Department of Mathematics, LSU Baton Rouge
Math 4005: Geometry
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I.A. The Area of a Triangle

Readings with Source and Comment

Euclid's approach (dissection)

Euclid Proposition 35, Book I. Source: Euclid's Elements (Thomas L. Heath, translator). Green Lion Press, Santa Fe, 2002. Comment: This proposition marks the beginning of Euclid's treatment of area in Book I.

From an 1830 reference book. Source: Society for the Diffusion of Useful Knowledge. Geometry, Plane, Solid and Spherical, in Six Books. Library of Useful Knowledge. London: Baldwin and Cradock, Paternoster-Row. 1830. Comment: This is Euclid I.35, but with a different proof that Euclid gave. The proof seems slightly better than Euclid's, since it does not require us to consider as special the case in which the upper edges of the parallelograms overlap.

Hartshorne on Euclid. Source: R. Hartshorne. Companion to Euclid. American Mathematical Society, Providence, 1997. Comment: In this passage, Hartshorne describes the approach to area that Euclid takes. Euclid does not view area as a number that one attaches to a figure. He has no area function. Euclid's treatment is based on the notion of equidecomposibility.

Moise on area. Source: Edwin Moise. Elementary Geometry from and Advanced Standpoint, 2nd. Ed. Reading: Addison-Wesley1974. Comment: This is a rigorous modern presentation of the theory of area based on equidecomposability. Many details are very different from Euclid, especially the introduction of an area function. By the way, this reading is only a part of the full treatment in Moise's book. The postulates presented here describe how the area function should behave, but Moise does not show that an area function actually exists until later. (To understand the problem, think of this. You might work hard to set down the properties that you desire in a house. So good. You've described the house you want. Actually finding a house that has those properties is a different problem. There might not be one.) The first edition of this book appeared in 1963. Note the similarity of the treatment given by Moise with the treatment in the high-school texts from around this time.

Modern approach (measure theory)

Lebesgue on area. Source: H. Lebesgue, "Sur la mesure des grandeurs," Enseignement mathématique, 31-34, 1933-1936. (This text is from a translation appearing in H. Lebesgue, Measure and the integral. Edited by Kenneth O. May. San Francisco, London, Amsterdam: Holden-Day. 1966.) Comment: This essay was composed as a result of Lebesgue's involvement in educating secondary teachers. It presents perspective on the idea of area that is entirely different from what we find in Euclid. Roughly speaking, we begin the process of measuring the area of a region by drawing a grid of unit squares over it. (See the illustration in Connected Math.) We might not be able to cover the entire region with intact grid squares, but the number of squares that are entirely inside at least gives us an underestimate. To get a better estimate, we subdivide the grid and count how many of the smaller squares are entirely inside. Lebesgue suggests using squares of 1/10 the side length of the original grid. We need to correct the count by dividing by 100, since each unit square consists of 100 smaller squares. This number is a better (under)estimate for the area. We can continue using finer and finer grids. In this way we may hope to get better and better estimates. The details require more explanation than room here permits; more comments later. The purpose of including this reading is to show you that the appearance of the idea of measuring area with grids, as in Connected Math, is connected to ideas that appear in more advanced contexts. This way of thinking about area is one of the foundational ideas of calculus.

Various High School Treatments

1963 High School Text. Source: Morgan and Zartman. Geometry. Houghton Mifflin, 1963. Comment: The treatment is incomplete, since some cases are ignored. The proof given actually works in some of the ignored cases. Compare with the next two texts.

1966 High School Text. Source: Anderson, Garon & Gremillion. School mathematics Geometry. Houghton Mifflin, 1966. Comment: This treatment is based on certain postulates, which appear earlier in the text. Roughly stated, these assumptions are: Each polygonal region has an area (P16, page 380). Congruent regions have the same area (P17). If two regions overlap only on edges, then the area of the union is the sum of the areas of the pieces (P18). The area of a square is the square of the length of a side (P19). The first author is Dick Anderson, LSU Boyd Professor of Mathematics. Dick is still very active as an advisor to the Louisiana Department of Education.

1972 High School Text. Source: Jurgensen, Donelly and Dolciani. Modern School Mathematics. Houghton Mifflin, 1972

Discovering Geometry. (1997). Source: Michael Serra. Discovering Geometry: An Inductive Approach, 2nd Ed. Berkeley: Key Curriculum Press. 1997. Comment: A popular "reform" high school text. The treatment of area begins with grids, but is based on decomposition. Seems intended to help develop intuitions; ignores finer points of logic.

UCSMP. (1997). Zalman Usiskin, et al. (University of Chicago School Mathematics Project). Geometry, Teachers' Ed. Glenview IL: ScottForesman. 1997. Comment: Remarks in teachers' edition are highly relevant.

Glencoe Geometry. (1998). Source: Geometry, Teachers' edition. New York: Glencoe McGraw-Hill. 1998. Comment: This is the book currently in use in East Baton Rouge. The presentation of the folding construction on page 529 seems obscure. It's actually very interesting idea, once you figure out what is intended. Many of the comments in the teachers' edition are irrelevant to the mathematics.

Math Connections. (2000). Source: W. Berlinghoff, C. Sloyer and R. Hayden. Math Connections 2a, Teacher Ed. Armonk, NY: It's About Time, 2000. Comment: Part of one of the so-called standards-based high-school curricula. Also see: Math Connections.

Connected Geometry. (2000). Source: Education Development Center, Inc. Connected Geometry. Chicago: Everyday Learning Corp. 2000. Comment: Uses an approach based on decomposition. Much less formal than texts from the 1960s. High school.

Middle School Treatments

NCTM Standards 2000. Comment: This is an excerpt from the middle-school standards.

Connected Mathematics. (2004). Source: Lappan, G. Fey, J., Fitzgerald, W. Friel, S. and Phillips, E. Connected mathematics: Covering and Surrounding. Glenview: Pearson/Prentice Hall. 2004. Comment: This is the only middle school text on the reading list. Part of one of the so-called standards-based middle-school curricula.