(Francis), a very celebrated French mathematician, was born in 1540 at Fontenai, or Fontenaille-Comté, in Lower Poitou, a province of France. He was Master of requests at Paris, where he died in 1603, being the 63rd year of his age. Among other branches of learning in which he excelled, he was one of the most respectable mathematicians of the 16th century, or indeed of any age. His writings abound with marks of great originality, and the finest genius, as well as intense application. His application was such, that he has sometimes remained in his study for three days together without eating or sleeping. His inventions and improvements in all parts of the mathematics were very considerable. He was in a manner the inventor and introducer of Specious Algebra, in which letters are used instead of numbers, as well as of many beautiful theorems in that science. He made also considerable improvements in geometry and trigonometry. His angular sections are a very ingenious and masterly performance: by these he was enabled to resolve the problem of Adrian Romanus, proposed to all mathematicians, amounting to an equation of the 45th degree. Romanus was so struck with his sagacity, that he immediately quitted his residence of Wurtzbourg in Franconia, and came to France to visit him, and solicit his friendship. His Apollonius Galles, being a restoration of Apollonius's tract on Tangencies, and many other geometrical pieces to be found in his works, shew the finest taste and genius for true geometrical speculations.—He gave some masterly tracts on Trigonometry, both plane and spherical, which may be found in the collection of his works, published at Leyden in 1646, by Schooten, besides another large and separate volume in folio, published in the author's life-time, at Paris, in 1579, containing extensive trigonometrical tables, with the construction and use of the same, which are particularly described in the introduction to Dr Hutton's Logarithms, p. 4. &c. To this complete treatise on trigonometry, plane and spherical, are joined several miscellaneous problems and observations; such as, the quadrature of the circle, the duplication of the cube, &c. Computations are here given of the ratio of, the diameter of a circle to the circumference, and of the length of the fine of 1 minute, both to a great many places of figures; by which he found that the fine of 1 minute is

between $2908881959$ and $2908882056$;

also the diameter of a circle being 1000, &c. that the perimeter of the inscribed and circumscribed polygon of 393216 sides will be as follows, viz. the perim. of the inscribed polygon $= 31415926535$ perim. of the circumscribed polygon $= 31415926537$ and that therefore the circumference of the circle lies between those two numbers.

Vieta having observed that there were many faults in the Gregorian Calendar, as it then existed, composed a new form of it, to which he added perpetual calendars, and an explication of it, with remarks and objections against Clavius, whom he accused of having deformed the true Lelian reformation, by not rightly understanding it.

Besides these, it seems a work, greatly esteemed, and the loss of which cannot be sufficiently deplored, was his Harmonicon Calyle, which, being communicated to Father Merlenne, was, by some pernicious acquaintance of that honest minded person, surreptitiously taken from him and irrecoverably lost, or suppressed, to the great detriment of the learned world. There were also, it is said, other works of an astronomical kind, that have been buried in the ruins of time.

Vieta was also a profound decipherer, an accomplishment that proved very useful to his country. As the different parts of the Spanish monarchy lay very distant from one another, when they had occasion to communicate any secret designs, they wrote them in ciphers and unknown characters during the disorders of the league. The cipher was composed of more than 500 different characters, which yielded their hidden contents to the penetrating genius of Vieta alone. His skill so disconcerted the Spanish councils for two years, that they published it at Rome, and other parts of Europe, that the French king had only discovered their ciphers by means of magic.