his execution is merely the separation of his soul from his body… and not his actual ‘death’.
Death of Socrates

The Death of Socrates, by Jacques-Louis David, 1787

But how does Plato/Socrates prove that there is an immortal soul? It’s not an easy task, no doubt, and so he employs the idea of recollection (or anamnesis). However, like with most of Plato’s concepts when fully investigated, it reveals far more than what is immediately obvious.
The theory of anamnesis was, in fact, first introduced in one of Plato’s earlier works, the Meno. In this dialogue, Socrates informs Meno that nothing can be either taught or learnt as we already possess all the knowledge in the world. Socrates explains that, through the lifetime of our soul, we have already learnt all there is to learn and that we can answer every question, provided we are asked in the correct manner.
He goes on to prove this by getting an uneducated slave to figure out a math problem by asking him a series of extremely leading questions. ie. “Is your personal opinion that the square on the diagonal of the original square is double its area?” Socrates seems convinced that he has done nothing to ‘educate’ the slave, but merely asked him the appropriate questions that allowed him to recollect.
This argument for recollection is taken a step further in the Phaedo, as Plato claims there are two aspects of recollection. The first involves no lapse of time and is less a recollection of something, but more a reminder of it: “you know what happens to lovers, whenever they see a lyre or cloak or anything else their loves are accustomed to use: they recognize the lyre, and they get in their mind, don’t they, the form of the boy whose lyre it is?”
The second aspect of recollection is one that does involve the lapse of time and is more familiar to the theory of recollection in the Meno. Additionally, it relates to Socrates’ goal of establishing the immortality of the soul. The argument that he lays out is that we are neither capable of learning anything new, nor were we born with the knowledge of things, but that we knew these things before our birth.
Bust of Plato

Bust of Plato

But before we proceed with the Theory of Recollection, we must first examine Plato’s Theory of Forms. As many will no doubt recall, Plato believed that the Forms were ethereal entities of extremely general terms, ie. sameness, difference, justice, purity, vice, beauty, etc. The reason these things were entities, rather than concepts, was due to the fact that Plato perceived them as something very real indeed, even though it seems they were invisible… at least to our eyes.
Returning to our foremost theory, Plato uses the Form of ‘equality’ to try and transmit his views on recollection. He states that in viewing two sticks of equal length, we recognize that they posses ‘equality’. However, he also makes clear that two sticks of unequal length can also cause us to recognize ‘equality’ by its absence. He also conveys that even what appears to be perfectly equal, can, in fact, fall short of ‘equality’, for the simple reason that only the Form of ‘equality’ can be truly, purely equal.
So how do these ideas bring us to the primary aim of the dialogue, the immortality of the soul?
Essentially, in order for the theory of recollection to work, our souls would have had to exist before our earthly incarnation, as well as go on existing after it. Additionally, if the soul is immortal then it must also be eternal, because if something can never come to an end, then it must never have had a beginning in the first place.
The fact that we can identify ‘equality’ (or any Form) is due to the fact that we have experienced the true Form ‘equality’ during a time when our souls were apart from our bodies and at one with (or at least closer to) the Forms.
Socrates portrait

Bust of Socrates

So, when we see double yellow lines, we can recognize the equality that they posses by recollecting the Form ‘equality’ and concluding they are the same length, width and distance apart. The conclusion then is that because we can recognize/remember the Form of ‘equality’, our soul existed before our bodies, and consequently it will exist afterwards. Hence, the soul is immortal.
Interestingly, despite the willingness of Plato to change his opinions throughout his works, the Theory of Recollection seems to be the one he particularly cares to develop, rather than disregard. What is more of a passing thought in the Meno becomes an intrinsic part of his dialogues in the Phaedo. The elaboration of the concept almost appears to be a consequence of Plato himself re-reading the Meno in search of inspiration.
That said and despite Plato’s inclinations to put words into Socrates’ mouth, the concept of recollection might actually be one we can assign to the older thinker. Cebes, in fact, comments to Socrates in the Phaedo, ‘there’s also that theory you’re always putting forward, that our learning is actually nothing but recollection.”
Although the idea of recollection is vital for the Phaedo, the Phaedo itself is not purely a dialogue about recollection, but about the soul’s immortality. As it was obviously written after the death of Socrates, it could be Plato’s attempt to not only convince the philosophic community that Socrates, and his great mind, lives on, but also his endeavor to make one of the great theories of his friend and mentor persevere throughout time. And maybe that’s how Socrates really achieved his immortality…

You can read Plato’s Phaedo for yourself for free here:

Incredible Construction: Greek Acropolis Built by Ancient Engineers to Resist Earthquakes

by February 21, 2018

By Liz Leafloor, Contributing Writer, Ancient Origins
Throughout its 2,500-year history, the ancient ruins of the Acropolis in Greece managed to survive many earthquakes where other, more modern constructions have fallen. How is this possible? Experts now conclude it comes down to skillful construction and accomplished engineering.
Kyriazis Pitilakis, Professor of Civil Engineering Department of the Aristotle University of Thessaloniki tells news site Greek Reporter, “This is an incredible construction, using ingenious solutions to insurmountable engineering and construction problems.”
Scientists and engineers, puzzled by how the ancient buildings survived the many regional earthquakes, examined the construction of the famous Parthenon and the Athenian citadel in total. Based on their findings, they concluded that the buildings were designed specifically in order to be protected from earthquakes.
At a workshop on “Contemporary Interventions in the Athenian Acropolis Monuments” organized by the Department of Civil Engineering, Pitilakis said “The modular columns, other than the fact that they were made to be constructed and transported more easily, they are designed so that they have excellent seismic performance properties.” In effect, the columns were built to withstand earthquakes.

The Erechtheion, ancient Greek temple at the Acropolis of Athens. It was dedicated to Poseidon and Athena. Juan Manuel Caicedo/ Flickr

It would seem the ancient engineers knew what they were doing in terms of ensuring their creations would last, which is part of the reason we still see them gracing the high, rocky outcrop in Athens.
The Acropolis of Athens, proclaimed the “preeminent monument” on the European Cultural Heritage list of monuments, is a sprawling citadel composed of many structures, including the famous Parthenon. Evidence suggests the site was inhabited as early as the fourth millennium B.C., and it has suffered damage due to wars and fire in its long history. Incredibly, the Parthenon was being used to store gunpowder, and a cannonball strike caused a blast that severely damaged the structure in 1687.

Famous ancient Greek structure, the Parthenon at night. Andreas Kontokanis/ Flickr


The Acropolis of Athens, as seen from Philopappou Hill. A. Savin/Wikimedia Commons

Pitilakis explained the importance of the enduring historic site in Athens, saying “The Parthenon condenses all that Greece is and all that it has offered to the Western World in the best way. It stands as a symbol of European culture, a symbol of the principle of measure, of art, technology and human capability. This is because that other than the highest artistic creation, it is also a marvel of mechanical engineering.”
Efforts began in 1964 by the Greek government in restoring the Acropolis, and teams of archaeologists, architects, civil engineers and chemists work to preserve the important cultural and historical icon.
The World Cultural Council writes of the award-winning restoration by the Athens Acropolis Preservation Group of Greece, observing, “Most of the marble comes from the Greek islands, where there is an age-old tradition of working marble; they now continue the work of their ancestors, using the same methods and same tools, not indeed to create but rather to save a masterpiece which belongs not only to the Greeks but to all humanity.”