How to Find the Logarithm of Any Number

Using Nothing But a Piece of String

Viktor Bl

˚

asj

¨

o

Viktor Bl

˚

asj

¨

o (v .n.e.b lasjo@uu.nl) studied at Stoc kholm

University and the London School of Economics and

received his Ph.D . in the histor y of mathematics at Utrecht

University , with a disser tation on the representation of

cur ves in the 17th centur y . In his spare time, Bl

˚

asj

¨

o enjoys

photograph y; a f av orite motif is the geometr y of industrial

landscapes.

The shape of a freely hanging chain suspended from two points is called the catenary ,

from the Latin word for chain. In principle, any piece of string would do, but one

speaks of a chain since a chain with ﬁne links embodies in beautifully concrete form

the ideal physical assumptions that the string is nonstretchable and that its elements

hav e complete ﬂexibility independent of each other .

In modern terms, the catenary can be expressed by the equation y = ( e

x

+ e

− x

)/ 2.

As we shall see, Leibniz did not state this formula explicitly , but he understood well

the relation it expresses, calling it a “wonderful and elegant harmony of the curve of

the chain with logarithms” [ 6 , p. 436]. (English translations of [ 5 ] and [ 6 ] are giv en

in [ 10 ].) Indeed, he continued, the close link between the catenary and the exponen-

tial function means that logarithms can be determined by simple measurements on an

actual catenary . “This may be helpful since during long journeys one may lose one’ s

table of logarithms ...I n case of need the catenary can then serve in its place” [ 7 ,

p. 152]. Leibniz’ s recipe for determining logarithms in this way is delightfully simple

and can easily be carried out in practice using, for example, a cheap necklace pinned

to a cardboard box with sewing needles.

Leibniz’ s recipe

Refer to Figure 1 and the follo wing description.

(a) Suspend a chain from two horizontally aligned nails. Draw the horizontal

through the endpoints, and the vertical axis through the lowest point.

(b) Put a third nail through the lowest point and extend one half of the catenary

horizontally .

(c) Connect the endpoint to the midpoint of the drawn horizontal, and bisect the

line segment. Drop the perpendicular through this point, draw the horizontal

axis through the point where the perpendicular intersects the vertical axis, and

take the distance from the origin of the coordinate system to the lowest point of

the catenary to be the unit length. W e will show below that the catenary no w has

the equation y = ( e

x

+ e

− x

)/ 2 in this coordinate system.

http://dx.doi.org/10.4169/colle ge.math.j.47.2.95

MSC: 01A45

V OL. 47, NO. 2, MARCH 2016 THE COLLEGE MA THEMA TICS JOURNAL

95