MATHEMATICIANS

MATHEMATICIANS OF ANTIQUITY

Larini Alessandro 2A5Calzolari Greta 07/01/23

THALES OF MILETUS

Thales was a Greek mathematician, surveyor, astronomer and philosopher from Miletus in Ionia, who lived between the sixth and fifth centuries BC. He was the founder of the school of Miletus and from Aristotle onwards he was recognised as the progenitor of Western philosophy. Thales is called one of the Seven Sages of Greece and he is recognised as the first philosopher who devoted himself to mathematics, natural science and astronomy.

His famous statement is: ”Water is the originating principle of all things”.

Thales used the observation of phenomenas to confirm his experiments.

He used geometry to measure the height of the pyramids from their shadow.

Many useful applications derive from his discoveries. ​ The most famous is Thales' THEOREM.

Thales' Theorem

If two straight lines, not necessarily parallel , are cut by a system of parallel lines,then the segments, determined on one of the two lines are proportional to the respective segments, obtained on the other line.

Beyond this theorem, Thales is remembered for five other theorems of basic geometry

pythagoras

Pythagoras of Samos (570 b.C. - 490 b.C.) was a Greek philosopher, mathematician, and founder of the Pythagorean brotherhood that, although religious in nature, formulated principles that influenced the thought of Plato and Aristotle and contributed to the development of mathematics and Western rational philosophy. For Pythagoras everything in the universe can be explained with numbers, specifically whole numbers.

From a young age he received the teachings of Thales of Miletus And his pupil, Anaximander. He was the most influential thinker of the called Presocratic, is considered the first pure mathematician and he was the first man to call himself a philosopher ("lover of wisdom"). He contributed significantly in the development of the mathematical principles of his time, of arithmetic, geometry, cosmology, astronomy and musical theory.

He is known best for the proof of the important Pythagorean theorem, that Although it was used in ancient Babylon and China, the credit for proving the theorem has been given to Pythagoras and his followers.

The Pythagorean Theorem

Curiosity

The Pythagorean theorem is the theorem with the largest number of proofs in Euclidean geometry, to date it has at least 370 proofs and is not limited to squares but can be used for any polygon built on the sides.

It states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of its other two sides, or a^2 + b^2 = c^2.

Pythagoras' Beliefs and Theories

Five Regular Solids

Pythagoras Reasonings

The Tetractys

Music and Lifestyle

Numerology

Cosmology

Metempsychosis

Euclid

Euclid was a Greek mathematician and geometer, considered one of the great mathematicians of antiquity and "father of geometry". He was born in Alexandria in 435 b.C and would die in 265 b.C. Starting with a few evident principles, such as that all right-angles are equal, Euclid deduced and proved a large number of ever more sophisticated mathematical theorems placing them in the Elements’ 13 books.

Euclid’s Elements

Euclid’s "Elements" contains 465 propositions, 93 problems and 372 theorems in thirteen volumes, it is a masterpiece, a work of genius that It inspired ancient Greeks, such as Archimedes; Persians, and, following the Renaissance, thousands of individual scientists such as Nicolaus Copernicus, Galileo Galilei, Isaac Newton, James Clerk Maxwell, Albert Einstein, and Thomas Gold.Elements describes axioms, theorems, and constructions and it also provides mathematical proofs for all of the definitions.

The most important theorems of Euclid’s work are:

Euclid's First Theorem “In a right-angled triangle ABC, right in A, the square built on a leg is equivalent to the rectangle whose dimensions are the hypotenuse and the projection of the leg on the hypotenuse”. The theorem relates three elements of the triangle: the hypotenuse, a cathetus and its projection onto the hypotenuse.

Euclid's second theorem “In a right triangle, the square built on the height relative to the hypotenuse is equivalent to the rectangle whose dimensions are the projections of the legs on the hypotenuse”. The second theorem gives us the relations that link the height relative to the hypotenuse of a right triangle with the projections.

Euclid’s axioms

1. Things which are equal to the same thing are equal to one another.2. If equals are added to equals, the wholes are equal. 3. If equals are subtracted from equals, the remainders are equal. 4. Things which coincide with one another are equal to one another. 5. The whole is greater than the part. 6. Things which are double of the same things are equal to one another. 7. Things which are halves of the same things are equal to one another.

Euclid's postulates

Postulate 1: A straight line may be drawn from any one point to any other point.

Postulate 4: All right angles are equal to one another.

Postulate 5: If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.

Postulate 2: A terminated line can be produced indefinitely.

Postulate 3: A circle can be drawn with any centre and any radius.

ARCHIMEDES

Mathematician and physicist (Syracuse 287 - 212 BC). He was one of the greatest mathematicians of antiquity. Probably a pupil of Euclid. He devoted himself constantly to research and to the achievement of his own inventions. About 40 inventions are attributed to him including: the hydrostatic principle, the laws of mechanics and optics, the lever and the screw. He was killed in 212 BC, during the sack of Syracuse.

THE HYDROSTATIC PRINCIPLE

A body immersed in a fluid (liquid or gas) undergoes a force directed from the bottom upwards of intensity equal to comparable to the weight of the fluid displaced by the body.

Fg=d(fluid)*Vg

Of the sphere and of the cylinder

What is Pi? – The game of Archimedes' constant

• This symbol is a Greek letter called Pi. It is a mathematical constant that indicates the ratio between the length of a circle and its diameter. In fact, if we denote with C the length of a circle and with d its diameter, we know that C = d • π. The length of a circumference with a diameter of 1 is worth π .

• Archimedes had discovered that the volume of the sphere was always equal to 2/3 of the volume of the cylinder and that there was the same ratio between the surfaces of the two solids, a discovery that brought him very close to square the circle.

Apollonius OF PergA: "The great surveyor"

Apollonius of Perga, was a Greek mathematician (c. 262 - c. 180 BC) who is considered the most original one after Archimedes. ​ Apollonius and Archimedes were nearly contemporaries.​ He had a long life, and studied in Alexandria in Egypt, in the school of the successors of Euclid. Apollonius' most famous work, studied and commented by his successors is known as"Coniche"; it is a classic work, which should be placed next to those of Archimedes. It contains definitions, essential properties, and theorems relating to the conical curves that Apollonius called ellipses, parabola, and hyperbola.

Conical intersections represent the intersections of a plane with a cone at different angles. Such intersections create a circle (in red), an ellipse (in green), a parabola (in lilac) and a hyperbola (in orange).

Let us cite two theorems of Apollonius:

A) given the ellipse , the area of a parellelogram circumscribed to it is constant and is equal to the product of the lengths of the two diameters of the ellipse.

B) Given a hyperbola, the area of a triangle formed by two asymptotes of the hyperbola and a tangent to it is such.

Circles of Apollonius

Apollonius demonstrated that a circle can be defined as the set of points in a plane that have a certain relationship between distances and two fixed points, known as foci. The Apollonian circles of a triangle are three circles, each of which passes through a vertex of the triangle and maintains a constant relationship of distances with the other two .