&c. 1+ + + + 100000' &c.; in which way of resolution the law of the progression of the series is not visible, as it is when found by division. the square root of the algebraic quantity a2+c2 gives And Hence it appears that the signs of the terms may be either all plus, or alternately plus and minus, though they may be varied in many other ways. It also appears that the terms may be either continually smaller and smaller, or larger and larger, or else all equal. In the first case, therefore, the series is said to be a decreasing one; in the 2d case, an increasing one; and in the 3d case, an equal one. Also the first series is called a converging one, because that by collecting its terms successively, taking in always one term more, the successive sums approximate or converge to the value or sum of the whole infinite series. So, in the series C4 8 a3 √a2+c2=a+ And a 3d way is by Newton's binomial theorem, which is an universal method, that serves for all sorts of quantities, whether fractional or radical ones: and by this means the same root of the last given quantity becomes a2 + c2= 1.3 c it + 2 a 1.3.5 ca + 2.4 a 2.4.6 a' 2.4.6.8 a &c. where the law of continuation is visible. three terms are too great; four terms + 3 1 27 81 = .246913, &c. are too great, and so on, alternately, too great and too small, but every succeeding sum still nearer than. the former, or converging. In the second case, or when the terms the first term + 1 is too great; two terms 1 -2-1 are too little ; three terms 1 2+4+3 are too great; four terms 1−2+4-8=-5 are too little; and so on, continually, after the 2d term, diverging more and more from the true value or radix, but alternately too great and too little, or positive and negative. But the alternate sums would be always more and more too great if the terms were all positive, and always too little if nega gative. But in the third case, or when the terms are all equal, the series of equals, with alternate signs, is called a neutral one, because the successive sums, found by a continual collection of the terms, are always at the same distance from the true value or radix, but alternately positive and negative, or too great and too little. Thus, in the gives bined with the first term, = 1 the true sum of the series. Again, af ter the first two terms der is +1, which, divided by the same di visor, 3, gives for the supplement, and 12 a", which, being divided by the same divisor, a + c, gives for the supplement as above. u+1(a+c) "The Law of Continuation."-A series being proposed, one of the chief questions concerning it is to find the law of its continuation. Indeed, no universal rule can be given for this; but it often happens, that the terms of the series, taken two and two, or three and three, or in greater numbers, have an obvious and simple relation, by which the series may be determined and produced indefinitely. Thus, if 1 be divided by 1-x, the quotient will be a geometrical progression, viz. 1+x+x2+x3, &c. where the succeeding terms are produced by the continual multiplication by x. In like manner, in other cases of division, other progressions are produced. But in most cases, the relation of the terms of a series is not constant, as it is in those that arise by division. Yet their relation often varies according to a certain law, which is sometimes obvious on inspection, and sometimes it is found by dividing the successive terms one by another, &c. Thus, in the series 1 + 1/2x+1= x2+ x3+ x, &c. by 35 dividing the 2d term by the 1st, the 3d by the 2d, the 4th by the 3d, and so on, the quotients will be 20 42 72 1.1 3.3 5.5 7.7 x, and therefore the terms of the series may be continued by the multiplication of these fractions. SERIES, summation of. We have before seen the method of determining the sums of quantities in arithmetical and geometrical progression, but when the terms increase, or decrease, according to other laws, different artifices must be used, to obtain general expressions for their sum, by subtraction, The methods chiefly adopted, and which may be considered as belonging to algebra, are, 1. The method of subtraction. 2. The + &c. (to n terms) + summation of recurring series, by the scale of relation. 3. The differential method. 4. The method of increments. We shall hence, r m.m+r m+r.m+ 2 r m.m+r content ourselves with an example or two, (to n terms) = 1 in the first of these methods. "The investigation of series, whose sums are known by subtraction." Er. 1. Let 1 ++++, &c. in inf. =S, + 1 m m+ar m r mr + n r2° If n be increased without limit, 1 m r + n r22 vanishes, and the sum of the series is 1 m r 1.2 2.3 1 6 12 20 3.4 Similar to the method of subtraction is the following, given by De Moivre. "Assume a series, whose terms converge to a, involving the powers of an indeterminate quantity, x; call the sum of the series S, and multiply both sides of the equation by a binomial, trinomial, &c. which involves the powers of x, and invariable coefficients; then, if x be so assumed that the binomial, trinomial, &c. may vanish, and some of the first terms be transposed, the sum of the remaining series is equal to the islands. It takes a good polish, and is turnterms so transposed.". SERIOLA, in botany, a genus of the Syngenesia Polygamia Æqualis class and order. Natural order of Compositæ Semiflosculosa. Cichoracea, Jussien. Essential character: calyx, simple; pappus subplumose; receptacle chaffy. There are four species. SERIPHIUM, in botany, a genus of the Syngenesia Polygamia Segregata class and order. Natural order of Compositæ Nucamentaceæ. Corymbiferæ, Jussieu. Essential character: calyx, imbricate; corolla, one petalled, regular, seed one; oblong, below the corolla. There are four species, all natives of the Cape of Good Hope. SERPENTINE, in mineralogy, a species of the Talc genus: divided by Werner into the common and precious: the common is chiefly green, though passing into various other colours, which are seldom uniform. There are generally several colours together, and these are arranged in striped, dotted, and clouded delineations. It occurs massive internally it is faintly glimmering, which passes into dull when there are no foreign particles to give a slight degree of lustre. It is soft, not very brittle, and frangible. Feels a little greasy, not very heavy. It is infusible before the blow-pipe without addition. It consists of ed into vessels and ornaments of a great variety of shapes. In upper Saxony, several hundred people are employed in quarrying, cutting, turning, and polishing the serpentine which occurs in that neighbourhood, and the articles into which it is manufactured are carried all over Germany. The precious is found in Silesia. SERPICULA, in botany, a genus of the Monoecia Tetrandria class and order. Natural order of Inundatæ. Ouagræ, Jussieu. Essential character: male, calyx, fourtoothed; corolla, four petalled; female, calyx, four-parted; pericarpium, nut tomentose. There are two species, viz. S. verticillata and S. repens. SERPULA, in natural history, a genus of the Vermes Testacea class and order: ani mal a terebella: shell univalve, generally adhering to other substances: often separated internally by divisions at uncertain distances. About fifty species have been enumerated. SERPENTES, in natural history, an order of the Amphibia, containing seven genera: viz. Cæcilia Coluber Crotalus. Achrochordus Amphisbæna Anguis Boa Serpents are distinguished as footless am phibia: their eggs are connected in a chain: penis frequently double: they breathe through the mouth. The amphibia were divided by Linnæus into four orders; viz. Reptilia, Serpentes, Meantes, and Nantes. Of the meantes or gliders, which were characterized as breathing by means of gills and lungs together: feet branchiated and furnished with claws: there was but a single genus, viz. the siren: this has since been classed with the reptiles. See REP TILIA and SIREN. The nantes, or swimming amphibia, characterized by their having fins; and by breathing by means of lateral gills, were afterwards distributed into the orders of fishes denominated branchiostigi, and chondropterygii, which have since been ranked by Dr. Shaw and others under the general term cartilaginous fishes. See CHRON DROPTERIGIOUS. We have thought it right to give this account of the changes in the Linnæan system, which we have generally adopted, having omitted any mention of the facts under the former articles. "Serpents," says the trans lator of Gmelin," are cast naked upon the earth, without limbs, exposed to every injury, but frequently armed with a poison, the most deadly and horrible: this is contained in tubular fangs resembling teeth, placed without the upper jaw, protruded or retracted at pleasure, and surrounded with. a glandular vesicle by which this fatal fluid is secreted: but lest this tribe should too much encroach upon the limits of other animals, the benevolent Author of nature has armed about a fifth part only in this dreadful manner, and has ordained that all should cast their skins, in order to inspire a necessary suspicion of the whole. The jaws are dilatable, and not articulate, and the œsophagus so lax that they can swallow, without any mastication, an animal twice or thrice as large as the neck: the colour is variable, and changes according to the season, age, or mode of living, and frequently vanishes, or turns to another in the dead body tongue filiform, bifid; skin reticulate." The distinction between the poisonous and innoxious serpents, is only to be known by an accurate examination of their teeth; those which are poisonous being always tubular, and calculated for the injection of the poisonous fluid, from a peculiar reservoir communicating with the fang on each side the head. These teeth or fangs are situated in the upper jaw they are frequently accompanied by smaller fangs, seemingly intended to supply the place of the others, if lost by age or accident. The fangs are situated in a peculiar bone, so articulated with the rest of the jaw as to elevate or depress them at the pleasure of the animal: in a quiescent state they are recumbent, with their points directed in wards or backwards; but when the animal is inclined to use them as weapons of offence, their position is altered by the peculiar mechanism of the bone in which they are rooted, and they become almost perpendicular. Serpents in cold and temperate climates conceal themselves during winter, in cavities beneath the surface of the ground, or in any other convenient places of retirement, where they become nearly or wholly in a state of torpidity. Some serpents are viviparous, as the rattle-snake; the viper, &c. while the innoxious species are oviparous, depositing, as we have observed, their eggs in a kind of chain in any warm and close situation, where they are afterwards hatched. The broad undivided laminæ on the bellies of serpents, are termed scuta, and the smaller or divided ones beneath the tail are called subcaudal scales, and from these different kinds of laminæ, the Linnæan genera are characterized. SERRATULA, in botany, saw-wort, a genus of the Syngenesia Polygamia Equalis class and order. Natural order of Compositæ Capitatæ. Cinarocephalæ, Jussieu. Essential character: calyx, subcylindrical, imbricate, awnless. There are twenty species. SERROPALPUS, in natural history, a genus of insects of the order Coleoptera: antennæ setaceous; four feelers unequal; the anterior ones longer, deeply serrate, composed of four joints, the last joint very large, truncate, compressed, patelliform ; the posterior one subclavate; thorax margined, concealing the head, with a prominent angle on each side; head deflected; feet formed for digging. There are two species, viz. S. striatus, which is of a brown colour, with striated shells, found in autumn in old buildings: S. lævigatus, which is black and smooth. SERTULARIA, in natural history, a genus of the Vermes Zoophyta class and order: animal growing in the form of a plant; stem branched, producing polypes, from cup-shaped denticles or minute cells. Nearly fourscore species have been enumerated. These are divided into two classes: A. stem horny, tubular, fixed by the base, beset with cup-shaped denticles, and furnished with vesicles or ovaries containing polypes, eggs, or the living young. B. stem crustaceous, inclining to stone, and composed of rows of cells: there are no vesicles, but in the place of them are small globules. SERUM. See BLOOD. ог SESAMUM, in botany, sesamum oilygrain, a genus of the Didynamia Angiospermia class and order. Natural order of Luridæ. Bignoniæ, Jussieu. Essential character: calyx five-parted; corolla bellshaped, five-cleft, the lower lobe larger; rudiment of a fifth filament; stigma lanceolate; capsule four-celled. There are three species, viz. the orientale, the indicum, and the luteum. S. orientale has ovate, oblong, entire leaves. It is an annual, and grows naturally on the coast of Malabar and in the island of Ceylon; rising with an herbaceous four-cornered stalk, two feet high, sending out a few short side branches. After the flowers are past, the germen turns to an oval acute pointed capsule, with four cells, filled with oval compressed seeds, which |