
248 helix
1
–
6
1
–
5
1
–
4
1
–
3
1
–
2
=
n
1
21+
Note that in the first leg of the journey, the ant
covers 1/12th of the length of the band, and that this
proportion remains the same as the band is stretched.
During the second leg of the journey, the ant covers
now only 1/24th of the length of the band (1 in. of 24
in.), 1/36th of the length during the third length, and so
on. Thus after nlegs of the journey, the ant covers the
fraction of the band. The
ant reaches the finish only if Hnever surpasses the
value 12. As the harmonic series diverges, this must
indeed be the case.
Although the harmonic series diverges, the
ALTER
-
NATING SERIES
test from the study of
CONVERGENT SERIES
shows that the alternating harmonic series, given by
1 – + – + – +…
converges. We can use Euler’s approximation formula
to find its value. First note:
where the error is no larger than 2/n and so converges
to zero as ngrows. Consequently, the even partial
sums of the series approach the value In(2). The sum
of an odd number of terms equals
ln2 + error + and so too
approaches ln2 as nbecomes large. Thus we have:
1 – + – + – + … = ln(2)
This example is often used to illustrate the difference
between conditional and
ABSOLUTE CONVERGENCE
of a
series. In that setting, it can also be used to provide an
amusing “proof” that 1 equals 2.
See also M
ERCATOR
’
S EXPANSION
.
helix A spiral-shaped curve sitting in three-dimensional
space is called a helix. The name is the Greek word for
a “spiral” or a “twist.”
A cylindrical helix lies on a cylinder and cuts
across straight lines drawn along the length of the
cylinder at a constant angle α. A spiral staircase and
the thread on a straight screw are examples of cylindri-
cal helices. A conical helix is a spiral curve on a
CONE
,
and a spherical helix is a spiral on a
SPHERE
that cuts
lines of longitude at a constant angle.
A cylindrical helix has
PARAMETRIC EQUATIONS
: x
= acost, y = asint, and z= bt, where aand bare con-
stants, and tis the parameter. A conical helix is given
by: x= aetcost, y = aetsint, and z= et.
Heron of Alexandria (ca.10–75
C
.
E
.) Greek Geome-
try, Number theory, Physics, Engineering Sometimes
called Hero, Heron of Alexandria is remembered in
mathematics for his three-volume text Metrica, redis-
covered in 1896. The work discusses and develops in
great detail the principles of geometry, number, and
numerical approximation. It also contains the earliest
known proof of the famous formula that bears Heron’s
name. Outside of mathematics, Heron is best known
for his contributions to mechanics and fluid mechanics.
Demonstrating a wide range of scientific interests,
Heron wrote studies in optics, pneumatics (the study
and use of gas and fluid pressures), astronomy, survey-
ing techniques, and planar and solid geometry, but it is
the work presented in Metrica that proves his genius as
a mathematical intellect. Book I of this famous piece
computes the areas of triangles, quadrilaterals, and reg-
ular polygons, as well as the surface areas of cones,
prisms, spheres, and other three-dimensional shapes.
His famous formula for the area of a triangle solely in
terms of its side-lengths is presented in this section,
along with a general method for computing the square
root of a number to any prescribed degree of accuracy.
(This procedure is today called H
ERON
’
S METHOD
.
Book II of Metrica derives formulae for the volume
of each P
LATONIC SOLID
, as well as the volumes of
cones and spherical segments, and Book III extends
E
UCLID
’s study of geometric division. Heron also pre-
sents a method for determining the cube root of a num-
ber in this third volume.
Heron also wrote a number of important treatises
on mechanics, many of which survive today. His text
Pneumatica represents a careful (but, in places, inaccu-
1
–
6
1
–
5
1
–
4
1
–
3
1
–
2
SS
n
nn21 2 1
21
+=+
+=
SS
n
nn21 2 1
21
+=+
+
Snn
nn n
nn n
n211
2
1
3
1
4
1
21
1
2
11
2
1
3
1
21
1
221
2
1
4
1
2
11
2
1
3
1
21
1
2
1
1
1
2
1
2
=− + − + + −−
=++++ −+
−+++
=++++ −+
−+++
=
L
LL
LL
ln( nnn) ln( )+− −+
=+
γγ
error
ln2 error
1
12
1
1
1
2
1
3
1
12
++++
=Ln
Hn