Why Euclid?



	

	Ralph Abraham
Visual Math Institute
POB 7920
Santa Cruz, CA 95061
Ph. 408-425-7436
Fx. 408-425-8612
abraham@vismath.org
http://thales.vismath.org/euclid




Introduction

Repeatedly I have urged the importance of a return to Euclid as the primary text for math in the schools. Here, briefly, I try to indicate my reasons.

The Sheldrake principle

According this guiding idea, our subject should be taught in historical order. This ensures that concepts which are essentially prerequisite to a particular mathematical notion (according to the actual historical record) are available to the student when needed. Without them, the student may be presented with an impossible learning task, and then blamed for failing.
Lardner's opinion, 1846

ªEuclid once superceded, every school teacher would esteem his own system the best, and every school would have its own class book. All that rigor and exactitude which have so long excited the admiration of men of science would be at an end.º
De Morgan's opinion, 1848

ªThere never has been, and till we see it we never shall believe that there can be, a system of geometry worthy of the name, which has any material departures (we do not speak of corrections or extensions or developments) from the plan laid down by Euclid.º
Heath's opinion, 1908

ªIt is, perhaps, too early yet to prophesy what will be the ultimate outcome of the new order of things; but it would at least seem possible that history will repeat itself and that, when chaos has come again in geometrical teaching, there will be a return to Euclid more or less complete for the purpose of standardizsing it once more. --- Euclid's work will live long after all the text-books of the present day are superseded and forgotten. It is one of the noblest monuments of antiquity; no mathematician worthy of the name can afford not to know Euclid, the real Euclid as distinct from any revise or rewritten versions which sill serve for schoolboys or engineers.
Newton's path

For example, the crucial role of Euclid in the math curriculum of Sir Isaac Newton is emphasied in Newton and the Culture of Newtonianism by Betty Jo Teeter Dobbs, pp. 3-15:
1642, born
1650s, educated in village schools
1661, age 18, to Trinity College, Cambridge
1661-1663, studied Aristotle
1664, begins to study math (Euclid)
1665, B.A., returns home
1665-1667, invents calculus
Math anxiety

Today our society suffers an epidemic of math anxiety, a crippling disease. In our NSF supported research into the causes of this disease during the early 1980s, we identified two main factors:
1.	Teaching symbolic manipulation too early.
2.	Lack of essential visual representations.
Euclid's elements avoids #1 by means of geometric algebra, while skirting #2 is normally up to the teacher (see VEE below.)
Geometric algebra

This is Euclid's strategy for avoiding the first cause of math anxiety. Each algebraic idea is introduced geometrically at first. Books I and II are devoted to this slow and careful introduction to algebraic thinking.
VEE: the Visual Elements of Euclid

This is a new edition of Euclid's Elements, from the Visual Math Institute. So far, the first six of Euclid's thirteen books have been expanded visually, in an extensive sequence of color drawings. Soon, all of this material will be available free on the World Wide Web. This is intended to assist teachers in presenting the Elements successfully in their classes, avoiding the second cause of math anxiety.
Constructions

The Elements of Euclid contain Definitions, Postulates, Common Notions, Propositions, Porisms, and Lemmas. Most of the elements are Propositions, which may be regarded as the primary content. These are of two types:
·	constructions (which end with Q.E.F.), and
·	theorems (ending with Q.E.D.)
In the context of ancient geometry, the constructions are undoubtedly the most important goal of the work. Each construction has two parts: the construction itself, and the proof that what has been constructed is precisely what was wanted. This is why the proof section of the constructive propositions ends with Q.E.F. (that which was to have been done.) Thus, we may think of the Definitions, and theorems as technical tools with which to establish constructions.

Here is a list of all 60 constructions of the thirteen books:

Book I: 14 (Props. 1,2,3,9,10,11,12,22,23,31,42,44,45,46)
Book II: 2 (Props. 11,14)
Book III: 6 (Props. 1,17,25,30,33,34)
Book IV: 16 (Props. 1 through 16)
Book VI: 10 (Props. 9,10,11,12,13,18,25,28,29,30)
Book XI: 5 (Props. 11,12,23,26,27)
Book XII: 2 (Props. 16,17)
Book XIII: 5 (Props. 13,14,15,16,17)

And what is constructed in these 60 most important elements? Each uses the preceding ones, so certainly the final constructions must be the goal of the whole work. And these are none other than the five regular solids. (See Proclus, A Commentary on the First Book of Euclid's Elements, Princeton, 1970/1992, p. xxv.)

Some of the constructions are simply subroutines, that is, subconstructions, to be used elsewhere. Extracting the most important of the 60 constructions, which may be regarded as important ends in themselves, may be accomplished various ways. Such a listing we call a thread. We have chosen the golden thread.
The golden thread

In this thread, the main milestone is achieved in Book IV, with the construction of a pentagon, in a given circle. And the critical technical subconstruction for this, in turn, is the DEMR (Division in Extreme and Mean Ratio) or golden section. Here are the 15 main constructive steps of the golden thread, from the first construction of Book I to the pentagon of Book IV.

1.	I.1, construct an equilateral triangle
2.	I.2, move a straight line
3.	I.3, cut off a given straight line
4.	I.9, bisect an angle
5.	I.10, bisect a straight line segment
6.	I.11, draw a right angle
7.	I.23, draw a given angle
8.	I.31, draw a straight line parallel to a given straight line
9.	I.46, make a square on a given straight line
10.	II.11, DEMR
11.	IV.1, fit a straight line in a given circle
12.	IV.2, inscribe a triangle in a given circle
13.	IV.5, circumscribe a circle about a given triangle
14.	IV.10, construct a special triangle
15.	IV.11, inscribe a pentagon in a given circle