Urbana, Ill., Nov. 17, 1934
My dear Professor Sanders
I was much pleased to see that the National Mathematics Magazine aims to publish papers on the history of mathematics since it seems to me that this subject represents at the present time one of the weakest parts of American mathematics. During the last decade very rapid advances have been made in this subject, especially as regards very ancient mathematics. Hence the text books on this subject usually give inadequate accounts relating thereto even if they were approximately up to date at the time of publication. For instance, recent discoveries relating to the finding of at least one root by the ancient Babylonians of certain numerical quadratic and cubic equations throws new light on the history of algebra and on the contributions made by the Greeks and the Arabians towards the solution of algebraic equations.
The methods used by the ancient Babylonians to solve their quadratic equations seem to have been practically the same as those employed at the present time but as regards cubic equations they proceeded in a very different manner than we do to-day. They constructed tables of numbers of the form n2 + n3 for the different values of n and then reduced their cubic equations to the form x3 + x2 = a. The given tables then enabled them to find a real definite root, at least approximately, when such a root exists. It is to be emphasized that the complete solution of general quadratic and of general cubic equations could not be attained until our ordinary complex numbers began to be understood at about the beginning of the nineteenth century, although complete formal solutions were used earlier in Europe. It is therefore far from the truth to say that "the general quadratic as we know it today was thus fully mastered by Greek mathematicians" - D E Smith, History of Mathematics, Volume 1 (1923), page 126.
The student of the history of mathematics naturally desires to correct his own books on this subject as errors are reported and many of them doubtless regret that what is now the most extensive work on this general subject in the English language was not reviewed in the Bulletin of the American Mathematical Society, where one might have expected to find references to desirable modifications. Historical knowledge cannot be expected to grow vigorously unless statements which appear to be erroneous are considered on their own merits irrespective of where they may have first appeared. Such considerations may sometimes exhibit the fact that they are not as unsound as they at first appeared to be. With respect to the extensive historical writings of the late Florian Cajori it seems to me that R C Archibald brought out an important feature when he said: "Many of Professor Cajori's publications, especially in the pre-Californian days, show evidence both of haste in composition and of lack of checking in proof with the sources of information." Isis, volume 17 (1932), page 388.
This really means that the reader should carefully check the statements which he finds in Cajori's writings before he accepts them as commonly accepted historical facts even at the time when they were published. Probably the most reliable place to do this at the present time is the Geschichte der Elementar-Mathematik by J Tropfke, in seven volumes, of which the first two have appeared in the third edition 1930 and 1933, respectively. The main object of this letter is to emphasize the fact that historical mathematics writings should be prepared with the most utmost care in order to be really useful. Too many of them are based on statements which are misleading. One paper of this type, which appeared in the American Mathematical Monthly, was recently reviewed by the following sentence: "A series of assertions are not correct." Zentralblatt für Mathematik, vol. 9 (1934), page 97.
G A MILLER, University of Illinois
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