224 V I E V I G cultural machines, also foundries, porcelain and earthen ware works, and glass-works. A very fine technical school has just been opened. VIETA, or VIETE, FKANgois, SEIGNEUR DE LA BIGOTIERE (1540-1603), more generally known as FRANCISCUS VIETA, mathematician, was born in 1540 at Fontenay-le-Comte, in Poitou. According to F. Eitter s investigations, 1 Vieta was brought up as a Catholic, and died in the same creed but there can be no doubt that he belonged to the Hugue nots for several years. On the completion of his studies in law at Poitiers Vieta began his career as an advocate in his native town. This he left about 1567, and some what later we find him at Rennes as a councillor of the parlement of Brittany. The religious troubles drove him thence, and Rohan, the well-known chief of the Huguenots, took him under his special protection. He recommended him in 1580 as a "maitre des requetes" (master of requests) ; and Henry of Navarre, at the instance of Rohan, addressed two letters to Henry III. of France on 3d March and 26th April 1585, to obtain Vieta s restoration to his former office, but without result. After the accession of Henry of Navarre to the throne of France Vieta filled in 1589 the position of councillor of the parlement at Tours. He afterwards became a royal privy councillor, and re mained so till his death, which took place suddenly at Paris in February 1603, but in what manner we do not know ; Anderson, the editor of his scientific remains, speaks only of a "prreceps et immaturum autoris fatum." We know of one important service rendered by Vieta as a royal officer. While at Tours lie 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. Yieta 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 what we may call 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 imknown 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 those 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 quan tities 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 Isagogc 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 sub stitution of new quantities having a certain connexion with the primitive imknown quantities. Another of his works, llecensio canonica cffcctionum geometricarum, bears a stamp not less modern, being what we now call an algebraic geometiy, 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 import ance, 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 Bolletino Boncompagni, vol. i. p. 227, note 1, Rome, 1868. 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 supersolids, an equation between mere numbers being inad missible. During the three centuries that have elapsed between Vieta s days and our own several changes of opinion have taken place on this subject, till the principle has at last proved so far victori ous 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 approxi mate 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 coefficients 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 1593. In that year Adriaan van Roomen gave out as a problem to all mathema ticians an equation of the 45th degree, which, being recognized by Vieta as depending on the equation between sin <j) and sin - _ , was. 4j resolved by him at once, all the twenty- three positive roots of which the said equation was capable being given at the same time. 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 sohition 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 TT (see SQUARING THE CIRCLE, vol. xxii. p. 434). Vieta s collected works were issued under the title of Opera Mathematica by P. van Schooten at Leyden in 1640. (M. CA.) VIGEVANO, a town of Italy, in the province of Pavia, on the right bank of the Ticino, 24 miles south-west from Milan and about the same distance north-west from Pavia. It is a mediasval walled town, with an arcaded market place, a cathedral, and a castle of the Sforza family, dating from the 14th century and adorned with a loggia by Bramante. It is a place of some importance in the silk trade, and also produces excellent macaroni. The popu lation in 1881 was 14,794 (commune, 20,096). VIGIL, in its ecclesiastical sense, means the day preceding a festival. In the liturgy of the Roman Catholic Church there are special offices for the vigils of the greater feasts, and the vigil is regarded as more or less of a fast day. In the early church there was a widespread practice of celebrating festivals, of martyrs especially, by actual vigils (mr/ilise, or pernoctationea, Travvvxt Ses) in the churches ; but this led to such serious evils that the custom had to be discouraged, and now sur vives, if at all, only in rare, unauthorized, and considerably disguised forms. In some of the stricter religious orders the daily office is recited at various hours of the night as well as during the day ; but, if these nocturnal services are spoken of as vigils, it is only in the popular meaning of that word. VIGILANTITJS, presbyter, celebrated as the author of a work, no longer extant, against superstitious practices, which called forth one of the most violent and scurrilous of Jerome s polemical treatises, was born about 370 at Calagurris in Aquitania (the modern Saint Bertrand de Comminges in the department of Haute-Garorme), where his father kept a " static " or inn on the great Roman road from Aquitania to Spain. While still a youth his talent became known to Sulpicius Severus, who had estates,
in that neighbourhood, and in 395 Sulpicius, who probably