# History of Trigonometry

## Timeline:

1,800 BCE Egyptians and Babylonians

The ancient Egyptians and Babylonians had known of theorems on the ratios of the sides of similar triangles for many centuries. But pre-Hellenic societies lacked the concept of an angle measure and consequently, the sides of triangles were the study instead, an area that would be better called “trialaterometry.” The Babylonian astronomers kept detailed records on the rising and setting of stars, the motion of planets, and the solar and lunar eclipses, all of which require familiarity with angular distances measured on the celestial sphere. Many of the records contained procedures or instructions on how to calculate intervals between astronomical events using the properties of arithmetic progressions. The Babylonians wrote down lists of numbers, in what we would call an arithmetic progression and recognized that numbers repeated themselves over periods of time. (NRICH)

500 BCE Euclid and Archimedes Contribute

Although there is no actual trigonometry in the works of Euclid and Archimedes, in the strict sense of the word, there are theorems presented in a geometric way (rather than a trigonometric way) that are equivalent to specific trigonometric laws or formulas. For instance, propositions twelve, and thirteen of book two of the Elements are the laws of cosine for obtuse and acute angles, respectively. (Wikipedia)

190-120 BCE Hipparchus “The Father of Trigonometry”

Hipparchus a Greek mathematician is often referred to as the “Father of Trigonometry.” He is “credited with constructing the first known table of chords. The importance of Hipparchus' achievement in doing this was to change the mathematical tools involved from the arithmetical 'procedures' of the Babylonian scholars to a geometrical device' , namely the use of arcs of a circle imagined to be on the surface of the 'heavenly sphere'. Even so, he was still using the new technique to investigate the location of heavenly bodies, and the process was still clearly embedded in astronomy. Trigonometry, as a separately identifiable science in its own right, does not appear before the Arab scholars developed it much further in the eleventh and twelfth centuries.” (NRICH) Using an original form of trigonometry using circles and chords, he compiled the first known star catalog and developed a system of the Earth by its latitude and longitude.

70-130 BCE Menelaos

Around 100 BCE Menelaos complied a book of spherical proportions “Sphaerica” in which he set up the basis for treating spherical triangles by using arcs of circles instead of arcs of parallel circles on the sphere. The plane triangle version of the theorem states: “If a straight lines crosses the three sides of a triangle (one the sides has to be produced) then the product of the three of the non-adjacent segments this formed is equal to product of the three segments of the triangle.” (NRICH)

c85-c165 CE Claudius Ptolemy

Some 300 hundred years after Hipparchus developed the basic trigonometry, Ptolemy expanded on his work. Due to a great deal of Hipparchus’ work being lost it was up to Ptolemy that the fundamental ideas of trigonometry were presented to the rest of the world. He was the author of table of chords, which was based on the same principles inherited form the Babylonians, and using more sophisticated observational techniques, more accurate data, and the new mathematics of Euclid, Archimedes, and Menelaos. “Using what is now called "Ptolemy's theorem" (this property of cyclic quadrilaterals was known much earlier) he set up a system where one of the sides of the quadrilateral was a diameter that enabled him to calculate sums and products of chords.” (NRICH)

Passing on the Knowledge

“By this time, mathematicians and astronomers had developed a complex mathematically based science, had a wide range of geometrical techniques whereby they measured the Earth, estimated the distances of the Moon and the Sun, developed a theory of the movement of the planets, and precisely catalogued hundreds of stars. A substantial body of geometrically based mathematics had been developed and scholars made commentaries on the works of Euclid, Apollonius, Archimedes, and others. In the next centuries, Diophantos wrote his Arithmetica, which was to inspire Fermat centuries later, Pappus recorded much of the earlier learning for later generations, and contacts along the trade routes began to be made with people in India and China. In September 622 Mohammed made his famous escape from persecution in Mecca to safety in Medina, and within two hundred years, the Arab culture had established an empire from India through the Middle East and North Africa and into Spain.” (NRICH)

300-400 CE India: The sine, cosine, and versine

Indian scholars had their own way of dealing with astronomical problems and they had great skill in calculation. They did not use the “chord.” Instead they used early versions of trigonometric tables using sines. They divided the 90 degrees arc into 24 sections, thus obtaining values of sines for every 3◦45' of arc. By the 5th century two other functions had been defined; cosine and versine. (NRICH)

770’s Trigonometry in the Arab Civilization

Many works in Greek, Sanskirt, and Syriac were brought by scholars to Al-Mansur’s House of Wisdom and translated. Among these were works of Euclid, Archimedes, and Ptolemy.

940-998 Abul Wafa

Abul Wafa made important contributions to both geometry and arithmetic and was the first to study trigonometric identities. This 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. (NRICH)

The sine, versine, and cosine had been developed in the context of astronomical problems, whereas the tangent and cotangent were developed from the study of shadows of the gnomon. Abal Wafa brought them together and established the relations between the six fundamental trigonometric functions for the first time. He was also the first Arab astronomer to develop ways of measuring the distance between stars using his new system of trigonometric functions including versine.

1292-1336 Richard of Wallingford

Richards’s early works were a series of instructions for the use of astronomical tables that had been drawn up by John Maudith, the Merton College Astronomer. Later he wrote an important work, the Quadripartitum, on the fundamentals of trigonometry needed for the solution of problems of spherical astronomy. The first part of this work is a theory of trigonometrical identities, and was regarded as a basis for the calculation of sines, cosines, cords, and versed sines. The next two parts of the Quadripartitum dealt with a systematic and rigorous exposition of Menelao’s theorem. The work ends with an application of these principles to astronomy. The main sources of the work appear to Ptolemy’s Almagest, and Thabit ibn Qurra (826-901 CE). (NRICH)

1423-1461 Georg von Peuerbach

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. (NRICH)

1436-1476 Johannes Muller von Konigsberg or Regiomontanus

Regiomontanus had become a pupil of Peuerbach at the University of Vienna in 1450. Later, he undertook with Peuerbach to correct the errors found in the Alfonsine Tables. He had a printing press where he produced tables of sines and tangents and continued Puerbach's innovation of using Hindu-Arabic numerals. As promised, he finished Peuerbach's Epitome of the Almagest, which he completed in 1462 and was printed in Venice. The Epitome was not just a translation, it added new observations, revised calculations and made critical comments about Ptolemy's work. Realizing 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) 1464. In his preface to the Reader he says,

"For no one can bypass the science of triangles and reach a satisfying knowledge of the stars .... 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 astronomy 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." (NRICH)

1473-1543 Nicolaus Copernicus

Copernicus wrote a brief outline of his proposed system called the Commentariolus that he circulated to friends somewhere between 1510 and 1514. By this time he had used observations of the planet Mercury and the Alfonsine Tables to convince himself that he could explain the motion of the Earth as one of the planets. The manuscript of Copernicus' work has survived and it is thought that by the 1530s most of his work had been completed, but he delayed publishing the book. (NRICH)

1514-1574 Georg Joachim von Lauchen called Rheticus

Rheticus had facilitated the publication of Copernicus' work, and had clearly understood the basic principles of the new planetary theory.

In 1551, 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. (NRICH)

“By the beginning of the seventeenth century, the science of trigonometry had become a sophisticated technique used in calculating more and more accurate tables for use in astronomy and navigation, and had been instrumental in fundamentally changing man's concept of his world.” (NRICH)

## The History of Trigonometry Integrated into the Classroom:

The information gained from the research of trigonometry could be used in a lesson when introducing trigonometric functions. I am sure students would be as curious as I was to find out where these unfamiliar functions and calculations derived from. I will introduce the main contributors as Hipparchus, Ptolemy, and Abul Wafa. I will have the students develop their own research of the topic in groups and come up with a series of events that took place to create what we know as trigonometry and then have them present it to the class. Each student in the group will have a role to contribute in the research. When teaching trigonometry it will also give me a chance to incorporate other subject areas into the lesson, like science and how trigonometry all began with astronomical calculations. I will be incorporating history, science, and math into the lesson.

## Abul Wafa's findings:

Abal Wafa brought them together and established the relations between the six fundamental trigonometric functions for the first time. He was also the first Arab astronomer to develop ways of measuring the distance between stars using his new system of trigonometric functions including versine.

## TRIG

“By the beginning of the seventeenth century, the science of trigonometry had become a sophisticated technique used in calculating more and more accurate tables for use in astronomy and navigation, and had been instrumental in fundamentally changing man's concept of his world.” (NRICH)