To refine search, see subtopic Surface Area. To expand search, see Measurement. Laterally related topics: The Balance and the Measurement of Weight, The Measurement of Distance, Leveling, Angular Measure, and The Astrolabe and Related Instruments.
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.
Dilke, O. A. W. Mathematics and measurement. Reading the Past, 2. University of California Press, Berkeley, CA; British Museum Publications, Ltd., London, 1987. 64 pp. ISBN: 0-520-06072-5. (Reviewer: Richard L. Francis.) SC: 01A05 (01A15 01A20), MR: 89f:01003.
This very interesting book discusses many aspects of mathematics in the Roman empire, Egypt, Babylonia, Greece, and sometimes other cultures. The book discusses systems of measurement of length, area, volume, and weight, mathematical or para-mathematical subjects such as surveying, cartography, interest rates, taxes, time keeping, games, and numerology. Also discusses number systems. Much of the discussion on number systems may be familiar, but here there is also a little that may be a little less familiar, such as the use of Etruscan letters in the early Roman numerals. In a work of this scope, the author of the book is not to be faulted that there may be some disagreement with occasional facts. The discussions on the mathematics of the Romans are particularly interesting; there are few other studies touching on Roman mathematical practices at all. Closely related topics: The Roman Empire, Ancient Egypt, Sumerians and Babylonians, Greece, The Measurement of Distance, The Balance and the Measurement of Weight, Surveying, Cartography, Banking, Taxation, The Reckoning of Time, Games, Numerology, and Number Systems.
Dwornik, Henryk. A $2\sp{n}$-number system in the arithmetic of prehistoric cultures. Organon No. 16-17 (1980/81), 199--222 (1983). (Reviewer: Garry J. Tee.) SC: 01A10, MR: 85f:01006.
The author attempts to explain use of base 12 or base 60 in otherwise primarily base 10 cultures as an attempt to reconcile a base 10 and a base 2n system. As evidence of such a base 2n system, the author discusses the use of "base" 2 worldwide in systems for measuring distance, area, volume, and weight. He also discusses how Indo-European languages show evidence of an ancient base 4 or 8 system in the words for nine, such as in the well-known example of the Latin novem for both new and nine. The numbers 4+1, 4-1, 16+1, and 16-1 are all represented neatly in base 60. The author discusses some advantages of a number system where numbers are represented by bn...b2b1b0 as in base two, except where bi=1, 0, or -1. As the author admits, all of this is highly speculative. The author also makes the interesting observation that some of the numbers used in Mayan cosmology become very symmetric when expressed in base 2 on a 3x3 board. The suggestion seems to be that base 2 computation may have been a motivating force for the Mayans. As we still have little knowledge about Mayan arithmetic, it may be awhile before we have a definitive answer. Closely related topics: Number Systems, The Measurement of Distance, The Balance and the Measurement of Weight, Number Words, and The Maya.
Fletcher, E. N. R. The area of the curved surface of a hemisphere in ancient Egypt. Math. Gaz. 54 (1970), no. 389, 227--229. SC: 01A15, MR: 58 #9987.
Problem 10 of the Moscow papyrus discusses the surface area of a basket and is thought by some to compute the surface area of a hemisphere. The author analyzes which units may have been used in the problem, and advances the theory that the basket in question was, in fact, hemispherical, and was designed to hold 100 Hekat of corn. He notes that the units used in ancient Egypt appear to have some interesting geometrical properties. For example, a circle with a radius of 1 pes (or "foot", equal to 16 digits) was approximately equal in area to a square with sides measuring 1 royal cubit. These are all fascinating possibilities. Closely related topics: Ancient Egypt, Surface Area, and The Sphere.
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 Pyramid, and The Method of Exhaustion.
Harvey, H. R. and Williams, B. J. Aztec arithmetic: positional notation and area calculation. Science 210 (1980), no. 4469, 499--505. (Reviewer: M. P. Closs.) SC: 01A10, MR: 81k:01002.
It has long been thought that the Mayans were the only Mesoamerican people to have developed a positional number system. However, as the authors have noted, the Aztecs also had such a system (using lines and dots). The treatment of zero may be less consistent than it was with the Mayans. The authors discuss Aztec calculations of area as well. The Aztecs clearly used some sort of algorithm to compute these areas. (It's difficult to assess the calculations perfectly since areas of quadrilaterals are only determined by the lengths of the sides in the special case of triangles.) The authors discuss why the mathematics discussed in this article was unlikely to have come from the Spanish. The authors also discuss an interesting feature of the Nahua language which was spoken by the Aztecs, where a system of classifiers was used; the language included classifiers for round objects, for objects where length is a primary factor, and for objects that can be stacked. Closely related topics: The Aztec, Number Systems, Number Words, and The Maya.
Jones, Phillip S. Irrationals or Incommensurables. III. The Greek solution. Mathematics Teacher 49 (1956), 282--85.
Shows how Eudoxus' Method of Exhaustion is used to prove that circles are to one another as the squares on their diameters. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Method of Exhaustion, Eudoxus, and The Circle.
Jones, Phillip S. Recent Discoveries in Babylonian Mathematics. I. Zero, Pi, and Polygons. Mathematics Teacher 50 (1957), 162--65.
Supplements Archibald, Raymond Clare, Babylonian Mathematics, discussing some work by Neugebauer and others 1936 and 1957. Discusses the invention of the zero in (later) Babylonia and its appearance in Greece. (Zero was apparently first regarded as a true number by Aristotle.) Also discusses a value of 3 1/8 for pi (reported by M.E.M. Bruins, anticipated by Neugebauer), a problem to determine the radius of a circle circumscribing an isosceles triangle with two sides of 50 and one of 60 (an often discussed example, originally discovered by Bruins, that is still a good algebra problem, using only the Pythagorean theorem), and a table giving areas of pentagons, hexagons, and heptagons from the square of a side. Not all are accurate, but agree with analogous values given later by Heron (c. 75 AD). Heron's table included the regular nonagon as well. The article is continued in Jones, Phillip S., Recent Discoveries in Babylonian Mathematics. II., which however, has a somewhat smaller scope. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Sumerians and Babylonians, The Circle, Zero, Aristotle, and Heron.
Jones, Phillip S. Recent Discoveries in Babylonian Mathematics. III. Trapezoids and Quadratics. Mathematics Teacher 50 (1957), 570--71.
Continues Jones, Phillip S., Recent Discoveries in Babylonian Mathematics. II.. The author discusses a single Babylonian problem. The problem is interesting more as a representative of a "typical" Babylonian problem than as a discovery that gives new insights into Babylonian mathematics. The problem involves the solution to a quadratic. The scribe uses an incorrect "formula" for the area of a trapezoid. The author discusses the solution both using modern notation and in a translation of the scribes actual language. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Sumerians and Babylonians and The Quadratic Formula.
Miller, G. A. Gerbert's Letter to Adelbold. School Science and Mathematics 21 (1921), 649--53.
Gerbert puts circles and squares inside an equilateral triangle, and attempts to explain why they give different answers for the area. We think of these answers as estimates, but Gerbert's letter contains no hint of a limiting process. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Abacus, Gerbert, Pope Sylvester II, and Limit.
Swetz, Frank J. The Method of Archimedes. In Swetz, Frank J. From Five Fingers to Infinity. A Journey through the History of Mathematics. Open Court, Chicago, 1994. . 180--181.
Shows how Archimedes used his Method to discover the formula for the volume of a sphere. (Of course Archimedes also gave a rigorous proof using Eudoxus' Method of Exhaustion.) Closely related topics: Archimedes' Method, Archimedes, and The Sphere.