• June 24, 2019

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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The positive evidence on the subject can be given very shortly. Thus, for example, in V. Let us now state Bachet’s conditions generally. He lived in Alexandria. Thus the anonymous author of prolegomena to Nicomachus” Introductio Arithmetica speaks of Diophantus’ ” thirteen Books of Arithmetic 5.

The number of units is expressed as a coefficient. ScorialensisR-II-3, end of 1 6th c. The great advantage of my hypothesis is that it makes the sign for aptfytov exactly parallel to those for the powers of the unknown, e. Veritatis porro apud me est autoritas, ut ei coniunctum etiam cum dedecore meo testimonium lubentissime perhibeam. As a result he fell into what Heraclitus called oiija-iv, lepav VOGOV, that is, into the conceit of ” being somebody ” in the field of Arithmetic and “Logistic”; others too, themselves learned men, thought him an arithmetician of exceptional ability.

These cases Diophantus treats more imperfectly. Notwithstanding that attention was thus called to the work, it 1 Printed in the work Rudimenta astronomica Alfragani, Niirnberg, Vaticanus graecus was copied from A before it had suffered the general alteration by means of a MS.

This led to tremendous advances in number theoryand the study of Diophantine equations “Diophantine geometry” and of Diophantine approximations remain important areas of mathematical research.


Arithmetica is the major work of Diophantus and the most prominent work on algebra in Greek mathematics. But, assuming that Diophantus’ resources are at an end in the sixth Book, Nesselmann has to suggest possible topics which would have formed approximately adequate material for the equivalent of seven Books of the Arithmetical. Schulz’s translation is based upon the edition of Bachet’s text published in But the denominators are nearly always omitted 1 Published by Baillet in Memoire s publih par Its Membres de la Mission archeologique franfaise au Caire, T.

It is to be observed that the first of these conditions can be obtained by considering the equation obtained on page 74 above.


Did k 23 To multiply by in this case would not give us a solution. In Book 3, Diophantus solves problems of finding values which make two linear expressions simultaneously into squares or cubes. This passage, which is good reading, but too long to reproduce here, I give in full in the note 1.

There is at least one Arabian algebraist, al-Karkhi died probably aboutthe author of the Fakkri, who uses the Diophantine system of powers of the unknown depending on the addition of exponents.

This is the most curious case of all, and the way in which Diophantus, after having worked with this ” I ” along with other numerals, is yet able to put his finger upon the particular place where it has passed to, so as to substitute 9 for it, is very remark- able. I shall not attempt to class as “methods” certain headings in Nesselmann’s classification of the problems, such as a ” Solution by mere reflection,” ” Solution in general expressions,” of which there are few instances definitely so described by Diophantus, or c “Arbitrary determinations and assumptions.

The reason why there were three cases to Diophantus, while today we have only one case, is that he did not have any notion for zero and he avoided negative coefficients by considering the given numbers abc to all be positive in each of the three cases above.

The first Book is confined to determinate algebraic equations; Books II.

Any further edition will neces- sarily be based on Tannery, who has added all that is required in the shape of introductions, etc.

The hypothesis that the Porisms arithmetiva part of the Arithmet- ica being thus given up, we can hardly hold any longer to Nesselmann’s view of the contents of the lost Books and their place in the treatise; and I am now much more inclined to the opinion of Tannery that it is the last and the most difficult Books which are lost.

This is, however, not so in the case of another pair of in- equalities, used later in V.

Yes, the cost is high, but the profit margin is undoubtedly much less than on a routine calculus book. Thus the ” first undescribed ” and the “second undescribed” correspond to “Relato i” and “Relato 2” respectively, but the “quadruple-square” exhibits the additive principle.

But we may at least be certain that Diophantus came as near to the proof of it as did Bachet, who takes all the natural numbers up to and finds by trial that all of them can actually be expressed as squares, or as the sum of two, three or four squares in whole numbers. Sometimes called “the father of algebra “, his texts deal with solving algebraic equations. These imperfections have been already noticed by Nesselmann 2.


In chapter 9, entitled “Diophantus’ methods of solution 3 ,” he classifies these ” methods ” as follows 4: Note the omission of Diophantus and Pappus, authors who evidently were not at first known in Arabia, although the Diophantine Arithmetica became familiar before the end of the tenth century.

According to Montucla 2″the historian of the French Academy tells us ” that Bachet worked at this edition during the course of a quartan fever, and that he himself said that, disheartened as he was by the difficulty of the work, he would never have completed it, had it not been for the stubbornness which his malady generated in him. I was therefore delighted at my good fortune in finding in the Library of Trinity College, Cambridge, a copy of Xylander, and so being able to judge for myself of the relation of the later to the earlier work.

Title page of the first book. Censo di Cubo, o Cubo di Censo. Now 10 and 3, f, are the sides of three squares the sum of which is The solution is as follows.

He also lacked a symbol for a general number n.

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After Maerten de Vos, Netherlandish, Antwerp ? But the same form ijip which Rodet gives is actually diophxntus in three places in Bachet’s own edition, i In his note to IV. Thus 16 U i 5 must be equal to a square.

But it is certain that up to this time the common symbols had been R Radix or ResZ Zensus, i. Filippo Calandri Italian, 15th century ; Publisher: One solution was all he looked for in a quadratic equation. Thus Diophantus’ method corresponds here again to the ordi- nary method of solving a mixed quadratic, by which we make ariithmetica into a pure quadratic with a different x.