History of Mathematics Project | Guldin's Factor Table

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1641

Guldin's Factor Table

One of the oldest factor tables, giving factorizations up to 10,000

Paul Guldin made an early factor table up to 10,000 in his 1641 four-book work on mathematics and astronomy De Centro Gravitatis, extending the earlier table from Cataldi.

Swiss Jesuit mathematician and astronomer Paul Guldin made a factor table of odd numbers not divisible by 10 up to 10,000 in the fourth book of his series De Centro Gravitatis in 1641. The table was captioned "tabula ultima," referring to the fact that it is the last table in the last book of his four-volume work. The motivation for the computation and inclusion of this table is not clear based on Guldin's own explanation.

Artifact dimensions

7.4 in. × 11.4 in.

Artifact origin

Vienna, Austria

Current artifact location

Linda Hall Library of Science, Engineering & Technology

Catalog number

QA444.G84 1635

Timeline

Interactive Content

Computational Explanation

Guldin's factor table spans pages 384–401 of his 1641 work De Centro Gravitatis. The table includes odd numbers and tabulates partial factorizations of numbers up to 10,000 as lists of distinct prime factors (but omitting the factor 5). For example, 105  3×5×7, so Guldin's table lists its factors as 3·7. Guldin's table indicates prime numbers (as well as 1) by leaving the column next to them blank:

As can be seen above, the table shows no factorization for 1, 3, 5 or 7, which is exactly as expected given that special case 1 has no factorization, 3 is prime, the (trivial) factorization of 5 is omitted and 7 is prime. The first number in the table with a value in the factorization column is therefore 9, which has a factorization of 3 since

9

has a single distinct factor 3.

Similarly, the first number with a list of two values in its 5-free factorization is 21  3×7.

The following table summarizes the computation of the entries Guldin's table in more detail. In particular, each odd number from 1 to 119 is given, followed by its full factorization, and finally by its list of 5-free distinct factors. Groups of numbers and their factorizations are partitioned into three-column blocks for convenience:

In[]:=

centerdotform[l_]:=Row[Riffle[Superscript@@@l,"·"]]row[n_]:=Module[{factorization=FactorInteger[n]},{n,centerdotform[factorization],If[n===1||PrimeQ[n],"",centerdotform[DeleteCases[factorization[[All,1]],5]]]}]

In[]:=

TextGridFlatten/@Transpose[Partition[Table[row[n],{n,1,120,2}],10]],



Out[]=

1	1 1		21	1 3 · 1 7	3·7	41	1 41		61	1 61		81	4 3	3	101	1 101
3	1 3		23	1 23		43	1 43		63	2 3 · 1 7	3·7	83	1 83		103	1 103
5	1 5		25	2 5		45	2 3 · 1 5	3	65	1 5 · 1 13	13	85	1 5 · 1 17	17	105	1 3 · 1 5 · 1 7	3·7
7	1 7		27	3 3	3	47	1 47		67	1 67		87	1 3 · 1 29	3·29	107	1 107
9	2 3	3	29	1 29		49	2 7	7	69	1 3 · 1 23	3·23	89	1 89		109	1 109
11	1 11		31	1 31		51	1 3 · 1 17	3·17	71	1 71		91	1 7 · 1 13	7·13	111	1 3 · 1 37	3·37
13	1 13		33	1 3 · 1 11	3·11	53	1 53		73	1 73		93	1 3 · 1 31	3·31	113	1 113
15	1 3 · 1 5	3	35	1 5 · 1 7	7	55	1 5 · 1 11	11	75	1 3 · 2 5	3	95	1 5 · 1 19	19	115	1 5 · 1 23	23
17	1 17		37	1 37		57	1 3 · 1 19	3·19	77	1 7 · 1 11	7·11	97	1 97		117	2 3 · 1 13	3·13
19	1 19		39	1 3 · 1 13	3·13	59	1 59		79	1 79		99	2 3 · 1 11	3·11	119	1 7 · 1 17	7·17

While computing the factor table for all numbers up to 10,000 took Guldin days or weeks, a modern off-the-shelf computer can complete the job in a couple hundredths of a second:

In[]:=

factortable=FactorInteger[Range[2,10000]];//Timing

Out[]=

{0.019583,Null}

The factorization process obviously generated the prime numbers as a byproduct:

In[]:=

Take[singlefactors=Cases[factortable,{{d_,1}}d],20]

Out[]=

{2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71}

In[]:=

singlefactors===Prime[Range[PrimePi[10^4]]]

Out[]=

True

The number of distinct prime factors and number of prime factors counting multiplicity also follow immediately:

In[]:=

With[{data=Take[factortable,99]},{ListPlot[Length/@data],ListPlot[Total/@data[[All,All,2]]]}]

Out[]=





The above plots can also be obtained directly using the built-in Wolfram Language functions PrimeNu and PrimeOmega:

In[]:=

{ListPlot[PrimeNu[Range[2,100]]],ListPlot[PrimeOmega[Range[2,100]]]}

Out[]=





Leaving these interesting digressions aside, return now to Guldin's table. It is quite straightforward to construct the table for any given number of columns and rows. Below, we reproduce the first page of the table as printed on page 384, consisting of seven columns and with the last row in the first column ending at the number 109:

In[]:=

GuldinFactorTable[cols_:7,rowmax_:109]:=Module{num},TextGridFlatten[#,1]&/@Table[{num=n+(rowmax+Boole[OddQ[rowmax]])k,If[num===1||PrimeQ[num],"",Row[Riffle[DeleteCases[FactorInteger[num][[All,1]],5],"·"]]]},{n,Range[1,rowmax,2]},{k,0,cols-1}],



In[]:=

RowStyle[GuldinFactorTable[7,109],8],



Out[]=

Other Resources

History of Mathematics
Cataldi's Divisor Table

History of Mathematics
Van Schooten's Prime Table

Wolfram Demonstration
Prime Factorization Table

Additional Reading

Bullynck, M. "Factor Tables 1657–1817, with Notes on the Birth of Number Theory." Revue d'Histoire des Mathematiques, Vol. 16, No. 2, pp. 133–216, 2010.
Guldin, P. Tabla Ultima following Book IV in De centro gravitatis: De usu centri gravitatis binarum specierum quantitatis continuae, sive de compositione et resolutione potestatum rotundarum. Vienna, Austria: Cosmerovius, pp. 393–401, 1641.

Additional Links

Image Credits

Linda Hall Library of Science, Engineering & Technology, Kansas City, Missouri