1

x2

1

x

1

x

constructible 95

function differ only by a constant. For example, any

antiderivative of the function f(x)= 2xmust be of the

form x2+ Cfor some constant C, called the constant

of integration. An antiderivative is usually expressed

as an

INDEFINITE INTEGRAL

. In our example we have

∫2xdx = x2+ C.

Care must be taken when working with an arbi-

trary constant of integration. For example, consider

computing the integral

∫

dx via the method of

INTEGRA

-

TION BY PARTS

. Set u= and v′= 1 (so that u′= –

and v= x) to obtain:

Subtracting the integral under consideration suggests the

absurdity: 0 = 1. Of course, this argument failed to take

care of the constants of integration that should appear.

See also

ANTIDIFFERENTIATION

;

INTEGRAL CALCULUS

.

constant width A circular wheel has the property

that it has constant height as it rolls along the ground.

Alternatively, one could say that the width of the curve

is the same no matter which way one orients the figure

to measure it. Any shape with this property is called a

curve of constant width. The so-called Reuleaux trian-

gle, constructed by drawing arcs of circles along each

side of an equilateral triangle (with the opposite vertex

as center of each circular arc) is another example of

such a curve. Wheels of this shape also roll along the

ground with constant height.

One can construct wheels of constant height with

the aid of a computer. One begins with the

PARAMETRIC

EQUATIONS

of a circle of radius 1 and center (0,0) given

by x(t) = 0 + cos(t) and y(t) = 0 + sin(t),for 0 ≤t <360°.

Certainly the distance between any two points on this

circle that are separated by an angle of 180°is always

2. By changing the location of the center of the circle

slightly, we can preserve this distance property, as long

as we ensure that the center returns to the same loca-

tion every 180°. Thus, for example, the equations:

are the parametric equations of another curve with

the same constant-width property. (The fractional

coefficients were chosen to ensure that the resulting

figure is

CONVEX

.)

French mathematician Joseph Barbier (1839–89)

proved that all curves of constant width dhave the

same perimeter, πd. It is also known that, for a given

width, Reuleaux’s triangle is the curve of constant

width of smallest area.

constructible A geometric figure is said to be con-

structible if it can be drawn using only the tools of a

straightedge (that is, a ruler with no markings) and a

compass. The straightedge allows one to draw line seg-

ments between points (but not measure the lengths of

those segments), and the compass provides the means

to draw circles with a given point as center and a given

line segment from that point as radius.

The Greek scholars of antiquity were the first to

explore the issue of which geometric constructs could

be produced with the aid of these primitive tools alone.

The geometer E

UCLID

(ca. 300

B

.

C

.

E

.) explicitly stated

these limitations in his famous text T

HE

E

LEMENTS

.

Despite the fact that his exercise has no real practical

application (it is much easier to draw figures with rulers

to measure lengths and protractors to measure angles),

the problem of constructibility captured the fascination

of scholars for the two millennia that followed. This

illustrates the power of intellectual curiosity alone for

the motivation of mathematical investigation. Students

in high schools today are still required to study issues of

constructibility.

The compass used by the Greeks was different from

the one we use today; it would not stay open at a fixed

angle when lifted from the page and would collapse.

Thus it was not directly possible to draw several circles

of the same radius, for instance, simply by taking the

compass to different positions on the page. However, in

his work The Elements, Euclid demonstrated how to

accomplish this feat with the Greek collapsible compass.

This shows that any construction that can be accom-

plished with a modern compass can also be accom-

plished with a collapsible compass. For this reason, it is

assumed today that the compass used is a modern one.

A surprising number of constructions can be

accomplished with the aid of a straightedge and com-

pass alone. We list here just a few demonstrations.