History of Trigonometry

24 Jan

It is not all islamic, the people of the past translated into their own language what have been found out from years before their time and developed on it.

As for trigonometry, it started to be known in India for many centuries before someone from Baghdad got hold of the knowledge from the Indians and translated it to Arabic. Similarly for philosophy and astrology, it was translated from the greeks and developed on and sometimes to correct the incorrect notions and theories.

It is a three part article from nrich.maths.org.

part one (read the full version here) Below is a summarized version.

part two : Eudoxus to Ptolemy’s times and the passing of 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.

part three :

The next part of the story can be found here. It tells how from the 10th century CE Arab scholars collected all kinds of knowledge from ‘even as far as China’. They consolidated the techniques of astronomy, and established trigonometry as a distinct branch of mathematics. During the 12th and 13th centuries many of the Arab works were translated into Latin and the ideas passed into the hands of European scholars. Further contributions were made and new material added, and by 1464, when Regiomontanus had read the available material in Greek and Latin, he completed his major work, On Triangles of Every Kind where trigonometry was set out as a series of logically connected theorems, in the style of Euclid.

1. Ancient Instruments and Measuring the Stars

The most ancient device found in all early civilisations, is a “shadow stick”. The shadow cast from a shadow stick was used to observe the motion of the Sun and thus to tell time. Today we call this instrument a Gnomon. The name gnomon comes from the Greek and refers to any L-shaped instrument, originally used to draw a right angle.

Mason's Square

GNomon 1 Gnomon 2Sundial

For more about sundials go to Leo’s article – Brief History of Time Measurement.

Hindu construction for E-W line

Egyptian Merkhet

The Merkhet is one of the oldest known astronomical instruments. It was developed around 600 BCE and uses a plumb line to obtain a true vertical, as in the picture. The other object is the rib of a palm leaf, split at one end to make a thin slit for a sight. Babylonian and Egyptian astronomers were able to measure the altitude and lateral displacement of heavenly objects from a particular direction by using a Merkhet, thus giving the earliest ideas of turning, or angle.

Wall carving of Egyptians using a Merkhet

The Egyptians divided the 360 degrees of the ecliptic into 36 sections of 10 degrees each. [see Note 1 below]. This division was known before 2300 BCE. Each ten degree section (called a decan from the Greek for ten) contained a constellation of stars lined up along the ecliptic. Since the Earth makes a full rotation in 24 hours, the stars in a new decan will rise above the horizon about every 40 minutes. The system of decans was used for determining the night hours and the seasons.
Egyptian Decan Star Chart
The divisions in the top part of the chart represent decans. The chart was read from right to left and the pictures represent Mars (the boat and the bull), Orion with the three stars including the Sun and Moon, Sirius, Jupiter, Saturn, Mercury and Venus. The lower section contains pictures of star gods or demons. They represent some of the most important days of the year. The chart is largely symbolic and functional but does contain pictures of some significant groups of stars.

2. Babylonian Astronomers

lUnar table

As you can see, Neugebauer published the sexagesimal values for twelve measurements of the position of the Moon taken from a clay tablet dated 133/132 BCE.

3. The Hindu Sulbasutras

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 800 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 Vedic people knew how to find the cardinal directions (NSEW). 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.

4. Chinese Astronomy

The Chinese were the most accurate observers of celestial phenomena before the Arabs. “Oracle Bones” with star names engraved on them dating back to the Chinese Bronze Age (about 2,000 BCE) have been found, and very old star maps have been found on pottery, engraved on stones, and painted on the walls of caves.

Surviving records of astronomical observations made by two astronomers Shi Shen and Gan De date from the 4th century BCE.

Shi Shen wrote a book on astronomy, and made a star map and a star catalogue. In 364 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 (1020-1101) and drawn on paper in 1092 represents the whole sky with the positions of some 1,350 stars.

Chinese Star Map

The equator is represented by the horizontal straight line running through the star chart, while the ecliptic curves above it.

Some elements of Indian astronomy reached China with the expansion of Buddhism (25-220 CE). Later, during the period (618-907 CE) a number of Indian astronomers came to live in China and Islamic astronomers collaborated closely with their Chinese counterparts particularly during (1271-1368).

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