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 under Macarthy (for James II.), Lisnaskea (26th July) and Newtownbutler (30th July). The chief place of interest to the antiquary is Devenish Island in Lough Erne, about 2 m. N.W. from (q.v.), with its ruined abbey, round tower and cross. In various places throughout the county may be seen the ruins of several ancient castles, Danish raths or encampments, and tumuli, in the last of which urns and stone coffins have sometimes been found. The round tower on Devenish Island is one of the finest examples in the country.

FERMAT, PIERRE DE (1601–1665), French mathematician, was born on the 17th of August 1601, at Beaumont-de-Lomagne near Montauban. While still young, he, along with Blaise Pascal, made some discoveries in regard to the properties of numbers, on which he afterwards built his method of calculating probabilities. He discovered a simpler method of quadrating parabolas than that of Archimedes, and a method of finding the greatest and the smallest ordinates of curved lines analogous to that of the then unknown differential calculus. His great work De maximis et minimis brought him into conflict with René Descartes, but the dispute was chiefly due to a want of explicitness in the statement of Fermat (see ). His brilliant researches in the theory of numbers entitle him to rank as the founder of the modern theory. They originally took the form of marginal notes in a copy of Bachet’s Diophantus, and were published in 1670 by his son Samuel, who incorporated them in a new edition of this Greek writer. Other theorems were published in his Opera Varia, and in John Wallis’s Commercium epistolicum (1658). He died in the belief that he had found a relation which every prime number must satisfy, namely 22n + 1 = a prime. This was afterwards disproved by Leonhard Euler for the case when n = 5. Fermat’s Theorem, if p is prime and a is prime to p then ap−1−1 is divisible by p, was first given in a letter of 1640. Fermat’s Problem is that xn + yn = zn is impossible for integral values of x, y and z when n is greater than 2.

Fermat was for some time councillor for the parliament of Toulouse, and in the discharge of the duties of that office he was distinguished both for legal knowledge and for strict integrity of conduct. Though the sciences were the principal objects of his private studies, he was also an accomplished general scholar and an excellent linguist. He died at Toulouse on the 12th of January 1665. He left a son, Samuel de Fermat (1630–1690) who published translations of several Greek authors and wrote certain books on law in addition to editing his father’s works.

The Opera mathematica of Fermat were published at Toulouse, in 2 vols. folio, 1670 and 1679. The first contains the “Arithmetic of Diophantus,” with notes and additions. The second includes a “Method for the Quadrature of Parabolas,” and a treatise “on Maxima and Minima, on Tangents, and on Centres of Gravity,” containing the same solutions of a variety of problems as were afterwards incorporated into the more extensive method of fluxions by Newton and Leibnitz. In the same volume are treatises on “Geometric Loci, or Spherical Tangencies,” and on the “Rectification of Curves,” besides a restoration of “Apollonius’s Plane Loci,” together with the author’s correspondence addressed to Descartes, Pascal, Roberval, Huygens and others. The Œuvres of Fermat have been re-edited by P. Tannery and C. Henry (Paris, 1891–1894).

See Paul Tannery, “Sur la date des principales découvertes de Fermat,” in the Bulletin Darboux (1883); and “Les Manuscrits de Fermat,” in the Annales de la faculté des lettres de Bordeaux.

 FERMENTATION. The process of fermentation in the preparation of wine, vinegar, beer and bread was known and practised in prehistoric times. The alchemists used the terms fermentation, digestion and putrefaction indiscriminately; any reaction in which chemical energy was displayed in some form or other—such, for instance, as the effervescence occasioned by the addition of an acid to an alkaline solution—was described as a fermentation (Lat. fervere, to boil); and the idea of the “Philosopher’s Stone” setting up a fermentation in the common metals and developing the essence or germ, which should transmute them into silver or gold, further complicated the conception of fermentation. As an outcome of this alchemical doctrine the process of fermentation was supposed to have a purifying and elevating effect on the bodies which had been submitted to its influence. Basil Valentine wrote that when yeast was added to wort “an internal inflammation is communicated to the liquid, so that it raises in itself, and thus the segregation and separation of the feculent from the clear takes place.” Johann Becher, in 1669, first found that alcohol was formed during the fermentation of solutions of sugar; he distinguished also between fermentation and putrefaction. In 1697 Georg Stahl admitted that fermentation and putrefaction were analogous processes, but that the former was a particular case of the latter.

The beginning of definite knowledge on the phenomenon of fermentation may be dated from the time of Antony Leeuwenhoek, who in 1680 designed a microscope sufficiently powerful to render yeast cells and bacteria visible; and a description of these organisms, accompanied by diagrams, was sent to the Royal Society of London. This investigator just missed a great discovery, for he did not consider the spherical forms to be living organisms but compared them with starch granules. It was not until 1803 that L. J. Thénard stated that yeast was the cause of fermentation, and held it to be of an animal nature, since it contained nitrogen and yielded ammonia on distillation, nor was it conclusively proved that the yeast cell was the originator of fermentation until the researches of C. Cagniard de la Tour, T. Schwann and F. Kützing from 1836 to 1839 settled the point. These investigators regarded yeast as a plant, and Meyer gave to the germs the systematic name of “Saccharomyces” (sugar fungus). In 1839–1840 J. von Liebig attacked the doctrine that fermentation was caused by micro-organisms, and enunciated his theory of mechanical decomposition. He held that every fermentation consisted of molecular motion which is transmitted from a substance in a state of chemical motion—that is, of decomposition—to other substances, the elements of which are loosely held together. It is clear from Liebig’s publications that he first regarded yeast as a lifeless, albuminoid mass; but, although later he considered they were living cells, he would never admit that fermentation was a physiological process, the chemical aspect being paramount in the mind of this distinguished investigator.

In 1857 Pasteur decisively proved that fermentation was a physiological process, for he showed that the yeast which produced fermentation was no dead mass, as assumed by Liebig, but consisted of living organisms capable of growth and multiplication. His own words are: “The chemical action of fermentation is essentially a correlative phenomenon of a vital act, beginning and ending with it. I think that there is never any alcoholic fermentation without there being at the same time organization, development and multiplication of globules, or the continued consecutive life of globules already formed.” Fermentation, according to Pasteur, was caused by the growth and multiplication of unicellular organisms out of contact with free oxygen, under which circumstance they acquire the power of taking oxygen from chemical compounds in the medium in which they are growing. In other words “fermentation is life without air, or life without oxygen.” This theory of fermentation was materially modified in 1892 and 1894 by A. J. Brown, who described experiments which were in disagreement with Pasteur’s dictum. A. J. Brown writes: “If for the theory ‘life without air’ is substituted the consideration that yeast cells can use oxygen in the manner of ordinary aërobic fungi, and probably do require it for the full completion of their life-history, but that the exhibition of their fermentative functions is independent of their environment with regard to free oxygen, it will be found that there is nothing contradictory in Pasteur’s experiments to such a hypothesis.”

Liebig and Pasteur were in agreement on the point that fermentation is intimately connected with the presence of yeast in the fermenting liquid, but their explanations concerning the mechanism of fermentation were quite opposed. According to M. Traube (1858), the active cause of fermentation is due to the action of different enzymes contained in yeast and not to the yeast cell itself. As will be seen later this theory was confirmed by subsequent researches of E. Fischer and E. Buchner.

In 1879 C. Nägeli formulated his well-known molecular-physical theory, which supported Liebig’s chemical theory on the one hand and Pasteur’s physiological hypothesis on the