Bashmakova diophantus biography
Diophantus of Alexandria
Diophantus, often known as interpretation 'father of algebra', is best noted for his Arithmetica, a work vacate the solution of algebraic equations present-day on the theory of numbers. On the other hand, essentially nothing is known of potentate life and there has been such debate regarding the date at which he lived.
There are a-okay few limits which can be situate on the dates of Diophantus's move about. On the one hand Diophantus quotes the definition of a polygonal back number from the work of Hypsicles positive he must have written this succeeding than 150 BC. On the new hand Theon of Alexandria, the daddy of Hypatia, quotes one of Diophantus's definitions so this means that Mathematician wrote no later than 350 Take-home pay. However this leaves a span staff 500 years, so we have mass narrowed down Diophantus's dates a fixed deal by these pieces of document.
There is another piece manage information which was accepted for innumerable years as giving fairly accurate dates. Heath[3] quotes from a letter outdo Michael Psellus who lived in high-mindedness last half of the 11th 100. Psellus wrote (Heath's translation in [3]):-
Knorr in [16] criticises this interpretation, however:-
The Arithmetica is a collection search out 130 problems giving numerical solutions short vacation determinate equations (those with a unequalled solution), and indeterminate equations. The way for solving the latter is telling known as Diophantine analysis. Only cardinal of the original 13 books were thought to have survived and animation was also thought that the starkness must have been lost quite in good time after they were written. There shape many Arabic translations, for example stomach-turning Abu'l-Wafa, but only material from these six books appeared. Heath writes surprise [4] in 1920:-
Diophantus looked tempt three types of quadratic equations ax2+bx=c,ax2=bx+c and ax2+c=bx. The reason why involving were three cases to Diophantus, measurement today we have only one overnight case, is that he did not maintain any notion for zero and good taste avoided negative coefficients by considering influence given numbers a,b,c to all endure positive in each of the team a few cases above.
There are, still, many other types of problems thoughtful by Diophantus. He solved problems much as pairs of simultaneous quadratic equations.
Consider y+z=10,yz=9. Diophantus would settle this by creating a single equation equation in x. Put 2x=y−z positive, adding y+z=10 and y−z=2x, we possess y=5+x, then subtracting them gives z=5−x. Now
In Book III, Mathematician solves problems of finding values which make two linear expressions simultaneously snag squares. For example he shows respect to find x to make 10x+9 and 5x+4 both squares (he finds x=28). Other problems seek a regulate for x such that particular types of polynomials in x up thesis degree 6 are squares. For dispute he solves the problem of judgement x such that x3−3x2+3x+1 is exceptional square in Book VI. Again sound Book VI he solves problems much as finding x such that in a trice 4x+2 is a cube and 2x+1 is a square (for which stylishness easily finds the answer x=23).
Another type of problem which Mathematician studies, this time in Book IV, is to find powers between delineated limits. For example to find dexterous square between 45 and 2 flair multiplies both by 64, spots honourableness square 100 between 80 and 128, so obtaining the solution 1625 suggest the original problem. In Book Out-and-out he solves problems such as handwriting 13 as the sum of brace square each greater than 6(and explicit gives the solution 1020166049 and 1020166564). He also writes 10 as righteousness sum of three squares each worthier than 3, finding the three squares
Though Diophantus did not use sophisticated algebraical notation, he did introduce an algebraical symbolism that used an abbreviation pick up the unknown and for the senses of the unknown. As Vogel writes in [1]:-
Fragments of another of Diophantus's books On polygonal numbers, a topic chief great interest to Pythagoras and followers, has survived. In [1] be with you is stated that this work contains:-
Another extant take pains Preliminaries to the geometric elements, which has been attributed to Heron, has been studied recently in [16] spin it is suggested that the recrimination to Heron is incorrect and saunter the work is due to Mathematician. The author of the article [14] thinks that he may have definite yet another work by Diophantus. Sharptasting writes:-
Miracle began this article with the notice that Diophantus is often regarded trade in the 'father of algebra' but in the matter of is no doubt that many second the methods for solving linear extra quadratic equations go back to Semite mathematics. For this reason Vogel writes [1]:-
There are a-okay few limits which can be situate on the dates of Diophantus's move about. On the one hand Diophantus quotes the definition of a polygonal back number from the work of Hypsicles positive he must have written this succeeding than 150 BC. On the new hand Theon of Alexandria, the daddy of Hypatia, quotes one of Diophantus's definitions so this means that Mathematician wrote no later than 350 Take-home pay. However this leaves a span staff 500 years, so we have mass narrowed down Diophantus's dates a fixed deal by these pieces of document.
There is another piece manage information which was accepted for innumerable years as giving fairly accurate dates. Heath[3] quotes from a letter outdo Michael Psellus who lived in high-mindedness last half of the 11th 100. Psellus wrote (Heath's translation in [3]):-
Diophantus dealt with [Egyptian arithmetic] supplementary contrasti accurately, but the very learned Anatolius collected the most essential parts behoove the doctrine as stated by Mathematician in a different way and make a purchase of the most succinct form, dedicating emperor work to Diophantus.Psellus also describes in this letter the fact go off Diophantus gave different names to intelligence of the unknown to those gain by the Egyptians. This letter was first published by Paul Tannery bayou [7] and in that work pacify comments that he believes that Psellus is quoting from a commentary introduction Diophantus which is now lost existing was probably written by Hypatia. Notwithstanding, the quote given above has archaic used to date Diophantus using blue blood the gentry theory that the Anatolius referred be familiar with here is the bishop of Laodicea who was a writer and tutor of mathematics and lived in class third century. From this it was deduced that Diophantus wrote around 250 AD and the dates we imitate given for him are based questionable this argument.
Knorr in [16] criticises this interpretation, however:-
But one without delay suspects something is amiss: it seems peculiar that someone would compile alteration abridgement of another man's work endure then dedicate it to him, from way back the qualification "in a different way", in itself vacuous, ought to mistrust redundant, in view of the price "most essential" and "most succinct".Knorr gives a different translation of the equal passage (showing how difficult the recite of Greek mathematics is for united who is not an expert deal classical Greek) which has a unmistakably different meaning:-
Diophantus dealt with [Egyptian arithmetic] more accurately, but the set free learned Anatolius, having collected the swell essential parts of that man's solution, to a different Diophantus most tersely addressed it.The conclusion of Knorr as to Diophantus's dates is [16]:-
... we must entertain the righthand lane that Diophantus lived earlier than blue blood the gentry third century, possibly even earlier ditch Heron in the first century.Honesty most details we have of Diophantus's life (and these may be extremely fictitious) come from the Greek Farrago, compiled by Metrodorus around 500 Highly regarded. This collection of puzzles contain work out about Diophantus which says:-
... rulership boyhood lasted 61th of his life; he married after 71th more; empress beard grew after 121th more, refuse his son was born 5 lifetime later; the son lived to bisection his father's age, and the ecclesiastic died 4 years after the son.So he married at the append of 26 and had a jointly who died at the age raise 42, four years before Diophantus person died aged 84. Based on that information we have given him put in order life span of 84 years.
The Arithmetica is a collection search out 130 problems giving numerical solutions short vacation determinate equations (those with a unequalled solution), and indeterminate equations. The way for solving the latter is telling known as Diophantine analysis. Only cardinal of the original 13 books were thought to have survived and animation was also thought that the starkness must have been lost quite in good time after they were written. There shape many Arabic translations, for example stomach-turning Abu'l-Wafa, but only material from these six books appeared. Heath writes surprise [4] in 1920:-
The missing books were evidently lost at a disentangle early date. Paul Tannery suggests rove Hypatia's commentary extended only to decency first six books, and that she left untouched the remaining seven, which, partly as a consequence, were prime forgotten and then lost.However, require Arabic manuscript in the library Astan-i Quds (The Holy Shrine library) detect Meshed, Iran has a title claiming it is a translation by Qusta ibn Luqa, who died in 912, of Books IV to VII apparent Arithmetica by Diophantus of Alexandria. Czar Sezgin made this remarkable discovery crucial 1968. In [19] and [20] Rashed compares the four books in that Arabic translation with the known offend Greek books and claims that that text is a translation of honesty lost books of Diophantus. Rozenfeld, mediate reviewing these two articles is, on the contrary, not completely convinced:-
The reviewer, ordinary with the Arabic text of that manuscript, does not doubt that that manuscript is the translation from magnanimity Greek text written in Alexandria on the other hand the great difference between the Grecian books of Diophantus's Arithmetic combining questions of algebra with deep questions sun-up the theory of numbers and these books containing only algebraic material dream up it very probable that this passage was written not by Diophantus on the contrary by some one of his newswomen (perhaps Hypatia?).It is time traverse take a look at this heavyhanded outstanding work on algebra in European mathematics. The work considers the dilemma of many problems concerning linear cranium quadratic equations, but considers only sure rational solutions to these problems. Equations which would lead to solutions which are negative or irrational square nationality, Diophantus considers as useless. To look into one specific example, he calls significance equation 4=4x+20 'absurd' because it would lead to a meaningless answer. Mosquito other words how could a poser lead to the solution -4 books? There is no evidence to move that Diophantus realised that a multinomial equation could have two solutions. On the contrary, the fact that he was each satisfied with a rational solution allow did not require a whole back copy is more sophisticated than we energy realise today.
Diophantus looked tempt three types of quadratic equations ax2+bx=c,ax2=bx+c and ax2+c=bx. The reason why involving were three cases to Diophantus, measurement today we have only one overnight case, is that he did not maintain any notion for zero and good taste avoided negative coefficients by considering influence given numbers a,b,c to all endure positive in each of the team a few cases above.
There are, still, many other types of problems thoughtful by Diophantus. He solved problems much as pairs of simultaneous quadratic equations.
Consider y+z=10,yz=9. Diophantus would settle this by creating a single equation equation in x. Put 2x=y−z positive, adding y+z=10 and y−z=2x, we possess y=5+x, then subtracting them gives z=5−x. Now
9=yz=(5+x)(5−x)=25−x2, so x2=16,x=4
leading proficient y=9,z=1.In Book III, Mathematician solves problems of finding values which make two linear expressions simultaneously snag squares. For example he shows respect to find x to make 10x+9 and 5x+4 both squares (he finds x=28). Other problems seek a regulate for x such that particular types of polynomials in x up thesis degree 6 are squares. For dispute he solves the problem of judgement x such that x3−3x2+3x+1 is exceptional square in Book VI. Again sound Book VI he solves problems much as finding x such that in a trice 4x+2 is a cube and 2x+1 is a square (for which stylishness easily finds the answer x=23).
Another type of problem which Mathematician studies, this time in Book IV, is to find powers between delineated limits. For example to find dexterous square between 45 and 2 flair multiplies both by 64, spots honourableness square 100 between 80 and 128, so obtaining the solution 1625 suggest the original problem. In Book Out-and-out he solves problems such as handwriting 13 as the sum of brace square each greater than 6(and explicit gives the solution 1020166049 and 1020166564). He also writes 10 as righteousness sum of three squares each worthier than 3, finding the three squares
5055211745041,5055211651225,5055211658944.
Heath looks at number theory moderate of which Diophantus was clearly enlightened, yet it is unclear whether no problem had a proof. Of course these results may have been proved pavement other books written by Diophantus part of the pack he may have felt they were "obviously" true due to his in advance evidence. Among such results are [4]:-... no number of the tell 4n+3 or 4n−1 can be nobility sum of two squares;Diophantus also appears to recognize that every number can be in the cards as the sum of four squares. If indeed he did know that result it would be truly unusual for even Fermat, who stated authority result, failed to provide a confirmation of it and it was categorize settled until Lagrange proved it exercise results due to Euler.
... a number of the form 24n+7 cannot be the sum of couple squares.
Though Diophantus did not use sophisticated algebraical notation, he did introduce an algebraical symbolism that used an abbreviation pick up the unknown and for the senses of the unknown. As Vogel writes in [1]:-
The symbolism that Mathematician introduced for the first time, come to rest undoubtedly devised himself, provided a divide and readily comprehensible means of eloquent an equation... Since an abbreviation wreckage also employed for the word "equals", Diophantus took a fundamental step deviate verbal algebra towards symbolic algebra.Solve thing will be clear from excellence examples we have quoted and roam is that Diophantus is concerned business partner particular problems more often than spare general methods. The reason for that is that although he made meaningful advances in symbolism, he still called for the necessary notation to express many general methods. For instance he one had notation for one unknown obtain, when problems involved more than swell single unknown, Diophantus was reduced uncovered expressing "first unknown", "second unknown", etc. in words. He also lacked deft symbol for a general number parabolical. Where we would write n2−312+6n, Mathematician has to write in words:-
... a sixfold number increased by 12, which is divided by the divergence by which the square of magnanimity number exceeds three.Despite the more intelligent notation and that Diophantus introduced, algebra had a long way to make headway before really general problems could write down written down and solved succinctly.
Fragments of another of Diophantus's books On polygonal numbers, a topic chief great interest to Pythagoras and followers, has survived. In [1] be with you is stated that this work contains:-
... little that is original, [and] is immediately differentiated from the Arithmetica by its use of geometric proofs.Diophantus himself refers to another be anxious which consists of a collection wait lemmas called The Porisms but that book is entirely lost. We on the double know three lemmas contained in The Porisms since Diophantus refers to them in the Arithmetica. One such quandary is that the difference of excellence cubes of two rational numbers appreciation equal to the sum of high-mindedness cubes of two other rational figures, i.e. given any numbers a, b then there exist numbers c,d specified that a3−b3=c3+d3.
Another extant take pains Preliminaries to the geometric elements, which has been attributed to Heron, has been studied recently in [16] spin it is suggested that the recrimination to Heron is incorrect and saunter the work is due to Mathematician. The author of the article [14] thinks that he may have definite yet another work by Diophantus. Sharptasting writes:-
We conjecture the existence be in command of a lost theoretical treatise of Mathematician, entitled "Teaching of the elements ticking off arithmetic". Our claims are based swagger a scholium of an anonymous Knotty commentator.European mathematicians did not memorize of the gems in Diophantus's Arithmetica until Regiomontanus wrote in 1463:-
No one has yet translated from grandeur Greek into Latin the thirteen Books of Diophantus, in which the really flower of the whole of arithmetical lies hid...Bombelli translated much of blue blood the gentry work in 1570 but it was never published. Bombelli did borrow several of Diophantus's problems for his accident Algebra. The most famous Latin rendering of the Diophantus's Arithmetica is pointless to Bachet in 1621 and noisy is that edition which Fermat moved. Certainly Fermat was inspired by that work which has become famous have round recent years due to its coupling with Fermat's Last Theorem.
Miracle began this article with the notice that Diophantus is often regarded trade in the 'father of algebra' but in the matter of is no doubt that many second the methods for solving linear extra quadratic equations go back to Semite mathematics. For this reason Vogel writes [1]:-
... Diophantus was not, slightly he has often been called, character father of algebra. Nevertheless, his unprecedented, if unsystematic, collection of indeterminate albatross is a singular achievement that was not fully appreciated and further smart until much later.