Page:The Rhind Mathematical Papyrus, Volume I.pdf/26

This page has been proofread, but needs to be validated.

10

EGYPTIAN ARITHMETIC

such fractions.^[1] In fact, he claims that the theories of Rodet and Hultsch are essentially the same as the modern theory, “bloc extractif” and “auxiliary unit” being only other names for common denominator. Perhaps he would say that my “particular number” is only another name for common denominator, but it does not seem so to me. The idea of taking a number, solving the problem for this number, and assuming that the result so obtained holds true for any number, is exactly what the boy in school is inclined to do for all sorts of problems, and what the author of our handbook does in much of his work. In fact, he always takes a particular number, and when he has solved a problem he does not hesitate to take a more complicated one of the same kind and use the same method. I do not think that the idea of a common denominator or of a fraction with numerator greater than 1 is involved in the theory as I have explained it, even though the number used is the same number as our common denominator, and some of the numerical work is the same as when a common denominator is used.^[2] When I say that ¹⁄₃ of 105 is 35 and ¹⁄₅ is 21, together making 56 things, there is no suggestion that ¹⁄₃ is equal to ³⁵⁄₁₀₅ and ¹⁄₅ to ²¹⁄₁₀₅.

2. The process of false position (positio falsa) consists in assuming a numerical answer and then by performing the operations of the problem getting a number which can be compared with a given number, the true answer having the same relation to the assumed answer that the given number has to the number thus obtained.^[3] In this method we see one point of distinction between arithmetic and algebra. In algebra a letter $\scriptstyle x$ represents exactly the answer, and its value is obtained by solving an equation. In this method we take a number

↑ Sethe, in his review of Peet (See Bibliography under Peet, 1923, 2) expresses emphatic dissent from this view. He says “Wherever there was the notion there would have been already established the word and the sign. Indeed, it would have been entirely incomprehensible that the Egyptians should have written ¹⁄₂ ¹⁄₄ or ¹⁄₃ ¹⁄₁₂ if they did not also read these words, and that the Greeks, Romans, and Arabs should have held on so long to the really circuitous reckoning with fundamental fractions if the so much simpler method of mixed fractions [fractions with numerator greater than 1] had already so early been known.” See also Sethe, 1916, pages 60 and 62.
↑ It is just as the modern schoolboy in solving simultaneous equations by different methods will have the same numbers to multiply or divide although the theories of the different methods are different.
↑ We might say that this is proportion, but the words “false position,” with their suggestion of the days when we had only arithmetic in our schools, give us a better realization of the Egyptian point of view. The rule was called by the Arabs, ḥisab al-Khataayn, and hence it appears in medieval Europe as Elchataym. It was also known by such names as Regula falsi, Regula positionum, False positie. See D. E. Smith, 1923, volume 2, pages 437-438.

[1] Sethe, in his review of Peet (See Bibliography under Peet, 1923, 2) expresses emphatic dissent from this view. He says “Wherever there was the notion there would have been already established the word and the sign. Indeed, it would have been entirely incomprehensible that the Egyptians should have written ¹⁄₂ ¹⁄₄ or ¹⁄₃ ¹⁄₁₂ if they did not also read these words, and that the Greeks, Romans, and Arabs should have held on so long to the really circuitous reckoning with fundamental fractions if the so much simpler method of mixed fractions [fractions with numerator greater than 1] had already so early been known.” See also Sethe, 1916, pages 60 and 62.

[2] It is just as the modern schoolboy in solving simultaneous equations by different methods will have the same numbers to multiply or divide although the theories of the different methods are different.

[3] We might say that this is proportion, but the words “false position,” with their suggestion of the days when we had only arithmetic in our schools, give us a better realization of the Egyptian point of view. The rule was called by the Arabs, ḥisab al-Khataayn, and hence it appears in medieval Europe as Elchataym. It was also known by such names as Regula falsi, Regula positionum, False positie. See D. E. Smith, 1923, volume 2, pages 437-438.

[1]

[2]

[3]