332 AV A L W A L et Semicirculi Tractatus," 1656; " Ejusdem Tractatus Defensio," 1685; " De Postulate Quiuto, et Quinta Definitione, Lib. VI. Euclidis, Disceptatio Geometrica," ?1663; " Cuno-Cuneus, seu Corpus partim Gonum partim Cuncuni Kcprescntans Gcometrice Consideratum," English, 1685; "De Gravitate et Gravitatione Disquisitio Geometrica," 1662 (English, 1674); "De JEstu Mails Hypothesis Nova," 1666-69. The Arithmctica Infinitorum relates chiefly to the quadrature of curves by the so-called method of indivisibles established by Cavalleri, 1629. and cultivated in the interval by him, Format, Descartes, and Roberval. The method is substantially that of the integral calculus; thus, e.g., for the curve y^x* to find the area from x = to x=l, the base is divided into n equal parts, and the area is obtained as=- 3 (! 2 + 2 2 . . . +n-), = ^n(n which, taking n indefinitely large, is = J. The case of the general parabola y = x m (m a positive integer or fraction), where the area is - , had been previously solved. "Wallis made the important remark that the reciprocal of such a power of x could be regarded as a power with a negative exponent^ = x~ m , and he was thus x m J enabled to extend the theorem to certain hyperbolic curves, but the case m a negative value larger than 1 presented a difficulty which he did not succeed in overcoming. It should be noticed that Wallis, although not using the notation a;"* in the case of a positive or negative fractional value, nor indeed in the case of a negative integer value of m, deals continually with such powers, and speaks of the positive or negative integer or fractional value of m as the index of the power. The area of a curve, 7/ = sum of a finite number of terms Ax m , was at once obtained from that for the case of a single term ; and Wallis, after thus establishing the several results which would now be written f (x-x*}dx=l,/?(x-x z ) 1 dx /"i /"i = & ,y o (x - x 2 ) 2 dx = 7 V ,y (x - x-) 3 dx = T J T , &c. , proposed to himself to interpolate from these the value otf l (x-x^dx, which is the expression for the area ( = g7r) of a semicircle, diameter =1 ; making a slight transformation, the actual problem was to find the value ( 4 of CH ( =- 1, the term halfway between 1 and 2, in the series of terms 1, 2, 6, 20, 70, . . . ; and he thus obtained the remarkable 2.4.4.6.6.8.8. . . expression 7T-= 3 g 5 g~y- y~g - , together with a succession of superior and inferior limits for the number TT. In the same work AVallis obtained the expression which would now be written ds^dx . /l + (^. } for the length of the element V dx / of a curve, thus reducing the problem of rectification to that of quadrature. An application of this formula to an algebraical curve was first made a few years later by W. Neil ; the investiga tion is reproduced in the "Tractatus de Cissoide, &c." (1659, as above), and Wallis adds the remark that the curve thus rectified is in fact the semicubical parabola. The Mathcsis Univcrsalis is a more elementary work intended for learners. It contains copious dissertations on fundamental points of algebra, arithmetic, and geometry, and critical remarks. The De Algebra Tractatus contains (chapters 66-69) the idea of the interpretation of imaginary quantities in geometry. This is given somewhat as follows : the distance represented by the square root of a negative quantity cannot be measured in the line backwards or forwards, but can be measured in the same plane above the line, or (as appears elsewhere) at right angles to the line either in the plane, or in the plane at right angles thereto. Con sidered as a history of algebra, this work is strongly objected to by Montucla on the ground of its unfairness as against the early Italian algebraists and also Vieta and Descartes, and in favour of Harriot ; but De Morgan, while admitting this, attributes to it considerable merit. The two treatises on the cycloid and on the cissoid, &c., and the Mechanica contain many results which were then new and valuable. The latter work contains elaborate investigations in regard to the centre of gravity, and it is remarkable also for the employment of the principle of virtual velocities. The cuno-cuneus is a highly interesting surface ; it is a ruled quartic surface, the equation of which may be written c"y - = (c - z}-(a" - x-). Among the letters in volume iii., we have one to the editor of the Leipsic Ads, giving the decipherment of two letters in secret characters. The ciphers are different, but on the same principle : the characters in each are either single digits or combinations of two or three digits, standing some of them for letters, others for syllables or words, the number of distinct characters which had to be deciphered being thus very considerable. For the prolonged conflict between Hobbes and Wallis, see HOBBES, vol. xii. pp. 36-38. (A. CA.) WALLON, or WALLOON, the collective name of the inhabitants of the south-eastern division of Belgium, who are distinguished from the rest of the population chiefly by their Romance speech and darker complexion. The Wallon domain comprises the four provinces of Hainault, Namur, Liege, and Luxemburg, besides about one-third of Brabant. It forms a nearly regular right-angled triangle, with apex at Maestricht within the Dutch frontier, and base stretching along the French frontier in a south-easterly direction, from the neighbourhood of Lille to Longwy at the south-west corner of German Luxemburg. It coincides almost exactly with the section of the Meuse basin com prised within Belgian territory, and has a total area of 6000 square miles, or about one-half of the kingdom, with a population (1886) of 2,780,000, or considerably less than half of the entire Belgian population. But from the following figures it is evident that the Romance is steadily gaining on the Flemish (Teutonic) section and will soon be in a majority. In 1830 the Wallon population numbered 1,360,000 as against 1,860,000 Flemish; in 1866 the corresponding figures were 2,040,000 and 2,406,000 ; and in 1886 they were 2,780,000 and 3,060,000. This north-eastern extremity of ancient Gaul, originally inhabited by the Aduatici, Eburones, Tungri, and other Belgic nations, was reduced by Caesar (51 B.C.), and early brought within the sphere of Roman culture. Hence the Latin language was already too firmly established to be supplanted by the Teutonic when the country was occupied in the 5th century by the Franks previous to their conquest of the more central provinces. But about the same time Roman institutions and speech were entirely supplanted in the more open low-lying districts of western Belgium by the same Franks penetrating from the east, and by the roving Saxon and Frisian tribes arriving by sea on the Litas Saxonicum. Since that time the two elements have remained, without much further change, in close proximity, or separated only by intervening strips of " marches " or border-lands formed by the uncleared forest formerly known as the Silva Carbonaria. The Wallons, allowing for inevitable intermingling, especially towards the German frontier, are thus Romanized Gauls, lineal representatives of the ancient Belgne, in a much truer sense than their Flemish neighbours, although accepting from them this name of " Wallon," that is, Welsh or Foreign, just as the Saxon intruders in Britain imposed the same designation on its primitive inhabit ants. Their Gallic descent is shown specially in their much darker complexion, as clearly established by the anthropological statistics collected by Dr Beddoe. 1 In other respects the Wallons contrast favourably with the Teutonic-speaking populations of Belgium. They are physically a stronger race, more bony, angular, and taller, and also more long-lived and exempt from disease, as shown by the much lower death-rate in the province of Namur (18 per 1000) than in West Flanders (25 per 1000). The cause of this superiority has been attributed by some writers to a greater inherent vitality of the Wallons, but by others with more probability to their greater general comfort, and particularly to the more salubrious climate of their elevated and more hilly territory. In mediaeval times the Flemings were certainly far superior in wealth, culture, and public spirit. But the great centres of industry have since been shifted from Bruges and Ghent (Flanders) to Liege and Namur, and the Wallons now take the lead in all these respects, in fact, in every thing except music, painting, and the fine arts generally. The Wallon language is a very marked dialect of the Langue d Oil (Northern French), betraying strong affinities to the neigh bouring idioms of Picardy and Lorraine, but still distinct from both. Being little cultivated, and possessing no literary standard, it has developed several varieties, the chief of which are (1) the Liegcois, current throughout the districts bordering on Rhenish Prussia, and largely affected by German influences ; (2) the Namiirois of the central districts and the Ardennes, with several local patois; (3) the Hennuyer of Hainault and the western districts, so named from the town of Hennuyeres, where it is spoken with the greatest purity. But since French has been accepted as the written standard and the official language of the whole kingdom Wallon is gradually receding from the large towns and assuming more and more the character of a purely rural patois. It possesses no literary remains of any consequence, and its earliest extant monument appears to be the "Declaration des Provost, Juret, Eskievin de Valenchienes" of the year 1256. 1 In Namur, a central point of the Wallon domain, the proportion of dark eyes is 47 and of dark hair 57 5 per cent, as compared with 24 and 20 respectively in Mechlin, a corresponding central point in
the Flemish domain.