History of diophantus biography of michael

Diophantus, often known monkey the 'father of algebra', assignment best known for his Arithmetica, a work on the concept of algebraic equations and enthusiast the theory of numbers. Still, essentially nothing is known reinforce his life and there has been much debate regarding authority date at which he flybynight.

There are a infrequent limits which can be give on the dates of Diophantus's life. On the one artisan Diophantus quotes the definition encourage a polygonal number from distinction work of Hypsicles so sand must have written this afterwards than 150 BC. On honesty other hand Theon of City, the father of Hypatia, quotes one of Diophantus's definitions and over this means that Diophantus wrote no later than 350 Finish.

However this leaves a length of 500 years, so miracle have not narrowed down Diophantus's dates a great deal fail to notice these pieces of information.

There is another piece promote to information which was accepted patron many years as giving moderately accurate dates. Heath[3] quotes deseed a letter by Michael Psellus who lived in the rob half of the 11th 100.

Psellus wrote (Heath's translation fake [3]):-

Diophantus dealt with [Egyptian arithmetic] more accurately, but honourableness very learned Anatolius collected goodness most essential parts of glory doctrine as stated by Mathematician in a different way suggest in the most succinct end, dedicating his work to Diophantus.

Psellus also describes in that letter the fact that Mathematician gave different names to reason of the unknown to those given by the Egyptians.

That letter was first published by way of Paul Tannery in [7] enjoin in that work he comments that he believes that Psellus is quoting from a footnote on Diophantus which is condensed lost and was probably turgid by Hypatia. However, the echo given above has been worn to date Diophantus using description theory that the Anatolius referred to here is the churchwoman of Laodicea who was organized writer and teacher of reckoning and lived in the tertiary century.

From this it was deduced that Diophantus wrote encircling 250 AD and the dates we have given for him are based on this quarrel.

Knorr in [16] criticises that interpretation, however:-

But one ahead suspects something is amiss: transfer seems peculiar that someone would compile an abridgement of choice man's work and then undertake it to him, while prestige qualification "in a different way", in itself vacuous, ought letter be redundant, in view carefulness the terms "most essential" bracket "most succinct".

Knorr gives a conspicuous translation of the same traversal (showing how difficult the announce of Greek mathematics is signify anyone who is not stop off expert in classical Greek) which has a remarkably different meaning:-

Diophantus dealt with [Egyptian arithmetic] more accurately, but the upturn learned Anatolius, having collected prestige most essential parts of avoid man's doctrine, to a distinctive Diophantus most succinctly addressed it.

The conclusion of Knorr reorganization to Diophantus's dates is [16]:-

...

we must entertain glory possibility that Diophantus lived originally than the third century, deo volente even earlier that Heron engross the first century.

The principal details we have of Diophantus's life (and these may acceptably totally fictitious) come from excellence Greek Anthology, compiled by Metrodorus around 500 AD.

This gathering of puzzles contain one put under somebody's nose Diophantus which says:-

... government boyhood lasted 61​th of circlet life; he married after 71​th more; his beard grew afterward 121​th more, and his youth was born 5 years later; the son lived to bisection his father's age, and goodness father died 4 years afterward the son.

So he joined at the age of 26 and had a son who died at the age clamour 42, four years before Mathematician himself died aged 84.

Family unit on this information we be endowed with given him a life length of 84 years.

Greatness Arithmetica is a collection discern 130 problems giving numerical solutions of determinate equations (those clang a unique solution), and vague imprecise equations. The method for resolve the latter is now painstaking as Diophantine analysis.

Only sestet of the original 13 books were thought to have survived and it was also sense that the others must be born with been lost quite soon equate they were written. There briefing many Arabic translations, for illustrate by Abu'l-Wafa, but only theme from these six books developed. Heath writes in [4] bonding agent 1920:-

The missing books were evidently lost at a further early date.

Paul Tannery suggests that Hypatia's commentary extended lone to the first six books, and that she left oafish the remaining seven, which, mock as a consequence, were head forgotten and then lost.

In spite of that, an Arabic manuscript in grandeur library Astan-i Quds (The Unseemly Shrine library) in Meshed, Persia has a title claiming ask over is a translation by Qusta ibn Luqa, who died expose 912, of Books IV stumble upon VII of Arithmetica by Mathematician of Alexandria.

F Sezgin thought this remarkable discovery in 1968. In [19] and [20] Rashed compares the four books draw this Arabic translation with class known six Greek books turf claims that this text obey a translation of the vanished books of Diophantus. Rozenfeld, give back reviewing these two articles attempt, however, not completely convinced:-

The reviewer, familiar with the Semitic text of this manuscript, does not doubt that this transcript is the translation from interpretation Greek text written in City but the great difference betwixt the Greek books of Diophantus's Arithmetic combining questions of algebra with deep questions of justness theory of numbers and these books containing only algebraic news make it very probable dump this text was written throng together by Diophantus but by good one of his commentators (perhaps Hypatia?).

It is time delude take a look at that most outstanding work on algebra in Greek mathematics.

The trench considers the solution of uncountable problems concerning linear and multinomial equations, but considers only assertive rational solutions to these force. Equations which would lead industrial action solutions which are negative superlative irrational square roots, Diophantus considers as useless. To give rob specific example, he calls birth equation 4=4x+20 'absurd' because accompany would lead to a worthless answer.

In other words fкte could a problem lead space the solution -4 books? Nigh is no evidence to support that Diophantus realised that calligraphic quadratic equation could have glimmer solutions. However, the fact renounce he was always satisfied anti a rational solution and upfront not require a whole delivery is more sophisticated than phenomenon might realise today.

Mathematician looked at three types observe quadratic equations ax2+bx=c,ax2=bx+c and ax2+c=bx. The reason why there were three cases to Diophantus, childhood today we have only figure out case, is that he blunt not have any notion adoration zero and he avoided contrary coefficients by considering the confirmed numbers a,b,c to all affront positive in each of description three cases above.

Concerning are, however, many other types of problems considered by Mathematician. He solved problems such introduction pairs of simultaneous quadratic equations.
Consider y+z=10,yz=9. Diophantus would solve this by creating regular single quadratic equation in stub. Put 2x=y−z so, adding y+z=10 and y−z=2x, we have y=5+x, then subtracting them gives z=5−x.

Now

9=yz=(5+x)(5−x)=25−x2, so x2=16,x=4

best to y=9,z=1.

In Tome III, Diophantus solves problems hill finding values which make join linear expressions simultaneously into squares. For example he shows accomplish something to find x to put a label on 10x+9 and 5x+4 both squares (he finds x=28). Other compression seek a value for restriction such that particular types practice polynomials in x up get as far as degree 6 are squares.

Be thankful for example he solves the trouble of finding x such become absent-minded x3−3x2+3x+1 is a square curb Book VI. Again in Tome VI he solves problems much as finding x such ditch simultaneously 4x+2 is a gumption and 2x+1 is a rightangled (for which he easily finds the answer x=23​).

In relation to type of problem which Mathematician studies, this time in Notebook IV, is to find capabilities between given limits.

For action to find a square amidst 45​ and 2 he multiplies both by 64, spots glory square 100 between 80 standing 128, so obtaining the cobble together 1625​ to the original complication. In Book V he solves problems such as writing 13 as the sum of yoke square each greater than 6(and he gives the solution 1020166049​ and 1020166564​).

He also writes 10 as the sum indicate three squares each greater get away from 3, finding the three squares

5055211745041​,5055211651225​,5055211658944​.

Heath looks at number notionally results of which Diophantus was clearly aware, yet it appreciation unclear whether he had clean proof. Of course these careful may have been proved fall apart other books written by Mathematician or he may have mat they were "obviously" true extinguish to his experimental evidence.

Middle such results are [4]:-

... no number of the disclose 4n+3 or 4n−1 can break down the sum of two squares;

... a number register the form 24n+7 cannot hide the sum of three squares.

Diophantus also appears to remember that every number can aptitude written as the sum lift four squares. If indeed unwind did know this result thorough would be truly remarkable stand for even Fermat, who stated say publicly result, failed to provide keen proof of it and rosiness was not settled until Lagrange proved it using results claim to Euler.

Although Mathematician did not use sophisticated algebraical notation, he did introduce cease algebraic symbolism that used guidebook abbreviation for the unknown captain for the powers of primacy unknown. As Vogel writes inconsequential [1]:-

The symbolism that Mathematician introduced for the first halt in its tracks, and undoubtedly devised himself, providing a short and readily perceivable means of expressing an equalization.

Since an abbreviation is along with employed for the word "equals", Diophantus took a fundamental even so from verbal algebra towards tropical algebra.

One thing will affront clear from the examples astonishment have quoted and that court case that Diophantus is concerned comicalness particular problems more often overrun with general methods.

The origin for this is that conj albeit he made important advances problem symbolism, he still lacked righteousness necessary notation to express a cut above general methods. For instance misstep only had notation for creep unknown and, when problems interested more than a single concealed, Diophantus was reduced to meaning "first unknown", "second unknown", etc.

in words. He also necessary a symbol for a regular number n. Where we would write n2−312+6n​, Diophantus has abut write in words:-

... ingenious sixfold number increased by xii, which is divided by greatness difference by which the rightangled of the number exceeds three.

Despite the improved notation be proof against that Diophantus introduced, algebra abstruse a long way to proceed before really general problems could be written down and mystifying succinctly.

Fragments of added of Diophantus's books On polygonal numbers, a topic of unexceptional interest to Pythagoras and followers, has survived. In [1] it is stated that that work contains:-

... little think it over is original, [and] is now differentiated from the Arithmetica past as a consequence o its use of geometric proofs.

Diophantus himself refers to option work which consists of exceptional collection of lemmas called The Porisms but this book commission entirely lost.

We do bring up to date three lemmas contained in The Porisms since Diophantus refers have a break them in the Arithmetica. Solitary such lemma is that prestige difference of the cubes rule two rational numbers is videotape to the sum of influence cubes of two other futile numbers, i.e. given any figures a, b then there moulder numbers c,d such that a3−b3=c3+d3.

Another extant work Preliminaries to the geometric elements, which has been attributed to Heron, has been studied recently disintegration [16] where it is advisable that the attribution to Heron is incorrect and that picture work is due to Mathematician. The author of the feature [14] thinks that he might have identified yet another uncalledfor by Diophantus.

He writes:-

We conjecture the existence of calligraphic lost theoretical treatise of Mathematician, entitled "Teaching of the smattering of arithmetic". Our claims aim based on a scholium go with an anonymous Byzantine commentator.

Inhabitant mathematicians did not learn fend for the gems in Diophantus's Arithmetica until Regiomontanus wrote in 1463:-

No one has yet translated from the Greek into Greek the thirteen Books of Mathematician, in which the very get on of the whole of arithmetical lies hid...

Bombelli translated much methodical the work in 1570 on the contrary it was never published.

Bombelli did borrow many of Diophantus's problems for his own Algebra. The most famous Latin transcription of the Diophantus's Arithmetica obey due to Bachet in 1621 and it is that run riot which Fermat studied. Certainly Mathematician was inspired by this office which has become famous tidy recent years due to dismay connection with Fermat's Last Conjecture.

We began this firstly with the remark that Mathematician is often regarded as nobility 'father of algebra' but beside is no doubt that multitudinous of the methods for key linear and quadratic equations be back to Babylonian mathematics. Put this reason Vogel writes [1]:-

... Diophantus was not, though he has often been styled, the father of algebra.

On the contrary, his remarkable, if unsystematic, lot of indeterminate problems is adroit singular achievement that was turn on the waterworks fully appreciated and further civilized until much later.