some famous scientists

7/28/2019 some famous scientists

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Johann Carl Friedrich Gauss1777-1855

Johann Carl Friedrich Gauss reportedly believed that there had been only three 'epoch-making'mathematicians: Archimedes,Newton, and one of his own students. While there is puzzlement asto why Gauss would accord this singular honor to his student, many would Gauss himself,sometimes referred to as the "prince of mathematics", the rightful third member of his list.

Gauss, a stickler for perfection, lived by the motto "pauca sed matura" (few but ripe). Hepublished only a small portion of his work. It is from a scant 19 page diary, published only afterGauss's death, that many of the results he established during his lifetime were posthumouslygleaned.

Gauss is portrayed with one of his most important results -- the refutation ofEuclid's fifthpostulate, the 'parallel postulate', which posited that parallel lines would never meet.

Gauss discovered that valid self-consistent geometries could be created in which theparallelpostulate did not hold. These geometries came to be known as 'non-Euclidean geometries".

The parallel lines which begin in the portrait ofEuclid, end and meet in the portrait ofGauss, the two portraits also sharing a common color palette.

Gauss chose not to publish his results in alternative geometries, and credit for the discovery of'non-Euclidean geometry' was accorded to others (Bolyai, and Lobachevski) who arrived atsimilar results independently.

In the Gauss portrait, above the parallel lines which meet, what has come to be known as"Gauss's Equation" (for the second derivatives of the radius vectorr), is inscribed.

Gauss did pioneering work on differential geometry (the specialized study of manifolds), which hedid publish in Disquisition cica superticies curvas.

Overlying Gauss's portrait the Gaussian distribution curve is incised. This probabilitydistribution curve is commonly referred to as the "normal distribution" by statisticians, and,because of its curved flaring shape, as the "bell curve" by social scientists. The Gaussiandistribution has found wide application in numerous experimental situations, where it describesthe deviations of repeated measurements from the mean. It has the characteristics that positiveand negative deviations are equally likely, and small deviations are much more likely than largedeviations.

Gauss is also known for Gaussian primes, Gaussian integers, Gaussian integration, andGaussian elimination -- to name only a few of the achievements directly named after an individualwho was, perhaps, the most gifted mathematician of all time.
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Georg Cantor 1845 - 1918

Georg Cantorundertook the exploration of the "infinite", and developed modern theory on infinitesets. which remains conceptually challenging.

Cantor's work provided an approach to problems that had beset mathematicians for centuries,including Zeno's ancient paradoxes.

He gave the first clear and consistent definition of an infinite set.

Cantor's image is flanked by the "Aleph", the first letter of the Hebrew alphabet, which Cantorused (accompanied by subscripts) in his descriptions oftransfinite numbers -- quite simplynumbers which were not finite.

Cantor recognized and demonstrated that infinite sets can be of different sizes. He distinguishedbetween countable and uncountable sets, and was able to prove that the set of all rationalnumbersQis countable, while the set off all real numbersRis uncountable, and therefore,though both were infinite, Rwas strictly larger.

Backing Cantor's image is a graphic generated from a "Cantor Set".

A "Cantor Set" is an infinite setconstructed using onlythe numbers between0and 1.

A Cantor set is constructed by starting with a line of length 1, and removing the middle 1/3. Next,the middle 1/3 of each of the pieces that are left are removed, and then the middle 1/3 of thepieces that remain after that are removed.

The set that remains after continuing this process forever is called the Cantor set.

The Cantor set contains uncountably many points.

The graphic set which backs Cantor's image began with an algorithm to generate the Cantor set,to which color was applied, and then universal operators related to color transition andmagnification, ultimately resulting in a unique image whose essence was the Cantor set.

The final problem which Cantor grappled with was the realization that there could be no setcontaining everything, since, given any set, there is always a larger set -- its set of subsets.

Cantor came came to the conclusion that the Absolute was beyond man's reach, and identifiedthis concept with God.

In one of his last letters Cantor wrote:

I have never proceeded from any 'Genus supermum' of the actual infinite. Quite the contrary, I

have rigorously proved that there is absolutely no "Genus supremum' of the actual infinite. Whatsurpasses all that is finite and transfinite is no 'Genus'; it is the single, completely individual unityin which everything is included, which includes the Absolute, incomprehensible to the humanunderstanding. This is theActus Purissimus, which by many is called God."

- Georg Cantor
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Ada Byron, Lady Lovelace

1815 - 1852

Ada Byron, Lady Lovelace aspired to be "an Analyst (&Metaphysician)", a title she presciently invented for herselfat a time when the notion of "professional scientist" had

not even taken full form. She not only met herexpectations, but is generally regarded as the first personto anticipate the general purpose computer, and in manysenses the world's first "computer programmer".

A complex intellect, Ada was the daughter of the romanticpoet Lord Byron -- who separated from her mother onlyweeks after Ada's birth, and never met his daughter Ada --and Annabella (Lady Byron), who was herself educated asboth a mathematician and a poet.

By the age of 8 Ada was adept at building detailed modelboats. By the age of 13 she had produced the design for a

flying machine. At the same time she was becoming anaccomplished musician, learning to play piano, violin, andharp, and had a passion for gymnastics, dancing, andriding.

Ada set her sights on meeting Mary Somerville, amathematician who had translated the works ofLaplaceinto English. And it was through her acquaintance withMary Sommerville that, in 1834, Ada met CharlesBabbage, Lucasian professor of mathematics atCambridge -- a post once held by Sir Isaac Newton.

Babbage was the inventor of a calculating machine known

as the "Difference Engine", so-named because it operatedbased on the method of finite differences.

Ada was struck by the "universality" of Babbage's ideas --something few others saw at the time. What was tobecome a life-long friendship blossomed, withcorrespondence that started with the topics ofmathematics and logic, and burgeoned to include allmanner of subjects.

In 1834 Babbage had already begun planning for a newtype of calculating machine -- the "Analytical Engine",conjecturing a calculating machine that could not only

foresee, but act.

When Babbage reported on his plans for this new"Analytical Engine" at a conference in Turin in 1841, oneof the attendees, Luigi Menabrea, was so impressed thathe wrote an account of Babbage's at lectures. Ada, then27, married to the Earl of Lovelace, and the mother ofthree children under the age of eight, translatedMenabrea's article from French into English. Babbagesuggested she add her own explanatory notes.

What emerged was "The Sketch of the AnalyticalEngine", published as an article in 1843, with Ada's notesbeing twice as long as the original material. It became thedefinitive work on the subject of what was to eventually

" "
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Ada's notes were divided into sections. Note A was notsimply technical, but philosophical than technical, and itwas in Note A that Ada anticipated what we would call ageneral purpose computer, suited to:

"The Analytical Engine . is not merely adapted fortabulating ... but for developing and tabulating any

function whatever. In fact the engine may be describedas being the material expression of any indefinite

function of any degree of generality and complexity ... "-from Note A

In Note A, Ada writes about the Analytical Engine'spotential to do anything we are able to instruct it to do --including, if it were properly provided with rules ofharmony and composition, produce "scientific" music.

"Again, it [the Analytical Engine] might act upon otherthings besides number , were objects found whose mutual

fundamental relations could be expressed by those of theabstract science of operations, and which should be alsosusceptible of adaptations to the action of the operatingnotation and mechanism of the engine ... Supposing forinstance, that the fundamental relations of pitchedsounds in the science of harmony and of musical

composition were susceptible of such expression andadaptations, the engine might compose elaborate andscientific pieces of music of any degree of complexity orextent."

- from Note A

There is a poetry in Ada's comparison of the AnalyticalEngine and the Jaquard loom:

"We may say most aptly that the Analytical Engineweaves algebraic patterns just as the Jacquard-loomweaves flowers and leaves."

Notes B through F delve into the functions and capabilitiesof the Analytical engine.

Note D is particularly prescient. It sets out the method forcalculating the Bernoulli number sequence, and isgenerally regarded as the first "computer program".

Note G which includes a discussion of the futurecapabilities of the Analytical Engine, is a remarkableanticipation of the modern day computer:

"The Analytical Engine has no pretensions whatever tooriginate any thing. It can do whatever we know how toorder it to perform. It can follow analysis; but it has no

power of anticipating any analytical relations or truths.Its province is to assist us in making available what weare already acquainted with. ... It is likely to exert anindirect and reciprocal influence on science itself inanother manner. For in so distributing and combiningthe truths and formulas of analysis ... the relations andthe nature of many subjects in that science arenecessarily thrown into new lights, and more profoundlyinvestigated."

- from Note G

In 1852, Ada Byron, Lady Lovelace, died from cervical

cancer. She was 36 years old.


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, .

Ada Byron's father, Lord Byron , had also died at age 36.It is reported that one of the last things he said was

"Oh my poor dear child! My dear Ada! My god, could I buthave seen her!"

-attributed to Lord Byron

Charles Babbage never completed a working model of theDifference Engine or the Analytical Engine.

One year after Ada's death George and Edward Scheutzbuilt a working model of the "Difference Engine" from oneof Babbage's original early designs.

In 1980, the United States Department of Defensecompleted a new computer language.

This advanced new computer language was named "Ada".

The Portrait of Ada Byron, Lady Lovelace

Ada is portrayed in a simple contemporary engraving.

Ada's portrait is overlaid with a notation from the Notesshe wrote in"Sketch of the Analytical Engine", whichanticipates the modern-day general purpose computer andmodern computer programming.

The color scheme of the portrait of Ada echoes that ofPascal, the other mathematician credited with being aprecursor of modern computing.

Gottfried Wilhelm Leibniz 1646 - 1716

Gottfried Wilhelm Leibniz was a philosopher,mathematician, physicist, jurist, and contemporary ofNewton. He is considered one of the great thinkers of the

17th century. He believed in a universe which followed a"pre-established harmony" between mind and matter, andattempted to reconcile the existence of a material worldwith the existence of a supreme being.

The twentieth century philosopher and mathematicianBertrand Russell considered Leibniz's greatest claim tofame to be his invention of the infinitesimal calculus -- aremarkable achievement considering that Leibniz was self-taught in mathematics.

Leibniz is portrayed overlaid with integral notation fromhis calculus which he developed coincident with but

independently ofNewton's development of calculus.

Although the historical record suggest that Newtondeveloped his version of calculus first, Leibniz was the firstto publish. Unfortunately, what emerged was not fruitfulcollaboration, but a rancorous dispute that raged fordecades and pitted English continental mathematicianssupporting Newton as the true inventor of the calculus,against continental mathematicians supporting Leibniz.

Today, Leibniz and Newton are generally recognized as'co-inventors' of the calculus.

But Leibniz' notation for calculus was far superior to that ofNewton, and it is the notation developed by Leibniz,including the integral sign and derivative notation, that isstill in use today.

Leibniz considered symbols to be critical for humanunderstanding of all things. So much so that he attemptedto develop an entire 'alphabet of human thought', in whichall fundamental concepts would be represented bysymbols which could be combined to represent morecomplex thoughts. Leibniz never finished this work.

Leibniz, who had strong conceptual differences withNewton in other areas, notably with Newton's concept ofabsolute space, also develop bitter conceptual differenceswith Descartes over what was then referred to as the"fundamental quantity of motion", a precursor of the Lawof Conservation of Energy.

Much of Leibniz' work went unpublished during hislifetime. He died embittered, in ill health, and withoutachieving the considerable wealth, fame, and honoraccorded to Newton.

Leibniz' diverse writings -- philosophical, mathematical,

historical, and political -- were resurrected and publishedin the late 19th and 20th centuries.
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Archimedes of Syracuse287 - 212 B.C.E.

Archimedes of Syracuse is generally regarded as the greatest mathematician and scientist ofantiquity, and widely considered, along with Newton and Gauss, as one of the greatestmathematicians of all time.

Archimedes' inventions were diverse -- compound pulley systems, war machines used in thedefense of Syracuse, and even an early planetarium.

His major writings on mathematics included contributions on plane equilibriums, the sphere, thecylinder, spirals, conoids and spheroids, the parabola, "Archimedes Principle" of buoyancy, andremarkable work on the measurement of a circle.

Archimedes is pictured with the methods he used to find an approximation to the area of acircle and the value of pi. Archimedes was the first to give a scientific method for calculating pi. to

arbitrary accuracy. The method used by Archimedes -- the measurement of inscribed andcircumscribed polygons approaching a 'limit" (described as 'exhaustion') -- was one of the earliestapproaches to "integration". It preceded by more than a millennia Newton, Leibniz, and moderncalculus.

Archimedes was killed in the aftermath of the Battle of Syracuse -- a siege won by the Romansusing war machines many of which had been invented by Archimedes himself. Archimedes waskilled by a Roman soldier who likely had no idea who Archimedes was. At the time of his death

Archimedes was reputedly sketching a geometry problem in the sand, his last words to theRoman soldier being "don't disturb my circles".

Little is known ofHipparchus's life, but he is known to have been born in Nicaea inBithynia. The town of Nicaea is now called Iznik and is situated in north-western Turkey.Founded in the 4th Century BC, Nicaea lies on the eastern shore of Lake Iznik.Reasonably enough Hipparchus is often referred to as Hipparchus of Nicaea orHipparchus of Bithynia and he is listed among the famous men of Bithynia by Strabo, the

Greek geographer and historian who lived from about 64 BC to about 24 AD. There arecoins from Nicaea which depict Hipparchus sitting looking at a globe and his imageappears on coins minted under five different Roman emperors between 138 AD and 253AD.

This seems to firmly place Hipparchus in Nicaea and indeedPtolemy does describeHipparchus as observing in Bithynia, and one would naturally assume that in fact he wasobserving in Nicaea. However, of the observations which are said to have been made by

rema ns s ower ng egacy.
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Hipparchus, some were made in the north of the island of Rhodes and several (althoughonly one is definitely due to Hipparchus himself) were made in Alexandria. If these areindeed as they appear we can say with certainty that Hipparchus was in Alexandria in 146BC and in Rhodes near the end of his career in 127 BC and 126 BC.

It is not too unusual to have few details of the life of a Greek mathematician, but withHipparchus the position is a little unusual for, despite Hipparchus being a mathematicianand astronomer of major importance, we have disappointingly few definite details of hiswork. Only one work by Hipparchus has survived, namely Commentary on Aratus andEudoxus and this is certainly not one of his major works. It is however important in that itgives us the only source of Hipparchus's own writings.

Most of the information which we have about the work of Hipparchus comes fromPtolemy'sAlmagestbut, as Toomer writes in [1]:-

... althoughPtolemy obviously had studied Hipparchus's writings thoroughly and had a

deep respect for his work, his main concern was not to transmit it to posterity but to useit and, where possible, improve upon it in constructing his own astronomical system.

Where one might hope for more information about Hipparchus would be in thecommentaries on Ptolemy'sAlmagest. There are two in particular by the excellentcommentators Theon of Alexandria and by Pappus, but unfortunately these followPtolemy's text fairly closely and fail to add the expected information about Hipparchus.Since when Ptolemy refers to results of Hipparchus he does so often in an obscure way,at least he seems to assume that the reader will have access to the original writings byHipparchus, and it is certainly surprising that neitherTheonorPappus fills in the details.One can only assume that neither of them had access to the information about Hipparchus

on which we would have liked them to report.Let us first summarise the main contribution of Hipparchus and then examine them inmore detail. He made an early contribution to trigonometry producing a table of chords,an early example of a trigonometric table; indeed some historians go so far as to say thattrigonometry was invented by him. The purpose of this table of chords was to give amethod for solving triangles which avoided solving each triangle from first principles. Healso introduced the division of a circle into 360 degrees into Greece.

Hipparchus calculated the length of the year to within 6.5 minutes and discovered theprecession of the equinoxes. Hipparchus's value of 46" for the annual precession is good

compared with the modern value of 50.26" and much better than the figure of 36" thatPtolemy was to obtain nearly 300 years later. We believe that Hipparchus's star cataloguecontained about 850 stars, probably not listed in a systematic coordinate system but usingvarious different ways to designate the position of a star. His star catalogue, probablycompleted in 129 BC, has been claimed to have been used byPtolemy as the basis of hisown star catalogue. However, Vogt shows clearly in his important paper [26] that byconsidering the Commentary on Aratus and Eudoxus and making the reasonableassumption that the data given there agreed with his star catalogue, then Ptolemy's star
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catalogue cannot have been produced from the positions of the stars as given byHipparchus.

This last point shows that in any detailed discussion of the achievements of Hipparchuswe have to delve more deeply than just assuming that everything in the Ptolemy's

Almagestwhich he does not claim as his own must be due to Hipparchus. This view wastaken for many years but since Vogt's 1925 paper [26] there has been much research donetrying to ascertain exactly what Hipparchus achieved. So major shifts have taken place inour understanding of Hipparchus, first it was assumed that his discoveries were all set outby Ptolemy, then once it was realised that this was not so there was a feeling that it wouldbe impossible to ever have detailed knowledge of his achievements, but now we are in athird stage where it is realised that it is possible to gain a good knowledge of his work butonly with much effort and research.

Let us begin our detailed description of Hipparchus's achievements by looking at the onlywork which has survived. Hipparchus's Commentary on Aratus and Eudoxus was written

in three books as a commentary on three different writings. Firstly there was a treatise byEudoxus (unfortunately now lost) in which he named and described the constellations.Aratus wrote a poem calledPhaenomena which was based on the treatise by Eudoxusand proved to be a work of great popularity. This poem has survived and we have its text.Thirdly there was commentary on Aratus by Attalus of Rhodes, written shortly before thetime of Hipparchus.

It is certainly unfortunate that of all of the writings of Hipparchus this was the one tosurvive since the three books on which Hipparchus was writing a commentary containedno mathematical astronomy. As a result of this Hipparchus chose to write at the samequalitative level in the first book and also for much of the second of his three book.

However towards the end of the second book, continuing through the whole of the thirdbook, Hipparchus gives his own account of the rising and setting of the constellations.Towards the end of Book 3 Hipparchus gives a list of bright stars always visible for thepurpose of enabling the time at night to be accurately determined. As we noted aboveHipparchus does not use a single consistent coordinate system to denote stellar positions,rather using a mixture of different coordinates. He uses some equatorial coordinates,although often in a rather strange way as for example saying that a star (see [ 1]):-

... occupies three degrees of Leo along its parallel circle...

He has therefore divided each small circle parallel to the equator into 12 portions of 30each and this means that the right ascensionof the star referred to in the quotation is123. The data in the Commentary on Aratus and Eudoxus has been analysed by manyauthors. In particular the authors of [15] argue that Hipparchus used a mobile celestialspherewith the stars pictured on the sphere. They claim that the data was taken from on astar catalogue constructed around 140 BC based on observations accurate to a third of adegree or even better. In the earlier work [16] by the same authors, they suggest that theobservations were made at a latitude of 36 15' which corresponds to that of northern
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Rhodes. This would tend to confirm that this work by Hipparchus was done near the endof his career. As Toomer writes in [1]:-

Far from being a "work of his youth", as it is frequently described, the commentary onAratus reveals Hipparchus as one who had already compiled a large number of

observations, invented methods for solving problems in spherical astronomy, anddeveloped the highly significant idea of mathematically fixing the positions of the stars...

There is of course no agreement on many of the points discussed here. For exampleMaeyama in [13] sees major differences between the accuracy of the data in Commentaryon Aratus and Eudoxus (claimed to be written around 140 BC) and Hipparchus's starcatalogue (claimed to be produced around 130 BC). Maeyama writes [13]:-

... Hipparchus's "Commentary" contains his own observations of the stellar positions,great in number but inaccurate in operation, despite all his ability for accurateobservations. ... the observational accuracy [of] his two different epochs have nothing in

common, as if they dealt with two different observers. Within an interval of10 yearseverything can happen, particularly in the case of a man like Hipparchus. Those viewswhich consider Hipparchus's astronomical activities at his two different epochs assimilar are completely unfounded.

Perhaps the discovery for which Hipparchus is most famous is the discovery ofprecession which is due to the slow change in direction of the axis of rotation of the earth.This work came from Hipparchus's attempts to calculate the length of the year with ahigh degree of accuracy. There are two different definitions of a 'year' for one might takethe time that the sun takes to return to the same place amongst the fixed stars or one couldtake the length of time before the seasons repeated which is a length of time defined by

considering the equinoxes. The first of these is called the sidereal yearwhile the second iscalled the tropical year.

Of course the data needed by Hipparchus to calculate the length of these two differentyears was not something that he could find over a few years of observations. Swerdlow[20] suggests that Hipparchus calculated the length of the tropical year using Babyloniandata to arrive at the value of1/300 of a day less than 3651/4 days. He then checked thisagainst observations of equinoxes andsolsticesincluding his own data and those ofAristarchusin 280 BC and Meton in 432 BC. Hipparchus also calculated the length of thesidereal year, again using older Babylonian data, and arrived at the highly accurate figureof1/144 days longer than 3651/4 days. This gives his rate of precession of 1 per century.

Hipparchus also made a careful study of the motion of the moon. There are difficultproblems in such a study for there are three different periods which one could determine.There is the time taken for the moon to return to the same longitude, the time taken for itto return to the same velocity (the anomaly) and the time taken for it to return to the samelatitude. In addition there is the synodic month, that is the time between successiveoppositions of the sun and moon. Toomer [22] writes:-
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For his lunar theory [Hipparchus] needed to establish the mean motions of the Moon inlongitude, anomaly and latitude. The best data available to him were the Babylonianparameters. But he was not content merely to accept them: he wanted to test themempirically, and so he constructed(purely arithmetically) the eclipse period of126007days 1 hour, then looked in the observational material available to him for pairs of

eclipses which would confirm that this was indeed an eclipse period. The observationsthus played a real role, but that role was not discovery, but confirmation.

In calculating the distance of the moon, Hipparchus not only made excellent use of bothmathematical techniques and observational techniques but he also gave a range of valueswithin which be calculated that the true distance must lie. Although Hipparchus's treatiseOn sizes and distances has not survived details given by Ptolemy, Pappus, and othersallow us to reconstruct his methods and results.

The reconstruction of Hipparchus's techniques is beautifully presented in [24] where theauthor shows that Hipparchus based his calculations on an eclipse which occurred on 14

March 190 BC. Hipparchus's calculations led him to a value for the distance to the moonof between 59 and 67 earth radii which is quite remarkable (the correct distance is 60earth radii). The main reason for his range of values was that he was unable to determinetheparallaxof the sun, only managing to give an upper value. Hipparchus appears toknow that 67 earth radii for the distance of the moon comes from this upper limit of solarparallax, while the lower value of 59 earth radii corresponds to the sun being at infinity.

Hipparchus not only gave observational data for the moon which enabled him to computeaccurately the various periods, but he developed a theoretical model of the motion of themoon based on epicycles. He showed that his model did not agree totally withobservations but it seems to be Ptolemy who was the first to correct the model to take

these discrepancies into account. Hipparchus was also able to give an epicycle model forthe motion of the sun (which is easier), but he did not attempt to give an epicycle modelfor the motion of the planets.

Finally let us examine the contributions which Hipparchus made to trigonometry. Heathwrites in [6]:-

Even if he did not invent it, Hipparchus is the first person whose systematic use oftrigonometry we have documentary evidence.

The documentary evidence comes from Ptolemy and Theon of Alexandria who explicitly

says that Hipparchus wrote a work on chords in 12 books. However, Neugebauer[7]points out that:-

... this number is obvious nonsense since 13 books sufficed for the whole of the"Almagest" or ofEuclid's "Elements"...

Toomer ([1] or [23]) reconstructs Hipparchus's table of chords, and the mathematicalmeans by which Hipparchus calculated it. The table was based on a circle divided into
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360 degrees with each degree divided into 60 minutes. The radius of the circle is then360.60/2 = 3438 minutes and the chord function Crd of Hipparchus is related to the sinefunction by

(Crd 2a)/2 = 3438 sin a.

Toomer claims that Hipparchus defined his Crd function at 7.5 intervals (1/48 of thecircle) and used linear interpolation to find the value at intermediate points. He then goeson to show that the table can be computed from some basic formulae which would beknown to Hipparchus, one of which is the supplementary angle theorem, essentiallyPythagoras's theorem, and the half-angle theorem. The only trace of Hipparchus's tablesthat survives is in Indian tables which are thought to have been based on that of that ofHipparchus.

Toomer summarises the contributions of Hipparchus in this area when he writes in [1]:-

... it seems highly probable that Hipparchus was the first to construct a table of chordsand thus provide a general solution for trigonometrical problems. A corollary of this isthat, before Hipparchus, astronomical tables based on Greek geometrical methods didnot exist. If this is so, Hipparchus was not only the founder of trigonometry but also theman who transformed Greek astronomy from a purely theoretical into a practicalpredictive science.

Article by:J J O'ConnorandE F Robertson

Click on this linkto see a list of the Glossary entries for this page
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Geometry, trigonometry, and other mathematical

techniques

Hipparchus is recognized as the first mathematician known to have possessed a

trigonometry table, which he needed when computing theeccentricity of the orbits of theMoon and Sun. He tabulated values for the chordfunction, which gives the length of thechord for each angle. He did this for a circle with a circumference of 21,600 and a radius(rounded) of 3438 units: this circle has a unit length of 1 arc minute along its perimeter.He tabulated the chords for angles with increments of 7.5. In modern terms, the chord ofan angle equals twice thesine of half of the angle, i.e.:

chord(A) = 2 sin(A/2).

He described the chord table in a work, now lost, called Tn en kukli euthein (Of LinesInside a Circle) by Theon of Alexandria(4th century) in his commentary on the

AlmagestI.10; some claim his table may have survived in astronomical treatises in India,for instance the Surya Siddhanta. Trigonometry was a significant innovation, because itallowed Greek astronomers to solve any triangle, and made it possible to makequantitative astronomical models and predictions using their preferred geometrictechniques.[8]

For his chord table Hipparchus must have used a better approximation forthan the onefrom Archimedes of between 3 + 1/7 and 3 + 10/71; perhaps he had the one later used byPtolemy: 3;8:30 (sexagesimal) (AlmagestVI.7); but it is not known if he computed animproved value himself.

Hipparchus could construct his chord table using the Pythagorean theorem and a theoremknown to Archimedes. He also might have developed and used the theorem inplanegeometrycalled Ptolemy's theorem, because it was proved by Ptolemy in hisAlmagest(I.10) (later elaborated on by Carnot).

Hipparchus was the first to show that the stereographic projection isconformal, and thatit transforms circles on the spherethat do not pass through the center of projection tocircles on theplane. This was the basis for the astrolabe.

Besides geometry, Hipparchus also used arithmetic techniques developed by theChaldeans. He was one of the first Greek mathematicians to do this, and in this way

expanded the techniques available to astronomers and geographers.There are several indications that Hipparchus knew spherical trigonometry, but the firstsurviving text of it is that ofMenelaus of Alexandria in the 1st century, who on that basisis now commonly credited with its discovery. (Previous to the finding of the proofs ofMenelaus a century ago, Ptolemy was credited with the invention of sphericaltrigonometry.) Ptolemy later used spherical trigonometry to compute things like the risingand setting points of the ecliptic, or to take account of the lunarparallax. Hipparchus may
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have used a globe for these tasks, reading values off coordinate grids drawn on it, or hemay have made approximations from planar geometry, or perhaps used arithmeticalapproximations developed by the Chaldeans. Or perhaps he used spherical trigonometry.

Bhaskara (1114 1185) (also known as Bhaskara II and Bhaskara Achrya("Bhaskara the teacher")) was an Indianmathematician and astronomer. He was bornnear Bijjada Bida (in present day Bijapur district,Karnataka state, South India) into theDeshastha Brahminfamily. Bhaskara was head of an astronomical observatory at Ujjain,the leading mathematical centre of ancient India. His predecessors in this post hadincluded both the noted Indian mathematician Brahmagupta (598c. 665) andVarahamihira. He lived in the Sahyadri region of Western Maharashtra.

It has been recorded that his great-great-great-grandfather held a hereditary post as a

court scholar, as did his son and other descendants. His father Mahesvara was as anastrologer, who taught him mathematics, which he later passed on to his son Loksamudra.Loksamudra's son helped to set up a school in 1207 for the study of Bhskara's writings. [1]

Bhaskara and his works represent a significant contribution to mathematical andastronomical knowledge in the 12th century. His main works were theLilavati(dealingwith arithmetic),Bijaganita (Algebra) and Siddhanta Shiromani (written in 1150) whichconsists of two parts: Goladhyaya (sphere) and Grahaganita (mathematics of theplanets).

[edit] Legends

His book on arithmetic is the source of interesting legends that assert that it was writtenfor his daughter, Lilavati. In one of these stories, which is found in aPersian translationofLilavati, Bhaskara II studied Lilavati's horoscope and predicted that her husbandwould die soon after the marriage if the marriage did not take place at a particular time.To alert his daughter at the correct time, he placed a cup with a small hole at the bottomof a vessel filled with water, arranged so that the cup would sink at the beginning of thepropitious hour. He put the device in a room with a warning to Lilavati to not go near it.In her curiosity though, she went to look at the device and a pearl from her nose ringaccidentally dropped into it, thus upsetting it. The marriage took place at the wrong timeand she was soon widowed.

Bhaskara II conceived the modern mathematical convention that when a finite number isdivided by zero, the result is infinity. In his bookLilavati, he reasons: "In this quantityalso which has zero as its divisor there is no change even when many [quantities] haveentered into it or come out [of it], just as at the time of destruction and creation whenthrongs of creatures enter into and come out of [him, there is no change in] the infiniteand unchanging [Vishnu]". (Ref. Arithmetic and mensuration of Brahmegupta andBhaskara, H.T Colebrooke, 1817).
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[edit] Mathematics

Some of Bhaskara's contributions to mathematics include the following:

A proof of the Pythagorean theorem by calculating the samearea in two different

ways and then canceling out terms to get a + b = c.

InLilavati, solutions ofquadratic,cubic and quarticindeterminate equations.

Solutions of indeterminate quadratic equations (of the type ax + b =y).

Integer solutions of linear and quadratic indeterminate equations (Kuttaka). Therules he gives are (in effect) the same as those given by the Renaissance Europeanmathematicians of the 17th century

A cyclic Chakravala method for solving indeterminate equations of the form ax +

bx + c =y. The solution to this equation was traditionally attributed to WilliamBrouncker in 1657, though his method was more difficult than the chakravalamethod.

His method for finding the solutions of the problemx ny = 1 (so-called "Pell'sequation") is of considerable interest and importance.

Solutions ofDiophantine equations of the second order, such as 61x + 1 =y.This very equation was posed as a problem in 1657 by the FrenchmathematicianPierre de Fermat, but its solution was unknown in Europe until the time ofEulerin the 18th century.

Solved quadratic equations with more than one unknown, and found negative andirrational solutions.

Preliminary concept ofmathematical analysis.

Preliminary concept ofinfinitesimalcalculus, along with notable contributionstowards integral calculus.

Conceived differential calculus, after discovering thederivative and differentialcoefficient.

Stated Rolle's theorem, a special case of one of the most important theorems inanalysis, the mean value theorem. Traces of the generalmean value theorem arealso found in his works.

Calculated the derivatives of trigonometric functions and formulae. (See Calculussection below.)
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In Siddhanta Shiromani, Bhaskara developedspherical trigonometry along with anumber of othertrigonometric results. (See Trigonometry section below.)

[edit] Arithmetic

Bhaskara'sarithmetic textLilavati covers the topics of definitions, arithmetical terms,interest computation, arithmetical and geometrical progressions,plane geometry,solidgeometry, the shadow of the gnomon, methods to solve indeterminate equations, andcombinations.

Lilavati is divided into 13 chapters and covers many branches of mathematics, arithmetic,algebra, geometry, and a little trigonometry and mensuration. More specifically thecontents include:

Definitions. Properties ofzero(including division, and rules of operations with zero).

Further extensive numerical work, including use ofnegative numbers and surds. Estimation of. Arithmetical terms, methods ofmultiplication, andsquaring. Inverse rule of three, and rules of 3, 5, 7, 9, and 11. Problems involving interest and interest computation. Arithmetical and geometrical progressions. Plane (geometry). Solid geometry. Permutations and combinations. Indeterminate equations (Kuttaka), integer solutions (first and second order). His

contributions to this topic are particularly important, since the rules he gives are

(in effect) the same as those given by therenaissance European mathematicians ofthe 17th century, yet his work was of the 12th century. Bhaskara's method ofsolving was an improvement of the methods found in the work ofAryabhataandsubsequent mathematicians.

His work is outstanding for its systemisation, improved methods and the new topics thathe has introduced. Furthermore theLilavati contained excellent recreative problems andit is thought that Bhaskara's intention may have been that a student of 'Lilavati' shouldconcern himself with the mechanical application of the method.

[edit] Algebra

HisBijaganita ("Algebra") was a work in twelve chapters. It was the first text torecognize that a positive number has twosquare roots (a positive and negative squareroot). His workBijaganita is effectively a treatise on algebra and contains the followingtopics:

Positive and negative numbers. Zero.
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The 'unknown' (includes determining unknown quantities). Determining unknown quantities. Surds (includes evaluating surds). Kuttaka (for solving indeterminate equationsand Diophantine equations). Simple equations (indeterminate of second, third and fourth degree).

Simple equations with more than one unknown. Indeterminate quadratic equations(of the type ax + b = y). Solutions of indeterminate equations of the second, third and fourth degree. Quadratic equations. Quadratic equations with more than one unknown. Operations with products of several unknowns.

Bhaskara derived a cyclic,chakravala method for solving indeterminate quadraticequations of the form ax + bx + c = y. Bhaskara's method for finding the solutions of theproblem Nx + 1 = y (the so-called "Pell's equation") is of considerable importance.

He gave the general solutions of:

Pell's equation using the chakravala method. The indeterminate quadratic equation using the chakravala method.

He also solved[citation needed]:

Cubic equations. Quartic equations. Indeterminate cubic equations. Indeterminate quartic equations. Indeterminate higher-orderpolynomialequations.

[edit] Trigonometry

The Siddhanta Shiromani (written in 1150) demonstrates Bhaskara's knowledge oftrigonometry, including the sine table and relationships between different trigonometricfunctions. He also discovered spherical trigonometry, along with other interestingtrigonometrical results. In particular Bhaskara seemed more interested in trigonometryfor its own sake than his predecessors who saw it only as a tool for calculation. Amongthe many interesting results given by Bhaskara, discoveries first found in his worksinclude the now well known results for and :

[edit] Calculus

His work, the Siddhanta Shiromani, is an astronomical treatise and contains manytheories not found in earlier works. Preliminary concepts of infinitesimal calculusandmathematical analysis, along with a number of results in trigonometry,differentialcalculusand integral calculus that are found in the work are of particular interest.
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Evidence suggests Bhaskara was acquainted with some ideas ofdifferential calculus. Itseems, however, that he did not understand the utility of his researches, and thushistorians of mathematics generally neglect this achievement. Bhaskara also goes deeperinto the 'differential calculus' and suggests the differential coefficient vanishes at anextremum value of the function, indicating knowledge of the concept of 'infinitesimals'.[2]

There is evidence of an early form ofRolle's theorem in his work:o If then for some with

He gave the result that if then , thereby finding the derivative of sine, although henever developed the general concept of differentiation.[3]

o Bhaskara uses this result to work out the position angle of theecliptic, aquantity required for accurately predicting the time of an eclipse.

In computing the instantaneous motion of a planet, the time interval betweensuccessive positions of the planets was no greater than a truti, or a 133750 of a

second, and his measure of velocity was expressed in this infinitesimal unit oftime.

He was aware that when a variable attains the maximum value, its differentialvanishes.

He also showed that when a planet is at its farthest from the earth, or at its closest,the equation of the centre (measure of how far a planet is from the position inwhich it is predicted to be, by assuming it is to move uniformly) vanishes. Hetherefore concluded that for some intermediate position the differential of theequation of the centre is equal to zero. In this result, there are traces of the general

mean value theorem, one of the most important theorems inanalysis, which todayis usually derived from Rolle's theorem. The mean value theorem was later foundby Parameshvarain the 15th century in theLilavati Bhasya, a commentary onBhaskara'sLilavati.

Madhava (1340-1425) and the Kerala School mathematicians (including Parameshvara)from the 14th century to the 16th century expanded on Bhaskara's work and furtheradvanced the development ofcalculusin India.

[edit] Astronomy

The study of astronomy in Bhaskara's works is based on a model of the solar systemwhich is heliocentric and whose movements are determined by gravitation. Heliocentrismhad been propounded in 499 by Aryabhata, who argued that the planets follow ellipticalorbits around the Sun. A law of gravity had been described by Brahmagupta in the 7thcentury. Using this model, Bhaskara accurately defined many astronomical quantities,including, for example, the length of the sidereal year, the time that is required for theEarth to orbit the Sun, as 365.2588 days which is same as in Suryasiddhanta. The modernaccepted measurement is 365.2563 days, a difference of just 3.5 minutes. This result was
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achieved using observations that had been made with only the naked eye, without anysophisticated instrument

bhaskara

Brahmagupta (listen(helpinfo)) (598668) was an Indianmathematicianandastronomer. Brahmagupta, whose father was Jisnugupta, wrote important works onmathematics and astronomy. In particular he wrote Brahmasphutasiddhanta (CorrectlyEstablished Doctrine of Brahma), in 628. The work was written in 25 chapters andBrahmagupta tells us in the text that he wrote it at Bhillamala which today is the city ofBhinmal. This was the capital of the lands ruled by the Gurjara dynasty.

Life and work

Brahmagupta was born in 598 CE in Bhinmal city in the state ofRajasthan of northwestIndia. He likely lived most of his life in Bhillamala (modern Bhinmalin Rajasthan) in theempire ofHarshaduring the reign (and possibly under the patronage) of KingVyaghramukha.[1] As a result, Brahmagupta is often referred to as Bhillamalacarya, that

is, the teacher from Bhillamala Bhinmal. He was the head of the astronomicalobservatory at Ujjain, and during his tenure there wrote four texts on mathematics andastronomy: the Cadamekela in 624, theBrahmasphutasiddhanta in 628, theKhandakhadyaka in 665, and theDurkeamynarda in 672. TheBrahmasphutasiddhanta(Corrected Treatise of Brahma) is arguably his most famous work. The historian al-Biruni (c. 1050) in his bookTariq al-Hindstates that theAbbasidcaliphal-Ma'mun hadan embassy in India and from India a book was brought to Baghdad which was translated
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into Arabic as Sindhind. It is generally presumed that Sindhindis none other thanBrahmagupta'sBrahmasphuta-siddhanta.[2]

Although Brahmagupta was familiar with the works of astronomers following thetradition ofAryabhatiya, it is not known if he was familiar with the work ofBhaskara I, a

contemporary.[1]

Brahmagupta had a plethora of criticism directed towards the work ofrival astronomers, and in hisBrahmasphutasiddhanta is found one of the earliest attestedschisms among Indian mathematicians. The division was primarily about the applicationof mathematics to the physical world, rather than about the mathematics itself. InBrahmagupta's case, the disagreements stemmed largely from the choice of astronomicalparameters and theories.[1] Critiques of rival theories appear throughout the first tenastronomical chapters and the eleventh chapter is entirely devoted to criticism of thesetheories, although no criticisms appear in the twelfth and eighteenth chapters. [1]

[edit] Mathematics

Brahmagupta's most famous work is hisBrahmasphutasiddhanta. It is composed inelliptic verse, as was common practice inIndian mathematics, and consequently has apoetic ring to it. As no proofs are given, it is not known how Brahmagupta's mathematicswas derived.[3]

[edit] Algebra

Brahmagupta gave the solution of the generallinear equationin chapter eighteen ofBrahmasphutasiddhanta,

18.43 The difference between rupas, when inverted and divided by the difference of the

unknowns, is the unknown in the equation. The rupas are [subtracted on the side] below that fromwhich the square and the unknown are to be subtracted.[4]

Which is a solution equivalent to , where rupas represents constants. He further gave twoequivalent solutions to the general quadratic equation,

18.44. Diminish by the middle [number] the square-root of the rupas multiplied by four times thesquare and increased by the square of the middle [number]; divide the remainder by twice thesquare. [The result is] the middle [number].18.45. Whatever is the square-root of the rupas multiplied by the square [and] increased by thesquare of half the unknown, diminish that by half the unknown [and] divide [the remainder] by itssquare. [The result is] the unknown.[4]

Which are, respectively, solutions equivalent to,

and

He went on to solve systems of simultaneous indeterminate equationsstating that thedesired variable must first be isolated, and then the equation must be divided by the
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desired variable's coefficient. In particular, he recommended using "the pulverizer" tosolve equations with multiple unknowns.

18.51. Subtract the colors different from the first color. [The remainder] divided by the first[color's coefficient] is the measure of the first. [Terms] two by two [are] considered [whenreduced to] similar divisors, [and so on] repeatedly. If there are many [colors], the pulverizer [isto be used].[4]

Like the algebra ofDiophantus, the algebra of Brahmagupta was syncopated. Additionwas indicated by placing the numbers side by side, subtraction by placing a dot over thesubtrahend, and division by placing the divisor below the dividend, similar to ournotation but without the bar. Multiplication, evolution, and unknown quantities wererepresented by abbreviations of appropriate terms.[5] The extent of Greek influence on thissyncopation, if any, is not known and it is possible that both Greek and Indiansyncopation may be derived from a common Babylonian source.[5]

[edit] Arithmetic

In the beginning of chapter twelve of hisBrahmasphutasiddhanta, entitled Calculation,Brahmagupta details operations on fractions. The reader is expected to know the basicarithmetic operations as far as taking the square root, although he explains how to findthe cube and cube-root of an integer and later gives rules facilitating the computation ofsquares and square roots. He then gives rules for dealing with five types of combinationsof fractions, , , , , and .[6]

[edit] Series

Brahmagupta then goes on to give the sum of the squares and cubes of the firstn

integers.12.20. The sum of the squares is that [sum] multiplied by twice the [number of] step[s] increasedby one [and] divided by three. The sum of the cubes is the square of that [sum] Piles of these withidentical balls [can also be computed].[7]

It is important to note here Brahmagupta found the result in terms of the sum of the first nintegers, rather than in terms ofn as is the modern practice.[8]

He gives the sum of the squares of the first n natural numbers as n(n+1)(2n+1)/6 and thesum of the cubes of the first n natural numbers as (n(n+1)/2).

[edit] Zero

Brahmagupta made use of an important concept in mathematics, thenumber zero. TheBrahmasphutasiddhanta is the earliest known text to treat zero as a number in its ownright, rather than as simply a placeholder digit in representing another number as wasdone by the Babylonians or as a symbol for a lack of quantity as was done by Ptolemy
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and the Romans. In chapter eighteen of hisBrahmasphutasiddhanta, Brahmaguptadescribes operations on negative numbers. He first de

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