N VERS TV OF CAL FORNIA AN DIEGC 1822 00204 5938 UMAK ivHAYYAM AS Anne A MATHEMATICIAN WILLIAM EDWARD STORY ialiforni gional cility 3 1822 00204 5938 Central University Library University of California, San Diego Please Note: This item is subject to recall. )ate Due fc JUN 3 1994 OMAR KHAYYAM AS A MATHEMATICIAN THE UNIVERSITY LIBRARY UNIVERSITY OF CALIFORNIA, SAN DIEGO LA JOLLA, CALIFORNIA WILLIAM EDWARD STORY OMAR KHAYYAM AS A MATHEMATICIAN By WILLIAM EDWARD STORY Professor of Mathematics CLARK UNIVERSITY Worcester. - Majwchuietts Read at the Annual Meeting of the OMAR KHAYYAM CLUB OF AMERICA 1918 April 5. 1919 Thit if one of an edition privately printed by the Rosemary Pm* for the member* of the Omar Khayyam Club of America. Limited to 200 copies, on American deckel-edged linen paper, bound with vellum back and antique paper ode*. Copies numbered 1 to 100 reserved to Professor Story. 100 copies numbered 1R to lOORreterved for the Rosemary Frew. This i* No. 60R OMAR KHAYYAM AS A MATHEMATICIAN. IT seems to be commonly assumed that Omar was by profession an astronomer and that with him pure mathematics was only a side issue. But it should be observed that all the earlier philosophers were, as the name "philosopher" implies, lovers of learn- ing of all kinds; such a lover of learn- ing Omar, indeed, seems to have been. The true philosopher takes the greatest pleasure in those forms of intellectual activity, within the field in -which his natural talents and education fit him to work, of course, that present the greatest difficulties. But the numbers of those to whom the results of these activities are intelligible are, in general, inversely proportional to the difficulties of obtaining them. Thus it comes about that many of the old philoso- phers are best known by those of their works in which they themselves did not take the greatest interest. Thales, the first of the Greek philosophers, the first of the "seven wise men" of Greece, was also the first Greek mathematician. Aristotle was a physicist, but he was also the first to enunciate the principle of continuity by the introduction of the idea of an "infinitesimal," which idea was developed by Cavalieri, Kepler, and others, and led, finally, in the hands of Leibnitz and Newton, to the invention of the infinitesimal calculus. Plato was a zealous promoter of mathe- matics among the Greeks. Archimedes, although a physicist, was called by his immediate successors the "great mathe- matician." Kepler was a mathema- tician as well as an astronomer. Fin- ally, Descartes, Leibnitz and Newton were pre-eminently mathematicians; in fact, from a certain point of view, I should b inclined to consider Des~ cartes the greatest mathematician that ever lived. I have said nothing of those who are known only as mathematicians and, I may almost say, are known only to mathematicians. My object has been to lay the foundation for my opinion that Omar was probably above all a pure mathematician. The distinction that is commonly made between pure and applied mathematics is somewhat inconsistent. Applied mathematics is not a branch of learning. It is mathe- matics as applied to practical purposes. The only conceivable reason for dis- tinguishing it from so-called pure mathematics is that the concepts to which the application is made are more or less necessarily associated with other concepts to which mathematics is not applicable. There is but one mathe- matics, namely, pure mathematics, which, however, ha* many forms. Most forms or branches of mathemat- fi ics have practical applications. Gauss, called by his contemp cries "princeps mathematicorum," himself an astron- omer by profession, praised the theory of numbers as having one great advan- tage over all other branches of mathe- matics in that it had no conceivable application to practical purposes. The only mathematical work of Omar with which we are acquainted is his "algebra." Algebra is the "soul" of modern mathematics; in its original form it is that branch of mathematics that deals with unknown numbers. The name algebra is derived from "al gibr w'al mukhabala" the title of every Saracen work on the subject since the time of Abu Jafar Muhammed ibn Musa al Khwarizmi (circa A.D. 825), who was long supposed to have in- vented the subject. But we now know that Al Khwarizmi' s work is simply a translation of the "dptOixeTtx-rj" of Dio- phantos of Alexandria (circa A.D. 275). Omar was one of a long series of Saracen algebraists who followed more or less closely in the track of Dio- phantos and Al Kwarizmi. Woepcke, in his French translation of Omar's al- gebra, finds in it traces of the influence of Diophantos, but, he says, "these are found also in Muhammed ibn Mousa, and there exists no historical datum that proves that at the time of this algebraist Diophantos was known to the Arabs." But we know better now. In algebra as the science of unknown numbers, it is necessary to have some method of designating the unknown in any particular question, as well as its positive and negative powers. Dio- phantos used symbols to represent these, but his symbols are simply ab- breviations of the names by which he called the corresponding numbers and in the text stand for these names rather than for the numbers. The Saracen mathematicians, including 8 Omar, adopted translations of Dio phantos's names and got along without symbols. Thus Omar gives a certain equation as "a cube and squares are equal to roots and a number," i.e. x 8 -f- ax 2 = bx -f- c. He calls the successive positive powers of the unknown "root" or "side," "square," "cube," "square- square," "square-cube," "cube-cube," etc. and the successive negative powers (reciprocals of the positive powers) "part of root," "part of square," etc., as Diophantos did. But all Omar's demonstrations are given in geomet- rical form, which was the standard form among the Greeks; in fact, the very names we have mentioned are borrowed from geometry. Moreover, Omar solves his equations by means of the intersections of conic sections; that is, he solves a typical form of the equation under consideration in this way and then modifies the solution to suit the particular equation. He is very systematic throughout, prefacing each section by such lemmas as he will have to use. Omar's greatest original contribution to algebra is the complete classification of the cubic equation, a classification that he recognizes as applicable to equations of every degree. He be- lieved that cubic equations could not be solved by calculation, but that one must be satisfied with the construction of solutions by intersecting conies. In the discussion of the several classes he sometimes overlooks particular cases. Thus, he fails to see that an equation of the form x 8 -j- bx = ax 2 -j- c may have three positive real roots. Again, he lost many roots by using only one branch of an hyperbola in his con- struction. And he was not very exact in the investigation of the numerical values that the several coefficients must have in order that the equation of one or other type should give real inter- 10 sections of the conies. He considered biquadratic equations to be unsolvable by geometric constructions. But these faults are of little conse- quence in comparison with the re- markably great advance Omar made in algebra by treating equations of degree higher than the second, and by having classified them. He was the only mathematician of any nation be- fore 1,100 who distinguished trinomial cubic equations from tetranomial, forming two groups of the former ac- cording as the term of the 2nd or 1st degree was wanting, and two groups of the latter according as the sum of 3 terms was equal to one term or the sum of 2 terms equal to the sum of two others. Apparently, also, he considered the binomial theorem for positive integral exponents. He says: "I have taught how to find the sides of the square- square, of the square-cube, of the cube- 11 cube, etc. to any extent, which no one had previously done." This theorem he used, apparently, for the purpose of extracting roots after the manner of the Hindus. Omar incidentally solved the geometrical problem: to construct an equilateral trapezoid whose base and sides are of the same given length and whose area is given, a problem that he reduced to the solution of the equation x 4 -f- bx = ax 8 -j- c. In the year 1 079 Omar corrected the calendar. He grouped the years in cycles of 33 years each, giving each common year 365 days and making every fourth year a leap-year of 366 days throughout each cycle; that is. each cycle of 33 years contained 8 leap-years and there was an interval of 5 years from the beginning of the last leap-year of any cycle to the beginning of the first leap-year of the next cycle. This makes the average length of Omar's solar year 365 5 h 49 5.45, 12 which is less by 6.55 seconds than the average length of the Gregorian year. According to the best modern calcula- tions, the Gregorian average year is too long by 25.557 seconds and Omar's average year is too long by only 19.007 seconds. That is, one leap year ought to be omitted from Omar's cal- endar every 4545 years, whereas the Gregorian calendar ought to omit one leap-year every 3381 years. This means that Omar's calendar was one- third more accurate than the calendar we use today. However, all people that use the solar year would probably find it more convenient to omit three leap-years out of 400 years than to group the years in cycles of 33. All things considered, I am inclined to think that Omar Khayyam was the most original and, therefore, the great- est of the Saracen mathematicians. 13 University of California SOUTHERN REGIONAL LIBRARY FACILITY 405 Hilgard Avenue, Los Angeles, CA 90024-1388 Return this material to the library from which it was borrowed. UC SOUTHERN REGIONAL LIBRARY FACILITY A A 000006485 7 Univen Soul Lit