History of Mathematics Project | Jones's Use of the Symbol π

History of
Mathematics

Project

1706

Jones's Use of the Symbol π

π gets its symbol

The Greek letter π (pi) was originally used as an abbreviation for the Greek word for "periphery" and combined with the Greek letter δ (delta; for diameter) or ρ (rho; for radius) to indicate relevant dimensions of a circle. William Oughtred made the first known use of such a notation (π/δ) to indicate the ratio of circumference to diameter in 1647, and Barrow and Gregory used similar notations (π/δ for 3.14… and π/ρ for 6.28, respectively). The first known use of the symbol π on its own to mean 3.14… was published by William Jones in 1706. This notation was universally adopted following advocacy by Euler, though disagreement on the convention of π = 3.14… vs. 6.28… persisted at least until 1761.

Self-taught Welsh mathematician William Jones published Synopsis Palmariorum Matheseos (A New Introduction to the Mathematics) in 1706. Page 263 of that work contains the earliest standalone use of the symbol π to denote the ratio of a circle's circumference to its diameter 3.14…. Interestingly, page 243 uses the same symbol to refer to twice that value, or 6.28…, so it appears Jones suffered from the same difficulty in deciding between those two conventions that persisted for more than 50 years subsequently.

Artifact dimensions

4.9 in. × 7.7 in.

Artifact origin

Wales, United Kingdom

Timeline

Interactive Content

Computational Explanation

The standalone symbol π was first used on page 243 of Jones's Synopsis Palmariorum Matheseos, reproduced above, where it occurs in the statement:

6×

,&c.



Periphery(π)

Here, the overbar indicates grouping (corresponding to parentheses in modern mathematical notation),

satisfies

and the series corresponds to the expansion of the inverse tangent function:

In[]:=

Series[ArcTan[t],{t,0,10}]

Out[]=

O[t]

Plugging in the given value of

gives:

In[]:=

6ArcTan1



Out[]=

Note that Jones stated the value should be

, but it is actually π. We therefore see that the first (in page-order sequence) occurrence of the symbol π on its own ironically is used to mean 6.28…, or twice the value now associated with π.

Page 263, shown above, contains a second use of the symbol π, in this case on the right-hand side of a series for computing π, which Jones attributes to John Machin. In this case, we are immediately reassured that π is being used in the sense to which we are accustomed, as not only does Jones write "3.14159, &c. π" in this equation, but also (further down)

dc÷π

, which we recognize as the modern definition

πC/d

, where

is the circumference and

the diameter of a circle.

The formula given states:

239

-,&c3.14159,&c.π

Here, the overbars again indicate grouping, so inserting parentheses and evaluating gives a rational number:

pi5=

239

Out[]=

38279241713339684

12184551018734375

… with numeric value:

In[]:=

N[pi5,10]

Out[]=

3.141621029

And indeed, this is a very close approximation to π:

In[]:=

N[pi5-Pi]

Out[]=

0.0000283757

The close agreement is not a coincidence, but follows from "Machin's formula" relating a specific difference of two multiples of the inverse tangents of unit fractions:

In[]:=

4ArcTan

-ArcTan

239



Out[]=

True

To see the correspondence with the formula given by Jones, consider the series expansion of the inverse tangent already noted above and use the Wolfram Language to confirm that explicitly summing terms of the form

for odd

with alternating signs out to infinity gives the inverse tangent function:

In[]:=

Sum

k+1

(-1)

2k-1

,{k,∞}

Out[]=

ArcTan[x]

To obtain Jones's formula, define a function that sums only the first

terms of the expansion:

In[]:=

arctan[x_,n_]:=Sum

k-1

(-1)

2k-1

,{k,n}

Taking three terms of the expansion for each term in Machin's formula gives:

In[]:=

With{n=3},16arctan

,n-4arctan

239

,n

Out[]=

38279241713339684

12184551018734375

… which is exactly the value given by Jones's formula.

It is easy to compute additional terms:

In[]:=

NTable16arctan

,n-4arctan

239

,n-π,{n,10},10

Out[]=

{0.04167094474,-0.0009956242637,0.00002837573524,-8.814076159×

-7

,2.881460628×

-8

,-9.744846303×

-10

,3.376152353×

-11

,-1.191009800×

-12

,4.260902306×

-14

,-1.541545364×

-15

}

Interestingly, the terms alternate between being slightly larger than and smaller than π. Rather amazingly, with the help of the computational power of the Wolfram Language, the error term can be determined exactly as a function of the number of terms

in terms of the special function known as the beta function:

In[]:=

FullSimplify16arctan

,n-4arctan

239

,n-π

Out[]=

24Beta-

+n,0-Beta-

57121

+n,0

(Note that while the imaginary unit

appears in the left-hand side, the expression inside the parentheses turns out to be purely imaginary, so the product of the two gives a real number.) As promised, tabulating and computing values for the first few

gives exactly the sequence of approximations previously found:

In[]:=

NTable24Beta-

+n,0-Beta-

57121

+n,0,{n,10}//FunctionExpand,10

Out[]=

{0.04167094474,-0.0009956242637,0.00002837573524,-8.814076159×

-7

,2.881460628×

-8

,-9.744846303×

-10

,3.376152353×

-11

,-1.191009800×

-12

,4.260902306×

-14

,-1.541545364×

-15

}

Other Resources

Additional Reading

Arndt, J. and Haenel, C. The Symbol Pi. In Pi Unleashed. Berlin: Springer-Verlag, pp. 165–166, 2001.
Beckmann, P. A History of Pi. New York: Barnes and Noble, p. 145, 1993.
Berggren, J.; Borwein, J.; and Borwein, P. "William Jones: The First Use of π for the Circle Ratio." In Pi: A Source Book. New York: Springer-Verlag, pp. 108–109, 1997.
Cajori, F. § 395–398 in A History of Mathematical Notations: Two Volumes Bound as One, Vol. 2. New York: Dover, pp. 9–13, 1993.
Jones, W. Synopsis Palmariorum Matheseos: Or, a New Introduction to the Mathematics: Containing the Principles of Arithmetic & Geometry, etc. London: Printed by J. Matthews for Jeff Wale, 1706.