For more material on this topic, see subtopic Exponentials. To expand search, see Arithmetic. For material on related topics, see Exponentials. Laterally related topics: Number Systems, Numerology, Magic Squares, Bookkeeping, Modular Arithmetic, Algorithms, The Number Concept, The Abacus, Exponentials, Interpolation, Zero, Fractions, The Real Number System, Irrationals, The Extraction of Roots, Mental Arithmetic, The Negative Numbers, and Imaginary and Complex Numbers.
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Archibald, Raymond Clare. Babylonian Mathematics. With Special Reference to Recent Discoveries. Mathematics Teacher 29 (1936), 209--19. (Originally delivered at a joint meeting of the National Council of Teachers of Mathematics, the American Mathematical Society, and The Mathematical Assocation of America, at St. Louis, Mo., on January 1, 1936.)
Surveys some of Neugebauer's remarkable discoveries on Babylonian mathematics, at a time when many of these discoveries were just made. Discusses notation, tables of squares, cubes, and n3+n2. Also exponentials, approximations to compound interest problems where we would use logarithms, a sum of a finite geometric series and a finite sum of squares. Geometric results, including the Pythagorean theorem, proportionality of sides in similar right triangles, a perpendicular bisecting the base in an isosceles triangle, the angle in a semicircle being a right angle, formulas for the circumference and area of a circle (using pi = 3), formulas for the frustum of a square pyramid (at least one incorrect). The relation between chords and sagitas in a circle. Approximations to the square root of a2+b2; both the well known a+b2/2a and the still hypothetical a+(2ab2)/(2a2+b2). An approximation to a square root by comparing with other solutions to an equation x2+D=y2. (The value isn't especially accurate, but the method is interesting.) Equations in five or more unknowns. Problems requiring solutions to apparently general cubic and biquadratic equations. Were the solutions just guessed, or, as Neugebauer suggests, did the Babylonians have some general methods? If so, the most likely theory is that the cubics were solved by effectively reducing them to the form x3+x2, and then using the n3+n2 table. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Sumerians and Babylonians, The Quadratic Formula, Cubics, Quartics, Solutions of Linear Equations, Exponentials, Square Roots, Interpolation, Geometric Theorems, The Circle, and The Pyramid.
Knuth, Donald E. Ancient Babylonian algorithms. Twenty-fifth anniversary of the Association for Computing Machinery. Comm. ACM 15 (1972), no. 7, 671--677; errata, ibid. 19 (1976), no. 2, 108; MR: 52#13133. SC: 01A15, MR: 52 #13132.
Were there computer scientists among the ancient Babylonians? Probably not. However, some of the ideas in computer science occurred to the ancient Babylonians as well. The author here discusses Babylonian algorithms in particular. Most algorithms are of course given as examples, but Knuth notes one text that is an exception: "Length and width is to be equal to the area. You should proceed as follows. Make two copies of one parameter. Subtract 1. Form the reciprocal. Multiply by the parameter you copied. This gives the width." Knuth explains, "In other words, if x+y=xy, it is possible to compute y by the procedure y=(x-1)-1x. The fact that no numbers are given made this passage particularly hard to decipher, and it was not properly understood for many years; hence we can see the advantages of numerical examples. The above procedure reads surprisingly like a program for a 'stack' machine like the Burroughs B5500!". Knuth finds a table involving compound interest where he finds evidence of a "DO I = 1 TO N" loop and something like a "WHILE" clause. He also discusses how one tablet may have been obtained by sorting a large set of numbers. "Thus, Inakibit seems to have the distinction of being the first man in history to solve a computational problem that takes longer than one second of time on a modern electronic computer!" [However, note that this statement was made in 1972.] Some tablets cited are available here in English for the first time (Knuth translated them using German and French translations, and at times Akkadian and Sumerian vocabularies as well). See errata in Knuth, Donald E., Errata: "Ancient Babylonian algorithms" (Comm. ACM 15 (1972), no. 7, 671--677). Closely related topics: Sumerians and Babylonians, Computation, and Algorithms.
Knuth, Donald E. Errata: "Ancient Babylonian algorithms" (Comm. ACM 15 (1972), no. 7, 671--677). Comm. ACM 19 (1976), no. 2, 108. SC: 01A15, MR: 52 #13133.
An errata to Knuth, Donald E., Ancient Babylonian algorithms. The table that was sorted was not as extensive as Knuth previously believed, and involved a "file" of about 500 instead of about 800. As Knuth notes "My italicized statement on p. 676 that 'this table contains every one' of the 231 regular sexagesimal numbers of six digits or less, is false; the table contains only 136 of those 231." The misunderstanding was due to a failure "to read the accompanying German commentary carefully enough, since [Neugebauer] departed from his usual custom in this particular case. Many of the lines in his rendition of the table were not on the original clay tablet at all, they were interpolated to show what the tablet would have looked like if it had been complete." Closely related topics: Sumerians and Babylonians, Computation, and Algorithms.