Hipparchos
Who was Hipparchos? (283 words)
Hipparchos is born around 190 BC in Turkey, and died around 120 BC in
Greece (Rhodes) 1). He is the earliest theoretical astronomer who
recorded observations of astronomical phenomena2). He wrote several
books but only one of them is preserved by later copyists. Most of his
work is quoted by Ptolemy in his book Almagest3) which was written
three centuries later.
In dr Struiks chapter on the Greek periode he has a whole sub chapter
on astronomy 4). This shows already the importance of the developments
in astronomy and how they are linked to mathematics. The sub chapter
starts with a general introduction and back ground and then goes via
Eudoxos with his planet theory ( a kinematic model of the planets
instead of using only tabels), to Aristarchos (heliocentric
hypothesis) to Hipparchos van Nicaea.
Hipparchos made lengthy observations, between 161 and 126 BC4), to
determine the length of a tropical year. He suspected that this length
was not a constant number of days. He discovered the precession of the
equinoxes. He made it therefor clear that the celestial equator was
not fixed among the stars, but precessed by a small amount2) . His
observations led to the use of eccentric circles and epicycles to
describe the movements of the sun, moon and planets
There are ideas that the whole concept of the geographic coordinates
that we use nowadays with longtitude and latitude is his idea. Also
that these coordinates could be determined using astronomical
measurements. However, it was not until much later, in the middle of
the eighteenth century, that it was discovered how to determine the
geographical length using good tabels of the position of the moon,
together with accurate chronometers4).
How did his world look like? (185 words)
The amount of highly educated people was not as big as it is nowadays
in Holland. There were only a few of them, and they lived scattered
around Europe and the Far East. However, most of them travelled a
lot. From the work of Hipparchos it is evident that there must have
been connections between the Greeks and the Babylonian astronomy.
Hipparchos lived in the Hellenistic period. This was a period with a
urban civilization. There were also rural places, but there was not
much contact between the rural places and the cities. In the cities,
which were before owned by the eastern people, now were inhabited by
the greeks. The Greeks thought their own norms and values were the
best so others had to follow. However, they also learned a lot from
the Babylonian mathematics 4).
Starting with Euclid (306-283) it was the period when the first
professional mathematicians begun to exist. This is the person who
dedicates his life to the praction of science. So it is convincing
that, observing the amount of data and thus work included, Hipparchos
was a professional scientist.
Literature
1) http://en.wikipedia.org/wiki/Hipparchus
2) Roger Cooke, The History of Mathematics, a brief course, 1997 by
John Wiley & sons, Inc . (The paragraph 7.3 in which Hipparchos is
described is part of chapter 7 on the Hellenistic Mathematical
Periode).
3) Ptolemy, The Almagest
4) D.J. Struik, 1965, Geschiedenis van de meetkunde, Aula boeken (dr
Struik notes that he uses the following book for his research:
O. Neugebauer, exact science in antiquity, studies in civilization,
university of Pennsylvania, Philadelphia, 1942 pages 22 – 31
[Marloes Kloosterboer, Carla Molenaar, Jacob Wietsma]
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Viktor J Katz
Hipparchus of Bithynia lived from 190 - 120 B.C.E. He was a Greek
astronomer, a major contributor to the development of mathematical
astronomy, between the times of Plato and Ptolemy. The primary reason
for the study of astronomy was the solving of problems connected with
the calendar, problems such as determination of the seasons, the
prediction of eclipses and the establishment of the beginning of the
lunar month. Hipparchus sytematically carried out numerous
observations of planetary positions, introduced a coordinate system
for the stellar sphere and began the tabulation of trigonometric
ratios necessary to enable one easily to solve right triangles.
Babylonians, before 300 B.C.E., initiated dividing the circle in 360
parts, called degrees. Hipparchus was one of the first to make use of
this measure. Although there are no explicit documents giving
Hipparchus' table or his method, enough has been pieced together from
various sources to give us a reasonable picture of his work.
Sources:
Thomas Heath, A History of Greek mathematics, 1921
B.L. Van der Waerden, Science Awakening, 1954
Otto Neugebauer, A History of Ancient Mathematical Astronomy (New
York: Springer, 1975)
Gerald J. Toomer, Ptolemy's Almagest (New York: Springer, 1984)
[Arina de Groot, Serieke Kloet]
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The Norton History of the Mathematical Sciences
Hipparchos lived from 190 BC until 120 BC. He probably lived in Asia
Minor (modern Turkey). He is represented in the list of principal
known figures in Greek mathematics. Of all ancient cultures, the
Greeks had by far the largest influence on the development of
mathematics in Europe. His work was about astronomy and
trigonometry. He was a major contributor of geometry of the
heaven. His main insight was to relate part of an orbit (epicyclic or
not) to an angle that it subtended at some point. One of his most
important theorems related an angle to its half-angle, from which he
was able to compile tables of chords. From his proportion he was able
to compile one of the first tables of values of chord for a sequence
of bisected angles. Hipparchos used this angle-chord trigonometry
extensively in his astronomy. Also he developed spherical as well as
planar trigonometry.
[Franziska van Dalen en Kees Vermaat]
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Cajori - History of Mathematics
Hipparchus is considered to be one of the greatest astronomers of
antiquity. He lived in Nicaea in Bithynia, his astronomical
observations date between 161 and 127 BC and is mentioned as the
(only) mathematician of real genius between Apollonius and
Ptolemy. Alexandria had become the capital of the region and
flourished until it became part of the Roman Empire (the first
Alexandrian school, 300 – 30 BC).
Hipparchus was interested in mathematics for its usefulness in
astronomy. There are no existing copies of his mathematical writings
but Theon of Alexandria gives him credit as the originator of the
science of trigonometry. He calculated in twelve books ‘a table of
chords’which presumed certain knowledge of arithmetic and
algebra. Ptolemy used his theory on epicycles and eccentrics for his
“Ptolemaic System" in which the earth is in the center of the
universe, and that the sun and planets revolve around the earth.
Sources:
No specific references in the parts where Hipparchus is mentioned. The
book has a list of extensively used sources without specific
references in its text. Probably relevant for the period of Hipparchus
are:
* Cantor, Moritz. Vorlesungen über Geschichte der
Mathematik. Leipzig. Bd. I., 1880; Bd. II., 1892.
* Gow, James. A Short History of Greek Mathematics. Cambridge,1884.
* Hankel, Hermann. Zur Geschichte der Mathematik im Alterthum und
Mittelalter. Leipzig, 1874.
* Allman, G. J. Greek Geometry from Thales to Euclid. Dublin, 1889.
* Loria, Gino. Die Hauptsächlichsten Theorien der Geometrie in ihrer
früheren und heutigen Entwickelung, transl. by Fritz
Schütte. Leipzig, 1888.
[Ed van Weegen and Emil Jamrich]
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Katz, Victor J., A History Of Mathematics – An Introduction, 3d
Publishers Inc., 1998, ISBN 978-0-321-38700-4
Hipparchos of Bithynia (190 – 120 B.C.E.)
When and where did he live? See above.
What did he do? He was an astronomer and continued the work of Plato
with the big difference that he was the first to apply trigonometry
and coordinates to explain the movement of stars and planets.
Mathematical culture back then? The earth was the centre of the
system. Many academics tried to explain the system of stars and
planets without hurting religious beliefs. Many worked on Plato’s
challenge before like Eudoxus who worked with single spheres and
Appolonius (who discovered that the seasons don’t have the same
length) and after Hipparchos died, Ptolemy calculated square roots
(like Archimedes did 4 centuries before) and he continued to further
work out Hipparchos’ theory.
Who paid him? A guess (No reference is made in our source): Very rich
people or religious institutions.
Why is he remembered? Hipparchos is remembered because he was the
first astronomer to use calculations of trigonometry as well as
coordinates to explain the movement of planets and stars.
How do THEY know? Unfortunately there is not a lot of original source
from Hipparchos’ work left but – after his death - Ptolemy used
Hipparchos’ theory in his studies which became the most influential
astronomical work back in those days which was copied and analysed
countless times.
[Niko Kuijper, Sabrina Berkenbosch and Zala Jamsek]
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Morris Kline
Hipparchos lived in Alexandrian Greek, died in 125 BC. Scientists
worked mostly on mechanics then. Most important disciplines for
mathematicians were: arithmetic, geometry and astronomy. They were
supported by state funds. His goal was to study quantitative
astronomy. They looked at paths and positions of heavenly bodies and
they tried to predict this. Moreover this science was used for time
telling and helped with calenders, navigation and geography. Part of
his work was studying circles and their circumference. He came up with
the idea to use ideas of Hypsides and the Babylonians for telling
time. He used the arc to calculate chords, this is equivalent with the
sine function we know. He is remembered by trigonometry, remembered
because of the work of Ptolemy, whom he worked with, as well as
Menelaus. Only one thing is left of his work, namely 'Commentary on
the Pheanomena of Eudoxus and Aratus'. He is mostly known for
calculating the length of a solar year.
[Eline Renskers, Rowan Reijtenbagh]
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Amy Dahan-Dalmédico and Jeanne Pfeiffer
In this book Hipparchus has been mentioned only on one page.
“The great geometers did their research in the framework created by
classical geometry and naturally turned toward application, in
favorite: Astronomy.”
Hipparchus (second century B.C.E.) is the founder of the spherical
trigonometry. It was a necessary conformable tool required because of
the geometric postulate of the spherical shape of the heavens and the
discovery of the spherical shape of the earth. “The construction of a
table of chords of a circle is attributed to him.”
Because, Strato organized a Museum in Alexandria, a community of
scholars paid by the king to devote themselves to scientific research,
we expect Hipparchus has been paid by the king for doing the research.
The school of Alexandria utilized the knowledge of the classical
period and enlarged the field of Greek mathematics by extending its
investigations to mechanics, astronomy and trigonometry. They also
reconnected it with the more algebraic tradition of the Babyloniers.
Sources:
Burton, D.M. The History of Mathematics, 4th ed. McGraw-Hill 1999
Heath, T.L. History of Greek Mathematics Oxford 1921
Katz, V. A History of Mathematics, An introduction, 3rd Edition
Addison-Wesley 2009
Stillwell, J. Mathematics and its History Springer Verlag 2002
[Nienke Keizer, Peter 't Hart]
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Hipparchus was born c. 190 B.C. in Nicaea, Bithynia (now Iznik,
Turkey) and he died c. 120 B.C. on the island of Rhodes. He was a
Greek astronomer and mathematician and lived in the Hellinistic
period.
He worked about 45 years as an astronomer, built on the work of the
Babylonians and is considered to be the greatest ancient astronomical
observer. He was the first to use quantitative and accurate models to
describe the eclipse of the Sun and the Moon and is also considered as
the founder of trigonometry. His most famous work is his discovery of
the precession of the equinoxes.
The Hellenistic period is the period of the ancient Greek, between the
death of Alexander the Great and the emergence of the Roman Empire. At
this time, Greek cultural influence and power was at its top in
Europa, Asia and Africa and resulted in prosperity and progress in the
arts, mathematics, philosophy, science, literature, architecture and
music.
Sources:
1 - A contextual History of Math, by Ronald Calinger
2 - The history of math mathematics, by Carl B. Boyer
3 - Wikipedia
[Nader Emani, Koen Jansen]
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History of Mathematics by David Eugene Smith, 1923
Hipparchos lived from 180-125 B.C. He was born in Nicaea, Bythigaia,
Asia Minor. He lived in a restless period with lots of war and a
declining Greek society. The fact that he was able to dedicate a part
of his life to science and mathematics, is therefore
exceptional. Smith leaves it unclear by whom Hypparchos was paid.
While living in Rhodes, Hipparchos wrote an important astronomical
work about basic principles of astronomy-science. Hipparchos noted 850
of the stars already. Out of necessity for measurements on behalf of
the astronomy, he developed the basics of trigonometry and spherical
trigonometry.
Smith considers him to be the founder of this branch of mathematics,
although Smith isn’t sure that he proved any major theorems or
formula’s in the field. He also laid the fundamentals for the theory
of stereographic projection and he wrote a commentory on the work of
Eudoxus.
The quotes of Smith come basicly from secundary sources, of which the
most important have been named below.
Sources of Smith:
Almagest; Halma's French Translation (Paris 1821)
Boll, F; Die Sternkataloge des Hipparchos und des Ptolemaios
Braummühl; Geschichte I
Heath; History II
Delambre; Histoire de l'astronomie ancienne I
Heath; Euclid Vol II
[Jan-Jelle de Wit en Annemarie van Randen]
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A History of Mathematics van Carl Boyer.
A History of Mathematics, C.B. Boyer, 3rd version, Dover Publications,
1968, New York
Note: this book does not provide any information about the Hipparchos
Test specifically, nor the payment of mathematicians in the
Alexandrian age, but only a description about the situation in which
Hipparchos was involved.
Hipparchus live from approximately 180-125 BC (page 179) and was part
of the mathematical culture in the Alexandrian Age (page 176).
Hippartchos created the first trigonometric table, for which he earned
the title “father of trigonometry”(page 179).
Until he came up with this table, “Greek mathematicians had studied
relationships between lines and circles, but no systematic
trigonometry had resulted. More and more, the astronomers of the
Alexandrian Age had handled problems pointing to a need for more
systematic relationships between angles and circles”(page 176). For
instance, Aristarchus (ca 310 – ca 230 BC) had known that in a given
circle the ratio of arc to chord decreased as the angle decreases from
180 to 0.
The main contributions of Hipparchus can be summarized as: the drawing
up a star catalogue, improvement in important astronomical constants (
such as the length of the year, month, the size of the moon) and the
discovery of the precession of the equinoxes.
It is not known how this table has been produced, but it is likely
that his method is similar to Ptolemy’s (page 180). This is described
in his book Ptolemy’s Almagest
The main source for this book is the so-called Historia Mathematica,
by Albert C. Lewis. Additionally, this book refers to several articles
written by several ancient mathematicians, such as Pappus from
Alexandria, and the book Ptolemy’s Almagest itself. with or without
additional comments. Furthermore the author refers to other historical
mathematicians, which might have translated or edited former works, or
produced historical works by themselves.
[Ruben Oosterman, Harun Cetin]
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Roger Cooke (The history of mathematics: A brief course, 3rd edition)
Hipparchus was a mathematician and astronomer that lived around
ca. 190 – ca. 120 BCE. From a geocentric point of view he studied the
orbit of the sun around the world. By interpreting lots of measured
data it was known that the speed of the assumed circular movement of
the sun was not constant over the year. This speed was referenced to
the constellation Gemini, which clearly could only be observed during
nights, so when the sun was out of sight. Besides speed variation the
latitude of the sun varies over the year where it reaches a higher
altitude during the summer than during the winter, at the northern
hemisphere. Hipparchos derived that the maximum angle corresponding
with this difference in altitude was about 24˚ with respect to the
annual plane orbit. Furthermore the book tells us that Hipparchus
cited observations from Eudoxus, Ptolemy used data from Hipparchus and
that Hipparchus gave lectures on philosophy.
[Margot van Es]
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