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This is the third of three articles on the History of Trigonometry. Part 2 Sections 5 - 7 can be found here. Part 1 Sections 1 - 4 can be found here. You, who wish to study great and wonderful things, who wonder about the movement of the stars, must read these theorems about triangles. Knowing these ideas will open the door to all of Dating age difference formula for tangents and to certain geometric problems.

For although certain figures must be transformed into triangles to be solved, the remaining questions of astronomy require these books. The first book gives the basic definitions of quantity, ratio, equality, circles, arcs, chords and the sine function. The formula for the area of a triangle is given followed by the sine rule giving examples of its application.

Books III to V cover the all-important theory of spherical trigonometry. The whole book is organised in the style of Euclid with propositions and theorems set out in a logical hierarchical manner. This work, published in was Dating age difference formula for tangents great value to Copernicus. Regiomontanus also built the first astronomical observatory in Germany at Nuremburg with a workshop where he built astronomical instruments.

He also took observations on a comet in that were accurate enough to allow it to be identified as Halley's Comet that reappeared years later. Copernicus wrote a brief outline of his proposed system called the Dating age difference formula for tangents that he circulated to friends somewhere between and By this time he had used observations of the planet Mercury and the Alfonsine Tables to convince himself that he could Dating age difference formula for tangents the motion of the Earth as one of the planets.

The manuscript of Copernicus' work has survived and Dating age difference formula for tangents is thought that by the s most of his work had been completed, but he delayed publishing the book.

His student, Rheticus read the manuscript and made a summary of Copernicus' theory and published it as the Narratio Prima the First Account in Since it Dating age difference formula for tangents that the Narratio had been well accepted by colleagues, Copernicus was persuaded to publish more of his main work, and in he published a section on his spherical trigonometry as De lateribus et angulis traingulorum On the sides and angles of triangles.

It appeared just before Copernicus' death in Pitiscus was commissioned to correct these errors and obtained a manuscript copy of Rheticus' work. Many of the results were recalculated and new pages were printed incorporating the corrections. Eventually, Pitiscus Dating age difference formula for tangents a new work in incorporating that of Rheticus with a table of sines calculated to fifteen decimal places entitled the Thesaurus Mathematicus.

By the beginning of the seventeenth century, the science of trigonometry had become a sophisticated Dating age difference formula for tangents used in calculating more and more accurate tables for use in astronomy and navigation, Dating age difference formula for tangents had been instrumental in fundamentally changing man's concept of his world. In the Indian texts, different astronomers took different values for R, and in most cases the value had to Dating age difference formula for tangents deduced from the context.

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A positive logarithm was necessary when calculations had to be done using tables. The most important Dating age difference formula for tangents was in navigation, for calculating the distance between two points on a sphere.

The perpendicular distance from the mid point of a chord to a curve is still used as a measure of 'deviation from straightness', for example, by railway engineers. It is used also in optics for measuring the curvature of lenses and mirrors, where he versine is sometimes called the sagitta from the Latin for arrow. The Hindu word jiya for the sine was adopted by the Arabs who called the sine jiba.

Eventually jiba became jaib and Dating age difference formula for tangents word actually meant a 'fold'. When Europeans translated the Arabic works into Latin they translated jaib into the word sinus meaning a fold in Latin. In his Practica Geometriae Fibonacci uses the term sinus rectus arcus which soon encouraged Dating age difference formula for tangents universal use of the word sine.

History of Trigonometry - Part 3. The Arabs collect knowledge from the known world The Arab civilisation traditionally marks its beginning from the year CE the date when Muhammad, threatened with assassination, fled from Mecca to Medina where Muhammad and his followers found safety and respect. Over a century later, the Arabs had established themselves as a powerful unified force across large parts of the Middle East and The Caliph Abu Ja'far Al-Mansour moved from Damascus to establish the city of Baghdad during the years to Al-Mansour sent his emissaries to search for and collect knowledge.

From China, they learnt how to produce paper, and using this new skill they started a programme of translation Dating age difference formula for tangents texts on mathematics, astronomy, science and philosophy into Arabic.

The quest for knowledge became a lasting and significant part of Arab Dating age difference formula for tangents. Al-Mansour had founded a scientific academy that became called 'The House of Wisdom'. This academy attracted scholars from many different countries and religions to Baghdad to work together and establish the traditions of Arabic science that were to continue well Dating age difference formula for tangents the Middle Ages.

Some of this work was later translated into Latin by Mediaeval scholars and passed on into Europe. The dominance of Baghdad and the influence of the Arab World was to last for the next years. The scholars in the House of Wisdom came from many cultures and translated the works of Egyptian, Babylonian, Greek, Indian and Chinese astronomers and mathematicians. The name ' Almagest ' has continued to this day and it is recognized as both the great synthesis and the culmination of mathematical astronomy of Dating age difference formula for tangents ancient Greek world.

It was translated into Arabic at least five times and constituted the basis of the mathematical astronomy carried out in the Islamic world. Dating age difference formula for tangents astronomy began to be known in India during the period CE.

However, Indian astronomers had long been using planetary data and calculation methods from the Babylonians, and even though it was well after Ptolemy had written the Almagest, 4th century Indian astronomers did not entirely take over Greek planetary theory.

Ancient works like the Panca-siddhantica now lost that had been transmitted through the version by Vrahamihira [ See Part 1 section 3 ] and Aryabhata's Aryabhatiya CE demonstrated that Indian scholars had their own ways of dealing with astronomical problems and that they had great skill in calculation.

Even in the oldest Indian texts, the Chord [to remind yourself about Chords see the section on Claudius Ptolemy in the previous article ] is not used, and instead there appear some very early versions of trigonometric tables using Sines.

So the relation between the jiya and our sine is:. By the 5th century, two other functions had been defined and used.

This was sometimes called the sama meaning an 'arrow', or sagitta in Latin. This became the standard for later works. Comparison with Varhamihira's Sines in sexagesimal numbers and Hipparchs' table in lengths of chords suggests a possible transmission of at least some of the Greek works to the Hindus.

However, we have no way of knowing this for certain, and it is quite possible that the Hindus calculated their values independently. He produced a remarkable method for approximating values for the Sines, by using the ratio of two quadratic functions. This was based entirely on comparing Dating age difference formula for tangents results of his calculations with earlier values.

The introduction and development of trigonometry into an independent science in the Arab Dating age difference formula for tangents took, in all, some years. In the early s Indian astronomical works reached Dating age difference formula for tangents C aliph Al-Mansur in Baghdad, and were translated as the Zij al-Sindhindand this introduced Indian calculation methods into Islam. Famous for his algebra book, Abu Ja'far Muhammad ibn Musa al-Khwarizmi see The Development of Algebra Part 1 had also written a book on Dating age difference formula for tangents methods of calculation al-hisab al-hindi and he produced an improved version of the Zij al-Sindhind.

Al-Khwarizmi's version of Zij used Sines and Versines, and developed procedures for tangents and cotangents to Dating age difference formula for tangents Dating age difference formula for tangents problems.

Al-Khwarizmi's Zij was copied many times and versions of it were used for a long time. Among these were the works of Euclid, Archimedes Apollonius and of course, Ptolemy.

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The Arabs now had two competing versions of astronomy, and soon the Almagest prevailed. The Indian use of the sine and its related functions were much easier to apply in calculations, and the Dating age difference formula for tangents system from the Babylonians continued to be used, so apart from these two changes, the early Arabic versions of the Almagest remained faithful to Ptolemy.

Abu al-Wafa al-Buzjani Abul Wafa made important contributions to both geometry and arithmetic and was the first to study trigonometric identities systematically. The study of identities was important because by establishing relationships between sums and differences, and fractions and multiples of angles, more efficient astronomical calculations could be conducted and more accurate tables could be established.

The sine, versine and cosine had been developed Dating age difference formula for tangents the context of astronomical problems, whereas the tangent and cotangent were developed from the study of shadows of the gnomon. In his AlmagestAbul Wafa brought them together and established the relations between the six fundamental trigonometric functions for the first time.

From these relations Abul Wafa was able to demonstrate a number of new identities using these new functions:. Greek astronomers had long since introduced Dating age difference formula for tangents model of the universe with the stars on the inside of a vast sphere. They had also worked with spherical triangles, but Abul Wafa was the Dating age difference formula for tangents Arab astronomer to develop ways of measuring the distance between stars using his new system of trigonometric functions including the versine.

By an ingenious application of Menelaos' Theorem [See History of Trigonometry Dating age difference formula for tangents 2 ] using special cases of great circles with two right angles, Abul Wafa Dating age difference formula for tangents how the theorem could be applied in spherical triangles.

This was a considerable advance in Spherical Trigonometry that enabled the calculation of the correct direction for prayer the quibla and was to have important applications in Navigation and Cartography. Abu al-Rayhan Muhammad ibn Ahmad Al-Biruni was an outstanding scholar reputed to have written over treatises on astronomy, science, mathematics, geography, history, geodesy and philosophy.

Only about twenty of these works now survive, and only about a dozen of these have been published. Al-Biruni's treatise entitled Maqalid 'ilm al-hay'a Keys to Dating age difference formula for tangents Science of Astronomy ran to over one thousand pages and contained extensive developments in on trigonometry. Among many theorems, he produced Dating age difference formula for tangents demonstration of the tangent formula, shown below.

While many new aspects of trigonometry were being discovered, the chord, sine, versine and cosine were developed in the investigation of astronomical problems, and conceived of as properties of angles at the centre of the heavenly sphere.

In contrast, tangent and cotangent properties were derived from the measurement of shadows of a gnomon and the problems of telling the time.

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In his Demarcation of the Coordinates of Cities he used spherical triangles for finding the coordinates of cities and other places to establish local meridian the Dating age difference formula for tangents and thereby finding the correct direction of Mecca, and in his Exhaustive Treatise on Shadows he showed how to use gnomons [See A Brief History of Time Measurement ] for finding the time of day. Abu Muhammad Jabir ibn Aflah Jabir ibn Aflah c - c probably worked in Seville during the first part of the 12th century.

His work is seen as significant in passing on knowledge to Europe. Jabir ibn Aflah was considered a vigorous critic of Ptolemy's astronomy. His treatise helped to spread trigonometry in Europe in the 13th Dating age difference formula for tangents, and his theorems were used by the astronomers who compiled the influential Libro del Cuadrante Sennero Book of the Sine Quadrant under the patronage of King Alfonso X the Dating age difference formula for tangents of Castille A result of this project was the creation of much more accurate astronomical tables for calculating the position of the Sun, Dating age difference formula for tangents and Planets, relative to the fixed stars, called the Alfonsine Tables made in Toledo somewhere between and These were the tables Columbus used to sail to the New World, and they remained the most accurate tables until the 16th century.

By the end of the 10th century trigonometry occupied an important place in astronomy texts with chapters on Dating age difference formula for tangents and chords, shadows tangents and cotangents and the formulae for spherical calculations. There was also considerable interest in the resolution of plane triangles. But a completely new type of Dating age difference formula for tangents by Nasir al-Din al-Tusi Al-Tusi entitled Kashf al-qina 'an asrar shakl al-qatta Treatise on the Secrets of the Sector Figurewas the first treatment of trigonometry in its own right, as a complete subject apart from Astronomy.

The work contained a systematic discussion on the application of proportional reasoning to solving plane and spherical triangles, and a thorough treatment of the formulae for solving triangles and trigonometric identities.

Al-Tusi originally wrote in Persian, but later wrote an Arabic version. The only surviving Persian version of his work is in the Bodleian Library in Oxford. This was a collection and major improvement on earlier knowledge.

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Book III deals with the basic geometry for spherical triangles and the resolution of plane triangles using the sine theorem:. Book V contains the principal chapters on trigonometry dealing with right-angled triangles and the six fundamental relations equivalent to those we use today; sine, cosine, tangent, cotangent, secant and cosecant.

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He provided many new proofs and showed how they could be used to solve many problems more easily. The Arab astronomers had learnt much from India, and there was contact with the Chinese along the Silk Road Dating age difference formula for tangents through the sea routes, so that Arab trading posts were established in India and in Dating age difference formula for tangents. Through these contacts Indian Buddhism spread into China and was well established by the 3rd century BCE, probably later Dating age difference formula for tangents with it some of the calculation techniques of Indian astronomy.

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