# Hipparchus’s Contribution in Mathematics

Hipparchus was born in 190 BC in the city of Bithynia, Nicaea. He was a famous astronomer, geographer, and mathematician. He was often referred to as the ‘father of astronomy’. Citizens of Nicaea were very proud of him, and they even used the coins having pictures of Hipparchus minted on them. There is no authentic information about his life as many of his writings are lost. It is believed, that he spent a lot of his time studying astronomy and recording the local weather pattern. Ptolemy mentions more than 20 observations made by Hipparchus from 147 BC to 127 BC, and three earlier observations made by him, from 162 BC to 158 BC. According to him, this was only a fraction of the work of Hipparchus. It was very unusual that his only book ‘Commentary on Aratus and Eudoxus’ survived, but cannot be considered as one of his major works. It is a commentary in two books, on a popular poem by Aratus based on the work by Eudoxus. His major works include his findings in astronomy out of which the most famous is the precession of equinoxes and the introduction of trigonometry in the world of mathematics. His results are considered more accurate than his predecessors. He died in 120 BC; the reason behind his death is not known.

## 1. Chord function

Hipparchus table of chords was based on a circle, divided into 360 degrees, where each degree is further divided into 60 minutes. Toomer retrieved this table by adopting the mathematical approach used by Hipparchus. Then, the radius of the circle is 60/2π = 3438 minutes, and the chord function of Hipparchus is related to sine function by 1/2(Crd 2a)= 3438(sin a). According to Toomer, Hipparchus calculated his cord function at 7.5º intervals and determined the value at intermediate points, using the method of linear interpolation. According to Toomer, this table is computable with some basic formulae that might be known to Hipparchus.

## 2. Father of Trigonometry

Hipparchus computed the first table of chords to support his discoveries in astronomy. This was considered to be his most significant discovery as it allowed other Greek mathematicians to solve any triangle and to make astronomical predictions using their geometric approach. To calculate this table, he inscribed a triangle into a circle so that each side becomes a chord. To complete this table, he used some formulae of plane trigonometry that were either derived by himself or borrowed from some other sources. He is believed to be the first astronomer who quoted the accurate time of rising and setting of zodiac signs. Hipparchus, in his book on the rising of zodiacal signs, demonstrated the calculation shows, it is not necessary that equal arcs of semicircle beginning with cancer, which sets in time having relationships with one another, shows the same relation between the times, in which they rise. Other mathematicians were never able to calculate the actual times. Having a close look at the facts delivered by him, we can say that he used proportions in spherical geometry. Like the rest of his work, his chord table also didn’t survive. Ptolemy’s Almagest proved to be the greatest source for retrieving the writings of Hipparchus on trigonometry. Almagest was a perfect example of clarity and formal perfection. The table of chords in Ptolemy’s Almagest may be similar to the Hipparchus table, but we cannot be sure since we don’t have the original table of Hipparchus.

## 3. Schroder–Hipparchus number

Schroder-Hipparchus numbers are also known as super-Catalan numbers, the little Schroder numbers, or the Hipparchus numbers. It is an integer sequence that can be used to count the number of ways to arrange parenthesis in a sequence, to count the number of ways of arranging 10 digits, etc. The Schroder number sequence is 1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, …. In 1870, Ernst Schroeder reintroduced the concept of Schroder numbers – his latest discovery of that time. Nobody had the idea that two thousand years ago, one of the greatest ancient astronomers, Hipparchus, was already using these numbers. In the picture below, we can see the subdivision of polygons in several ways. Here the nth number in the sequence counts the number of different ways of subdividing a polygon with n + 1 sides into smaller polygons by adding diagonals of the original polygon.

## 4. Combinatorics (Permutations and combinations) – The Lost World

It is a very important branch of science. It deals with the number of ways in which things can be arranged. Combinatorics is used in various fields like computer science, statics, probability, and pure mathematics. Hipparchus is known to introduce the concept of Combinatorics. All his work in this field survived because of the comments made by Plutarch in Table Talk:

Chrysippus said that the number of compound statements obtainable from ten simple statements is over one million. Hipparchus contradicted him, showing that affirmatively there are 103049 compound statements…’

In 1994, at George Washington University, David Hough realized that 103049 is the 10th Schroder number, which is equal to several ways in which 10 different things can be arranged inside parenthesis. This shows that Hipparchus calculations were so advanced that after so many years, in 1870, Ernst Schroder rediscovered these numbers.

## 5. Length of the Year

A year is a time taken by the Earth to complete an orbit around the Sun. For creating an accurate calendar, it is important to observe how long the tropical year is. A tropical year is a time between one summer solstice and the other. Hipparchus observed it carefully and got the most accurate result of all time, which is just 6 minutes more than the actual figure. No one before him was able to approximate this figure. Hipparchus wrote,

I have composed a book ‘On the length of the year’ in which I show that the tropical year contains 365 days plus a fraction of a day which is not exactly 14 day as the mathematicians-astronomers suppose, but which is less than 14 by about 1⁄300, i.e., 365.2429 days which is about 365 days 5 hours 55 min, which is different from actual value 365 days 5 hours 48 min 45 s by only about 6 minutes.”

## 6. The Earth-Moon Distance

Hipparchus observed that the moon shows parallax when viewed from different places on Earth. Parallax means an object appears to be at different places when viewed from different locations. Astronomers usually take the help of geometry when heavenly bodies show parallax. With the help of parallax, Hipparchus measured the Earth-Moon distance twice. In his first attempt, he measured distance as 77 times of Earth’s radii, providing an upper limit as 83 and a lower limit as 71. In the second time, he found the least distance of 62, a mean of 67+1/3, and consequently, the greatest distance of 72+2/3 Earth radii, showing a great deviation from his first result. Hipparchus, later on, recalculated the distance by putting Sun farther from the Earth and got the minimum limit for the mean distance is 59 Earth radii and the mean to be 60 1⁄2 radii of Earth. According to modern scientific results, the actual distance is 60 Earth radii.

## 7. The Earth-Sun Distance

In the second book, Hipparchus fixed a minimum distance to the Sun, i.e., 470 Earth radii. In this scenario, the shadow of the Earth is a cone, and it corresponds to the parallax of 7′ which is the maximum, that he can think of. He observed that during lunar eclipses, the diameter of the shadow cone was 2+½ lunar diameters. With the help of these values and geometry, he determined the mean distance, as it is computed for the minimum distance of the sun, which is the maximum mean distance to the moon. According to Theon of Smyrna, Hipparchus found that the Sun is 1,880 times the size of the Earth, and it is placed at 2,550 Earth radii. Ptolemy later used the Hipparchus method to find the distance between the Earth and Sun. He failed to understand Hipparchus’s strategy of providing bounds to every observation rather than providing a single value and criticized Hipparchus for giving inaccurate results. Hipparchus provided the most accurate observations and results of that time.

## 8. Heliocentric system

Hipparchus was the first to discover a heliocentric system, but he uninhibited his work because his calculations revealed that the orbits were not circular (in the science of that time, it was believed that orbits should be necessarily circular). Aristotle supported Hipparchus, in rejection of the heliocentric system and this idea prevailed for over 2000 years until Copernicus introduced Copernican heliocentrism. Thanks to Johannes Kepler who later proposed that the orbits are elliptical. However, Seleucus of Seleucia remained a supporter of the heliocentric system from the time of Hipparchus.

## 9. Stellar magnitude

The concept of magnitude scale was introduced by Hipparchus. He ranked stars on a numerical scale from 1 to 6, on the basis of their brightness. He placed the brightest star at 1 and the faintest star at 6. When Italian scientist Galileo Galilei observed that the stars were even fainter than the faintest star mentioned by Hipparchus, he put those stars at the seventh magnitude. In 1856, N.R. Pogson placed the magnitudes on a logarithmic scale making, a magnitude 1 star a hundred times brighter than the magnitude 6 stars, thus making each magnitude 2.512 times brighter than the next faintest magnitude.

## 10. Earth’s Precession

Hipparchus discovery of Earth’s precision was the most famous discovery of that time. Earth’s precession means a change in direction of the axis of rotation of Earth. His two books on precession, ‘On the Displacement of the Solsticial and Equinoctial Points’ and ‘On the Length of the Year’, are both mentioned in the Almagest of Ptolemy. This discovery of Hipparchus is a result of his efforts in calculating the length of the year. He took into account two different years, sidereal year, the time taken by the sun to return to the same place amongst the fixed stars, and the tropical year, length of time before the seasons repeated. It took several years to calculate the length of the years. It is believed that he used Babylonian sources to calculate the length of years. For the tropical year, he found the length to be 1⁄300 of a day less than 365 1⁄4 days. Again, using Babylonian data, Hipparchus also calculated the length of the sidereal year to be 1⁄144 days longer than 365 1⁄4 days. This gave him the rate of precession to 1° per century.