THALES OF MILETUS
CALCULATING THE HEIGHT OF CHEOPS PYRAMID

Thales of Miletus
• Around the 620 BC • Miletus was an ancient Greek city on the western coast of Anatolia in what is now Turkey • Philosopher, astronomer, geographer, mathematician… • Great observer and traveller.

• In one of his travels to Egypt he aimed to measure the height of the Great Pyramid of Cheops. • There are many theories about how he accomplished his goal but there is agreement that he used the concept of similarity of triangles.

Similar triangles
Triangles are similar if they have the same shape, but not necessarily the same size. Have all the same angles. Its correspondent sides are proportional.

PROPORTION?

PQ QR PR P 'Q ' Q ' R ' P ' R '

17 15 12 2 8,5 7,5 6

PROPERTY
Two right angled triangles are similar when one of their angles other than the rigth ones (90º) are equal in measure. Isoceles right angled triangles

The sun is so far away that we might consider its rays parallel and so, they hit the tops of all the objects at the same slant.

h = Thales’ height H = Height of the pyramid

H h h Pyramid S (shade)
s (shade)=h

What was really new?
• • Do you see what Thales did? He used an abstract right triangle! He pictured the height of the Great Pyramid as an imaginary post from its top straight down to its base. Such an imaginary post would cast an imaginary shadow, all the way from where it stood at the center of the pyramid to the tip of the pyramid's real shadow: so the length of this imaginary shadow would be the length of the base plus the actual projecting shadow!

...Math308
Biographical Sketch: Thales and Hypatia
Thales
While it is clear that Euclid definitely set a precedent for geometry and mathematics as a whole, he was not alone in his work, his endeavors, or his ideas. He certainly was not the first to come up with these theories or rules for geometry either. Before there was Euclid, there was Thales of Miletus. Thales, along with other mathematicians or “geometers” laid some of the foundation for Euclid to compile in order to write the Elements centuries later. Thales was a “renaissance man” well ahead of his time, dabbling in such subjects as astronomy, engineering, philosophy and of course, mathematics. Exploring his beginnings, and his accomplishments will afford a decent look at his impact on geometry and Euclid.
Thales was born in Miletus in Ionia ca 624 B.C. (O'Grady, n.d.). Often considered the first, or founder of, Ionian natural philosophy it is hard to be certain on many subjects of his beginnings. Thales’ past is checkered with uncertainties, not uncommon for people from this time period. Due to this, Thales’ exact birth year as well as ancestry are not certain as different sources offer conflicting answers. There is a lot of evidence that has survived to attest to the fact that he “was interested in almost everything, investigating almost all areas of knowledge,...

...Thales of Miletus (ca 624 - 546 BC) Greek domain
Thales was the Chief of the Seven Sages of ancient Greece, and has been called the "Father of Science," the "Founder of Abstract Geometry," and the "First Philosopher." Thales is believed to have studied mathematics under Egyptians, who in turn were aware of much older mathematics from Mesopotamia. Thales may have invented the notion of compass-and-straightedge construction. Several fundamental theorems about triangles are attributed to Thales, including the law of similar triangles (which Thales used famously to calculate the height of the Great Pyramid) and "Thales' Theorem" itself: the fact that any angle inscribed in a semicircle is a right angle. (The other "theorems" were probably more like well-known "axioms", but Thales proved Thales' Theorem using two of his other theorems; it is said that Thales then sacrificed an ox to celebrate what might have been the very first mathematical proof!) Thales noted that, given a line segment of length x, a segment of length x/k can be constructed by first constructing a segment of length kx.
Thales was also an astronomer; he invented the 365-day calendar, introduced the use of Ursa Minor for finding North, and is the first person believed to have correctly predicted a solar eclipse. His theories of...

...Thales (624 BC-546 BC) lived in Ancient Greece and was arguably the world's first philosopher and scientist.
Prior to Thales, supernaturalism was considered to be the cause of most if not all things. For example, the answer to a question such as, "Why does the wind blow?" or "What caused the epidemic?" would invariably involve a supernatural explanation such as, "It is the will of the gods."
Explanations based on supernaturalism, however, are not really explanations at all. They represent non-rational, untestable assertions (such as god's motives) and, consequently, make the possibility of scientific advance impossible.
Thales rejected the pseudo-science of supernaturalism and put forth a radically different explanation of causes, one from which an advance in knowledge could occur. Specifically, Thales believed that water is ultimately the cause of all things. While this belief is incorrect, Thales's idea was revolutionary because it implied that natural forces are the cause of events, not supernatural ones.
In other words, Thales is the father of theories -- objective statements about the world that can be subject to criticism, revision or rejection based on internal consistency and sensory evidence.
Largely because of the contributions of Thales, both philosophy and science soon developed on a grand scale, laying the intellectual foundation for the Greco-Roman...

...Thales’ Theorem
Thales’ Theorem simply states that if three points exist within a circle, and one of those points is the diameter of the circle, then the resulting triangle will always be a right triangle. This simple idea can become very useful for certain applications such as, identifying the center of a circle with its converse. On the triangle the vertex of the right angle always terminates at the ends of the diameter line. By locating the two points of the diameter line and drawing a line connecting them, the center can be found directly between them. This is the case if and only if it is a right triangle. Thales’ theorem is a special case of the inscribed angle theorem. It was attributed to Thales of Miletus and proved in the third book of Euclid’s Elements, 33rd proposition. According to history the Indian and Babylonian mathematicians knew this for special cases prior to Thales. It is even said that he learned of inscribed triangles during his journey to Babylon. It is mostly attributed to him due to his initial proving, utilizing his results of Isosceles base angles being equal and the total sum of angles within a triangle equally 180 degrees. Thales’ is the subject of two stories regarding astronomy and geometry. In the first, he predicted an eclipse in the year 585 BCE. In the next, Thales took observations from two land points and used his...

...Benjamin Foster
9-15-2011
Philosophy 101-501
"Thales"
In Brian Magee's " Story of Philosophy" he composed that Thales,the pre Socratic
Greek philospher of Miletus and founder of Milesian school,taught that everything in nature
is composed of one basic matter.Thales presumed that water the was the fundamental
building block of all things and the Earth floated on water.In Joseph Carter's lecture he states
that "Thales concluded that everything seem to need water,if not directly then indirectly,so
water was the source of all things"(Phil 101-501 Delgado Community College Fall 2011).
It was thought by Thales that the Earth must be supported by water,since all lands
are surrounded by water.He figured that all living things need a huge and constant intake of
water to go on living. There are numerous facts that seem to back up Thales' theories.The
human body is made up of 60 percent water( Story of Philosphy pg 13). The Earth is also made
up of 71 percent water to surface area.Water at very high temperatures become air and at
very low temperatures become rock( Story of Philosophy pg 13).
Magee explained that Thales also observed and studied the world ,using direct
obversation and reason to guide him on his quest for knowledge. He charted the
stars,moon,and planets as well as using geometry and logical sequences to measure shadow...

...In 499 BCE, Aristagoras made a mistake that greatly changed history and led to western ideas being spread throughout the world, a major turning point in shaping the culture of future generations. It resulted in many wars and conquests but ultimately this mistake is why today delegates to the United Nations wear suits and ties instead of turbans.
Aristagoras was the Persian governor of Miletus on the edge of the Persian Empire. The nearby island of Naxos rebelled against the Persian Empire and Aristagoras seized the opportunity to take it back and get a promotion to a better position from the Persian emperor Darius I. At the very least, even if it did not lead to advancement, he could add Naxos to his growing state and get money from the taxes on its citizens.
Since Naxos was an island, Aristagoras needed a navy to conquer it so he got the help of Artaphernes, the governor of Lydia and Darius’ brother, to supply his navy to take Naxos in exchange for some of the plunder. In addition, Artaphernes provided Aristagoras with his skilled and clever naval admiral, Megabates. Unfortunately, Aristagoras publically insulted Megabates leading him to warn the people of Naxos of the impending invasion. The invasion failed because the people of Naxos were prepared and Aristagoras was defeated. The problem was that Aristagoras promised Artaphernes a portion of his booty and since he had none, Artaphernes would take revenge. At the very least Aristagoras...

...didn't believe in Animism, an Ionian named Thales
adopted this idea in his own way. Thales was born in the Greek city-state of Ionia in the
mid 620's(BC.) Thales did not only study knowledge philosophy, but also practiced
science, history, engineering, geography, and politics. Thales was the first of his time to
propose theories of a primary substance that causes change, and supports the universe.
Thales believed that water was this substance, and the essence of life. He also believed
that it was made up of small Gods. Although his theories didn't prove to be right, such as
spontaneous generation, or the theory that earthquakes were caused by waves, Thales is
considered to be the father of philosophy.
Thales had a follower named Anaximander that wasn't just a philosopher, but he was also a scientist and inventer. He believed in the theory of 'apieron" apieron is the unlimited indeficit undestructible substance out of which individual things are created with and destroyed.
Anaximander is considered one of the most important of presocratic philosophy all of whom are concerned with the principle governing change in the natural world. He was the first to have developed a comprehensive scientific approach to the origin and guiding principles of the world. He conducted the colony of Apollonia on the black sea.One popular, and apparently successful explanation for...

...that everything in the
universe, especially material objects, have some kind of sole or is a living being.) These
people believed that rocks, trees, and water had some kind of sole. Animism can still be
seen today in Native American tribes as well as the Aboriginal people of Australia.
Although the Greek culture didn't believe in Animism, an Ionian named Thales
adopted this idea in his own way. Thales was born in the Greek city-state of Ionia in the
mid 620's(BC.) Thales did not only study knowledge philosophy, but also practiced
science, history, engineering, geography, and politics. Thales was the first of his time to
propose theories of a primary substance that causes change, and supports the universe.
Thales believed that water was this substance, and the essence of life. He also believed
that it was made up of small Gods. Although his theories didn't prove to be right, such as
spontaneous generation, or the theory that earthquakes were caused by waves, Thales is
considered to be the father of philosophy.
Anaximander was another Ionian who happened to be a philosopher. He knew
Thales but disagreed with him in his theories. Anaximander wasn't only a philosopher,
but a scientist and inventor. He is even credited for creating the first sundial.
Anaximander believed in the theory of "apieron." Apieron is the unlimited, indeficit,...