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ner, by dividing 1 by the algebraic sum Hence it appears that the signs of the a+c, or by a-e, the quotient will be in terms may be either all plus, or alternately these two cases, as below, viz.

plus and minus, though they may be varied

in many other ways. It also appears that 1__1 C,c c

the terms may be either continually smaller atcaa-äitä- ja &c.

and smaller, or larger and larger, or else all equal. In the first case, therefore, the se

ries is said to be a decreasing one; in the 2d where the terms of each series are the same,

case, an increasing one; and in the 3d case,

an equal one. Also the first series is called and they differ only in this, that the signs

a converging one, because that by collecting are alternately positive and negative in the

its terms successively, taking in always one former, but all positive in the latter.

term more, the successive sums approxi. And hence, by expounding a and c by

mate or converge to the value or sum of the any numbers whatever, we obtain an end.

whole infinite series. So, in the series less variety of infinite series, whose sums or

1 1 1 1 1 1 values are known. So, by taking a orc

| 3-1-=+ + +, ốc. equal to 1, or 2, or 3, or 4, &c. we obtain these series, and their values ;

the first term is too little, or below , 1-1+1-1

1—1, &c.

which is the value or sum of the whole infi

nite series proposed; the sum of the first 3=1= =+3+3+3,&c. two terms +" is =-444

•4444, &c. is also ,1,= =- + -., &c. too little, but nearer to or 5 than the for1= =1–2 +22 -- 2', &c. mer; and the sum of thr 1 1 1

1. 13 =3 32+z:- 34. &c. az is =.481481, &c. is nearer than the And hence it appears, that the same

last, but still too little; and the sum of four quantity or radix may be expressed by a

terms

tern great variety of infinite series, or that many

1,111+1 is different series may have the same radix, or

3f9f97f81 2817

= .493827, &c. sum.

which is again nearer than the former, but Another way in which an infinite series still too little; which is always the case arises, is by the extraction of roots. Thus, when the terms are all positive. But when by extracting the square root of the number the converging series has its terms alter. 3 in the cominon way, we obtain its value nately positive and negative, then the suc. in a series as follows, viz. 73=1.73205, cessive sumis are alternately too great and

, 7, 3, 2 , 5 too little, though still approaching nearer &c. = 1 +

Tot 100 + 1000 + 100000' and nearer to the final sum or value. Thus, &c.; in which way of resolution the law of in the series the progression of the series is not visible, 1__ as it is when found by division. And S+1-7- 5 972737C. the square root of the algebraic quantity the

the 1st ter a? + c gives

, &c. is too great; Na te=ate m otiv,&c. two terms -=222, &c. are too little;

And a 3d way is by Newton's binomial theorem, which is an universal method, that three terms möta= 959259, &c. serves for all sorts of quantities, whether are too great; fractional or radical ones : and by this means the same root of the last given quan. four terms - + - = -246915, tity becomes va+c=

&c. are too great, and so on, alternately, i od 1. 1.3 1.3.5 o too great and too small, but every succeed“Tza 2.4 at 2.1.04 T 2.4.6.8 a?! ing sum still nearer than the former, or &c. where the law of continuation is vis converging. sible.

In the second case, or when the terms

3+1

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grow larger and larger, the series is called a gives – ? for the supplement, which, com dicerging oue, because that by collecting the terms continually, the successive sums

bined with the first term, diverge, or go always further and further from the true value or radix of the series; = 1, the true sum of the series. Again, afbeing all too great when the terms are all positive, but alternately too great and too

ter the first two little when they are alternately positive and

msthe remainnegative. Thus, in the series

der is +, which, divided by the same di1 1 1+y=z=1—2 +4-8, &c.

visor, 3, gives for the supplement, and the first term + 1 is too great; two terms 12=1 are too little ; this combined with those two terms -three terms 1 -2+4=+3 are too great; four terms 1–2+1-8= -5 are too makes – +1=1+1 = 1 or the little; and so on, continually, after the 2d term, diverging more and more from the

same sum or value as before. And, in ge

neral, by dividing 1 by a tc, there is obtrue value or radix , but alternately too tained

1 great and too little, or positive and negative.

... Inti F But the alternate sums would be always mati more and more too great if the terms were qitila tej; where, stopping the division all positive, and always too little if nega. gative,

at any term, as nimi, the remainder after But in the third case, or when the terms are all equal, the series of equals, with al this term ist in which, being divided by terpate signs, is called a neutrul one, be

*+1

the same divisor, a + c, gives utilats. cause the successive sums, found by a continual collection of the terms, are always at for the supplement as above.. . the same distance from the true value or « The Law of Continuation." -- A series radix, but alternately positive and negative, being proposed, one of the chief questions or too great and too little. Thus, in the

concerning it is to find the law of its contiseries

nuation. Indeed, no universal rule can be

given for this; but it often happens, that ==1-1+1-1+1—1, &c.

the terms of the series, taken two and two, the first term 1 is too great;

or three and three, or in greater numbers, two terms 1-1=0 are too little;

bave an obvious and simple relation, by three terms 1-1+1=1 too great; which the series may be determined and four terms 1-1+1-1=0 too little; produced indefinitely. Thus, if i be diand so on, continually, the successive sums vided by 1 -- X, the quotient will be a geobeing alternately 1 and 0, which are equally metrical progression, viz. 1 + x + x2 + x,

. 1 &c. where the succeeding terms are pro. different from the true value, or radix, duced by the continual multiplication by r. the one as much above it as the other be. In like manner, in other cases of division, low it.

other progressions are produced. A series may be terminated and render. But in most cases, the relation of the ed finite, and accurately equal to the sum terms of a series is not constant, as it is in or value, by assuming the supplement, after those that arise by division. Yet their reany particular term, and combining it with lation often varies according to a certain

law, which is sometimes obvious on inspecthe foregoing terms. So, in the series - tion, and sometimes it is found by dividing 1,1_1. &c. which is equal to

the successive terms one by another, &c.

, and Thus, in the series found by dividing 1 by 2 +1, after the 1+2x+ y + 10x + 1 r*, &c. by first term, -, of the quotient, the remain- dividing the 2d term by the 1st, the 3d by

the 2d, the 4th by the 3d, and so on, the der is –, which, divided by 2 + 1, or S,

quotients will be

2 4 6 8

ž, zr, 77, ģx, &c.; and therefore the terms may be continued inf. = S, indefinitely by the successive multiplication

then by these fractions. Also in the following

it

“ т series

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95

Ex. 3. Let +xists.& +, &c. in

ft, &e. in inf. = S

++ 1152 2*, &c. by dividing the adjacent terms successively by subtraction, 7.4.3 +2.3.a t ena

+, &c. in inf. = , and

+3.4.5+, &c. in inf. =

Let + m +1+m+ert..

1

193

Er. 4. Let

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S

by each other, the series of quotients is

1 9 25 49 őt, 20%, 77x, 5x, &c.; or 1.1 3.3 5.5, 7.7. &c.;

2.3*, 4.5", 7. 7 t, 3.91, &c.; and therefore the terms of the series may be continued by the multiplication of these fractions. Series, summation of. We have before

-=S,
seen the method of determining the sams m -
of quantities in arithmetical and geometri- then
cal progression, but when the terms in then mti+m+er+ m +3,+ ....
crease, or decrease, according to other laws,
different artifices must be used, to obtain mtir m '
general expressions for their sum,
The methods chiefly adopted, and which

by subtraction, -
y S ACAOW, mon

mtr.7 torr
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. hence, - ts

t &c. 4. The method of increments. We shall m.in + rm trimtur content ourselves with an example or two, (to n teims) == in the first of these methods. “ The investigation of series, whose sums and

ta are known by subtraction."

m.m + ram trim ter

11 Er. 1. Let 1 +o+itit, &c. in (to n terms) = mi-mrtnimi inf. =s,

If n be increased without limit,

m.mtr

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then+*+*+*+,&c. in inf. =-1,

vanishes, and the sum of the series i by subtraction, i-.x+yns+s4 +, &c. Ilm=r=1, we have 1.2 +onts + in inf. = 1; ortátie + 0, &c. = 1. 3.4 +, &c. (to n terms) = 1- tn

Ex. 2. Let 1 ++*+*+,&c. in = 177 inf. = S. then +*+*+* +, &c. in inf. = s–, the following, given by De Moivre. by subtraction, to at t ait to a, involving the powers of an indetermi

Similar to the nethod of subtraction is

then

“ Assume a series, whose terms converge

&c. in inf. =

or is that its + otot, &e. in inf. =

wate quantity, r; call the sum of the series
S, and multiply both sides of the equation
by a binomial, trinomial, &c. which in-
volves the powers of x, and invariable co-
efficients; then, if r be so assumed that the
binomial, trinomial, &c. may vanish, and
some of the first terms be transposed, the

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1.2

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&c.

2.3

sum of the remaining series is equal to the islands. It takes a good polish, and is turpterms so transposed.”.

ed into vessels and ornaments of a great , I, I, II be in inf.-S.

variety of shapes. In upper Saxony, several Let i+ + -+- t,&c. in inf. = S. hundred people are employed in quarry, Multiplying both sides by x — 1, we bave ing, cutting, turning, and polishing the ser

pentine which occurs in that neighbour

hood, and the articles into which it is manu-low factured are carried all over Germany. The

precious is found in Silesia.

SERPICULA, in botany, a genus of the or – 1 + + + +, &c. = Monoecia Tetrandria class and order. Na 7-1.S; and if r=1,

tural order of Inundatæ. Ouagræ, Jus

sieo. Essential character: male, calyx, fourthen, - 1

toothed; corolla, four petalled; female,

calyx, four-parted; pericarpium, nut tomen=0; or, I + 1 + 1 t, &c. in tose. There are two species, viz. S. ver

ticillata and S. repens. inf. = 1.

SERPULA, in natural history, a genus SERIOLA, in botany, a genus of the of the Vermes Testacea class and order: ani Syngenesia Polygamia Æqualis class and mal a terebella: shell univalve, generally order. Natural order of Compositæ Semi- adhering to other substances : often sepaflosculosæ. Cichoraceæ, Jussien. Essen- rated internally by divisions at uncertain tial character: calyx, simple; pappus sub distances. About fifty species have been plumose; receptacle chaffy. There are enumerated. four species.

SERPENTES, in natural history, an SERIPHIUM, in botany, a genus of the order of the Amphibia, containing seven Syngenesia Polygamia Segregata class and genera : viz. order. Natural order of Compositæ Nuca

Achrochordus Cæcilia
mentaceæ. Corymbiferz, Jussien. Essen.
tial character: calyx, imbricate; corolla,

Amphisbæna Coluber
Anguis

Crotalus. one petalled, regular, seed one; oblong,

Boa below the corolla. There are four species, all natives of the Cape of Good Hope. Serpents are distingnislied as footless am.

SERPENTINE, in mineralogy, a species phibia: their eggs are connected in a of the Talc genus : divided by Werner into chain: penis frequently donble: they breathe the common and precious: the common through the mouth. The amphibia were is chiefly green, though passing into various divided by Linnæus into four orders; vis. other colours, which are seldom uniform. Reptilia, Serpentes, Meantes, and Nantes. There are generally several colours to Of the meantes or gliders, which were chagether, and these are arranged in striped, racterized as breathing by means of gills dotted, and clouded delineations. It occurs and Invgs together: feet branchiated and massive : internally it is faintly glimmering, furnished with claws: there was but a which passes into dull when there are no single genus, riz, the siren : this has since foreign particles to give a slight degree of been classed with the reptiles. See Rep. Instre. It is soft, not very brittle, and TILIA and SIREN. frangible. Feels a little greasy, not very T he nantes, or swimming amphibia, chaheavy. It is infusible before the blow.pipe racterized by their having fins; and by without addition. It consists of

breathing by means of lateral gills, were

afterwards distributed into the orders of Magnesia..................... 23

fishes denominated branchiostigi, and chonSilica .............

dropterygü, which have since been ranked Alumina .................

by Dr. Shaw and others under the general Iron .......................

term cartilaginous fishes. See CHRONWater ......................

DROPTERIGIOUS.

We have thought it right to give this ac

count of the changes in the Linnæan system, It is one of the primitive rocks: is found which we have generally adopted, having in many parts of Germany, Italy, Siberia, omitted any mention of the facts uuder the in this country, Scotland, and the Shetland former articles. “Serpents," says the trans,

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101

lator of Gmelin, “ are cast naked upon the scuta, and the smaller or divided ones bei earth, without lirubs, exposed to every in. Death the tail are called subcaudal scales, jury, but frequently armed with a poison, and from these different kinds of laminæ, the most deadly and horrible: this is con- the Linnæan genera are characterized. tained in tubular fangs resembling teeth, SERRATULA, in botany, saw-wort, a placed without the upper jaw, protruded or genus of the Syngenesia Polygamia Æqualis retracted at pleasure, and surrounded with. class and order. Natural order of Com. a glandular vesicle by which this fatal Auid positæ Capitatæ. Cinarocephalæ, Jussien. is secreted: but lest this tribe should too Essential character: calyx, subcylindrical, much encroach upon the limits of other imbricate, awnless. Tbere are twenty animals, the benevolent Author of nature species. bas armed about a fifth part only in this SERROPALPUS, in natural history, a dreadful manner, and has ordained that all genus of insects of the order Coleoptera: should cast their skins, in order to inspire antennæ setaceous; four feelers unequal; a necessary suspicion of the whole. The the anterior ones longer, deeply serrate, jaws are dilatable, and not articulate, and composed of four joints, the last joint very the æsophagus so lax that they can swallow, large, truncate, compressed, patelliform; without any mastication, an animal twice the posterior one subclavate; thorax mar. or thrice as large as the neck: the colour is gined, concealing the head, with a provariable, and changes according to the sea. minent angle on each side; head deflected; son, age, or mode of living, and frequently feet formed for digging. There are two vanishes, or turns to another in the dead species, viz. S. striatus, which is of a brown body: tongue filiform, bifid ; skin reticu- colour, with striated shells, found in autumn late.” The distinction between the poison- in old buildings: S. lævigatus, which is ous and innoxious serpents, is only to be black and smooth. known by an accurate examination of their SERTULARIA, in natural history, a teeth ; those which are poisonous being algenus of the Vermes Zoophyta class and ways tubular, and calculated for the injec. order: animal growing in the form of a tion of the poisonous fuid, from a peculiar plant; stem branched, producing polypes, reservoir communicating with the fang on from cup-shaped denticles or minute cells. each side the head. These teeth or fangs Nearly fourscore species have been enu. are situated in the upper jaw : they are merated. These are divided into two classes : frequently accompanied by smaller fangs, A. stem horny, tubular, fixed by the base, seemingly intended to supply the place of beset with cup-shaped denticles, and fur. the others, if lost by age or accident. The nished with vesicles or ovaries containing fangs are situated in a peculiar bone, so polypes, eggs, or the living young. B. stem articulated with the rest of the jaw as to crustaceous, inclining to stone, and comelevate or depress them at the pleasure of posed of rows of cells: there are no vesi. the animal : in a quiescent state they are cles, but in the place of them are small recumbent, with their points directed in- globnles. wards or backwards; but when the animal SERUM. See Blood. is inclined to use them as weapons of of- SESAMUM, in botany, sesamum or fence, their position is altered by the pecu- oilygrain, a genus of the Didynamia Angiliar mechanism of the bone in which they ospermia class and order. Natural order of are rooted, and they become almost per. Luridæ. Bignoniæ, Jussieu. Essential pendicular.

character : calyx five-parted; corolla bella Serpents in cold and temperate climates shaped, five-cleft, the lower lobe larger ; conceal themselves during winter, in cavi. rudiment of a fifth filament ; stigma lanties beneath the surface of the ground, or ceolate; capsule four-celled. There are in any other convenient places of retire. three species, viz. the orientale, the indicum, ment, where they become nearly or wholly and the luteum. S. orientale has ovate, obin a state of torpidity. Some serpents are long, entire leaves. It is an annual, and viviparous, as the rattle-spake; the viper, grows naturally on the coast of Malabar and &c.: while the innoxious species are ovi- in the island of Ceylon ; rising with an herparons, depositing, as we have observed, baceous four-coruered stalk, two feet bigh, their eggs in a kind of chain in any warm sendiog out a few short side branches. After and close situation, where they are after the flowers are past, the germen turns to an wards hatched. The broad undivided laoval acute pointed capsule, with four cells, ininæ on the bellies of serpents, are termed filled with oval compressed seeds, which

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