The latter is reflected in the Pythagorean motto: Number Rules the Universe. Pythagoreans consumed vegetarian dried and condensed food and unleavened bread as matzos, used by the Biblical Jewish priestly class the Kohanim , and used today during the Jewish holiday of Passover. One reason for the rarity of Pythagoras original sources was that Pythagorean knowledge was passed on from one generation to the next by word of mouth, as writing material was scarce.
Therefore, the true discovery of a particular Pythagorean result may never be known. Is seems that Pythagoras was the first person to define the consonant acoustic relationships between strings of proportional lengths. Specifically, strings of equal tension of proportional lengths create tones of proportional frequencies when plucked.
For example, a string that is 2 feet long will vibrate x times per second that is, hertz, a unit of frequency equal to one cycle per second , while a string that is 1 foot long will vibrate twice as fast: 2 x. Furthermore, those two frequencies create a perfect octave. The Pythagoreans were so troubled over the finding of irrational numbers that they swore each other to secrecy about its existence.
It is known that one Pythagorean did tell someone outside the school, and he was never to be found thereafter, that is, he was murdered, as Pythagoras himself was murdered by oppressors of the Semicircle of Pythagoras. Mesopotamia arrow 1 in Figure 2 was in the Near East in roughly the same geographical position as modern Iraq. Mesopotamia was one of the great civilizations of antiquity, rising to prominence years ago. Thousands of clay tablets, found over the past two centuries, confirm a people who kept accurate records of astronomical events, and who excelled in the arts and literature.
Only a small fraction of this vast archeological treasure trove has been studied by scholars. The great majority of tablets lie in the basements of museums around the world, awaiting their turn to be deciphered and to provide a glimpse into the daily life of ancient Babylon.
The marks are in wedge-shaped characters, carved with a stylus into a piece of soft clay that was then dried in the sun or baked in an oven. They turn out to be numbers, written in the Babylonian numeration system that used the base In this sexagestimal system, numbers up to 59 were written in essentially the modern base numeration system, but without a zero.
Units were written as vertical Y-shaped notches, while tens were marked with similar notches written horizontally. What is the breadth? Its size is not known. And 5 times 5 is You take 16 from 25 and there remains 9. What times what shall I take in order to get 9? The number along the upper left side is easily recognized as The conclusion is inescapable.
This was probably the first number known to be irrational. Two factors with regard to this tablet are particularly significant. First, it proves that the Babylonians knew how to compute the square root of a number with remarkable accuracy.
The unknown scribe who carved these numbers into a clay tablet nearly years ago showed a simple method of computing: multiply the side of the square by the square root of 2. But there remains one unanswered question: Why did the scribe choose a side of 30 for his example? Probably, 30 was used for convenience, as it was part of the Babylonian system of sexagesimal, a base numeral system.
To Pythagoras it was a geometric statement about areas. It was with the rise of modern algebra, circa CE , that the theorem assumed its familiar algebraic form. In any right triangle, the area of the square whose side is the hypotenuse the side opposite the right angle is equal to the sum of the areas of the squares whose sides are the two legs the two sides that meet at a right angle.
An area interpretation of this statement is shown in Figure 5. The square of the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides. Ancient Egyptians arrow 4, in Figure 2 , concentrated along the middle to lower reaches of the Nile River arrow 5, in Figure 2 , were a people in Northeastern Africa.
The ancient civilization of the Egyptians thrived miles to the southwest of Mesopotamia. The two nations coexisted in relative peace for over years, from circa BCE to the time of the Greeks. As to the claim that the Egyptians knew and used the Pythagorean Theorem in building the great pyramids, there is no evidence to support this claim. Egypt has over pyramids, most built as tombs for their country's Pharaohs.
Egypt arrow 4, in Figure 2 and its pyramids are as immortally linked to King Tut as are Pythagoras and his famous theorem. King Tut ruled from the age of 8 for 9 years, — BC. He was born in BC and died some believe he was murdered in BC at the age of Elisha Scott Loomis — Figure 7 , an eccentric mathematics teacher from Ohio, spent a lifetime collecting all known proofs of the Pythagorean Theorem and writing them up in The Pythagorean Proposition , a compendium of proofs.
The manuscript was prepared in and published in Loomis received literally hundreds of new proofs from after his book was released up until his death, but he could not keep up with his compendium. As for the exact number of proofs, no one is sure how many there are. Surprisingly, geometricians often find it quite difficult to determine whether certain proofs are in fact distinct proofs.
He died on 11 December , and the obituary was published as he had written it, except for the date of his death and the addresses of some of his survivors. According to his autobiography, a preteen Albert Einstein Figure 8. Many known proofs use similarity arguments, but this one is notable for its elegance, simplicity and the sense that it reveals the connection between length and area that is at the heart of the theorem.
At the age of 12, I experienced a second wonder of a totally different nature: in a little book dealing with Euclidean plane geometry, which came into my hands at the beginning of a school year. Here were assertions, as for example the intersection of the three altitudes of a triangle in one point, which — though by no means evident — could nevertheless be proved with such certainty that any doubt appeared to be out of the question.
This lucidity and certainty made an indescribable impression upon me. For example I remember that an uncle told me the Pythagorean Theorem before the holy geometry booklet had come into my hands. Einstein Figure 9 used the Pythagorean Theorem in the Special Theory of Relativity in a four-dimensional form , and in a vastly expanded form in the General Theory of Relatively.
The following excerpts are worthy of inclusion. Special relativity is still based directly on an empirical law, that of the constancy of the velocity of light. The fact that such a metric is called Euclidean is connected with the following. The postulation of such a metric in a three-dimensional continuum is fully equivalent to the postulation of the axioms of Euclidean Geometry. The defining equation of the metric is then nothing but the Pythagorean Theorem applied to the differentials of the co-ordinates.
Such transformations are called Lorentz transformations. From the latest results of the theory of relativity, it is probable that our three-dimensional space is also approximately spherical , that is, that the laws of disposition of rigid bodies in it are not given by Euclidean geometry, but approximately by spherical geometry.
According to the general theory of relativity , the geometrical properties of space are not independent, but they are determined by matter. I wished to show that space time is not necessarily something to which one can ascribe to a separate existence, independently of the actual objects of physical reality.
Physical objects are not in space, but these objects are spatially extended. The above excerpts — from the genius himself — precede any other person's narrative of the Theory of Relativity and the Pythagorean Theorem. Accordingly, I now provide a less demanding excerpt, albeit one that addresses the effects of the Special and General theories of relativity. The system of units in which the speed of light c is the unit of velocity allows to cast all formulas in a very simple form. The Pythagorean Theorem graphically relates energy, momentum and mass.
Multimedia OpenMind books Authors. Featured author. Manuel Mira Godinho. Lisbon, Portugal. Latest book. Work in the Age of Data. Technology Visionaries. Computing Leading Figures Mathematics. Ventana al Conocimiento Knowledge Window. Estimated reading time Time 4 to read. Blaise Pascal, seated at his desk. From games of chance to the search for faith This was the early start of a professional career steeped in achievements, discoveries and contributions, which was to reach its climax in thanks to a professional Parisian gambler, Antoine Gombaud.
Source: Wikimedia Pascal was so enthusiastic about the possibilities offered by this new mathematics that he even became convinced that with it he could justify the need to believe in God, the essence of faith. Pascaline was one of the first mechanical calculating machine. Credit: Rama Because of this invention, Blaise Pascal is considered the father of calculation machines, precursors of the first computers; and that is why, in , the Swiss electronic engineer Niklaus Wirth named the programming language he had just created PASCAL, which was the gateway to computing for many students at the end of the 20th century.
Non-Euclidean Geometry supports the first four postulates as discussed in the Euclidean Geometry, but the fifth postulate, which is also called the parallel postulate was not agreed upon. This paved the way to create two other types of Geometry called 'Hyperbolic Geometry' and 'Elliptical Geometry or Spherical Geometry '.
Spherical geometry is used to study spherical surfaces, whereas Hyperbolic geometry is used to study saddle surfaces. A saddle is a point that is low lying between two high points. Geometry deals with the study of shapes and sizes of objects. It is derived from the Greek word which denotes 'Earth Measurement'.
It also deals with rules to calculate length, area, and volumes. There was not any correct way of calculating these parameters. Many were based on trial and error. The concepts of geometry are concerned with our day-to-day life.
In the ancient days, people were willing to know the volume of solid shapes to store goods and also for construction purposes.
The theory of conic sections, which was a part of Greek geometry found its application in astronomy and optics. Another important and notable discovery was that of Pythagoras. Pythagoras was an ancient Greek mathematician, who discovered that there is a relationship between the sides of a right-angled triangle and gave it proof.
The Pythagoras theorem finds its application in the field of trigonometry. It uses the concepts of both algebra and geometry.
Euclidean geometry deals with the study of length, area, and volume of solid shapes based on certain axioms and theorems. It was developed by a Greek mathematician called Euclid. This branch of geometry deals with terms like points, lines, surfaces, dimensions of the solids, etc. Analytical geometry also referred to as coordinate geometry or cartesian geometry deals with the coordinate system to represent lines and points. In analytical geometry a point is represented by two or three numbers to denote its position on a plane, this is called a coordinate point.
It is written in the form of 3,4 , where 3 is the x-coordinate and 4 is the y-coordinate. We can also have a point 2, 3, 4 where 2 is the x-coordinate, 3 is the y-coordinate and 4 is the z-coordinate.
This branch of geometry uses algebraic equations and methods to solve problems. It also deals with midpoint, parallel and perpendicular lines, line equations, distances between two linear paths. The figure below shows a point 3,4 on a coordinate plane. A branch of geometry that deals with geometric images when they are projected into another surface.
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