Vega published a series of books containing
logarithmic tables. The first volume appeared in 1783. Much later, in 1797, it was followed by a second volume that included a collection of integrals and other useful formulae. His
Handbook, originally published in 1793, was later translated into several languages and issued in over 100 editions. His most significant work was
Thesaurus Logarithmorum Completus (
Treasury of All Logarithms), first published in 1794 in
Leipzig (its 90th edition appeared in 1924). Although based on the tables of
Adriaan Vlacq, Vega's version corrected numerous errors and extended the logarithms of trigonometric functions for small angles. An engineer, Franc Allmer, an honorary senator of the
Graz University of Technology, discovered a copy of Vega's 10-digit logarithmic tables in the Museum of
Carl Friedrich Gauss in
Göttingen. Gauss frequently used Vega's tables and even wrote calculations in the margins. He also found and marked some errors in the millions of values Vega had calculated. A copy of
Thesaurus Logarithmorum Completus from the private collection of mathematician and computing pioneer
Charles Babbage (1791–1871) is preserved at the
Royal Observatory, Edinburgh. Over the years, Vega also wrote a four-volume textbook titled
Vorlesungen über die Mathematik (
Lectures on Mathematics). Volume I was published in 1782 when Vega was 28 years old, followed by Volume II in 1784, Volume III in 1788, and Volume IV in 1800. These textbooks include valuable tables; for example, Volume II contains
closed-form expressions for the sines of multiples of 3 degrees, presented in a user-friendly format. Vega also authored at least six scientific papers. On August 20, 1789, he set a world record by calculating
pi to 140 decimal places, of which the first 126 were correct. He submitted this calculation to the
Russian Academy of Sciences in
Saint Petersburg in his booklet
V. razprava (
The Fifth Discussion), in which he identified an error at the 113th decimal place in the previous best estimate by
Thomas Fantet de Lagny (1660–1734), which had claimed 127 digits. Vega's record held for 52 years, until 1841, and his method remains notable to this day. The academy published his result only in 1795, six years after submission. Vega improved upon
John Machin's 1706 formula: : {\pi\over 4} = 4 \arctan \left({1\over 5}\right) - \arctan \left({1\over 239}\right) with his own formula, equivalent to one used by
Leonhard Euler in 1755: : {\pi\over 4} = 5 \arctan \left({1\over 7}\right) + 2 \arctan \left({3\over 79}\right), which converges faster than Machin's version. He also cross-checked his results using
Charles Hutton’s formula: : {\pi\over 4} = 2 \arctan \left({1\over 3}\right) + \arctan \left({1\over 7}\right). Vega only developed the second term of the series once in his calculations. Although he made contributions to
ballistics,
physics, and
astronomy, his most enduring legacy lies in the field of mathematics during the second half of the 18th century. In 1781, Vega advocated for the adoption of the decimal
metric system of units within the Austrian
Habsburg monarchy. While his proposal was not accepted at the time, the system was officially introduced later under Emperor
Franz Joseph I in 1871. Vega was a member of several scholarly societies, including the
Academy of Practical Sciences in Mainz, the Physical and Mathematical Society of
Erfurt, the Bohemian Scientific Society in
Prague, and the
Prussian Academy of Sciences in
Berlin. He was also an associate member of the British Scientific Society in
Göttingen. On May 11, 1796, he was awarded the Order of
Maria Theresa, and in 1800 he was granted the title of hereditary baron, along with the right to a personal coat of arms. ==Legacy==