To refine search, see subtopic The Method of False Position. To expand search, see Algebra. Laterally related topics: Solutions of Polynomial Equations, Indeterminate Equations, Group Theory, and Imaginary and Complex Numbers.
The Mathematics and the Liberal Arts pages are intended to be a resource for student research projects and for teachers interested in using the history of mathematics in their courses. Many pages focus on ethnomathematics and in the connections between mathematics and other disciplines. The notes in these pages are intended as much to evoke ideas as to indicate what the books and articles are about. They are not intended as reviews. However, some items have been reviewed in Mathematical Reviews, published by The American Mathematical Society. When the mathematical review (MR) number and reviewer are known to the author of these pages, they are given as part of the bibliographic citation. Subscribing institutions can access the more recent MR reviews online through MathSciNet.
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, Logarithms, Exponentials, Square Roots, Interpolation, Geometric Theorems, The Circle, and The Pyramid.
Eves, Howard. On the Practicality of the Rule of False Position. Mathematics Teacher 51 (1958), 606--8.
Eves shows how the method of false position can be simpler than our own methods by giving one example from the Ahmes Papyrus, three from the Greek Anthology of c. 500 AD, and two of his own. One of his examples is from surveying, and Eves says that it is the method a surveyor would probably use. In the other example of his own, he likens the rule of false position to the method of similitude in geometric constructions. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Method of False Position, Ancient Egypt, Medieval Europe, Surveying, and Geometry.
Lumpkin, Beatrice. From Egypt to Benjamin Banneker: African origins of false position solutions. Vita mathematica (Toronto, ON, 1992; Quebec City, PQ, 1992), 279--289, MAA Notes, 40, Math. Assoc. America, Washington, DC, 1996. SC: 01A05 (01A13), MR: 1 391 748.
Discusses the work of the Benjamin Banneker, who is perhaps the most interesting early American mathematician. The author gives a fine introduction to Banneker's life; this is necessarily brief, because as the author observes, his house burned down on the day of his funeral, destroying almost all his papers. She notes that there were hints of his genius starting with his building of a wood clock at the age of 22 (he used a borrowed pocket watch as a model; unfortunately, the clock was destroyed in the fire); he thereafter became famous for his ability to solve and create mathematical puzzles. "People sent him puzzles from all over the colonies and later from the new republic." His work became more serious when he was 57 and borrowed some books and astronomy instruments from a neighbor. He taught himself the mathematics he needed to become an astronomer, and published local almanacs including things such as the planetary positions and the times of sunrise, sunset, moonrise, moonset, eclipses, and tides. "Based on Banneker's work on his almanac, he was appointed an astronomer on the team of surveyors that drew up the outline for the new nation's capital, Washington, DC. Banneker was appointed because he was one of the few in the country capable of doing such work. Charles Leadbetter, author of an astronomy book that Banneker studied, wrote that knowledge of astronomy in London was 'so rare, ... not one of 20,000 hath attained to it.' Knowledge of astronomer", Lumpkin continues, "was even rarer in the new United States. Banneker's work so impressed Thomas Jefferson, then Secretary of State, that he wrote Banneker that he was sending a copy of the almanac to the Paris Academy of Sciences." Most amazing of all is that Banneker accomplished all this as an African American who had spent most of his life thus far hard physical labor. After this introduction, the author focuses on how Banneker and other mathematicians used the rule of false position. She notes, the rule of false position was used by the Egyptians in the time of the Rhind Papyrus and in a variety of other Egyptian sources (e.g., the Kahun and Berlin papyri), in the work of Alexandrian Greeks like Diophantus (c. 250 AD), in the work of Islamic mathematicians such as Abu Kamil (b. 850 AD), and in the work of the mathematician Leonardo of Pisa (Fibonacci) (who was also influenced by the work in Northern Africa). The author then discusses some interesting false position problems from Banneker's own work. Closely related topics: Benjamin Banneker, The Method of False Position, The Rhind/Ahmes Papyrus, Ancient Egypt, Diophantus, Abu Kamil (b. 850), and Leonardo of Pisa (Fibonacci).