Tales of Impossibility: The 2,000 Year Quest to Solve the Mathematical Problems of Antiquity by David S. Richeson

Author:David S. Richeson [Richeson, David S.]
Language: eng
Format: epub
ISBN: 9780691192963
Google: d-csEAAAQBAJ
Amazon: 0691192960
Barnesnoble: 0691192960
Goodreads: 44438361
Published: 2019-07-08T11:29:15+00:00

he states is z = a/ 2 + a 2 / 4 + b 2.

To do so, he uses a compass and straightedge to construct a

right triangle LMN with LN = a/ 2 and LM = b (see figure 15.5).

Then he draws a circle with radius LN and center N and extends

C O M PA S S - A N D - S T R A I G H T E D G E A R I T H M E T I C

241

FIGURE 15.4. Descartes’s method of constructing square roots.

FIGURE 15.5. The segment OM is the solution to z 2 = az + b 2.

NM to O. By the Pythagorean theorem, MN 2 = LN 2 + LM 2. Moreover, MN = MO − NO = MO − LN. Combining these equations we have (MO − LN) 2 = LN 2 + LM 2. Rearranging terms gives MO 2 = 2 MO · LN+

LM 2 = a MO + b 2, which implies that z = MO is the solution to the equation.

Constructible Points and Constructible Numbers

So that we can understand what came next in the investigation of

these problems, it is important to be very clear about what we can

conclude from Descartes’s work on constructibility. Unfortunately, his

Geometry isn’t set up in a modern way, with clearly articulated defi-

nitions, carefully stated theorems, and rock-solid proofs. Because we

want to present the conclusions in a way that our path forward is more

clear, in this section we will transport ourselves to the present day and

examine some of the conclusions we can draw from Descartes’s work.

We will do it in a clear, rigorous, and modern fashion—not in the way

Descartes wrote about them.

242

C H A P T E R 15

FIGURE 15.6. Two circles and a line yield constructible numbers 0, ±1,

√

√

±1 / 2, ± 3 / 2, and ± 3.

Although we see lines and circles when we look at a compass-

and-straightedge construction, it is the points of intersection of the

lines and circles that are important. To say in full generality what is

constructible, we assume that we begin with as blank a slate as pos-

sible. We assume that we begin with two points, and we make the

assumption that they are one unit apart (or equivalently, we begin

with a unit line segment). At times, it is convenient to assume we have

a coordinate system; in this case, we assume the two points are ( 0, 0 ) and ( 1, 0 ). (We could even draw the x- and y-axes using the compass and straightedge if we wanted to.) Now, from these we can define two

related notions: constructible points and constructible numbers.

A point P is a constructible point if, starting with our two points, there is a sequence of legal compass-and-straightedge moves (recall

we gave the “rules of the game” on page 49) such that P is the point of intersection of two constructed curves (lines or circles). A real number

a is a constructible number if there exist constructible points P and Q

(with P = Q a possibility) such that | a| is the length of segment PQ.

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