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The Sum of Interior Angles

Alternate angles: angles on opposite sides of a transversal line, which are equal in size

blue angle

green angle

Thales' Theorem - Proof

The red angle is identical in both forms. Since a straight angle consits of two right angles (=180°) we can conclude that the sum of the green, blue and red angle is 180°.

Sources

source: Thales of Miletus, PDF

  • A. A. Long (Hrsg.): Handbuch Frühe Griechische Philosophie. Von Thales bis zu den Sophisten. Metzler, 2001.
  • Alfred Hooper: Makers of Mathematics. Faber and Faber, London, 1965.
  • Diogenes Laertius: Leben und Meinungen berühmter Philosophen. Buch I-X. Verlag von Felix Meiner, Hamburg, 1967.
  • Hans-Joachim Fischer: Frühgriechische Philosophie. Den Anfang nachdenken. W. Kohlhammer, Stuttgart, 2022.
  • Klaus Mainzer: Geschichte der Geometrie. Bibliographisches Institut, 1980.
  • Keimpe Algra: Die Anfänge der Kosmologie. In: A. A. Long (Hrsg.): Handbuch Frühe Griechische Philosophie. Von Thales bis zu den Sophisten. Metzler, 2001.
  • OLD = Oxford Learner's Dictionary (https://www.oxfordlearnersdictionaries.com/definition/english/mythology?q=mythology)
  • Thales of Miletus, PDF
  • Wikipedia (https://en.wikipedia.org/wiki/Greek_mythology), (https://en.wikipedia.org/wiki/Miletus), (https://en.wikipedia.org/wiki/Ionian_school_(philosophy)), (https://en.wikipedia.org/wiki/Anaximander), (https://en.wikipedia.org/wiki/Anaximenes_of_Miletus), (https://en.wikipedia.org/wiki/Thales_of_Miletus#), (https://en.wikipedia.org/wiki/Intercept_theorem), (https://en.wikipedia.org/wiki/Thales%27s_theorem),

source: Thales of Miletus, PDF

The sum of the angles of a triangle is 180°

Thales' Theorem

Definition: If A, B, and C are distinct points on a circle where the line AC is a diameter, the angle <ABC is a right angle.

source: Wikipedia

Intercept Theorem

Legacy

  • Thales as the first philosopher and mathematician had an important influence on Greek thinkers.
  • Some consider him the teacher of Pythagoras, though it may be only be that he advised Pythagoras to travel to Egypt and Chaldea. In general, Pythagoras, was a student of Anaximander, who was a student of Thales.
  • He is unanimously ascribed to have introduced the mathematical and astronomical sciences into Greece, therefore, he had an influence on the Western history.
  • Mythology rational philosophy/ logic
  • Nevertheless, the extent of his influence debatable.

Initial Situation: "Suppose S is the common starting point of two rays, and two parallel lines are intersecting those two rays. Let A, B be the intersections of the first ray with the two parallels, such that B is further away from S than A, and similarly C, D are the intersections of the second ray with the two parallels such that D is further away from S than C."

source: Wikipedia

source: Wikipedia; Thales of Miletus, PDF

01

The ratio of any two segments on the first ray equals the ratio of the according segments on the second ray.

Thales as a Mathematician

The ratio of the two segments on the same ray starting at S equals the ratio of the segments on the parallels.

02

Deductive Reasoning

"Thales was known for introducing the theoretical and practical use of geometry to Greece, and has been described as the first person in the western world to apply deductive reasoning to geometry."

source: Wikipedia

Deduction = a form of pure reasoning

The thinker starts with certain first principles, and argues from them until he reaches, through the use of logic, some conclusion. [...] he uses only facts that are agreed first principles or have themselves been proved from such first priciples.

Thales' Pyramid

Intercept Theorem - Proof

source: Alfred Hooper

source: Wikipedia

5 Theorems of Elementary Geometry

Origin of his Mathematical Knowledge

The ratio of any two segments on the first ray equals the ratio of the according segments on the second ray.

01

A circle is bisected by any diameter.

B

A

S C D

Temple of Apollo in Didyma

02

The base angles of an isosceles triangle are equal.

Said to be found in his travels to Egypt, which he visited during his days as a merchant.

He tried to find an explanation for the geometric facts that were discovered empirically by the Egyptians.

In Egypt he calculated the height of the pyramids by measuring the shadows. Which means, that he measured the size of the pyramid at the exact time, when his own shadow was as big as himself.

Hans-Joachim Fischer

Wikipedia

03

The angles between two intersecting straight lines are equal.

Wikipedia

04

Two triangles are congruent if they have two angles and the included side equal.

source: Alfred Hooper

Wikipedia

How to Include Thales in your Classroom:

Miletus

05

An angle in a semicircle is a right angle (Thales' Theorem).

source: Thales of Miletus, PDF

Thales' geometry was committed to solve practical problems, e.g., to determine the distance from a ship at sea to land.

  • The Intercept Theorem is ordered into the 8th grade by the curriculum.
  • Since Thales tried to solve pratical problems through geometry, try to give your students the same task.
  • Give them a theoretical input and let them explore the topic on their own.
  • E.g., calculating the size of a tall object/building.
  • Similarity to how Thales measured the pyramids: using his knowledge about his own size and the exact time when his shadow was his own size.
  • In general: make them curious about the topic, by showing them that they can use it in their own life.
  • The birthplace of the Milesian School. It is seen as the origin of the Greek philosophical and scientific tradition.
  • An ancient Greek mercantile city in ancient Ionia at the western coast of Asia Minor.
  • Located near the mouth of the Maeander river.
  • Today: Turkey.
  • Considered as one of the wealthiest and greatest Greek cities of those days.
  • Had its own oracle of Apollo Didymeus Milesios in Didyma.

Dr. Klaus Mainzer

source: Hans-Joachim Fischer

source: Wikipedia

source: Lambacher Schweizer

source: A. A. Long

Anaximander

The Milesian School

School's Aim

source: Wikipedia

The aim of the Milesian School was to explain nature by finding its fundamental element (=arche).

  • Believed that all matter had something in common.
  • The members of the Milesian School did not agree on one arche.
  • The Milesian School was a Greek school of thought.
  • Its first three members were Thales, Anaximander, Anaximenes.
  • Used abstract reasoning instead of religion or mythology to explain their beliefs and are therefore seen as the first rational philosophers.
  • "Ideas of Thales and other Milesians created a bridge between the two worlds–the world of myth and the world of the mind.”

Anaximenes

source: Wikipedia

The Milesian School

  • Air is the arche, which means that he believed that all things are created trough air.
  • Probably instructed by Anaximander.

source: Wikipedia

source: Wikipedia; Thales of Miletus, PDF

Thales died at the age of 76/79.

!

Thales

None of Thales's writing survived and no contemporary sources exist; thus, his achievements are difficult to assess, particularly his philosophy and mathematical discoveries.

One must consider the ancient practice of crediting particular discoveries to men with a reputation for wisdom.

548/545 B.C.

today

624 B.C.

"How old did Thales become?"

source: Thales of Miletus, PDF

624 B.C. - 548/545 B.C.

  • Born 624 B.C.
  • "His parents were descendants of the original Greek settlers in this city."
  • Thales was a successful merchant and retired early.
  • After his retirement he "devoted his time to the study of astronomy and mathematics"
  • He is one of the Seven Sages (= Wise Men), which means that he is considered to be one of the wisest people of Ancient Greece.
  • Many sources give different names to name the Seven Wise Men, but Thales's name can be found on every list.
  • Believed to be the first one to have engaged in mathematics, science, and deductive reasoning.
  • Thales was the first known Greek philosopher, scientist and mathematician.
  • He is known for his intellectual achievments, which he contributed to the science of mathematics.
  • Died in 548/545 B.C.

(Greek) Mythology

"Know Thyself"

Thales' Cosmological Thesis

source: Alfred Hooper, Wikipedia

Thales of Miletus

When Thales got asked what he found difficult, he answered "knowing yourself". Adding to that, he answered the question, what he finds very easy, with "to give advice".

Wikipedia

In general, he believed that the earth is a flat disk, which floats on an endless expanse of water. He explained his theory by seeing water as the arche, which means that he believed that everything is born out of water.

He explained earthquakes through this theory, by saying that they occur when the earth is rocked by waves.

Thales investigated the world on a non-religious basis and tried to find naturalistic explanations of the world.

Astronomy

  • He described the position of Ursa Minor (=star constellation).
  • Calculated the duration of the year (365 days).
  • Calculated the timings of the equinoxes and solstices.

The ancient myths of Greek culture and society, build a theoretical foundation for the Milesian School.

  • Homer: The earth is flat and surrounded by an ocean.
  • Cosmic protagonists (e.g., earth, sun, moon) were seen and praised as gods.
  • Some gods were considered responsible for cosmic phenomena (e.g., Poseidon - earthquakes).

The (origin of the) world was explained through mythology.

Made up or "actual" history?

source: Wikipedia

Game-Pin: 151 1631

https://create.kahoot.it/creator/fdbe9c51-159f-48d4-bd84-fc4d526313bb

Made up or "actual" history? Results:

source: OLD, Keimpe Algra

Wikipedia

1. "Thales was studying the stars and gazing into the sky, when he fell into a well, and a servant girl made fun of him."

An actual story reported by Plato, Diogenes Laertius, and Hippolytus.

2. Thales predicted a solar eclipse.

It is said, that he predicted an eclipse that took place on May 28th, 585 B.C.

3. Thales demonstrated the use of natural philosophy by predicting the weather and renting olive presses to gain money.

By that he proved the usefulness of philosophy to the Milesians.

4. A crater on the moon is named "Thales" to honor him.

The crater Thales is located in the northeast part of the moon.

5. He died while watching the Olympic Games.

Reported by Diogenes Laertius.

Wikipedia

main source: Wikipedia

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