history of math during euclid to euler?

Answer 1

my advice is to look for a history book, but also you can check wikipedia:

Greek mathematics studied before the Hellenistic period refers only to the mathematics of Greece. Greek mathematics studied from the time of the Hellenistic period (from 323 BC) refers to all mathematics of those who wrote in the Greek language, since Greek mathematics was now not only written by Greeks but also non-Greek scholars throughout the Hellenistic world, which was spread across the Eastern end of the Mediterranean. Greek mathematics from this point merged with Egyptian and Babylonian mathematics to give rise to Hellenistic mathematics. Most mathematical texts written in Greek were found in Greece, Egypt, Mesopotamia, Asia Minor, Sicily and Southern Italy.

Although the earliest found Greek texts on mathematics were written after the Hellenistic period, many of these are considered to be copies of works written during and before the Hellenistic period. Nevertheless, the dates of Greek mathematics are more certain than the dates of earlier mathematical writing, since a large number of chronologies exist that, overlapping, record events year by year up to the present day. Even so, many dates are uncertain; but the doubt is a matter of decades rather than centuries.

Greek mathematics is thought to have begun with Thales (c. 624—c.546 BC) and Pythagoras (c. 582—c. 507 BC). Although the extent of the influence is disputed, they were probably influenced by the ideas of Egypt, Mesopotamia and less likely[citation needed], India. Thales used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. According to Proclus' commentary on Euclid, Pythagoras stated the Pythagorean theorem and constructed Pythagorean triples algebraically. It is generally conceded that Greek mathematics differed from that of its neighbors in its insistence on axiomatic proofs.[8]

Greek and Hellenistic mathematicians were the first to give a proof for irrational numbers (due to the Pythagoreans), and the first to develop Eudoxus's method of exhaustion, and the Sieve of Eratosthenes for uncovering prime numbers. They took the ad hoc methods of constructing a circle or an ellipse and developed a comprehensive theory of conics; they took many various formulas for areas and volumes and deduced methods to separate the correct from the incorrect and generate general formulas. The first recorded abstract proofs are in Greek, and all extant studies of logic proceed from the methods set down by Aristotle. Euclid, in the Elements, wrote a book that would be used as a mathematics textbook throughout Europe, the Near East and North Africa for almost two thousand years. In addition to the familiar theorems of geometry, such as the Pythagorean theorem, The Elements includes a proof that the square root of two is irrational and that there are infinitely many prime numbers.

Some say the greatest of Greek mathematicians, if not of all time, was Archimedes (287—212 BC) of Syracuse. According to Plutarch, at the age of 75, while drawing mathematical formulas in the dust, he was run through with a spear by a Roman soldier. Roman society has left little evidence of an interest in pure mathematics.


European Renaissance mathematics (c. 1200—1600)

In Europe at the dawn of the Renaissance, most of what is now called school mathematics -- addition, subtraction, multiplication, division, and geometry -- was known to educated people, though the notation was cumbersome: Roman numerals and words were used, but no symbols: no plus sign, no equal sign, and no use of x as an unknown. Most of the mathematics now taught at universities was either known only to the mathematical community in India or had yet to be investigated and developed in Europe.

Through Latin translations of Arabic texts, knowledge of the Hindu-Arabic numerals and other important developments of Islamic and Indian mathematics were brought to Europe. Robert of Chester's translation of Al-Khwarizmi's Al-Jabr wa-al-Muqabilah into Latin in the 12th century was particularly important. The earlier works of Aristotle were redeveloped in Europe, first in Arabic and later in Greek. Of particular importance was the rediscovery of a collection of Aristotle's logical writing, compiled in the 1st century, known as the Organon.

The reawakened desire for new knowledge sparked a renewed interest in mathematics. Fibonacci, in the early 13th century, produced the first significant mathematics in Europe since the time of Eratosthenes, a gap of more than a thousand years. But it was only from the late 16th century that European mathematicians began to make advances without precedent anywhere in the world, so far as is known today.

The first of these was the general solution of cubic equations, generally credited to Scipione del Ferro circa 1510, but first published in Gerolamo Cardano's Ars magna. It was quickly followed by Lodovico Ferrari's solution of the general quartic equation.

From this point on, mathematical developments came swiftly, and combined with advances in science, to their mutual benefit. In the landmark year 1543, Copernicus published De revolutionibus, asserting that the Earth traveled around the Sun, and Vesalius published De humani corporis fabrica, treating the human body as a collection of organs.

Driven by the demands of navigation and the growing need for accurate maps of large areas, trigonometry grew to be a major branch of mathematics. Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595. Regiomontanus' table of sines and cosines was published in 1533.[9]

By century's end, thanks to Regiomontanus (1436—1476) and François Vieta (1540—1603), among others, mathematics was written using Hindu-Arabic numerals and in a form not too different from the elegant notation used today.

[edit] 17th century

The 17th century saw an unprecedented explosion of mathematical and scientific ideas across Europe. Galileo, an Italian, observed the moons of Jupiter in orbit about that planet, using a telescope based on a toy imported from Holland. Tycho Brahe, a Dane, had gathered an enormous quantity of mathematical data describing the positions of the planets in the sky. His student, Johannes Kepler, a German, began to work with this data. In part because he wanted to help Kepler in his calculations, Lord Napier, in Scotland, was the first to investigate natural logarithms. Kepler succeeded in formulating mathematical laws of planetary motion. The analytic geometry developed by Descartes, a Frenchman, allowed those orbits to be plotted on a graph. And Isaac Newton, an Englishman, discovered the laws of physics that explained planetary orbits and also the mathematics of calculus that could be used to deduce Kepler's laws from Newton's principle of universal gravitation. Science and mathematics had become an international endeavor. Soon this activity would spread over the entire world.

In addition to the application of mathematics to the studies of the heavens, applied mathematics began to expand into new areas, with the correspondence of Pierre de Fermat and Blaise Pascal. Pascal and Fermat set the groundwork for the investigations of probability theory and the corresponding rules of combinatorics in their discussions over a game of gambling. Pascal, with his wager, attempted to use the newly developing probability theory to argue for a life devoted to religion, on the grounds that even if the probability of success was small, the rewards were infinite. In a sense this forshadowed the later 18th-19th century development of utility theory.


As we have seen, knowledge of the natural numbers, 1, 2, 3,..., as preserved in monolithic structures, is older than any surviving written text. The earliest civilizations -- in Mesopotamia, Egypt, India and China -- knew arithmetic.

One way to view the development of the various number systems of modern mathematics is to see new numbers studied and investigated to answer questions about arithmetic performed on older numbers. In prehistoric times, fractions answered the question: what number, when multiplied by 3, gives the answer 1. In India and China, and much later in Germany, negative numbers were developed to answer the question: what do you get when you subtract a larger number from a smaller. The invention of the zero may have followed from similar question: what do you get when you subtract a number from itself.

Another natural question is: what kind of a number is the square root of two? The Greeks knew that it was not a fraction, and this question may have played a role in the development of continued fractions. But a better answer came with the invention of decimals, developed by John Napier (1550 - 1617) and perfected later by Simon Stevin. Using decimals, and an idea that anticipated the concept of the limit, Napier also studied a new constant, which Leonhard Euler (1707 - 1783) named e.

Euler was very influential in the standardization of other mathematical terms and notations. He named the square root of minus 1 with the symbol i. He also popularized the use of the Greek letter π to stand for the ratio of a circle's circumference to its diameter. He then derived one of the most remarkable identities in all of mathematics:

e^{i \pi} +1 = 0 .

Answer 2

History Of Mathematics Wikipedia

Answer 3

euclid

Answer 4

one good site is http://www-groups.dcs.st-and.ac.uk/~history/Indexes/HistoryTopics.html