Ancient Greece
Ancient Greece was home toВ manyВ mathematicians, philosophers, sculptors, and, ofВ course, military leaders. It is also aВ place that gave democracy toВ the world.
G.1. Athens is the capital ofВ Greece
The history ofВ Athens dates back centuries. It is difficult toВ tell exactly who and when founded Athens. The period ofВ greatest prosperity ofВ Athens was the 5
century BC. Naming the exact number of population also proves to be a difficult task – the census was not conducted, and no one saw slaves as equal men: they were seen as property. Aside from slaves, there were wealthy citizens in Athens (slave owners, artisans), demos (poorer citizens), metics (migrants). Let’s assume the following: there were 90,000 slaves in Athens (historians believe this number to be 75—100 thousand) and settlers accounted for 20% of the population. There were as many citizens as slaves and metics combined. How many people lived in Athens?
G.2. Alexander the Great – a builder?
Let’s add together the number of cities founded by Alexander the Great[12 - Alexander III (Alexander the Great, 20/21 July 356 BC – 10 June 323 BC) of the Argead dynasty was a king and a military leader of the ancient Greek kingdom of Macedon, who created one of the largest empires in history that collapsed after his death.] and the number of modern countries where those cities are located. (Half the number of the cities) Then, add another 1/10 of the total number of cities and states, and we’ll get the number of years Alexander the Great lived. By the way, he used to name his cities Alexandria. 3/5 of the cities founded still exist, but have changed their names, all except for one. How many cities, founded by the King of Macedonia, have survived to this day? Alexander the Great was born in 356 BC and died in 323 BC.
G.3. How many students were in Pythagoras’s school?[13 - This is a very old task. The author found it necessary to mention in his book. He first came across it in a school math club a long time ago and then, in 2014, saw it in a History in Arithmetic in Science and Life.]
«Tell me, famous Pythagoras[14 - Pythagoras of Samos (c. 570 BC – c. 495 BC) was an ancient Ionian Greek philosopher and the eponymous founder of Pythagoreanism.], how many students attend your school and listen to your speeches?»
«I’ll tell you,» the philosopher replied. «Half of my students study mathematics, a fourth of them – music, and a seventh of them have taken a vow of silence. Additionally, there are three women.
How many students are there in Pythagoras’s school? They were also called «Pythagoreans».
G.4. The Battle of Gaugamela, and Darius III’s[15 - Darius III was the last Achaemenid King of Kings of Persia, reigning from 336 BC to his death in 330 BC.] miscalculation
Like many generals, Darius III had initially underestimated his enemy. Therefore, he hastily gathered war elephants, 200 chariots, infantry and cavalry and recklessly rushed into battle. The Macedonians immediately captured the elephants, and Alexander’s 7,000 horsemen and 40,000 infantry put the Persians to flight.
Indeed, some historians claim that Darius’s army outnumbered the Macedonians by ten to one. However, there are more conservative estimates. How many horses took part in the battle if a chariot was harnessed with two 2 horses, and Darius had 8 thousand more cavalry than Alexander had infantry? Remember that this event took place on October 1, 331 BC, after which the Achaemenid Empire ceased to exist.[16 - Let’s assume that the book was published in 2017.]
G.5. Calendars and sport
As you may have already guessed, we are talking about the Olympics. From the first Olympics the ancient Greeks began toВ count years using their calendar. Calculate inВ what year, according toВ the ancient Greek calendar, this book was published, if the first Olympics took place inВ 776В BC.
G.6. The Battle ofВ Thermopylae
The Battle ofВ Thermopylae took place inВ September 480В BC. This unique battle is notable for an almost 40-fold superiority ofВ the Persian forces. If the Greeks had 200В soldiers less, then, according toВ Herodotus[17 - Herodotus was an ancient Greek writer, geographer, and historian.], the Persian army would have been 40В times larger than the Greek. As you know, the Spartans, like the rest ofВ the Greeks, fought toВ the death. Only the betrayal ofВ aВ local resident allowed the Persians toВ attack the rear ofВ the Greek positions. Leonidas foresaw this and therefore sent part ofВ his army back. After 4,000В Greeks were killed and the Persians lost 20,000, the Persian army became 150В times larger than the Greek. How many soldiers were inВ the Persian army?
G.7. Pericles[18 - Pericles (c. 495 BC – 429 BC) was a Greek statesman and general during the Golden Age of Athens. One of the founders of the ancient democracy.], the father of Athenian democracy, or what did Pythagoras have to do with him?
In history, it is crucial to what came first: Ancient Rome or the Republic of Venice, Caesar or Dmitry Donskoy? After that, there should no confusion about who was born first: Archimedes, Pythagoras or Pericles. All three of them were Greeks and prominent personalities. Pythagoras was born first in 570 BC. He lived 5 years longer than Archimedes. Archimedes lived 10 years longer than Pericles. If we add up all the years they lived, we get 220 years. Pericles was born 4 years after the death of Pythagoras. How many years passed between Pericles’ death and Archimedes’ birth, if 358 years passed between the birth of Pythagoras and the death of Archimedes?
G.8. The Peloponnesian War or Sparta versus Athens
As usual, the Peloponnesian War started because Athens wanted the Spartans toВ be just like them, and the Spartans wanted the opposite. The war lasted from 431В toВ 404. BC. Historians distinguish two periods ofВ the war: the Archidamian War and the Deceleian War. Between them, there was the В«Peace ofВ NiciasВ» that lasted for six years. How long did the Archidamian War last, if it was aВ year longer than the DeceleianВ WarВ did?
G.9. The Parthenon as tremendous historical legacy
Dedicated to Athena, the Parthenon’s construction on the Acropolis[19 - An acropolis was the settlement of an upper part of an ancient Greek city, especially a citadel, and frequently a hill with precipitous sides, mainly chosen for purposes of defense.] lasted from 447 to 438 BC. Even though it was burned down and partially destroyed, it draws attention with its columns’ straight lines. To achieve this, ancient architects curved the straight lines and even tilted some of the columns around the temple’s perimeter. There are 9 more columns on the east side than on the south. In total, there are 16 columns on the south and north sides. How many columns are there along the temple’s perimeter?
G.10. The Battle ofВ MarathonВ or how the Marathon races wereВ born
On September 12, 490В BC, not far from Athens (42В km), aВ battle between the Persians and the Greeks took place. This was the first significant victory ofВ the Greeks over the Achaemenid Empire. The Persian forces were 160% larger than the Greek army. The Athenians made up 90% ofВ the Greek army. The Greeks won, and aВ messenger ran from Marathon toВ Athens. Upon his arrival inВ Athens, he fell dead inВ the square with the words, В«Rejoice, Athenians, we have won!В»[20 - Historians debate the truthfulness ofВ this event, as it was mentioned inВ the documents only five centuries later.] After his epic run, the distance he ran (42В km) became known as the В«MarathonВ».
How many Athenians were inВ the Greek army, if, inВ total, 36В thousand soldiers took part inВ the battle?
G.11. AВ little about Theseus[21 - Theseus was the mythical king and founder-hero ofВ Athens. He is sometimes described as the son ofВ Aegeus, King ofВ Athens, and sometimes as the son ofВ the god Poseidon.], Aegeus and the Minotaur[22 - InВ Greek mythology, the Minotaur is aВ mythical creature portrayed during classical antiquity with the head and tail ofВ aВ bull and the body ofВ aВ man. He dwelt at the center ofВ the Labyrinth and was eventually killed byВ the Athenian hero Theseus.]
Having defeated the Minotaur, Theseus, as depicted in the famous myth and cartoon, went home to Athens. Today, in Crete, a ferry from Heraklion, near the Palace of Knossos, runs to Athens in the evening at 21:00. It arrives at 6:00. Could Aegeus have seen the black sail of Theseus’ ship if Theseus sailed at the same time as the modern ferry? Theseus’ ship moved 6 times slower. The distance to Athens is 306 km.
AncientВ Rome
We could talk for hours about ancient Rome, and even today, many scholars still study the great history of the Roman Empire. We won’t go deeply into the details here. Instead, we’ll look at some tasks and think about the Empire’s main historical events.
THE FIRST PUNICВ WAR
The В«First Punic WarВ» is the war between Carthage and the Roman Republic. It started inВ 264В BC and lasted for 23В years ending with the victory ofВ the Romans.
R.1. Legions are longing for theВ sea
At the start of the First Punic War, Carthage reigned supreme as the dominant maritime power, and Roman light ships couldn’t inflict any significant damage to the Carthaginian quinqueremes[23 - A quinquereme is a boat with 5 rows of oars, up to 25 rowers in each row and 1 oar per rower. (Displacement – 200 tons, length – 70 m, width – 8 m). It could took up to 100 warriors.]. However, by an accident of fate, one quinquereme was thrown on land. The delighted Romans took the ship and carefully disassembled the vessel. They copied the design and constructed a mighty Roman fleet of 100 quinqueremes and 20 triremes[24 - A trireme is a boat with 3 rows of oars, up to 150 rowers, 12 sailors, and 80 warriors.].
How many: legions[25 - The Roman legion was the standard military unit of the Roman army. Its number changed over time. Let’s assume there were 4200 infantry without cavalry. It was composed of 30 maniples and they were composed of 2 centurias (60—120 men).], maniples,centurias the new fleet could carry?
R.2. Rowing is an honorable occupation
InВ the era ofВ the Roman Republic, rowers were highly respected. After the Roman fleet was built, rowers hired and trained tirelessly. How many rowers enrolled inВ the fleet? The size ofВ the fleet is mentioned inВ the previous task.
R.3. Toward each other
The distance between Carthage and Sicily[26 - Sicily is the largest island in the Mediterranean Sea.] was 213 km. The First Punic War was still going on, and a fast quinquereme sailed from Lilibey to Carthage with an urgent message. Meanwhile, a small fleet of various ships with reinforcements and provisions set sail for Sicily. How long will it take for the fast ship to meet the fleet if the fleet’s speed is 4 knots and the quinqireme’s is 6 knots? Sailors measure everything in miles. One mile is 1853 m, and a knot – measure of speed, – equals 1 mile per hour.
R.4. How many ships did disappear and where did theyВ go?
In ancient times they went, as a rule, to the seabed. There were several reasons why they sank. Firstly, they could have been damaged in battle, or, simply because of their poor state due to the crew’s negligence, could start leaking. However, the primary reason for ships’ disappearance was Neptune’s[27 - Neptune is the god of freshwater and the sea in Roman religion.] wrath or forces of nature, to put it in contemporary terms.
1,200 ships of the First Punic War didn’t escape this fate. Carthage lost five out of every twelve ships, while the Romans lost seven of every twelve. In total, who lost more ships and by how many?
The task can be dealt with inВ two ways, and, ofВ course, we might try toВ assess whether there were many ships inВ the ancient fleet compared toВ modern times.
THE SECOND PUNICВ WAR
The Second Punic War lasted since 218В toВ 201В BC inВ Italy, Spain and modern Tunisia. Many historians described the war, and many battles were included into military texts and may fascinate even today. For example, the eminent Roman historian Titus Livius[28 - Titus Livius was aВ Roman historian. He wrote aВ monumental history ofВ Rome and the Roman people, titled Ab Urbe Condita, From the Founding ofВ the City.] wrote aВ book about it called The War with Hannibal. 2,000В yearsВ ago.
R.5. The trophies ofВ Saguntum[29 - Today, it is aВ municipality ofВ Spain, Sagunto.]
The Second Punic war began inВ 219В BC when Hannibal[30 - Hannibal was aВ Carthaginian general. He is widely regarded as one ofВ the greatest military commanders inВ history.] sieged Saguntum. His victorious army captured 2,925В pounds ofВ silver.
Hannibal’s army consisted of Nubian horsemen, Spanish infantry, Carthaginian warriors, tribes from southern Spain, and Balearic slingers[31 - They were hired from the Balearic Islands in the Mediterranean Sea.]. The ratio of their numbers in the army was 1:2 2:1:0.5.
Hannibal made aВ just decision toВ divide everything equally between the soldiers. There were 1,000В Balearic slingers inВ the army. How many asses (ancient Roman currency) did each soldier get? One denarius was worthВ 10В asses, and one pound is 72В denarii.
R.6. Swift CarthaginianВ spy
The Roman consul decided to send several triremes to support the armies fighting in Spain, near Massalia, (modern Marseille in France). The fleet was not far from Pompeii[32 - Pompeii is an ancient city that was buried under volcanic ash and pumice in the eruption of Mount Vesuvius in AD 79.], and Gannon, a cunning Carthaginian spy, found out about this. To warn Hannibal’s army, he would need to travel faster than the Roman triremes. How quickly does Gannon need to travel by land if the distance between Pompeii and Massalia is 791 km by sea? By land, the spy needs to pass through Rome and Genoa. The distance between Pompeii and Rome is 173 km, from Rome to Genoa – 403 km, and from Genoa to Massalia – 346. The average speed of trireme is 5 km per hour. Well, one cannot always have a tailwind! The spy needs to sleep at least 6 hours a day. Otherwise, he might lose his vigilance and get captured by the valiant Praetorians[33 - The Praetorian Guard was a unit of the Imperial Roman army that served as personal bodyguards and intelligence agents for the Roman emperors.].
R.7. The first battle ofВ the Second PunicВ War
In November 218 BC, Hannibal’s troops approached the Ticinus River. Simultaneously, from Italy, Scipio’s[34 - Publius Cornelius Scipio was a general and statesman of the Roman Republic. He served as consul in 218 BC, the first year of the Second Punic War.] troops approached the same river. The first battle took place near this place. From Carthage’s side, only its cavalry took part in the battle. However, the Romans used both cavalry and light infantry, velites[35 - An infantry class in the Roman army.]. Hannibal had 6,000 cavalry units. From the Roman side, the cavalry was 100 units more than half of the Carthaginian forces. Moreover, the infantry was 20% more than Hannibal’s cavalry. In total, how many soldiers took part in the battle?
R.8. The Battle ofВ Trebbia or how toВ win allies
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notes
Примечания
1
Khufu was the second pharaoh of the Fourth Dynasty, in the Old Kingdom period. (2589—2566 BC or 2551—2528 BC)
