= + + &c. + + 3 4. 16 64' -- 1 4 4. 43 &c. And thus, by dividing 1 by 5-2, or 6 3, or 7 4, &c. the series answering to the fraction may be found in an endless variety of infinite series; and the finite quantity is called the value or radix of the series, or also its sum, being the number or sum to which the series would amount, or the limit to which it would tend or approximate, by summing up its terms, or by collecting them together one after another. In like manner, by dividing 1 by the algebraic sum a + c, or by ac, the quotient will be in these two cases as below, viz. where the terms of each series are the same, and they differ only in this, that the signs are alternately positive and negative in the former, but all positive in the latter. And hence, by expounding a and c by any numbers whatever, we obtain an endless variety of infinite series, whose sums or values are known. So by taking a or c equal to 1, or 2, or 3, or 4, &c. we obtain these series, and their values; the number 3 in the common way, we ob- 2 &c. C4 сб + 2a 8a3 16as 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 second case, an increasing one; and in the third case, an equal one. Also the first series is called a converging one, because that, by collecting its terms successively, taking in al 1+1-1+1-1, ways one term more, the successive sums 1 1 1 1 + + + &c. 3+ 1 1 1 1 1 So in 1 9 + &c. the first term is too little or below which is the value or sum of the whole infinite series proposed; the sum of the first two terms + is 4 9 =.4444, &c. &c. are too great. four terms 1 27 = .259259, 1 = 9 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 grow larger and larger, the series is called a diverging one, because that, by collecting the terms continually, the successive sums diverge, or go always further and further from the true value or radix of the series; being all too great when the terms are all positive, but alternately too great and too little when they are alternately positive and negative. Thus, in the series 1+2 3 the first term + two terms 1-2 three terms 1great; 8 - - 5 are four terms 1 2+4 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 negative. But in the third case, or when the terms are all equal, the series of equals, with alternate signs, is called a neutral VOL. XI. one because the successive sums, formed 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 series, 1 1+1 &c. gives above en + i an + 1 (a + c) for the supplement as 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 progres sion, 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 16 35 1+ 23 + 2 8 3x+ 128 315 3 x4, +, &c. in inf. = 1st, 2 4 6 8 &c.; 1 1 + + + 1.3 2.4 3.5 4.6 x4, S 1 2' 2 2 by subtraction, + 1.2.3 1 X, &c.; or +, &c. in inf. 2? 1 x, &c. 8.9 2.3.4 1 1 3 3 5.5 7.7 2.3 and, therefore, the terms of the series may be continued by the multiplication of these fractions. SERIES, Summation of fore seen the method of determining the sums of quantities in arithmetical and geometrical progression, but when the terms increase or decrease. according to ether laws, different artifices must be m + n r 1 = S, 1.r 1 + m+ 3r m + n→→ 1 + m+r m+2r then subplumose; receptacle chaffy. There are four species. SERIPHIUM, in botany, a genus of the Syngenesia Polygamia Segregata class and order. Natural order of Composite Nucamentaceæ. Corymbiferæ, Jussieu. Essential character: calyx imbricate; corolla one-petalled, regular: seed one, +, &c. oblong, below the corolla. There are four species, all natives of the Cape of Good Hope. m + &c. (to n terins) + hence, + m.m+r 1 (to n terms) = m m+ nr If n be increased without limit, 1 mrnr Similar to the method of subtraction is the following, given by De Moivre. "Assume a series, whose terms converge to o, involving the powers of an indeterminate quantity, ; 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 co-efficients; 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 terms so transposed." 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, winch are seldom uniform. There are generally several colours together, and these are arranged in striped, dotted, and clouded delineations. It occurs massive; insernally 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 It is one of the primitive rocks; is found in many parts of Germany, Italy, Siberia, in this country, Scotland, and the Shetland islands. It takes a good polish, and is turned 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. Onagræ. Jussieu. Essential character: male, calyx four-toothed; corolla four-petalled, female, calyx four-parted; pericarpium nut tumentose. There are two species, viz. S verticillata and S. repens. SERPULA, in natural history, a genus of the Vermes Testacea class and order: animal a terebella: shell univalve, gene Serpents are distinguished as footless amphibia: 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 gl. ders, 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 REPTILIA 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 CHRONDROPTERIGIOUS. 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. "Ser pents," says the translator of Gmelin, are cast naked upon the earth, without limbs, exposed to every injury, but fre quently 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 swal low without any mastication, an animal twice as thrice as large as the : neck: the colour is variable, and changes, according to the season, age, or mode of living, and frequently va nishes, 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 of 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 inwards 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 lamina 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 Equilis class and order. Natural order of Compositæ Capitatæ. Cinarocephala, 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, patelli |