# Jurij Vega

Jurij Vega
Baron Jurij Vega, portrait by Matej Sternen
Born
March 23, 1754
Zagorica near Dolsko, Slovenia
Died
September 26, 1802
Nussdorf near Vienna, Austria

Baron Jurij Bartolomej Vega (also correct Veha; official Latin Georgius Bartholomaei Vecha; German Georg Freiherr von Vega) (March 23, 1754September 26, 1802) was a Slovenian mathematician, physicist and artillery officer.

## Early life

Born in the small village of Zagorica, near Dolsko, east of Ljubljana in Slovenia, Jurij was 6 years old when his father Jernej Veha died. Jurij (or George in English) was educated first in Moravče and later in 1767 attended high school for six years in Ljubljana, studying Latin, Greek, religion, German, history, geography, science, and mathematics. At that time there were about 500 students there. He was a schoolfellow of Anton Tomaž Linhart, a Slovenian writer and historian. Jurij completed high school when he was 19, in 1773. After completing Lyceum in Ljubljana he became a navigational engineer. Tentamen philosophicum, a list of questions for his comprehensive examination, was preserved and is available in the Mathematical Library in Ljubljana. The problems cover logic, algebra, metaphysics, geometry, trigonometry, geodesy, stereometry, geometry of curves, ballistics, and general and special physics.

## Military service

Jurij left Ljubljana five years after graduation and entered military service in 1780 as Professor of Mathematics at the Artillery School in Vienna. At that time he started to sign his last name as Vega and no longer Veha. When Jurij was 33 he married Josefa Svoboda (Jožefa Swoboda) (17711800), a Czech noble from Ceske Budejovice who was 16 at that time.

Vega participated in several wars. In 1788 he served under Austrian Imperial Field-Marshal Gideon Ernst Laudon (17171790) in a campaign against the Turks at Belgrade. His command of several mortar batteries contributed considerably to the fall of the Belgrade fortress. Between 1793 and 1797 he fought French Revolutionaries under the command of Austrian General Dagobert-Sigismond de Wurmser (17241797) with the European coalition on the Austrian side. He fought at Fort Louis, Mannheim, Mainz, Wiesbaden, Kehl, and Dietz. In 1795 he had two 13.6 kg (30-pound) mortars cast, with conically drilled bases and a greater charge, for a firing range up to 2998 m (3280 yards). The old 27.2 kg (60-lb) mortars had a range of only 1791 m (1960 yd).

In September 1802 Jurij Vega was reported missing. After a few days' search his body was found. The police report concluded that it was an accident. However, the true cause of his death remains a mystery, but it is believed that he died on September 26, 1802 in Nussdorf upon Danube near the Austrian capital, Vienna.

## Mathematical accomplishments

Vega published a series of books of logarithm tables. The first one appeared in 1783. Much later, in 1797 it was followed by a second volume that contained a collection of integrals and other useful formulae. His Handbook, which was originally published in 1793, was later translated into several languages and appeared in over 100 issues. His major work was Zakladnica vseh logaritmov (Thesaurus Logarithmorum Completus or Treasury of all Logarithms) that was first published 1794 in Leipzig. An engineer, Franc Allmer, honourable senator of the Technical university of Graz, has found Vega's logarithmic tables with 10 decimal places in the Museum of Carl Friedrich Gauss in Göttingen. Gauss used this work frequently and he has written in it several calculations. Gauss has also found some of Vega's errors in the calculations in the range of numbers, of which there are more than a million. A copy of Vega's Thesaurus belonging to the private collection of the British mathematician and computing pioneer Charles Babbage (1791-1871) is preserved at the Royal Observatory, Edinburgh.

Over the years Vega wrote a four volume textbook Vorlesungen über die Mathematik (Lectures about Mathematics). Volume I appeared in 1782 when he was 28 years old, Volume II in 1784, Volume III in 1788 and Volume IV in 1800. His textbooks also contain interesting tables: for instance, in Volume II one can find closed form expressions for sines of multiples of 3 degrees, written in a form easy to work with.

Vega wrote at least six scientific papers. On August 20, 1789 Vega achieved a world record when he calculated pi to 140 places, of which 137 were correct. This calculation he proposed to the Russian Academy of Sciences in Saint Petersburg (Санкт Петербург) in the booklet V. razprava (The fifth discussion), where he had found with his calculating method an error on the 113th place from the estimation of Thomas Fantet de Lagny (16601734) from 1719 of 127 places. Vega retained his record 52 years until 1841 and his method is mentioned still today. His article was not published by the Academy until six years later, in 1795. Vega had improved John Machin's formula from 1706:

${\displaystyle {\pi \over 4}=4\arctan \left({1 \over 5}\right)-\arctan \left({1 \over 239}\right)}$

with his formula, which is equal to Euler's formula from 1755:

${\displaystyle {\pi \over 4}=5\arctan \left({1 \over 7}\right)+2\arctan \left({3 \over 79}\right)\;,}$

and which converges faster than Machin's formula. He had checked his result with the similar Hutton's formula:

${\displaystyle {\pi \over 4}=2\arctan \left({1 \over 3}\right)+\arctan \left({1 \over 7}\right)\;.}$

He had developed the second term in the series only once.

Japanese mathematicians of that time had used two approximations :

π = 3; {22/7}; {333/106}; {355/113}; {103993/33102}; {104348/33215}; {208341/66317}; {312689/99532}; {833719/265381}; {1146408/364913};
= [3;7,15,1,292,1,1,1,2,1,4] = {5419351/1725033}
= 3.14159265358981538324194377730744861

and

π = 3; {22/7}; {333/106}; {355/113}; {1043/ 332}; {304911/97057}; {305954/97389}; {610865/194446}; {916819/291835}; {4278141/1361786}; {5194960/1653621}; {14668061/4669028}; {19863021/6322649}; {34531082/10991677}; {503298169/160206127}; {15133476152/4817175487}; {30770250473/9794557101}; {599768235139/190913760406}; {630538485612/200708317507}; {1230306720751/391622077913}; {14163912413873/4508551174550}; {15394219134624/4900173252463}; {60346569817745/19209070931939}; {75740788952369/24109244184402}; {136087358770114/43318315116341}; {211828147722483/67427559300743}; {347915506492597/110745874417084};
= [3;7,15,1,2,292,1,1,1,4,1,2,1,1,14,30,2,19,1,1,11,1,3,1,1,1,1,3] = {1255574667200274/399665182551995}
= 3.14156629602561954577603945201650090

which were found in the same manner as John Wallis' method from 1655 with the development in the infinite continued fraction: the first approximation is the sixth even approximation of the infinite continued fraction for π and the second varies from the first in the ninth term. Both approximations differ in the 13th place. Among these Japanese mathematicians were presumably Shinsuke Seki Kowa, named also Takakazu (16401708) who in 1700 had found 10 places, Takebe Hikojiro Katahiro Kenko (16641739) who in 1722 had found 42 (41 correct) places of π, Kamata Yoshikiyo (16781744) who in 1730 found 25 places and Matsunaga Yoshisuke Ryohitsu, (circa 16391744) who in 1739 had found 51 places of π with the same method as Isaac Newton in 1665 with a series arcsin (1/2) = π/6:

π = 3 ( 1+{12 / (4 · 6)}+{12 · 32 / (4 · 6 · 8 · 10)} + {12 · 32 · 52 / ( 4 · 6 · 8 · 10 · 12 · 14)} + {12 · 32 · 52 · 72 / (4 · 6 · 8 · 10 · 12 · 14 · 16 · 18) } + ...),

where for such precision one has to take about 140 terms. First approximations of infinite continued fractions of this series are:

π1 = [3];
π2 = [3;8];
π3 = [3;7,5,4,4];
π4 = [3;7,11,1,5,1,1,3,9];
π5 = [3;7,15,51,7,1,1,1,3,3,2];
π6 = [3;7,15,1,3,1,8,1,1,32,1,14,1,5,1,7];
π7 = [3;7,15,1,21,2,7,1,1,1,11,1,1,1,1,1,5,1,3,1,2,2,24];
π8 = [3;7,15,1,82,1,1,4,5,1,1,1,12,1,6,3,1,6,1,2,3,2];

His method of calculating π is still mentioned today. Although he worked in the subjects of ballistics, physics and astronomy, his major contributions are to the mathematics of the second half of the 18th century.

In 1781 Vega tried to push further his idea in the Austrian Habsburg monarchy about the usage of the decimal metric system of units. His idea was not accepted, but it was introduced later under the emperor Franz Josef I in 1871.

Jurij Vega was a member of 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. He was awarded the Order of Maria Theresa on May 11, 1796. In 1800 Jurij Vega obtained a title of hereditary baron including the right to his own coat of arms.

The Slovenian PTT has issued a stamp honouring Jurij Vega and the National Bank of Slovenia has issued a 50 Tolar banknote in his honour.