Department of Mathematics, LSU Baton Rouge

**Math 4005: Geometry**

Course home page >> Topic
Page I.A. >> this page

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.

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.

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.

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.