isa-normaladvance-1903-00105

Description:	THE NORMAL ADVANCE.105The Oldest Mathematical Manuscript.Simple number ideas, a few number words toexpress these ideas, and the simplest mathematical act—that of counting, are common property of all people. But the power to expressnumbers in any systematic way, and to make calculations with them was possessed by the few inancient times.The Egyptian civilization at an early day produced some ripe scholars, known as priests andphilosophers. From these scholars the peoplelearned to read, to write, and to cipher.To the Egyptians must be ascribed the invention and development of ordinary arithmetic.An Egyptian priest—Ahmes by name—wrotethe oldest mathematical book. It was entitled Directions for Obtaining a Knowledge of all DarkThings, and was written nearly 2000 years before Christ. Ahmes manuscript is in the BritishMuseum at London, and it treated of arithmetic,algebra, geometry and trigonometry, not, however, as completely as they are treated at present.This manuscript was deciphered in 1877 by aGerman scholar, Eisenlohr.Ahmes did not give much attention to integralnumbers, but devoted more attention to fractionalnumbers, and this fact has led Gow and other historians to suppose that Ahmes probably wrotefor the elite mathematicians of his day.Ahmes seems to have confined himself to thetreatment of unit-fractions, fractions which haveunity for their numerators.The Babylonians treated of fractions which hada constant denominator, sixty and the Romanstreated fractions which had a constant denominator, twelve.Ahmes gave special attention to a class of frac-2tions of the general form -—-—- that is, to fractions which had 2 for their numerators, and whosedenominators were the series of odd numbers3-99. Notice that n was given values 1-49. Itseems to have been his purpose to resolve suchfractions into unit-fractions.If n.and so on.if n = 2,2n +1He said f = \|\| f =2n + 1ai l •3 TS)■ 1 1 T 28We would say \| -4- \| i + JT, and \ +butAhmes did not know of the use of the plus sign.Again, f = 1 ¥V 2iii •6 6 >22T1 _J •If 2 3 1)2_351 11 8 t s o and so on.From a study of the above reductions it wouldseem that2n + l1might be resolved intorig dro wllilei3¥6-n + 1and -7T— „t „ — —-T. But the writer does not(2n + 1) (n -4- 1)know whether Ahmes used this general formulaor not. He gives that -^ = -£%the above formula gives ¥\ = -£% andIf it was desired to multiply -§ by £, the equivalent of f, or (•£ £) was multiplied by -}t. Thus,£ of f = 4, of (i £) = jiT -fa, which was the formof Ahmes answer.To multiply -f by \, resolve \| into 4/ and -fa.Then J- of f = Ki A) = ttV ale-Since a fraction is an unexecuted division,Ahmes could work out the division of 2 by 5.2 divided by 5 = f = \| -fo 2 divided by 11 =-j\ = J- JT. History does not record that he dealtwith other than fractions which had 2 for their numerators. But the division of 5 by 7 may havebeen done as follows: 5 divided by 7 = f = \** = .+, (i A) and (tA) = t and (I A) = +and (i -h) = i -} tVFollowing his treatment of fractions Ahmestakes up the solutions of certain equations. Hewas limited to simple equations of one unknown.His algebra is rhetorical rather than symbolical.He represented the unknown by the word hau,heap. Thus, heap, its seventh, its whole, itmakes 19. From this he found the heap toequal 16-i ■§-. According to modern methods, let-=—r- x = 19, whenceting x = the unknown, ~=—f- x7= 19, and x16\|.From certain problems in Ahmes work, it is
Source:	http://indstate.contentdm.oclc.org/cdm/ref/collection/isuarchive/id/33870
Collection:	Indiana State University Archives

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