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This edition of Books IV to VII of Diophantus’ Arithmetica, which are extant only in a recently discovered Arabic translation, is the outgrowth of a doctoral. Diophantus’s Arithmetica1 is a list of about algebraic problems with so Like all Greeks at the time, Diophantus used the (extended) Greek. Diophantus begins his great work Arithmetica, the highest level of algebra in and for this reason we have chosen Eecke’s work to translate into English

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In the rare cases where the first hand in the oldest MS. Most of these appear under the name of Metrodorus, a grammarian, who probably lived about the time of the Emperors Anastasius I.

We find the geometrical equivalent of the solution of a quadratic assumed as early as the fifth century B. Nor is it likely that indeterminate equations of the first degree were treated of in the lost Books. The works on which the fame of Diophantus rests are: It was based on Xylander and was a free reproduction, not a translation, Stevin himself observing that the MS. Either equation gives the same value of x, and a.

Quanto autem hoc est praeclarius, in iis problematis, quae surdis etiam numeris uix posse uidentur explicari, rem eo deducere, ut quasi aritbmetica arithmeticum uertere iussi obsurdescant illi plane, et ne mentio quidem eorum in tractatione ingenio- sissimarum quaestionum admittatur. Besides the ” Porisms ” there are many other propositions assumed or implied by Diophantus which are not definitely called 1 Fermat note on IV. Even though the text is otherwise inferior to the edition, Fermat’s annotations—including the “Last Theorem”—were printed in this version.

Xylandro Augustano incredibili labore Latinc redditum, et Commentariis explanatum, inque lucem editum, ad Illustriss. But the latter circumstances are better explained, as Tannery explains them,by the supposition that the first problems of Books II.

The algebraical proof is easy.

He multiplies both by a square, 64 ; this gives 80 andand is clearly a square which lies between them; there- fore gf or f satisfies the prescribed condition. Le ennglish de ceux qui sont tels. We’ve sent an email to Please follow the instructions to reset your password.

Who were his predecessors, who his successors? The alternative is to solve by the method of Euler who does not use the ” double-equation “i. This can be rationally solved according to Diophantus a When A is positive and a square, say 2. Bombelli did not carry out his plan of publishing Diophantus in a translation, but he took all the problems of the first four Books and some of those of the fifth, and embodied them in his Algebra, interspersing them with his own problems.


The Greek term for the “double-equation” occurs variously as SwrXoi’- croT? We take diophanrus, as our first division, indeterminate equations of the second degree.

Here y is assumed to be ax or ejx, and in either case we have a rational value for x. One solution was all he looked for in a quadratic equation.


Historia Matematica, New York,Vol. Nor is there any sign that more of the arithmeticw than we possess was known 1 Dioph. Let it be proposed then to divide 16 into two squares. Tannery’s suggestion that Hypatia’s commentary was limited to the six Books, and the parallel of Eutocius’ commentary on Apollonius’ Conies, imply that it is the last seven Books, and the most difficult, which ‘are lost.

Still he is far from accounting for seven whole Books; he has therefore to press into the service the lost “Porisms” and the tract on Polygonal Numbers. The best known Latin translation of Arithmetica was made by Bachet in and became the first Latin edition that was widely available.

The Epistola Nuncupatoria is addressed to the Prince Ludwig, and Xylander neatly introduces it by the line ” Offerimus numeros, numeri diophanuts principe digni. It is clear that the epigram was written, not long after his death, by an intimate personal friend with knowledge of and taste for the science which Diophantus made his life-work 3. It is true that this uncertainty in the use of the sign in the MSS.

Almost more different in kind englidh the problems are their solutions, and we are completely unable to give an even tolerably exhaustive review of the different turns which his procedure takes. In Otto Schulz, professor in Berlin, arithmefica a very meritorious German translation with notes: The fourth power is with him mdl mal, the fifth mal kab, the sixth arithmdtica ka’b, the seventh mdl mal kab, the eighth mdl kab kab, the ninth kab kab kab, and so on.

The second letter is E, but AE is equally a number. If we do not know to what lengths Archimedes brought the theory of numbers to say nothing of other thingslet us admit our ignorance. From Wikipedia, the free encyclopedia. I believe that the true explanation is the following. Englisb was towardsfor in his Algebra 2 published in Bombelli tells airthmetica that he had in the years last past discovered a Greek book on Algebra written by ” a certain Diofantes, an Alexandrine Greek author, who lived in the time of Antoninus Pius ” ; that, thinking highly of the contents of the work, he and Antonio Arithmwtica Pazzi determined to translate it ; that they actually translated ejglish books out of the seven into which the MS.


Take half the number of roots, that is in this case five; then multiply this by itself, and the result is five and twenty. In this case there is the additional condition of a limit to the engilsh of x. This could only be possible in particular cases such as those which I have mentioned; but, even here; it seems scarcely possible now to work out the problem by using x and I for the variables as originally taken by Diophantus without falling into confusion. The form iji ot above mentioned is also given by Lehmann, with the remark that it seems to be only a modification of the other.

Under this heading come the pairs of equations I use Greek letters to distinguish the numbers which the problem requires us to find from the one unknown which Dio- phantus uses and which I shall call x.

Full text of “Diophantus of Alexandria; a study in the history of Greek algebra”

He lived in Alexandria. Therefore the factors are and we have finally which equation gives two possible values for x. But, when there are three or four quantities to be found, this reduction is much more difficult, and Diophantus shows great adroitness in effecting it: Method of ” doubU-tqitation.

Diophantus considered negative or irrational square root solutions “useless”, “meaningless”, and even “absurd”.

However, Bombelli borrowed many of the problems for his own book Algebra. A very simple case is the following: Precisely the same remark applies to “methods” 2567. The best that can be done is to give a number of typical instances. Diophantus contents himself with the enuncia- tion of the proposition and does diophwntus show how to prove it or how he effected the transformation in practice.

The first class of theorems or facts assumed without ex- planation by Diophantus are more or less of the nature of identical formulae.