Rejecting the usual arid descriptions of sine, cosine, and their trigonometric relatives, he brings the subject to life in a compelling blend of history, biography, and mathematics. He presents both a survey of the main elements of trigonometry and a unique account of its vital contribution to science and social development.

Woven together in a tapestry of entertaining stories, scientific curiosities, and educational insights, the book more than lives up to the title "Trigonometric Delights. He shows how Greek astronomers developed the first true trigonometry. He traces the slow emergence of modern, analytical trigonometry, recounting its colorful origins in Renaissance Europe's quest for more accurate artillery, more precise clocks, and more pleasing musical instruments. Along the way, we see trigonometry at work in, for example, the struggle of the famous mapmaker Gerardus Mercator to represent the curved earth on a flat sheet of paper; we see how M.

Escher used geometric progressions in his art; and we learn how the toy Spirograph uses epicycles and hypocycles.

### Princeton Science Library

Maor also sketches the lives of some of the intriguing figures who have shaped four thousand years of trigonometric history. We meet, for instance, the Renaissance scholar Regiomontanus, who is rumored to have been poisoned for insulting a colleague, and Maria Agnesi, an eighteenth-century Italian genius who gave up mathematics to work with the poor--but not before she investigated a special curve that, due to mistranslation, bears the unfortunate name "the witch of Agnesi.

Additional Product Features Dewey Edition. Maor's presentation of the historical development of the concepts and results deepens one's appreciation of them, and his discussion of the personalities involved and their politics and religions puts a human face on the subject. His exposition of mathematical arguments is thorough and remarkably easy to understand. There is a lot of material here that teachers can use to keep their students awake and interested.

In short, Trigonometric Delightsshould be required reading for everyone who teaches trigonometry and can be highly recommended for anyone who uses it. If you always wanted to know where trigonometry came from, and what it's good for, you'll find plenty here to enlighten you. I highly recommend [it] to teachers who would like to ground their lessons in the sort of mathematical investigations that were undertaken throughout history.

## man who loved

Eli Maor writes clearly, interestingly, and soberly. No dumb jokes to lighten up the subject matter! Just a teaser, really Interesting selection of math topics surrounding "a squared plus b squared equals c squared. The Pythagorean Theorem : A 4,year History. 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 century, 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 Wise of Castille A result of this project was the creation of much more accurate astronomical tables for calculating the position of the Sun, Moon 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 sines 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 work by Nasir al-Din al-Tusi Al-Tusi entitled Kashf al-qina 'an asrar shakl al-qatta Treatise on the Secrets of the Sector Figure , was 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. 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.

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 and through the sea routes, so that Arab trading posts were established in India and in China. Through these contacts Indian Buddhism spread into China and was well established by the 3rd century BCE, probably later carrying with it some of the calculation techniques of Indian astronomy. However few, if any, technological innovations seemed to have passed from China to India or Arabia.

By CE, the Arab empire had reached its furthest expansion in Europe, conquering most of the Iberian peninsula, an area called Al-Andalus by the Arabs. At this time many religions and races coexisted in Iberia, each contributing to the culture.

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The Muslim religion was generally very tolerant towards others, and literacy in Islamic Iberia was more widespread than any other country in Western Europe. By the 10th century Cordoba was said to have equally good libraries and educational establishments as Baghdad, and the cities of Cordoba and Toledo became centres of a flourishing translation business. Between and a series of religiously inspired military Crusades were waged by the Christians of Europe against the Arab Empire. The principal reason was the restoration of Christian control over the Holy Land, but there were also many other political and economic reasons.

In all this turmoil and conflict there were periods of calm and centres of stability, where scholars of all cultures were able to meet and knowledge was developed, translated and transmitted into Western Europe. The three principal routes through which Greek and Arab science became known were Constantinople now Istanbul Sicily and Spain.

Greek texts became known to European monks and scholars who travelled with the armies through Constantinople on their way South to the Holy Land. These people learnt Greek and were able to translate the classical works into Latin. From Sicily, Arabs traded with Italy, and translation took place there, but probably the major route by which Arabic science reached Europe was from the translation houses of Toledo and Cordoba, across the Pyrenees into south-western France.

During the twelfth and thirteenth century hundreds of works from Arabic, Greek and Hebrew sources were translated into Latin and the new knowledge was gradually disseminated across Christian Europe. Geometrical knowledge in early Mediaeval Europe was a very practical subject.

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It dealt with areas, heights, volumes and calculations with fractions for measuring fields and the building of large manors, churches, castles and cathedrals. Hugh of St. Victor in his Practica Geometriae divides the material into Theorica what is known and practised by a teacher and Practica what is done by a builder or mason. Theoretical geometry in the Euclidean sense was virtually unknown until the first translations of Euclid appeared in the West. The astrolabe was commonly used to measure heights by using its 'medicline' a sighting instrument fixed at the centre of the circle and the shadow square engraved in the centre of the instrument, and then comparing the similar triangles.

The horizontal distance from the centre of the astrolabe to the edge of the square was marked with twelve equal divisions. This is from Thomas Digges' Pantometria of The same system is still used, but the square in the quadrant is marked with six divisions. A popular twelfth century text, the Artis Cuiuslibet Consummatio shows the gradual insertion of more technical knowledge, where the measuring of heights altimetry was much more related to astronomy, showing how to construct gnomons and shadow squares. Gradually the translations made on the continent of Europe came to England.

After his ordination as a priest, his Abbot sent him back to Oxford where he studied for nine years. In he became Abbot of St Albans. The Quadripartitum was probably the first comprehensive mediaeval treatise on trigonometry to have been written in Europe, at least outside Spain and Islam.

When Richard was abbot of St. Albans, he revised the work, taking into account the Flores of Jabir ibn Afla. In to Richard also designed a calculation device, called an equatorium , a complex geared astrolabe with four faces. He described how this could be used to calculate lunar, solar and planetary longitudes and thereby predict eclipses in his Tractatus Albionis. It is possible that this led to his design for an astronomical clock described in his Tractatus Horologii Asronomici , Treatise on the Astronomical Clock of , which was the most complex clock mechanism known at the time.

The mechanism comprised a rotating star map that modeled the lunar eclipse and planets by gearing, presented as a geocentric model. It appeared at a transitional period in clock design, just before the advent of the escapement. This makes it one of the first true clocks, and certainly one of first self powered models of the heavens.

Peuerbach's work helped to pave the way for the Copernican conception of the world system; he created a new theory of the planets, made better calculations for eclipses and movements of the planets and introduced the use of the sine into his trigonometry. Peuerbach's Theoricae Novae Planetarum , New Theories of the Planets was composed about was published in by Regiomontanus' printing press in Nuremburg. While the book was involved in attempting a technical resolution of the theories of Eudoxus and Ptolemy, Peuerbach claimed that the movement of the planets was determined by the Sun, and this has been seen as a step towards the Copernican theory.

This book was read by Copernicus, Galileo and Kepler and became the standard astronomical text well into the seventeenth century. In he began working on a new translation of Ptolemy's Almagest, but he had only completed six of the projected thirteen books before died in Realising that there was a need for a systematic account of trigonometry, Regiomontanus began his major work, the De Triangulis Omnimodis Concerning Triangles of Every Kind In his preface to the Reader he says,.

## Trigonometric Delights : Eli Maor :

Rheticus had facilitated the publication of Copernicus' work, and had clearly understood the basic principles of the new planetary theory. In , with the help of six assistants, Rheticus recalculated and produced the Opus Palatinum de Triangulis Canon of the Science of Triangles which became the first publication of tables of all six trigonometric functions. This was intended to be an introduction to his greatest work, The Science of Triangles. When he died his work was still unfinished, but like Copernicus, Rheticus acquired a student, Valentinus Otho who supervised the calculation by hand of some one hundred thousand ratios to at least ten decimal places filling some 1, pages.

This was finally completed in These tables were accurate enough to be used as the basis for astronomical calculations up to the early 20th century. See the notes to this article to read some thoughts on the value of teachig the history of mathematics. Aveni, A, Stairways to the Stars.

Y and Chichester Wiley.

Skywatching in three ancient cultures: Megalithic Astronomy, the Maya and the Inca.