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. 
 
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 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
 
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