We know of one important service rendered by Vieta as a royal officer. While at Tours he discovered the key to a Spanish cipher, consisting of more than 500 characters, and thenceforward all the dispatches in that language which fell into the hands of the French could be easily read. His fame now rests, however, entirely upon his achievements in mathematics. Being a man of wealth, he printed at his own expense the numerous papers which he wrote on various branches of this science, and communicated them to scholars in almost every country of Europe. An evidence of the good use he made of his means, as well as of the kindliness of his character, is furnished by the fact that he entertained as a guest for a whole month a scientific adversary, Adriaan van Roomen, and then paid the expenses of his journey home. Vieta’s writings thus became very quickly known, but, when Franciscus van Schooten issued a general edition of his works in 1646, he failed to make a complete collection, although probably nothing of very great value has perished.
The form of Vieta’s writings is their weak side. He indulged freely in flourishes; and in devising technical terms derived from the Greek he seems to have aimed at making them as unintelligible as possible. None of them, in point of fact, has held its ground, and even his proposal to denote unknown quantities by the vowels A, E, I, O, U, Y—the consonants B, C, &c., being reserved for general known quantities—has not been taken up. In this denotation he followed, perhaps, some older contemporaries, as Ramus who designated the points in geometrical figures by vowels, making use of consonants, R, S, T, &c., only when these were exhausted. Vieta is wont to be called the father of modern algebra. This does not mean, what is often alleged, that nobody before him had ever thought of choosing symbols different from numerals, such as the letters of the alphabet, to denote the quantities of arithmetic, but that he made a general custom of what until his time had been only an exceptional attempt. All that is wanting in his writings, especially in his Isagoge in artem analyticam (1591), in order to make them look like a modern school algebra, is merely the sign of equality—a want which is the more striking because Robert Recorde had made use of our present symbol for this purpose since 1557, and Xylander had employed vertical parallel lines since 1575. On the other hand, Vieta was well skilled in most modern artifices, aiming at a simplification of equations by the substitution of new quantities having a certain connexion with the primitive unknown quantities. Another of his works, Recensio canonica effectionum geometricarum, bears a stamp not less modern, being what we now call an algebraic geometry—in other words, a collection of precepts how to construct algebraic expressions with the use of rule and compass only. While these writings were generally intelligible, and therefore of the greatest didactic importance, the principle of homogeneity, first enunciated by Vieta, was so far in advance of his times that most readers seem to have passed it over without adverting to its value. That principle had been made use of by the Greek authors of the classic age; but of later mathematicians only Hero, Diophantus, &c., ventured to regard lines and surfaces as mere numbers that could be joined to give a new number, their sum. It may be that the study of such sums, which he found in the works of Diophantus prompted him to lay it down as a principle that quantities occurring in an equation ought to be homogeneous, all of them lines, or surfaces, or solids, or super solids—an equation between mere numbers being inadmissible. During the three centuries that have elapsed between Vieta’s day and our own several changes of opinion have taken place on this subject, till the principle has at last proved so far victorious that modern mathematicians like to make homogeneous such equations as are not so from the beginning, in order to get values of a symmetrical shape. Vieta himself, of course, did not see so far as that; nevertheless the merit cannot be denied him of having indirectly suggested the thought. Nor are his writings lacking in actual inventions. He conceived methods for the general resolution of equations of the second, third and fourth degrees different from those of Ferro and Ferrari, with which, however, it is difficult to believe him to have been unacquainted. He devised an approximate numerical solution of equations of the second and third degrees, wherein Leonardo of Pisa must have preceded him, but by a method every vestige of which is completely lost. He knew the connexion existing between the positive roots of an equation (which, by the way, were alone thought of as roots) and the coefficient of the different powers of the unknown quantity. He found out the formula for deriving the sine of a multiple angle, knowing that of the simple angle with due regard to the periodicity of sines. This formula must have been known to Vieta in 1503. In that year Adriaan van Roomen gave out as a problem to all mathematicians an equation of the 45th degree, which, being recognized by Vieta as depending on the equation between sin φ and sin φ/45, was resolved by him at once, all the twenty-three positive roots of which the said equation was capable being given at the same time (see Trigonometry). Such was the first encounter of the two scholars. A second took place when Vieta pointed to Apollonius’s problem of taction as not yet being mastered, and Adriaan van Roomen gave a solution by the hyperbola. Vieta, however, did not accept it, as there existed a solution by means of the rule and the compass only, which he published himself in his Apollonius Gallus (1600). In this paper Vieta made use of the centre of similitude of two circles. Lastly he gave an infinite product for the number π (see Circle, Squaring of).
Vieta’s collected works were issued under the title of Opera Mathematica by F. van Schooten at Leiden in 1646. (M. Ca.)
VIEUXTEMPS, HENRI (1820–1881), Belgian violinist and composer, was born at Verviers, on the 20th of February 1820. Until his seventh year he was a pupil of Lecloux, but when De Bériot heard him he adopted him as his pupil, taking him to appear in Paris in 1828. From 1833 onwards he spent the greater part of his life in concert tours, visiting all parts of the world with uniform success. He first appeared in London at a Philharmonic concert on the 2nd of June 1834, and in the following year studied composition with Reicha in Paris, and began to produce a long series of works, full of formidably difficult passages, though also of pleasing themes and fine musical ideas, which are consequently highly appreciated by violinists. From 1846 to 1852 he was solo violinist to the tsar, and professor in the conservatorium in St Petersburg. From 1871 to 1873 he was teacher of the violin class in the Brussels Conservatoire, but was disabled by an attack of paralysis in the latter year, and from that time could only superintend the studies of favourite pupils. He died at Mustapha, in Algiers, on the 6th of June 1881. He had a perfect command of technique, faultless intonation and a marvellous command of the bow. His staccato was famous all over the world, and his tone was exceptionally rich and full.
VIGAN, a town and the capital of the province of Ilocos Sur, Luzon, Philippine Islands, at the mouth of the Abra river, about 200 m. N . by W. of Manila. Pop. of the municipality (1903) 14,945; after the census of 1903 was taken there were united to Vigan the municipalities of Bantay (pop. 7020), San Vicente (pop. 5060), Santa Catalina (pop. 5625) and Coayan (pop. 6201), making the total population of the municipality 38,851. Vigan is the residence of the bishop of Nueva Segovia and has a fine cathedral, a substantial court-house, other durable public buildings and a monument to Juan de Salcedo, its founder. It is engaged in farming, fishing, the manufacture of brick, tile, cotton fabrics and furniture, and the building of boats. The language is Ilocano.
VIGÉE-LEBRUN, MARIE-ANNE ELISABETH (1755–1842), French painter, was born in Paris, the daughter of a painter, from whom she received her first instruction, though she benefited more by the advice of Doyen, Greuze, Joseph Vernet and other masters of the period. When only about twenty years of age she had already risen to fame with her portraits of Count Orloff and the duchess of Orleans, her personal charm making her at the same time a favourite in society. In 1776 she married the painter and art-critic J. B. P. Lebrun, and in 1783 her picture of "Peace bringing back Abundance" (now at the Louvre) gained her the membership of the Academy. When the Revolution broke out in 1789 she escaped first to Italy, where she worked at Rome and Naples. At Rome she painted the portraits of Princesses Adelaide and Victoria, and at Naples the "Lady Hamilton as a Bacchante" now in the collection of Mr Tankerville Chamberlayne, and then journeyed to Vienna, Berlin and St Petersburg. She returned to Paris in 1781, but went in the following year to London, where she painted the portraits of Lord Byron and the prince of Wales, and in 1808 to Switzerland. Her numerous journeys, and the vogue she enjoyed wherever she went, account for the numerous portraits from her brush that are to be found in the great collections of many countries. Having returned to France from Switzerland, she lived first at her country house near Marly and then in Paris, where she died at the age of eighty-seven, in 1842, having been widowed for twenty-nine years. She published her own memoirs under the title of Souvenirs (Paris, 1835-37). Among her many sitters was
