account is as follows: Ptolemy having erected a fine library at Alexandria, which he took care to fill with the most curious and valuable books from all parts of the world, was informed that the Jews had one containing the laws of Moses, and the history of that people, and being desirous of enriching his library with a Greek translation of it, applied to the high priest of the Jews; and to engage him to comply with his request, set at liberty all the Jews, whom his father Ptolemy Soter had reduced to slavery. After such a step, he easily obtained what he desired; Eleazar, the Jewish high-priest, sent back his embassadors with an exact copy of the Mosaical law, written in letters of gold, and six elders of each tribe, in all seventy-two, who were received with marks of respect by the king, and then conducted into the Isle of Pharos, where they were lodged in a house prepared for their reception, and supplied with every thing necessary in abundance. They set about the translation without loss of time, and finished it in seventy-two days; and the whole being read in the presence of the king, he admired the profound wisdom of the laws of Moses, and sent back the deputies, laden with presents for themselves, the high-priest, and the temple. This version was in use to the time of our Saviour, and is that out of which all the citations in the New Testament, from the Old, are taken. It was also the ordinary and canonical translation made use of by the Christian church in the earliest ages; and it still subsists in the churches both of the east and west. It is, however, observable, that the chronology of the septuagint is different from the Hebrew text. SEQUENCE, in gaming, a set of cards immediately following each other, in the same suit as a king, queen, knave, &c. and thus we say, a sequence of three, four, or five cards; but at piquet these are called tierces, quarts, quints, &c. SEQUESTRATION, is the separating or setting aside of a thing in controversy from the possession of both those who contend for it; and it is of two kinds, voluntary or necessary: voluntary is that which is done by consent of each party; necessary is what the judge does of his authority, whether the parties will or not. It is used also for the act of the ordinary disposing of the goods and chattels of one deceased, whose estate no man will meddle with. A sequestration is also a kind of execution for debt, especially in the case of a beneficed VOL. VI. clerk, of the profits of the benefice, to be paid over to him that had the judgment till the debt is satisfied. SEQUIN, a gold coin struck at Venice, and in several parts of the Grand Seignior's dominions. SERAPIAS, in botany, helleborine, a genus of the Gynaudria Diandria class and order. Natural order of Orchidea. Es sential character: nectary ovate, gibbous, with an ovate lip. There are fourteen spe cies. SERGE, in commerce, a woollen stuff manufactured in a loom, of which there are various kinds, denominated either from their different qualities, or from the places where they are wrought; the most considerable of which is the English serge, which is highly valued abroad, and of which a manufacture had been for some years carried on in France. In the manufacture of serges, the longest wool is chosen for the warp, and the shortest for the woof. But before either kind is used, it is first scoured, by putting it in a copper of liquor, somewhat more than lukewarm, composed of three parts of fair water and one of urine. After it has staid in it long enough for the liquor to take off the grease, &c. it is stirred briskly about with a wooden peel, taken out, drained, washed in a running water, and dried in the shade; beaten with sticks on a wooden rack, to drive out the coarser dust and filth; and then picked clean with the hands. It is then greased with oil of olives, and the longest wool combed with large combs, heated in a little furnace for that purpose: to clear it from the oil, it is put into a vessel of hot soap-water, whence being taken out, wrung, and dried, it is spun on the wheel. As to the shorter wool, intended for the woof, it is only carded on the knee with small fine cards, and then spun on the wheel, without being scoured of its oil: and here it is to be observed, that the thread for the warp is always to be spun finer, and much better twisted, than that of the woof. The wool both for the warp and woof being spun, and the thread reeled into skeins; that of the woof is put on spools, fit for the cavity of the shuttle; and that for the warp is wound on a kind of wooden bobbins, to fit it for warping; and when warped, it is stiffened with a size, usually made of the shreds of parchment; and, when dried, put into the loom, and mounted so as to be raised by four treddles, placed under the F foom, which the workman makes to act transversely, equally, and alternately, one after another, with his feet; and as the threads are raised, throws the shuttle. WEAVING. See The serge, on being taken from the loom, is carried to the fuller, who fulls or scours it, in the trough of his mill, with fullers earth and after the first fulling, the knots, ends, straws, &c. sticking out on either side of the surface, are taken off with a kind of pliers, or iron pincers, after which it is returned into the fulling-trough, where it is worked with warm water, in which soap has been dissolved; when quite cleared, it is taken out, the knots are again pulled off; it is then put on the tenter to dry, taking care, as fast as it dries, to stretch it out both in length and breadth, till it be brought to its just dimensions; then being taken off the tenter, it is dyed, shorn, and pressed. SERJEANT at law, is the highest degree taken in that profession, as that of a doctor is in the civil law. To these serjeants, as men of great learning and experience, one court is set apart for them to plead in by themselves, which is the Court of Common Pleas, where the common law of England is most strictly observed; yet, though they have this court to themselves, they are not restrained from pleading in any other courts. The judges cannot be elevated to that dignity till they have taken the degree of Serjeant at Law. They are called brothers by the judges, who hear them next to the King's counsel; but a King's Serjeant has precedence of all but the Attorney and Solicitor General. These are made by the King's mandate, or writ. SERJEANT at arms, is one whose office is to attend on the person of the King, to arrest persons of condition offending. SERJEANTY, in law, signifies a service that cannot be due from a tenant to any Lord, but to the King only. Although the old tenures are abolished, yet the merely honorary services of grand and petit serjeanty remain. SERIES,' in general, denotes a continued succession of things in the same order, and having the same relation or connection with each other: in this sense we say, a series of emperors, kings, bishops, &c. In natural history, a series is used for an order or subdivision of some class of natural bodies; comprehending all such as are distinguished from the other bodies of that class, by certain characters, which they pos sess in common, and which the rest of the bodies of that class have not. SERIES, in mathematics, is a number of terms, whether of numbers or quantities, increasing or decreasing in a given proportion; the doctrine of which has already been given under the article PROGRESSION. SERIES, infinite, is a series consisting of an infinite number of terms, that is, to the end of which it is impossible ever to come; so that let the series be carried on to any assignable length, or number of terms, it can be carried yet further, without end or limitation. A number actually infinite, (that is, all whose units can be actually assigned, and yet is without limits) is a plain contradiction to all our ideas about numbers; for whatever number we can conceive, or have any proper idea of, is always determinate and finite; so that a greater after it may be assigned, and a greater after this; and so on, without a possibility of ever coming to an end of the addition or increase of numbers, assignable; which inexhaustibility, or endless progression in the nature of num. bers, is all we can distinctly understand by the infinity of number; and therefore to say that the number of any things is infinite, is not saying that we comprehend their number, but indeed the contrary; the only thing positive in this proposition being this, that the number of these things is greater than any number which we can actually conceive and assign. But then, whether in things that do really exist, it can be truly said that their number is greater than any assignable number; or, which is the same thing, that in the numeration of their units one after another, it is impossible ever to come to an end; this is a question about which there are different opinions, with which we have no business in this place; for all that we are concerned here to know, is this certain truth, that after one determinate number, we can conceive a greater, and after this a greater, and so on without end. And, therefore, whether the number of any things that do, or can really exist all at once, can be such that it exceeds any determinable number, or not, this is true, that of things which exist, or are produced successively one after another, the number may be greater than any assignable one; because though the number of things thus produced, that does actually exist at any time, is finite, yet it may be increased with. out end. And this is the distinct and true notion of the infinity of a series, that is, of the infinity of the number of its terms, as it is expressed in the definition. Hence it is plain that we cannot apply to an infinite series the common notion of a sum, viz. a collection of several particular numbers that are joined and added together one after another, for this supposes that these particulars are all known and determined; whereas the terms of an infinite series cannot be all separately assigned, there being no end in the numeration of its parts, and therefore it can have no sum in sense. But again, if we consider that the idea of an infinite series consists of two parts, viz. the idea of something positive and determined, in so far as we conceive the series to be actually carried on; and the idea of an inexhaustible remainder still behind, or an endless addition of terms that can be made to it one after another, which is as different from the idea of a finite series as two things can be: hence we may conceive it as a whole of its own kind, which, therefore, may be said to have a total value whether that be determinable or not. Now in some infinite series this value is finite or limited; that is, a number is assignable beyond which the sum of no assignable number of terms of the series can ever reach, nor indeed ever be equal to it, yet it may approach to it in such a manner as to want less than any assignable difference; and this we may call the value or sum of the series; not as being a number found by the common method of addition, but as being such a limitation of the value of the series, taken in all its infinite capacity, that if it were possible to add them all one after another, the sum would be equal to this number. Again, in other series the value has no limitation; and we may express this, by saying, the sum of the series is infinitely great; which, indeed, signifies no more than that it has no determinate and assignable value; and, that the series may be carried such a length as its sum, so far, shall be greater than any given number. In short, in the first case, we affirm there is a sum, yet not a sum taken in the common sense; in the other case, we plainly deny a determinate sam in any sense. Theorem 1. In an infinite series of numbers, increasing by an equal difference or ratio (that is, an arithmetical or geometrical increasing progression) from a given number, a term may be found greater than any assignable number. Hence, if the series increase by differences that continually increase, or by ratios that continually increase, comparing each term to the preceding, it is manifest that the same thing must be true, as if the differences or ratios continued equal. Theorem 2. In a series decreasing in infinitum in a given ratio, we can find a term less than any assignable fraction. Hence, if the terms decrease, so as the ratios of each term to the preceding do also continually decrease, then the same thing is also true, as when they continue equal. Theorem 3. The sum of an infinite series of numbers all equal, or increasing continually, by whatever differences or ratios, is infinitely great; that is, such a series has no determinate sum, but grows so as to exceed any assignable number. Demons. First, if the terms are all equal, as A: A: A, &c. then the sum of any finite number of them is the product of A by that number, as An; but the greater n is, the greater is An; and we can take n greater than any assignable number, therefore An will be still greater than any assignable number. Secondly, suppose the series increases continually, (whether it do so infinitely or limitedly) then its sum must be infinitely great, because it would be so if the terms continued all equal, and therefore will be more so, since they increase. But if we suppose the series increases infinitely, either by equal ratios or differences, or by increasing differences or ratios of each term to the preceding; then the reason of the sums being infinite will appear from the first theorem; for in such a series, a term can be found greater than any assignable number, and much more therefore the sum of that and all the preceding. Theorem 4. The sum of an infinite series of numbers decreasing in the same ratio is a finite number; equal to the quote arising from the division of the product of the ratio and first term, by the ratio less by unity; that is, the sum of an assignable number of terms of the series can ever be equal to that quote; and yet no number less than it is equal to the value of the series, or to what we can actually determine in it; so that we can carry the series so far, that the sum shall want of this quote less than any assignable difference. Demons. To whatever assigned number of terms the series is carried, it is so far finite; and if the greatest term is 7, the least Now, in a decreasing series from 1, the more terms we actually raise, the last of them, A, becomes the lesser, and the lesser A is rl A is the greater, and so also is : but rl-A being still less than rl-A r-1 is still less than 1 1 rl, therefore -1' that is, the sum of any assignable number of terms of the series is still less than the rl and this quote mentioned, which is is the first part of the theorem. Again, the series may be actually conrl-A tinued so far, that shall want of r-1 rl r 1 less than any assignable difference; for, as the series goes on, A becomes less and less in a certain ratio, and so the series may be actually continued till A becomes less than any assignable number, (by Theorl rl-A A rem 2) now and -1 T-1 r-1' A r-1 = is less than A; therefore let any num ber assigned be called N, we can carry the series so far till the last term A be less than rl - A rl wants of over that complete value; but this excess being unassignable, that consequent error must be so too; because still the less the excess is, the less will the error be that depends upon it. And for this reason we may justly enough look upon as expressing the adequate value of the infinite series. But we are further satisfied of the reasonableness of this, by finding, in fact, that a finite quantity does actually convert into an infinite series, which happens in the case of infinite decimals. For example, =. 6 6 6 6, &c. which is plainly a geometrical series from in the continual 10 6 N; and because -1 therefore rl= whence = = rl T-1 9 We have added here a table of all the varieties of determined problems of infinite, decreasing, geometrical progressions, which all depend upon these three things, viz. the greatest term 1, the ratio r, and the sum by any two of which the remaining one may be found: to which we have added some other problems, wherein S― Lis. considered as a thing distinct by itself, that is, without considering S and L sepas rately. Or supposing the ratio or the second term to be M, whereby the ratio is 8 i=。 of 1= ΜΙ ofs-1 Theorem 5. In the arithmetic progression 1, 2, 3, 4, &c. the sum is to the product of the last term, by the number of terms, that is, to the square of the last term; in a ratio always greater than 1:2, but approaching infinitely near it. But if the arithmetical series begins with 0, thus, 0, 1, 2, 3, 4, &c. then the sum is to the product of the last term, by the number of terms, exactly in every step as 1 to 2. 3 + + + 1 &c. where the law of continuation is manifest. Or, if the same and difraction be set in this form 2+ vision be performed in the algebraic man. the quotient will be ner, Theorem 6. Take the natural progression beginning with 0, thus, 0, 1, 2, 3, &c. and take the squares of any the like powers of the former series, as the squares, 0, 1, 4, 9, &c. or cubes, 0, 1, 8, 27; and then again take the sum of the series of powers to any number of terms, and also multiply the last of the terms summed by the number of terms, (reckoning always 0 for the first Or, if it be expressed in this form term) the ratio of that sum, to that product is more than 1 1 3 1 by a like division there will arise 1 (n being the index of the 4 the series, powers), that is, in the series of squares it is more than ; in the cubes more than ; and so on: but the series going on in infinitum, we may take in more and more terms without end into the sum; and the more we take, the ratio of the sum to the product mentioned grows less and less; yet so as it never can actually be equal to but approaches infinitely near to it, or within less than any assignable difference. 1 "The nature, origin, &c. of series." In finite series commonly arise, either from a |