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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Inquiries made for it at different times suggested to me that it was a pity that a treatise so unique and in many respects so attractive as the Arithmetica should once more have become practically inaccessible to the English reader.
At the same time I could not but recognise that, after twenty-five years in which so much has been done for the history of mathematics, the book needed to be brought up to date, Some matters which in were still subject of controversy, such as the date of Diophantus, may be regarded as settled, and some points which then had to be laboured can now be dismissed more briefly.
Practically the whole of the Introduction, except the chapters on the editions of Diophantus, his methods of solution, and the porisms and other assumptions found in his work, has been entirely rewritten and much shortened, while the chapters on the methods and on the porisms etc.
The new text of Tannery Teubnerhas enabled a number of obscure passages, particularly in Books V and VI, to be cleared up and, as a basis for a reproduction of the whole work, is much superior to the text of Bachet.
I have taken the opportunity to make my version of the actual treatise somewhat fuller and somewhat closer to the language of the original. In other respects also I thought I could improve upon a youthful work which was my first essay in the history of Greek mathematics. It is well known that it is in these notes that many of the great propositions discovered by Fermat in the theory of numbers are enshrined ; but, although the notes are literally translated in Wertheim’s edition, they do not seem to have appeared in English ; moreover they need to be supple- mented by passages from the correspondence of Fermat and from the Doctrinae analyticae Inventum Novum of Jacques de Billy.
The histories of mathematics furnish only a very inadequate description of Fermat’s work, and it seemed desirable to attempt to give as full an account of his theorems and problems in or connected with Diophantine analysis as it is possible to compile from the scattered material available in Tannery and Henry’s edition of the Oeuvres de Fermat So much of this material as could not be conveniently given in the notes to particular problems of Diophantus I have put together in the Supplement, which is thus intended to supply a missing chapter in the history of mathematics.
Lastly, in order to make the book more complete, I thought it right to add some of the more remarkable solutions of difficult Diophantine problems given by Euler, for whom such problems had a great fascination ; the last section of the Supplement is therefore devoted to these solutions.
Diophantus and his Works! Notation and definitions of Diophantus. Diophantus’ methods of solution. The Porisms and other assumptions in Diopbantus VI.
On numbers separable into integral squares. Other problems by Fermat V. There is now, however, no longer any doubt that the name was Diophantar, not Diophanter 2.
The question of his date is more difficult Abu’lfaraj, the Arabian historian, in his History of tlte Dynasties, places Diophantus under the Emperor Julian A. The following is the evidence, which is collected in the second volume of Tannery’s edition of Diophantus englieh to be quoted as “Dioph. Parisinus is right, wrote, in the plural, wj UvOaydpcu.
The positive evidence on the subject can be given very shortly. An arithmetiac limit is indicated by the fact that Diophantus, in his book on Polygonal Numbers, quotes from Hypsicles a definition of such a number 1. A lower limit is furnished by the fact that Diophantus is quoted by Theon of Alexandria 2 ; hence Diophantus wrote before, say, A.
There is a wide interval between B. We have a letter of Psellus nth c. Now Anatolius wrote about A. We may therefore safely say that Diophantus flourished about A. This agrees well with the aeithmetica that he is not quoted by Nicomachus about A.
On the basis of eratpy Tannery builds the further hypothesis that the Dionysius to whom the Arithmetica is dedicated is none other than Aritmhetica who was at the head of the Catechist school at Alexandria egnlish was Bishop there A.
Tannery conjectures then that Diophantus was a Christian and a pupil of Dionysius Tannery, “Sur la religion des derniers mathematicians de 1’antiquite,” Extrait des Annales de Philosophic Chretienne,p. It is however difficult to establish this Hultsch, art.
The solution gives dio;hantus as the age at which he died. His boyhood lasted 14 years, his beard grew at 21, he married at 33; a son was diophants to him five years later and died, at the age of 42, when his father was 80 years old. Diophantus’ own death followed four years later 2. 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.
The works on which the fame of Diophantus rests are: Six Books of the former and part of the latter survive. Allusions in the Aritkmetica imply the existence of 3 A collection of propositions under the title of Porisms; in three propositions 3, 5 and 16 of Book V. Arithmettica, as Hultsch 1 Anthology, Ep. I do not see any advantage in this solution. Diophantos in Pauly-Wissowa’s Real-Encyclopadie. It may have been a separate work by Diophantus giving rules for reckon- ing englsh fractions ; but I do not feel clear that diophhantus reference may not simply be to the definitions at the beginning of the Arithmetica.
Geminus also distinguishes the two terms 3. But in Diophantus the calculations take an abstract form except in V. We findjthe Arithmetica quoted under slightly different titles.
Thus the anonymous author of prolegomena to Nicomachus” Introductio Arithmetica speaks of Diophantus’ ” thirteen Eenglish of Arithmetic 5. Plato, Laws B, c, on the advantage of combining amusement with instruction in arithmetical calculation, e. The idea that Regiomontanus saw, or said he saw, a MS.
There is no doubt that the missing Books were ‘lost at a very early date. Tannery 4 suggests that Hypatia’s commentary extended only to the first six Books, and that she left untouched aritthmetica remaining seven, which accordingly were first forgotten and then lost ; he compares the case of Apollonius’ Conies, the first four Books of which were preserved by Eutocius, who wrote a commentary on them, while the rest, which he did not include in his commentary, were lost so far as the Greek text is concerned.
While, however, three of the last four Books of the Conies have fortunately reached us through the Arabic, there is no sign that even the Arabians ever possessed the missing Books of Diophantus. Thus the second part of an algebraic treatise called the Fakhrl by Abu Bekr Muh. In the fourth section of this work, which begins and ends with problems corresponding to problems in Diophanttus Books II. Nor diophanhus there any sign that more of the work than we possess was known 1 Dioph.
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. This view is in strong contrast to that which Jiad previously found most acceptance among diopjantus petent authorities. The latter view was most clearly put, and most ably supported, by Nesselmann 1though Colebrooke 2 had already put forward a conjecture to the same effect ; and historians of mathematics such as Hankel, Moritz Cantor, and Giinther diophantua accepted Nesselmann’s conclusions, which, stated in his own words, are as follows: Nesselmann’s general argument eng,ish that, if we carefully read the last arithmerica Books, from the third to the sixth, we find that Diophantus moves in a rigidly defined and limited circle of methods and artifices, and that any diopnantus which he makes to free himself are futile ; ” as often as he gives the impression that he wishes to spring over the magic circle drawn round him, he is invariably thrown back by an invisible hand on the old domain already known ; aritumetica see, similarly, in half-darkness, behind the clever artifices which he seeks to use in order to free himself, the chains which fetter his genius, we hear their rattling, whenever, in dealing with difficulties only too freely imposed upon himself, he knows of no other means of extricating himself except to cut through the knot instead of untying it.
The subject is diophaantus finding of right- angled triangles in rational numbers such that the sides and area satisfy given conditions, the geometrical property of the right-angled triangle being introduced ehglish a fresh condition additional to the purely arithmetical conditions which have to be satisfied in the 1 Algebra der Griechen, pp.
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.
The first step englizh to consider what is actually wanting which we should expect to find, either as foreshadowed by the author himself or as necessary for the elucidation or completion of the whole subject. Now the first Book contains problems leading to determinate equations of the first degree ; the remainder of the work is a collection of problems which, with few exceptions, lead to indeterminate equations of the second degree, beginning with simpler cases and advancing step by step to more complicated questions.
There would have been room therefore for problems involving i determinate equations of the second degree and 2 indeterminate equations of the first. There is indeed nothing to show that 2 formed part of the writer’s plan ; but on the other hand the writer’s own words in Def. Pure quadratics Diophantus diophantjs as simple equations, taking no account of the negative -root. Indeed it would seem that he adopted as his ground for the classification of quadratics, not the index of the highest power of the unknown quantity contained in it, diopjantus the number of terms left in it when reduced to its simplest form.
His words are 1: If there are on both sides, or on either side, any terms with negative coefficients ev englsih riva eiS? This should be the object aimed at in framing the hypotheses of propositions, that is to say, diophantys reduce the equations, if possible, until one term is left equated to one term.
But afterwards I will 1 Dioph. The exclusion of the latter case is natural, since it is of the essence of the work to find rational and positive solutions. Nesselmann might have added that Diophantus’ requirement that the equation, as finally stated, shall contain only positive terms, of which two are equated to the third, suggests that his solution would deal separately with the three possible cases just as Euclid makes separate cases of the equations in his propositions VI.
The suitable place for it would be between the first and second Books. There is no evidence tending to confirm Nesselmann’s further argument that the six Books may originally have diophsntus divided into even more than seven Books. He argues from the fact that there are often better natural divisions in the middle of the Books e. But the latter circumstances are better explained, as Tannery explains them,by the supposition that the first problems of Books II.
Next Nesselmann points out that there are a number of imperfections in the text, Book V. 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. If the phrase which, as we have said, occurs three times in Book V. Schulz’s argument is, indeed, not conclusive. If they had been, I think the expression ” we have it in the Porisms ” would have been inappropriate.
In the first place, the Greek mathe- maticians do not usually give references in such a form as this to propositions which they cite when they come from the same work as that in which they are cited ; as a rule the propositions are quoted without any references at all. The references in this case would, on the assumption that the Porisms were a portion of the thirteen Books, more naturally have been to particular pro- positions of particular Books cf.
Konnte man diese Schrift als einen Bestandtheil des grossen in dreizehn Biichern abgefassten arithmetischen Werkes ansehen, so ware es sehr erklarbar, dass gerade dieser Theil, der den blossen Liebhaber weniger anzog, verloren ging. And, as Hultsch says, it is hard, on Tannery’s supposition, to explain why the three partv. The hypothesis that the Porisms formed 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.
Tannery’s argument seems to me to be very attractive and to deserve quotation in full, as finally put in the preface to Vol. He replies first to the assumption that Diophantus could not have proceeded to problems more difficult than those of Book V. It would be the greatest error, in any case in which a 1 Dioph. If we do not know to what lengths Archimedes brought the theory of numbers to say nothing of other thingslet us admit our ignorance.
But, between the famous problem of the cattle and the most difficult of Diophantus’ problems, is there not a sufficient gap to require seven Books to fill it? And, without attributing to the ancients what modern mathematicians have discovered, may not a number of the things attributed to the Indians and Arabs have been drawn from Greek sources?
May not the same be said of a problem solved by Leonardo of Pisa, which is very similar to those of Diophantus but is nc now to be found in the Arithmetical In fact, it may fairly be said that, when Chasles made his reasonably probable restitution of the Porisms of Euclid, he, notwithstanding the fact that he had Pappus’ lemmas to help him, undertook a more difficult task than he would have undertaken if he had attempted to fill up seven Diophantine Books with numerical problems which the Greeks may reasonably be supposed to have solved.
By means of the double equation Diophantus shows how to find a value of the unknown which will make two expressions linear or quadratic containing it simultaneously squares. Schulz then thinks that he went on, in the lost Books, to make three such expressions simultaneously squares, i. But this explanation does not in any case take us very far.
Bombelli thought that Diophantus went on to solve deter- minate equations of the third and fourth degree 1 ; this view, however, though natural at that date, when the solution of cubic and biquadratic equations filled so large a space in contemporary investigations and in Bombelli’s own studies, has nothing to support djophantus.
Hultsch 2 seems to find the key to the question in the fragment of the treatise arthmetica Polygonal Numbers and the developments to 1 Cossali, I.