To expand search, see Geometry. Laterally related topics: Symmetry, Analytic Geometry, Trigonometry, Pattern, Geometric Theorems, Similarity, The Triangle, The Method of Exhaustion, Projective Geometry, Algebraic Geometry, Non-Euclidean Geometry, The Parallel Postulate, The Regular Solids, Irrationals, The Pentagram, The Sphere, The Conic Sections, Polygons, Topology, Spirals, Line-Point Duality, Geometric Fixed Point Principles, The Cycloid, Tilings, and The Square.
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, Solutions of Linear Equations, Logarithms, Exponentials, Square Roots, Interpolation, Geometric Theorems, and The Circle.
Gillings, R. J. The Volume of a Truncated Pyramid in Ancient Egytian Papryi. Mathematics Teacher 57 (1964), 552--55.
Gillings gives a clever way to derive the formula V=1/3(a2+ab+b2) for the volume of a truncated pyramid, using only the formula for the volume of a complete pyramid and other methods that the Egyptians had at their disposal. As he shows, fairly simple arguments suffice when b=a/2,a/3,..., and also when b=2/3a. Since to the Egyptians, every number could be represented as a finite sum of unit fractions, the demonstration is now complete. Of course we (or the Greeks) would require something like the method of exhaustion. (Even without it, the jump to a general number is a difficult step, and not trivial geometrically.) (Since in the Moscow papyrus, b=a/2, one might wonder if perhaps the Egyptians did not know the general case after all.) Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Ancient Egypt, The Measurement of Area and Volume, and The Method of Exhaustion.
Neugebauer, O. On the orientation of pyramids. Special issue dedicated to Olaf Pedersen on his sixtieth birthday. Centaurus 24 (1980), 1--3. (Reviewer: H. W. Guggenheimer.) SC: 01A15, MR: 81k:01004.
Neugebauer gives a theory that explains how the Egyptians could have oriented their pyramids without using the advanced astronomical knowledge sometimes attributed to them. The theory relies on the construction of an accurately shaped pyramidal model (for example the capstone of the future pyramid), and on watching the shadow of the model in the course of the day. The biggest question about this procedure may be the question of how the model can be made accurately enough. Nevertheless, this theory represents a great simplification over many other theories. Closely related topics: The Egyptian Pyramids and Astronomy.