The tangent is the ratio between the sinus and cosinus. This invention of the tangent inspire mathematicians who will succeed him and will cause in the seventeenth century the calculus that we owe to French mathematician Fermat. Europe rediscovered mathematics thanks to Fibonacci in the twelfth century.
He imported from North Africa, where his father was a businessman, zero, the Hindu-Arabic numerals and the decimal system. The Italian mathematics was first developed primarily for accounting and commercial reasons.
In seeking to calculate interest rates, Johann Bernoulli found the value of the constant of logarythm e. Mathematics diffused throughout Europe. In Germany in the fifteenth century, Regiomontanus an astronomer admirer of Ptolemy gave to trigonometry its credentials.
He « detached » trigonometry from Astronomy and made it a branch of wholly mathematics. The use of the term « sinus » « fold » in Latin became final. The rope was dropped. In India, found the relation between the sinus of an angle its projection on the y-axis and cosinus its projection on the x-axis :.
A basic application of the Pythagorean theorem on the triangle formed by the cosine, sine and radius of the circle which is also the hypotenuse of the triangle to easily find this simple but important result. We can not talk about Mathematics without mentioning Pi , the constant that has turned the head of the great thinkers of antiquity. The Indians, early, used to characterize Pi angles within a circle of radius 1. Sine and cosine were clearly linked.
Many other trigonometric formulas has been discovered, the best known are those of the angles:. In Italy, in the sixteenth century, Bombelli had solved the cubic equation by inventing imaginary numbers.
These numbers had the distinction of having a negative square. We must understand the importance of daring Italian mathematician. For the Greeks, a number was an object of nature. A negative number, for example, made no sense: nothing could indeed measure — 5 cubits!
The Indians of the Indus valley was not embarrassed by such considerations. They made a first giant step inventing negative numbers. Bombelli made the second one. That number had no physical meaning! What could be a basket i apples? Yet this magic number allowed blazing advanced mathematics. When falling on difficulty, calculations in imaginary wolrd were facilitated.
We define the complex numbers as follow :. A complex number z is a number which has:. A point M can be represented by its coordinates a and b in the Cartesian coordinate modified complex plane:. The complex plane is a plane in which a point can be defined with sizes from trigonometry. The persecution of Protestants started up again. He showed that the point N of affix z n was also on the circle and formed with the axis of real numbers an angle of nx radians , or more generally:.
When we look at the history of mathematics, whatever the subject, you come one time or another on Euler. This is the case also for trigonometry: The Devil Swiss mathematician establishes the famous formula:. This expression can greatly facilitate calculations. Vous devez vous connecter pour laisser un commentaire.
The string of Hipparchus of Nicea. Geocentric Ptolemaic. The unit circle. Johann Bernoulli. Louis the XIVth — or sun king. Dead see scrolls. Chargement des commentaires…. Impossible de partager les articles de votre blog par e-mail. They were able to predict paths of other objects across the sun, for example the transit of Venus, a description and explanation of which can be found here on Wikipedia.
The Babylonian astronomers recorded astronomical data systematically and by the Seleucid period BCE there were a great many astronomical tablets showing ephemerides for the moon and the major planets. Many of the tablets contain "procedures" or instructions for how to calculate intervals between astronomical events using the properties of simple arithmetic progressions.
These procedural processes were the earliest steps of a mathematical astronomy, and both the procedures and the data were used by those who came later. The Babylonians wrote down lists of numbers, in what we would call an arithmetic progression and recognised that numbers repeated themselves over periods of time. In the table above, the top line shows the end of the year BCE with the last month Aires, so the start of the Babylonian year was at the vernal equinox, and the bottom line represents the end of year BCE.
The height of the lines on the zig-zag graph below approximately represent the sequence of the numerical values in the table. There are two groups of numbers, one starting with 28, followed by another starting The results for Gemini and Cancer differ only in the third place of sexagesimals and the minimum on the graph is interpolated from the results in the table.
Similarly the results for Sagitarius and Capricorn indicate the maximum value for the longitude. Looking at the first three sets of sexagesimal numbers: 28, 55, 57, 58; 28, 37, 57, 58 and 28, 19, 57, 58 we can notice that the significant differences in the second place between 55, 37 and 19 are all giving a constant 18, which is the difference in height of the vertical lines on the zig-zag graph except at the minimum and maximum.
The graph was drawn to illustrate the periodicity of the data. It is important to realise that the Babylonians recognised the events repeated themselves after some time, but they did not see these results as a 'graph' as we can [see Note 3 below]. The use of graphs as a way of recording the data comes from Neugebauer's book The Exact Sciences in Antiquity.
The Babylonian astronomers recognised the events were periodic but they did not have a theory of planetary motion. The Sulbasutras are the only early sources of Hindu mathematical knowledge and originally come from the Vedic period during the second millennium BCE. The earliest written texts we have from this oral tradition date from about BCE. The Sulbasutras are the instructions for constructing various geometrical shapes to make 'fire-altars' using the "Peg and Cord" technique.
Each 'fire-altar' was a different shape and associated with unique gifts from the Gods. The Sulbasutras gave procedures for the construction of the altars by starting with a line marking the E-W direction sun rises in east and sinks in the west , thus the E-W direction had special religious significance. Around this time there was a collection of mathematical knowledge called jyotsia, a mixture of astronomy, calendar calculations and astrology.
The rulers still maintained trading links between western India and the Hellenistic culture of the Roman Empire. At this time, Indian horoscope astrology became popular needing precise calendar and astronomical calculations. The Panca-siddhantica is a collection of five astronomical works composed in the sixth century CE by Vrahamihira. Also, the use of similar calculation methods as the Babylonians suggest that this is the earliest surviving Indian sine table.
This suggests that the Indian invention of the trigonometry of Sines was inspired by replacing the Greek Chord geometry of right triangles in a semicircle by the simpler Sine geometry of right triangles in a quadrant [See Note 5 below]. This discovery is much earlier than the account usually given of the sine table derived from chords by Aryabhata the Elder CE who used the word jiya for sine. Brahmagupta reproduced the same table in CE and Bhaskara gave a detailed method for constructing a table of sines for any angle in CE.
The Chinese were the most accurate observers of celestial phenomena before the Arabs. Shi Shen wrote a book on astronomy, and made a star map and a star catalogue. In BCE Gan De made the first recorded observation of sunspots, and the moons of Jupiter and they both made accurate observations of the five major planets.
Their observations were based on the principle of the stars rotating about the pole equivalent to the earth rotating on its axis. A famous map due to Su Song and drawn on paper in represents the whole sky with the positions of some 1, stars. The equator is represented by the horizontal straight line running through the star chart, while the ecliptic curves above it.
The oldest star map found so far is from Dunhuang. Earlier thought to date from about CE, it was made with precise mathematical methods by the astronomer and mathematician Li Chunfeng and shows stars in Chinese star groups with a precision between 1.
In all there are 12 charts each in 30 degree sections displaying the full sky visible from the Northern hemisphere. Up to now it is the oldest complete preserved star atlas discovered from any civilisation. It has been on display this year in the British Library to celebrate as the International Year of Astronomy. Later, during the period CE a number of Indian astronomers came to live in China and Islamic astronomers collaborated closely with their Chinese counterparts particularly during Very little of the knowledge of the Indians and the Chinese was known in Europe before the Portuguese navigators and the Jesuit scientist Matteo Ricci in the fifteenth century.
Babylonian astronomy contributed direct empirical data as a foundation for Greek theory and exactly the same data which provided the information for the "zig-zag" data results in Babylonian theory were used to calculate the mean motions of the sun and moon by Hipparchus. Pedagogical notes to support this article can be found in the Teachers' Notes accompanying this resource.
Explanations for some of the astronomical terms used in this article can be found here. Part 2 of the History of Trigonometry will take you from Eudoxus to Ptolemy. Katz, V. New York. Addison Wesley. Recommended as the best general history of mathematics currently available. There is good coverage of aspects of astronomy in antiquity, and the discussion on 'functions' p. Princeton University Press. This book contains a wealth of up-to-date information on mathematics and some aspects of astronomy in these ancient civilisations.
Linton, C. Cambridge University Press The first chapter deals with ancient people and early Greek astronomy. Needham, J. Mathematics and the Sciences of the Heavens and the Earth. Cambridge University Press.
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