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. Euclid of Alexandria was a Greek mathematician Figure 10 , and is often referred to as the Father of Geometry. The date and place of Euclid's birth, and the date and circumstances of his death, are unknown, but it is thought that he lived circa BCE. His work Elements , which includes books and propositions, is the most successful textbook in the history of mathematics.
In it, the principles of what is now called Euclidean Geometry were deduced from a small set of axioms. When Euclid wrote his Elements around BCE , he gave two proofs of the Pythagorean Theorem: The first, Proposition 47 of Book I, relies entirely on the area relations and is quite sophisticated; the second, Proposition 31 of Book VI, is based on the concept of proportion and is much simpler.
He may have used Book VI Proposition 31, but, if so, his proof was deficient, because the complete theory of Proportions was only developed by Eudoxus, who lived almost two centuries after Pythagoras.
Euclid's Elements furnishes the first and, later, the standard reference in geometry. It is a mathematical and geometric treatise consisting of 13 books.
It comprises a collection of definitions, postulates axioms , propositions theorems and constructions and mathematical proofs of the propositions. Euclid provided two very different proofs, stated below, of the Pythagorean Theorem. This is probably the most famous of all the proofs of the Pythagorean proposition. In right-angled triangles the figure on the side opposite the right angle equals the sum of the similar and similarly described figures on the sides containing the right angle.
In right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle. Euclid I 47 is often called the Pythagorean Theorem , called so by Proclus — a Greek philosopher who became head of Plato's Academy and is important mathematically for his commentaries on the work of other mathematicians — and others centuries after Pythagoras and even centuries after Euclid.
Although many of the results in Elements originated with earlier mathematicians, one of Euclid's accomplishments was to present them in a single, logically coherent framework, making them easy to use and easy to reference, including a system of rigorous mathematical proofs that remains the basis of mathematics twenty-three centuries later.
Although best known for its geometric results, Elements also includes number theory. It considers the connection between perfect numbers and Mersenne primes, the infinitude of prime numbers and the Euclidean algorithm for finding the greatest common divisor of two numbers.
The geometrical system described in the Elements was long known simply as geometry , and was considered to be the only geometry possible. Today, however, this system is often referred to as Euclidean Geometry to distinguish it from other so-called Non-Euclidean geometries that mathematicians discovered in the nineteenth century. At this point in my plotting of the year-old story of Pythagoras, I feel it is fitting to present one proof of the famous theorem.
For me, the simplest proof among the dozens of proofs that I read in preparing this article is that shown in Figure Start with four copies of the same triangle. See upper part of Figure See lower part of Figure In the seventeenth century, Pierre de Fermat — Figure 14 investigated the following problem: for which values of n are there integer solutions to the equation. Fermat conjectured that there were no non-zero integer solutions for x and y and z when n was greater than 2.
He did not leave a proof, though. Instead, in the margin of a textbook, he wrote that he knew that this relationship was not possible, but he did not have enough room on the page to write it down. His conjecture became known as Fermat's Last Theorem. This may appear to be a simple problem on the surface, but it was not until when Andrew Wiles of Princeton University finally proved the year-old marginalized theorem, which appeared on the front page of the New York Times.
Today, Fermat is thought of as a number theorist, in fact perhaps the most famous number theorist who ever lived. It is therefore surprising to find that Fermat was a lawyer , and only an amateur mathematician. Also surprising is the fact that he published only one mathematical paper in his life, and that was an anonymous paper written as an appendix to a colleague's book.
Because Fermat refused to publish his work, his friends feared that it would soon be forgotten unless something was done about it. His son Samuel undertook the task of collecting Fermat's letters and other mathematical papers, comments written in books and so on with the goal of publishing his father's mathematical ideas. Samuel found the marginal note the proof could not fit on the page in his father's copy of Diophantus's Arithmetica.
In this way the famous Last Theorem came to be published. His graduate research was guided by John Coates beginning in the summer of Let us assume the length of all sides is "a", semi-perimeter is "s" and the area of the equilateral triangle is "A". A scalene triangle has all lengths of different sides. Let us assume the length of sides is a, b, c, semi-perimeter is "s" and the area of the scalene triangle is "A".
An isosceles triangle has two sides of equal length. Let us assume the length of the two sides is a and one side is b, semi-perimeter is "s" and the area of the isosceles triangle is "A". We can use Heron's formula to determine the formula for the area of the quadrilateral by dividing it into two triangles. Let us say we have a quadrilateral ABCD with the length of its sides measuring a, b, c, and d. Let us say A and B are joined to show the diagonal of the quadrilateral having length e.
The application of Heron's formula in finding the area of the quadrilateral is that it can be used to determine the area of any irregular quadrilateral by converting the quadrilateral into triangles. Example 1: If the length of the sides of a triangle ABC are 4 in, 3 in, and 5 in. Calculate its area. Find the length of the sides of the triangle. Solution: To find: The length of the sides of the triangle. Example 3: Calculate the area of an isosceles triangle using Heron's formula if the lengths of its sides are 4 units, 8 units, and 8 units.
Solution: To find: Area of triangle Given the lengths of sides are 4 units, 8 units, and 8 units. Solution: Note that we have only been given the lengths of the four sides, but not the length of any diagonal. Heron's formula is used to find the area of the triangle when the lengths of all triangles are given.
It can be used to determine areas of different types of triangles, equilateral, isosceles, or scalene triangles.
There were two main schools of thought on this, one believing that he lived around BC and the second believing that he lived around AD. The first of these was based mainly on the fact that Heron does not quote from any work later than Archimedes.
The second was based on an argument which purported to show that he lived later that Ptolemy , and, since Pappus refers to Heron, before Pappus. Both of these arguments have been shown to be wrong. There was a third date proposed which was based on the belief that Heron was a contemporary of Columella.
Columella was a Roman soldier and farmer who wrote extensively on agriculture and similar subjects, hoping to foster in people a love for farming and a liking for the simple life. Columella, in a text written in about 62 AD [ 5 ] However, most historians believed that both Columella and Heron were using an earlier source and claimed that the similarity did not prove any dependence. We now know that those who believed that Heron lived around the time of Columella were in fact correct, for Neugebauer in discovered that Heron referred to a recent eclipse in one of his works which, from the information given by Heron, he was able to identify with one which took place in Alexandria at From Heron's writings it is reasonable to deduce that he taught at the Museum in Alexandria.
His works look like lecture notes from courses he must have given there on mathematics, physics, pneumatics, and mechanics.
Some are clearly textbooks while others are perhaps drafts of lecture notes not yet worked into final form for a student textbook. Pappus writes see for example [ 8 ] :- The mechanicians of Heron's school say that mechanics can be divided into a theoretical and a manual part; the theoretical part is composed of geometry, arithmetic, astronomy and physics, the manual of work in metals, architecture, carpentering and painting and anything involving skill with the hands.
A large number of works by Heron have survived, although the authorship of some is disputed. Measure ad performance. Select basic ads. Create a personalised ads profile. Select personalised ads. Apply market research to generate audience insights.
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