History of Mathematics Project | π Tombstone of Ludolph van Ceulen

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around 1610

π Tombstone of Ludolph van Ceulen

Approximations on the memorial of a π calculator

The restored tombstone of π calculator Ludolph van Ceulen features engravings of digits of π placed in a circle with bounds to π (3.14159265358979323846264338327950288 < π < 3.14159265358979323846264338327950289) appearing in the middle.

Ludolph van Ceulen spent a major part of his life calculating the numerical value of the mathematical constant π. After his death in 1610, digits of π were engraved on his tombstone in Leiden, together with upper and lower limits given as fractions. The tombstone was eventually lost, but later restored in 2000. The text on the tombstone reads, "Buried here lies Ludolph van Ceulen, Dutch professor of mathematical sciences in the vocational school [Engineering School in the Faliedenbegijnkerk] of this city. [He was] born in Hildesheim in the year 1540 on the 28th of January [and] died the 31st of December 1610. In his life, with great effort, he found the ratio between the circumference and diameter of a circle as here follows."

Artifact origin

Leiden, Netherlands

Current artifact location

Pieterskerk, Leiden

Timeline

Interactive Content

Computational Explanation

Ludolph van Ceulen, who died in the year 1610, spent a major part of his life calculating the numerical value of π to (almost) 36 decimal digits (i.e. 35 digits after the initial "3"). More than four centuries later, an off-the-shelf electronic device can compute these digits:

In[]:=

RealDigits[Pi,10,36][[1]]

Out[]=

{3,1,4,1,5,9,2,6,5,3,5,8,9,7,9,3,2,3,8,4,6,2,6,4,3,3,8,3,2,7,9,5,0,2,8,8}

as well as many more:

In[]:=

N[Pi,1000]//AbsoluteTiming

Out[]=

{0.000012,3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420199}

in at most a few tens of microseconds.

Nevertheless, at the time, van Ceulen's achievement in manual computation was unsurpassed, and to commemorate it, a monument was erected in his honor giving the digits he computed, namely the first 35 digits together with the determination that the 36th was either an 8 or a 9. His bounds were therefore given as follows:

In[]:=

digits1=RealDigits[Pi,10,36][[1]]

Out[]=

{3,1,4,1,5,9,2,6,5,3,5,8,9,7,9,3,2,3,8,4,6,2,6,4,3,3,8,3,2,7,9,5,0,2,8,8}

In[]:=

digits2=RealDigits[Pi+

-(36-1)

,10,36][[1]]

Out[]=

{3,1,4,1,5,9,2,6,5,3,5,8,9,7,9,3,2,3,8,4,6,2,6,4,3,3,8,3,2,7,9,5,0,2,8,9}

Expressing these as unevaluated fractions:

In[]:=

bounds=FromDigits/@{digits1,digits2}/HoldForm[Evaluate@Expand[10^35]]

Out[]=



314159265358979323846264338327950288

100000000000000000000000000000000000

314159265358979323846264338327950289

100000000000000000000000000000000000



and writing as an inequality gives:

In[]:=

Less@@Riffle[bounds,Pi]

Out[]=

314159265358979323846264338327950288

100000000000000000000000000000000000

<π<

314159265358979323846264338327950289

100000000000000000000000000000000000

We now check that the above values indeed bracket π:

In[]:=

314159265358979323846264338327950288

100000000000000000000000000000000000

<π<

314159265358979323846264338327950289

100000000000000000000000000000000000

Out[]=

True

The inscription on van Ceulen's tomb consists of text in the interior of a circle that states (in Dutch), "If the diameter is 1, then the circumference is more than 3.1415… and less than 3.1415… But if the diameter is 10000…", and text written around the circumference of the circle states, "then the circumference is more than 31415… and less than 31415 …".

To construct this text and its arrangement, splice the digits computed by van Ceulen into the appropriate Dutch text and separate into individual characters:

In[]:=

circumferentialcharacters=Join[digits1,Characters[" EN MINDER ALS "],digits2,Characters[" DAN IS DEN OMLOOP MEERDER ALS "]]

Out[]=

{3,1,4,1,5,9,2,6,5,3,5,8,9,7,9,3,2,3,8,4,6,2,6,4,3,3,8,3,2,7,9,5,0,2,8,8, ,E,N, ,M,I,N,D,E,R, ,A,L,S, ,3,1,4,1,5,9,2,6,5,3,5,8,9,7,9,3,2,3,8,4,6,2,6,4,3,3,8,3,2,7,9,5,0,2,8,9, ,D,A,N, ,I,S, ,D,E,N, ,O,M,L,O,O,P, ,M,E,E,R,D,E,R, ,A,L,S, }

Then make a list of the lines of the interior text, again splicing in the bounds on π from van Ceulen's calculation:

In[]:=

interiortext={"ALS","DE MIDDELLYN IS","1","DAN IS DEN OMLOOP MEERDER ALS",Row[{3,bounds〚1〛}],"EN MINDER ALS",Row[{3,bounds〚2〛}],"MAER ALS DE MIDDELLYN IS",

};

Assembling these components in their proper places together with the initial portion of the epitaph text finally reproduces the full memorial:

In[]:=

CosSin[i_,n_,off_:0]:=Through[{Cos,Sin}[2π(i-1)/n-off]]

With{n=Length[circumferentialcharacters]},Framed@Graphics

Text[

]

,Table[Text[circumferentialcharacters[[i]],CosSin[i,n,Pi/2],{0,0},CosSin[i,n]],{i,n}],Text[Column[interiortext,AlignmentCenter],{0,0.2}],

Rule[

]



Out[]=

Other Resources

Wolfram Demonstration
Euler's Estimate of Pi

Wolfram Demonstration
Three Calculations of Pi Before 1800

Additional Reading

Anonymous. "5 juli 2000: Pi in de Pieterskerk." Nieuw Archief voor Wiskunde 5/1, pp. 114–115, 2000.
Beckmann, P. A History of Pi. New York: Barnes & Noble, p. 102, 1993.
Cipra, B. Digits of Pi. In What's Happening in the Mathematical Sciences, Vol. 6. Providence, RI: American Mathematical Society, pp. 28–38, 2006.
Vajta, M. "Pi, Fourier Transform and Ludolph van Ceulen." In 3rd TEMPUS-INTCOM Symposium, September 9-14, 2000, Veszprém, Hungary. pp. 1–6, 2000.
Vajta, M. "Fourier Transform and Ludolph van Ceulen." 9th IEEE Mediterranean Conference on Control and Automation, MED 2001 - Hotel Excelsior, Dubrovnik, Croatia. 2001.
Van Ceulen, L. Vanden Circkel. Tot Delft : Gedruckt by Jan Andriesz, 1596.
Van Ceulen, L. De Arithmetischen en Ggeometrische fondamenten. 1615.

Additional Links

Image Credits

Wikimedia (padded, background adjusted, contrast enhanced)

π Tombstone of Ludolph van Ceulen

Timeline

Interactive Content

Computational Explanation

Other Resources

Wolfram MathWorld

Wolfram Demonstration

Wolfram Demonstration

Additional Reading

Additional Links

Image Credits