The LOCOMAT Project: Recomputing Mathematical and Astronomical Tables

On a Saturday in early September 2009, I had have some idle time, and looking at the conversion table published by Edward Sang in 1868 that I found on Google books,1 I decided to recompute it. As it turned out, it was fairly easy and I was able to reproduce the whole book before the midnight chimes. This almost insignificant event was the start of the LOCOMAT Project (http://locomat.loria.fr). (LOCOMAT stands for LORIA Collection of Mathematical Tables, but it could also be read as the LORIA Collection of Mathematical and Astronomical Tables. LORIA is a computer science research center, located in Nancy, France.)

The LOCOMAT site is constantly growing. By the end of 2011, it contained reconstructions of approximately 60 historical table projects (nearly 200 PDF files) and links to about 2,000 digitized tables on Google Books and elsewhere.

Mathematical and Astronomical Tables

Mathematical tables have a fascinating history. We know of Babylonian tables of multiplication, Pythagorean triples, and reciprocals that are nearly 4,000 years old. Ptolemy computed a table of chords, and later Indian and Arab mathematicians computed various trigonometrical and astronomical tables called zijes. Al-Khwarīzmī computed large astronomical tables. Then came the Toledan tables, the Alfonsine tables, and many others. Trigonometrical tables took on a life of their own in the 15th century, with Georg von Peuerbach (1423-1461) and Regiomontanus (1436-1476). The 16th century then saw the first large trigonometrical tables, by Rheticus, François Viète, and others. At the beginning of the 17th century, John Napier invented logarithms, and this spawned a considerable number of tables, in particular those of Henry Briggs and Adriaan Vlacq. Trigonometrical and logarithmic tables continued to be computed, adapted, and refined until the early 20th century, when mechanical and then electronic computing took over.

The first purely mathematical tables were computing aids and therefore tables of multiplication (squares, cubes, and so forth) as well as tables of factors. Some of the first tables were sizeable and must have taken years to compute. For instance, in 1610 Hans Georg Herwart von Hohenburg published a large volume giving all products up to 1,000 × 1,000.²

The first large tables of factors were published in 1659 (up to 24,000)³ and 1668 (up to 100,000).⁴ Not much happened for the next 100 years, but new developments took place in the 1770s through the impulse of Johann Heinrich Lambert⁵ and Leonhard Euler.⁶ Eventually, this led to the first table of factors giving complete decompositions up to 1 million.⁷ This was the work of Ladislaus Chernac, a Hungarian who moved to the Netherlands. An important effort was made in the 19th century, and by 1883, the first nine millions had been published. It was only in 1909 that the factors of all numbers up to 10 million were published by Derrick N. Lehmer.⁸

At the same time, there were also many failed attempts, such as those of Anton Felkel, Jakob Philipp Kulik, and other computers whose work either lays forgotten in libraries or archives or perhaps even vanished completely.

The 19th and 20th centuries saw a huge number of new tables computed, as testified by the index of tables compiled in 1962 by Alan Fletcher and his colleagues.⁹ Many of these tables are no longer needed, at least not in book form. We can often quickly compute some of the tabulated functions with functions built in mathematical software. Sometimes, software applications use tables internally, in a way reminiscent of the use of books. Thus, in a way, as Liesbeth De Mol recently observed, tables are still alive today, but in a different frame.¹⁰

Reconstructing Tables

Mathematical or astronomical tables are particular cases of computed tables. Computed by humans,¹¹ these tables almost always contain errors. Of course, tables computed by machine can also contain errors, but most of the historical tables that are found online were computed by hand.

Historians of mathematics or astronomy who take such a table as the object of their study are interested in an array of questions. They want...