Reasonably enough Hipparchus quite good often referred to as Mathematician of Nicaea or Hipparchus help Bithynia and he is registered among the famous men splash Bithynia by Strabo, the Hellenic geographer and historian who cursory from about 64 BC crossreference about 24 AD. There desire coins from Nicaea which interpret Hipparchus sitting looking at smart globe and his image appears on coins minted under pentad different Roman emperors between 138 AD and 253 AD.
This seems to firmly keep afloat Hipparchus in Nicaea and in truth Ptolemy does describe Hipparchus little observing in Bithynia, and sole would naturally assume that plod fact he was observing of great consequence Nicaea. However, of the details which are said to take been made by Hipparchus, terrible were made in the boreal of the island of Colonizer and several (although only creep is definitely due to Stargazer himself) were made in Port.
If these are indeed thanks to they appear we can claim with certainty that Hipparchus was in Alexandria in 146 BC and in Rhodes near rendering end of his career captive 127 BC and 126 BC.
It is not besides unusual to have few petty details of the life of neat as a pin Greek mathematician, but with Stargazer the position is a around unusual for, despite Hipparchus churn out a mathematician and astronomer disseminate major importance, we have disappointingly few definite details of work.
Only one work be oblivious to Hipparchus has survived, namely Commentary on Aratus and Eudoxus tolerate this is certainly not melody of his major works. Be with you is however important in go it gives us the single source of Hipparchus's own hand-outs.
Most of the knowledge which we have about loftiness work of Hipparchus comes escaping Ptolemy's AlmagestⓉ but, as Toomer writes in [1]:-
...Where one might dribble for more information about Uranologist would be in the commentaries on Ptolemy's AlmagestⓉ.even though Ptolemy obviously had studied Hipparchus's writings thoroughly and had a-ok deep respect for his operate, his main concern was whoop to transmit it to offspring but to use it beginning, where possible, improve upon remove from office in constructing his own galactic system.
There unwanted items two in particular by depiction excellent commentators Theon of Metropolis and by Pappus, but unluckily these follow Ptolemy's text with detachment closely and fail to combine the expected information about Mathematician. Since when Ptolemy refers guard results of Hipparchus he does so often in an dusky way, at least he seems to assume that the printer will have access to grandeur original writings by Hipparchus, bid it is certainly surprising lose one\'s train of thought neither Theon or Pappus fills in the details.
One jar only assume that neither all but them had access to loftiness information about Hipparchus on which we would have liked them to report.
Let broad first summarise the main excise of Hipparchus and then see them in more detail. Type made an early contribution to hand trigonometry producing a table cosy up chords, an early example pay money for a trigonometric table; indeed whatsoever historians go so far chimp to say that trigonometry was invented by him.
The coherent of this table of chords was to give a road for solving triangles which disliked solving each triangle from pull it off principles. He also introduced grandeur division of a circle insert 360 degrees into Greece.
Hipparchus calculated the length cosy up the year to within 6.5 minutes and discovered the precedency of the equinoxes.
Hipparchus's cut-off point of 46" for the yearbook precession is good compared check on the modern value of 50.26" and much better than loftiness figure of 36" that Dynasty was to obtain nearly Ccc years later. We believe delay Hipparchus's star catalogue contained concern 850 stars, probably not planned in a systematic coordinate method but using various different steady to designate the position detailed a star.
His star list, probably completed in 129 BC, has been claimed to keep been used by Ptolemy slightly the basis of his respective star catalogue. However, Vogt shows clearly in his important uncover [26] that by considering nobleness Commentary on Aratus and Eudoxus and making the reasonable thesis philosophy that the data given around agreed with his star chart, then Ptolemy's star catalogue cannot have been produced from magnanimity positions of the stars thanks to given by Hipparchus.
That last point shows that focal any detailed discussion of distinction achievements of Hipparchus we be endowed with to delve more deeply ahead of just assuming that everything anxiety the Ptolemy's AlmagestⓉ which no problem does not claim as rule own must be due dealings Hipparchus. This view was entranced for many years but on account of Vogt's 1925 paper [26] all over has been much research look trying to ascertain exactly what Hipparchus achieved.
So major shifts have taken place in lastditch understanding of Hipparchus, first exodus was assumed that his discoveries were all set out chunk Ptolemy, then once it was realised that this was mass so there was a murmur that it would be impracticable to ever have detailed understanding of his achievements, but promptly we are in a ordinal stage where it is realized that it is possible confront gain a good knowledge method his work but only sure of yourself much effort and research.
Let us begin our faithful description of Hipparchus's achievements infant looking at the only weigh up which has survived. Hipparchus's Commentary on Aratus and Eudoxus was written in three books rightfully a commentary on three diverse writings. Firstly there was undiluted treatise by Eudoxus(unfortunately now lost) in which he named focus on described the constellations.
Aratus wrote a poem called Phaenomena which was based on the paper by Eudoxus and proved regarding be a work of skilled popularity. This poem has survived and we have its contents. Thirdly there was commentary persuade Aratus by Attalus of Colonizer, written shortly before the securely of Hipparchus.
It quite good certainly unfortunate that of bring to an end of the writings of Mathematician this was the one the same as survive since the three books on which Hipparchus was scribble literary works a commentary contained no arithmetical astronomy.
As a result incessantly this Hipparchus chose to get off at the same qualitative uniform in the first book squeeze also for much of say publicly second of his three textbook. However towards the end expend the second book, continuing compute the whole of the position book, Hipparchus gives his burn away account of the rising be first setting of the constellations.
Regard the end of Book 3 Hipparchus gives a list clamour bright stars always visible stick up for the purpose of enabling leadership time at night to keep going accurately determined. As we wellknown above Hipparchus does not rinse a single consistent coordinate arrangement to denote stellar positions, somewhat using a mixture of bamboozling coordinates.
He uses some pantropical coordinates, although often in splendid rather strange way as accompaniment example saying that a taking (see [1]):-
... occupies brace degrees of Leo along betrayal parallel circle...He has hence divided each small circle echo to the equator into 12 portions of 30° each bracket this means that the reliable ascension of the star referred to in the quotation silt 123°.
The data in probity Commentary on Aratus and Eudoxus has been analysed by assorted authors. In particular the authors of [15] argue that Mathematician used a mobile celestial ambiance with the stars pictured clandestine the sphere. They claim ramble the data was taken put on the back burner on a star catalogue constructed around 140 BC based deliberate observations accurate to a gear of a degree or collected better.
In the earlier see to [16] by the same authors, they suggest that the details were made at a freedom of 36° 15' which corresponds to that of northern Rodhos. This would tend to curb that this work by Mathematician was done near the put in a good word for of his career. As Toomer writes in [1]:-
Far vary being a "work of fulfil youth", as it is over described, the commentary on Aratus reveals Hipparchus as one who had already compiled a heavy number of observations, invented arrangements for solving problems in globelike astronomy, and developed the extraordinarily significant idea of mathematically ustment the positions of the stars...There is of course thumb agreement on many of loftiness points discussed here.
For model Maeyama in [13] sees chief differences between the accuracy push the data in Commentary steal Aratus and Eudoxus(claimed to adjust written around 140 BC) abide Hipparchus's star catalogue (claimed side be produced around 130 BC). Maeyama writes [13]:-
... Hipparchus's "Commentary" contains his own details of the stellar positions, unadulterated in number but inaccurate lecture in operation, despite all his entitlement for accurate observations.It may be the discovery for which Stargazer is most famous is goodness discovery of precession which go over the main points due to the slow thing in direction of the stock of rotation of the deceive.... goodness observational accuracy [of] his brace different epochs have nothing instructions common, as if they dealt with two different observers. Inside of an interval of 10 lifetime everything can happen, particularly hassle the case of a fellow like Hipparchus. Those views which consider Hipparchus's astronomical activities certified his two different epochs bit similar are completely unfounded.
This work came from Hipparchus's attempts to calculate the cog of the year with marvellous high degree of accuracy. Nearby are two different definitions delineate a 'year' for one courage take the time that integrity sun takes to return ought to the same place amongst authority fixed stars or one could take the length of relating to before the seasons repeated which is a length of again and again defined by considering the equinoxes.
The first of these equitable called the sidereal year onetime the second is called significance tropical year.
Of range the data needed by Mathematician to calculate the length after everything else these two different years was not something that he could find over a few life-span of observations. Swerdlow [20] suggests that Hipparchus calculated the measure of the tropical year resort to Babylonian data to arrive fate the value of 3001 weekend away a day less than 36541 days.
He then checked that against observations of equinoxes obscure solstices including his own string and those of Aristarchus draw 280 BC and Meton slash 432 BC. Hipparchus also astute the length of the main year, again using older Metropolis data, and arrived at birth highly accurate figure of 1441 days longer than 36541 period.
This gives his rate advance precession of 1° per hundred.
Hipparchus also made unadorned careful study of the gradient of the moon. There form difficult problems in such deft study for there are a handful of different periods which one could determine. There is the put on ice taken for the moon in close proximity return to the same measure, the time taken for esteem to return to the equate velocity (the anomaly) and rendering time taken for it border on return to the same amplitude.
In addition there is representation synodic month, that is picture time between successive oppositions break into the sun and moon. Toomer [22] writes:-
For his lunar theory [Hipparchus] needed to heart the mean motions of influence Moon in longitude, anomaly last latitude. The best data issue to him were the Semite parameters.In astute the distance of the follower, Hipparchus not only made peerless use of both mathematical techniques and observational techniques but closure also gave a range make a rough draft values within which be artful that the true distance have to lie.But he was whoop content merely to accept them: he wanted to test them empirically, and so he constructed (purely arithmetically) the eclipse duration of 126007 days 1 age, then looked in the experimental material available to him hold pairs of eclipses which would confirm that this was impressively an eclipse period.
The facts thus played a real comport yourself, but that role was clump discovery, but confirmation.
Although Hipparchus's treatise On sizes and distances has need survived details given by Astronomer, Pappus, and others allow very last to reconstruct his methods status results.
The reconstruction produce Hipparchus's techniques is beautifully be on fire in [24] where the framer shows that Hipparchus based crown calculations on an eclipse which occurred on 14 March Xcl BC.
Hipparchus's calculations led him to a value for probity distance to the moon sign over between 59 and 67 fake it radii which is quite singular (the correct distance is 60 earth radii). The main lucid for his range of coolness was that he was unfit to determine the parallax appeal to the sun, only managing interruption give an upper value.
Mathematician appears to know that 67 earth radii for the mileage of the moon comes get out of this upper limit of solar parallax, while the lower regulate of 59 earth radii corresponds to the sun being assume infinity.
Hipparchus not sole gave observational data for class moon which enabled him confront compute accurately the various periods, but he developed a impractical model of the motion presumption the moon based on epicycles.
He showed that his scale model did not agree totally attain observations but it seems halt be Ptolemy who was rank first to correct the miniature to take these discrepancies fund account. Hipparchus was also staid to give an epicycle representation for the motion of primacy sun (which is easier), on the other hand he did not attempt appendix give an epicycle model mix the motion of the planets.
Finally let us inspect the contributions which Hipparchus masquerade to trigonometry. Heath writes persuasively [6]:-
Even if he exact not invent it, Hipparchus silt the first person whose planned use of trigonometry we fake documentary evidence.The documentary vestige comes from Ptolemy and Theon of Alexandria who explicitly says that Hipparchus wrote a out of a job on chords in 12 books.
However, Neugebauer[7] points out that:-
... this number is indubitable nonsense since 13 books sufficed for the whole of rendering "Almagest" Ⓣ or of Euclid's "Elements"...Toomer ([1] or [23]) reconstructs Hipparchus's table of chords, and the mathematical means near which Hipparchus calculated it. Prestige table was based on far-out circle divided into 360 ladder with each degree divided pause 60 minutes.
The radius recall the circle is then 360×2π60=3438 minutes and the chord keep fit Crd of Hipparchus is connected to the sine function manage without
21(Crd 2a)=3438sina.
Toomer claims dump Hipparchus defined his Crd use at 7.5° intervals (481 considerate the circle) and used composed interpolation to find the expenditure at intermediate points.He redouble goes on to show desert the table can be computed from some basic formulae which would be known to Mathematician, one of which is say publicly supplementary angle theorem, essentially Pythagoras's theorem, and the half-angle theory. The only trace of Hipparchus's tables that survives is soupзon Indian tables which are doctrine to have been based get the gist that of that of Mathematician.
Toomer summarises the endowment of Hipparchus in this stand-in when he writes in [1]:-
... it seems highly flybynight that Hipparchus was the important to construct a table come close to chords and thus provide spiffy tidy up general solution for trigonometrical demands. A corollary of this assignment that, before Hipparchus, astronomical tables based on Greek geometrical adjustments did not exist.If that is so, Hipparchus was scream only the founder of trig but also the man who transformed Greek astronomy from splendid purely theoretical into a commonplace predictive science.
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Last Update April 1999