History and math. Нand in hand. Book 1. Ancient World. 50 mathematical tasks for students based on historical events. Ancient Rome, Greece, Egypt and Persia

History and math. Нand in hand

Book 1. Ancient World. 50 mathematical tasks for students based on historical events. Ancient Rome, Greece, Egypt and Persia

Dmitry Moskovets

Дмитрий Московец

Translator Vasily Ermolin

Translator Dmitry Moskovets

© Dmitry Moskovets, 2024

© Дмитрий Московец, 2024

© Vasily Ermolin, translation, 2024

© Dmitry Moskovets, translation, 2024

ISBN 978-5-0059-0760-8 (т. 1)

ISBN 978-5-0059-0761-5

Created with Ridero smart publishing system

Dedicated to my father Vladimir Nikolaevich Moskovets, who loved to teach me the history of the Ancient World and discuss it in the evenings of the now distant 80s…

Introduction

Let’s remember the tasks that we all had at school. Sometimes they were interesting, sometimes tedious. Someone was always travelling from point A to point B. Why? There was no answer. Still, they always reached their destination. Time flew, and we always forgot about this brave traveller. The purpose of the book and the tasks inside is to make students remember such travellers, and, while doing mathematics, also learn about history.

Some of the characters are fictional, but the larger part really existed and left a mark in history (their names can be found on the list at the end of the book).

Just like the events: the real ones have a date next to them.

A list of the used terms is located at the end of the book. It goes along with a list of all historical personalities mentioned in the book. All the tasks (and more) can be found on www.mathistory. xyz.

Ancient Egypt

Ancient Egypt still beckons tourists with its amazing pyramids, mystical Sphinx, mysterious hieroglyphs…

E.1.Time scares everyone and fears pyramids

The largest pyramid in Egypt (probably, in the world) is the Great Pyramid of Giza (the tomb of Fourth Dynasty pharaoh Khufu1).

Scientists still have not reached a consensus and can only guess how and when it was erected. In Egypt, the official celebrated date of the start of the construction is August 26, 2540 BC. It is believed that the construction took 20 years. How many years has the pyramid been standing if the current date is 2017?

E.2. The first peace treaty

The first peace treaty was made before Christ by the great Pharaoh Ramesses II2 with the Hittite3 King. If Ramesses II had been born 9 years later, then by doubling the date of his birth, we would have gotten the date of the start of the construction of the pyramids (see the previous task). When did Ramesses II die, if, according to some sources, he lived for 66 years?

E.3. Gardens and cubits – what do they have in common?

After the war with the Nubian tribes, the Amenhotinox – one of the Egyptian military leaders, – gave his squire Hannumat with a small garden measuring 45 by 45 cubits4.

Can you count how many carrots he planted in the garden, if each cubit stood for 45 carrots? (Some may wonder why this problem is here, but it helps to practice mental counting. All actions can be done in mind using simple techniques)

E.4. What came first – Alexandria or Rome?

Certainly, Rome was founded earlier, the official date of its foundation is considered to be 753 BC. Alexandria was founded by Alexander the Great5 in Egypt nine years prior to his death. Both cities still exist. Determine which year Rome was twice as old as Alexandria.

E.5. Stadium and the Pyramids

Can pyramids be used for sports? The pharaohs hardly would have liked it.

Calculate how many laps one needs to make around the Great Pyramid of Giza to run a marathon distance of 42 km (see more about the marathon distance in task G. 10). The length of the pyramid’s side is 440 royal cubits6.

E.6. On a trip along the river Nile

At one time Thebes used to be the capital of Upper Egypt. The Pharaoh set in a boat downstream the Nile. Half the time servants rowed at 4 km/h, then a quarter of the time the passengers sailed at 6 km/h, the rest of the way the boat moved under sail and oars at 8km/h. Finally, the Pharaoh arrived to Memphis7, having travelled 726 km along the Nile. How many kilometers did the boat go under oars, if the daytime lasted for 12 hours?

Ancient Persia

Ancient Persia (or the «Achaemenid Empire) was a state that, at its height, stretched from Libya and Greece to India. According to some historians, the population of this ancient Empire reached 50 million: almost half the population of the entire planet at that time.

P.1.Grass tribute

The Persian Empire in 521 BC was divided into 20 satrapi8 and each of them paid tribute. However, there were areas that were not satrapies, but still had to pay tribute. For example, the Arabs in the territory of modern Jordan sent 1,000 talents9 of thyme creeper annually. Dried plants were brought to the Empire.

How many donkeys are needed daily to transport all the thyme for drying if there is 15% of moisture in dried thyme and 95% in newly harvested? One donkey transports a load of 5 talents once a day.

P.2. Gold and silver

Of course, a tribute in grass is an exotic type of tribute and, in fact, most of the tribute was paid in gold or silver. The greatest tribute in silver was paid by Babylonia and Assyria. The amount was 1,000 silver talents. Besides horses, grass and other «non-metal» tributes, Babylonia and Assyria paid 1/7 of the entire tribute in silver.

How many gold talents did India pay, if the value of silver was 3/4 of the tribute, and the gold was paid only by India. The ratio of silver to gold at that time was 1:10. (i.e., 1 gold coin was worth 10 silver coins, equal in weight). Determine the weight of all the gold tribute by India in kilograms. 1 Persian gold talent equals 1 silver in weight: 33.65 kg.

P.3. The population of the planet

Some historians suggest that at the height the Achaemenid Empire half the world population lived there. The most numerous at that time was the Indian nation, and 1/7 of the Indians was part of Achaemenid Empire. How many people lived outside India and Persia at that time, if the number of Hindus in the Achaemenid Empire was 10%? (Hint: read the beginning of the chapter).

P.4. Army size

The most important battles in the history of Ancient Persia are the Battle of Thermopylae, in September 480 BC, in which Leonidas10 and his 300 fearless Spartans went down in history for centuries, and the Battle of Gaugamela on October 1, 331 BC, after which the Achaemenid Empire ceased to exist, conquered by young Alexander the Great (see more in task D.4). In both cases, the Persians had huge armies: both armies added together constituted 0.9% of the total population of the Empire. How many Persian soldiers took part in the Battle of Gaugamela, if we know that there were 1/4 more of them than in the Battle of Thermopylae? The population of the Empire equaled 50 million.

P.5. The King`s «eyes» and «ears»

A whole army of spies worked for Cyras II the Great11. They were keeping an eye on everything that was going on in the Empire. Some even believe that the security servicemen lined up in a chain from the capital to the outskirts and shouted the latest news to the next in the chain.

It took thirty days for an army to march from the border of the Empire, but the king could know it was approaching in just one day because of his line of spies.

See whether this is possible if the army moves 8 hours a day, the king`s «ears» stand every 200m and the speed of sound 330 m/s. (for convenience sake, let it be 300 m/s). The speed is 5 km/h.

Let’s assume that it takes 10 second to convey the message from one spy to another.

P.6. Anniversary of Cyrus II.

Have you ever attended the king’s anniversary? No? But they are still held. Iran, for example, celebrated the 2,500 anniversary of the death of King Cyrus II in 1971. How long did Cyrus live, if he came to the throme in 559 BC, when he was 31 years old?

Ancient Greece

Ancient Greece was home to many mathematicians, philosophers, sculptors, and, of course, military leaders. It is also a place that gave democracy to the world.

G.1. Athens is the capital of Greece

The history of Athens dates back centuries. It is difficult to tell exactly who and when founded Athens. The period of greatest prosperity of Athens was the 5th 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 Great12 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

«Tell me, famous Pythagoras14, 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’s15 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

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 to count years using their calendar. Calculate in what year, according to the ancient Greek calendar, this book was published, if the first Olympics took place in 776 BC.

G.6. The Battle of Thermopylae

The Battle of Thermopylae took place in September 480 BC. This unique battle is notable for an almost 40-fold superiority of the Persian forces. If the Greeks had 200 soldiers less, then, according to Herodotus17, the Persian army would have been 40 times larger than the Greek. As you know, the Spartans, like the rest of the Greeks, fought to the death. Only the betrayal of a local resident allowed the Persians to attack the rear of the Greek positions. Leonidas foresaw this and therefore sent part of his army back. After 4,000 Greeks were killed and the Persians lost 20,000, the Persian army became 150 times larger than the Greek. How many soldiers were in the Persian army?

G.7. Pericles18, 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 to be just like them, and the Spartans wanted the opposite. The war lasted from 431 to 404. BC. Historians distinguish two periods of the war: the Archidamian War and the Deceleian War. Between them, there was the «Peace of Nicias» that lasted for six years. How long did the Archidamian War last, if it was a year longer than the Deceleian War did?

G.9. The Parthenon as tremendous historical legacy

Dedicated to Athena, the Parthenon’s construction on the Acropolis19 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 of Marathon or how the Marathon races were born

On September 12, 490 BC, not far from Athens (42 km), a battle between the Persians and the Greeks took place. This was the first significant victory of the Greeks over the Achaemenid Empire. The Persian forces were 160% larger than the Greek army. The Athenians made up 90% of the Greek army. The Greeks won, and a messenger ran from Marathon to Athens. Upon his arrival in Athens, he fell dead in the square with the words, «Rejoice, Athenians, we have won!»20 After his epic run, the distance he ran (42 km) became known as the «Marathon».

How many Athenians were in the Greek army, if, in total, 36 thousand soldiers took part in the battle?

G.11. A little about Theseus21, Aegeus and the Minotaur22

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.

Ancient 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 PUNIC WAR

The «First Punic War» is the war between Carthage and the Roman Republic. It started in 264 BC and lasted for 23 years ending with the victory of the Romans.

R.1. Legions are longing for the 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 quinqueremes23. 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 triremes24.

How many: legions25, maniples, centurias the new fleet could carry?

R.2. Rowing is an honorable occupation

In the era of the Roman Republic, rowers were highly respected. After the Roman fleet was built, rowers hired and trained tirelessly. How many rowers enrolled in the fleet? The size of the fleet is mentioned in the previous task.

R.3. Toward each other

The distance between Carthage and Sicily26 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 they 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’s27 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 in two ways, and, of course, we might try to assess whether there were many ships in the ancient fleet compared to modern times.

THE SECOND PUNIC WAR

The Second Punic War lasted since 218 to 201 BC in 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 Livius28 wrote a book about it called The War with Hannibal. 2,000 years ago.

R.5. The trophies of Saguntum29

The Second Punic war began in 219 BC when Hannibal30 sieged Saguntum. His victorious army captured 2,925 pounds of silver.

Hannibal’s army consisted of Nubian horsemen, Spanish infantry, Carthaginian warriors, tribes from southern Spain, and Balearic slingers31. The ratio of their numbers in the army was 1:2 2:1:0.5.

Hannibal made a just decision to divide everything equally between the soldiers. There were 1,000 Balearic slingers in the army. How many asses (ancient Roman currency) did each soldier get? One denarius was worth 10 asses, and one pound is 72 denarii.

R.6. Swift Carthaginian 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 Pompeii32, 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 Praetorians33.

R.7. The first battle of the Second Punic War

In November 218 BC, Hannibal’s troops approached the Ticinus River. Simultaneously, from Italy, Scipio’s34 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, velites35. 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?

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Примечания

1

Khufu was the second pharaoh of the Fourth Dynasty, in the Old Kingdom period. (2589—2566 BC or 2551—2528 BC)

2

Ramesses II was the third pharaoh of the Nineteenth Dynasty of Egypt that reigned around the 13 century BC.

3

The Hitties were an Anatolian people that lived in the Bronze Age.

4

A cubit is a unit of measurement that equals 45 cm.

5

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. During his life, he came to be known as one of the greatest military leaders of all time.

6

A Royal cubit is a unit of measurement that equals 52.5 cm.

7

Memphis was an ancient Egyptian city. It existed from the early 3 century BC until the second half of the 1 century AD.

8

A satrapy is the territory in the Achaemenid Empire governed by a satrap.

9

A talent was a unit of weight and currency in the ancient times.

10

Leonidas I was a king of the Greek city-state of Sparta from 491 BC until 480 BC.

11

Cyrus II was the founder of the Achaemenid Empire, the first Persian empire.

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.

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.

14

Pythagoras of Samos (c. 570 BC – c. 495 BC) was an ancient Ionian Greek philosopher and the eponymous founder of Pythagoreanism.

15

Darius III was the last Achaemenid King of Kings of Persia, reigning from 336 BC to his death in 330 BC.

16

Let’s assume that the book was published in 2017.