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11 the number 3 in the common way, we ob
tain its value in a series as follows, viz. arise the series,
✓ 3 = 1.73205, &c. =1+ 1_1111len_11_1
tpot =āt 16+ 64, &c. =ātaitā 2 . 5 &c. And thus, by dividing 1 by 5-2, or 6 1000 + 100000; &c. in which way of - 3, or 7 - 4, &c. the series answering resolution the law of the progression of to the fraction may be found in an end the series is not visible, as it is when less variety of infinite series; and the fi- found by division. And the square root nite quantity 1 is called the value or ra. of the algebraic quantity a: + c gives dix of the series, or also its sum, being whi c , the number or sum to which the series
16asi would amount, or the limit to which it &c. would tend or approximate, by summing And a third way is by Newton's binoup its terms, or by collecting them toge- mial theorem, which is an universal me. ther one after another. In like manner,
ver, thod that serves for all sorts of quantiby dividing I by the algebraic sum at C, ties, whether fractional or radical ones : or by a - c, the quotient will be in these
and by this means the same root of the two cases as below, viz.
last given quantity becomes Va+c= 1_1_Circ en
12c+ . 1.3 cố. 1.3.5c8 atora-äitä сat. &c.
a + 2a - 2.4a3 + 2.4.6 as + 126.96.36.1997'
&c. where the law of continuation is vi1 1.c c , c3 arc=ät äität&c. sible.
Hence it appears that the signs of the where the terms of each series are the terms may be either all plus, or altersame, and they differ only in this, that the nately plus and minus, though they may signs are alternately positive and nega- be varied in many other ways. It also live in the former, but all positive in the appears that the terms may be either latter.
continually smaller and smaller, or larger And hence, by expounding a and c by and larger, or else all equal. In the first any nuinbers whatever, we obtain an end. case, therefore, the series is said to be a less variety of infinite series, whose sums decreasing one ; in the second case, an inor values are known. So by taking a or creasing one ; and in the third case, an c equal to 1, or 2, or 3, or 4, &c. we ob- equal one. Also the first series is called tain these series, and their values; a converging one, because that, by collect
ing its terms successively, taking in aln -j=1–1+1-1+1 – 1, ways one term more, the successive sums
approximate or converge to the value or sum of the whole infinite series. So in the series
1 1 1,1, 1 1 3—1=2=5+5+27 +51: &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 S+1=4 =337 33+ 57&c.
first two terms +5 is = .4444, &c. And hence it appears, that the same quantity or radix may be expressed by a is also too little, but nearer to or .5 great variety of infinite series, or that many different series may have the same than the former; and the sum of three Jadix, or sum.
1 , 1 , 1. 13 Another way in which an infinite series
terms ž + + 27 is 27 =
= .481481, arises, is by the extraction of roots. &c. is nearer than the last, but still too Thus, by extracting the square root of little; and the sum of four terms
1 1 1 1. 40
one because the successive sums, formed ato +5 +5 is g = .493827, &c. by a continual collection of the terms, which is again nearer than the former, are always at the same distance from the but still too little ; which is always the true value or radix, but alternately posi. case when the terms are all positive. tive and negative, or too great and too But when the converging series has its little. Thus, in the series, terms alternately positive and negative, 1
=1-1+1-1+1--1, then the successive sums are alternately 1+1T2 too great and too little, though still ap. &c. proaching nearer and nearer to the final the first term is 1 is too great; sum or value. Thus, in the series two terms 1-1=0 are too little ; 1_09 _1 1 , 1 three terms 1-1+1=1 too great ;
four terms 1-1+1-1=0 too little ; and so on, continually, the successive sums being alternately 1 and 0, which are
equally different from the true value, or the 1st term = = .333, &c. is too great; radis, I the one as much above it as the two terms
222, &c. are too other below it.
A series may be terminated and ren
dered finite, and accurately equal to the three ter
sum or value, by assuming the supple. &c. are too
ment, after any particular term, and com.
bining it with the foregoing terms. So, four terms l_1111 four terms - + 27 - 81 = in the series -- &c. whichi .246913, &c. are too great, and so on, al. ternately too great and too small, but
found by dividing 1 by 2 every succeeding sum still nearer than the former, or converging:
+1, after the first term, , of the quoIn the second case, or when the terms grow larger and larger, the series is call- tient, the remainder is
ainder is - which, died a diverging one, because that, by col. lecting the terms continually, the succes. vided by 2 + 1, or 3, gives - for the sive sums diverge, or go always further and further from the true value or radix supplement, which combined with the of the series; being all too great when the terms are all positive, but alternately first term, ā gives ;--- = , the true too great and too little when they are al sum of the series. Again, after the first ternately positive and negative. Thus, in the series
the remainder is + 1= =1–2 +4 – 8,&c. 1+2 – 3
, which, divided by the same divisor, 3, the first term + 1 is too great; two terms 1-2= - 1 are too little ;
for the supplement, and this 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 alternate. or the same sum or value as before. ly too great and too little, or positive and And, in general, by dividing 1 by a toc, negative. But the alternate sums would there is obtained be always more and more too great if the 1 1 terms were all positive, and always too ata
.... #anti F little if negative.
Isi where stopping the diBut in the third case, or when the an+1(a + c)'." terms are all equal, the series of equals, vision at any term, as with alternate signs, is called a neutral
, the remainVOL. XI.
combined with those two terms
der after this term is
used to obtain general expressions for
, which, be their sum. ing winter! !he same divisor, a + c,
The methods chiefly adopted, and
which may be considered as belonging to " antilator for the supplement as algebra..
algebra, are, 1. The method of subtracabove
tion. 2. The summation of recurring • The Law of Continuation.”-A series
series by the scale of relation. 3. The being proposed, one of the chief ques.
differential method. 4. The method of tions concerning it is to find the law of its
increments. We shall content ourselves continuation. Indeed no universal rule
with an example or two in the first of can be given for this; but it often han. these methods. pens, that the terms of the series, taken “The investigation of series, whose two and two, or three and three, or in sums are known by subtraction." greater numbers, have an obvious and Er. 1. Let 1+ -+-+ simple relation, by which the series may
-t, &c. in be determined and produced indefinite. inf. = S, then . ly. Thus, if I be divided by 1 - x, the 1.1.1 quotient will be a geometrical progres. 5+ ta+ +, &c. in inf. = S-1 sion, viz. 1tr t r + x3, &c. where the succeeding terms are produced by by subtraction, the continual multiplication by x. In like
1.2+2 3+3 +,&c. manner, in other cases of division, other in infa
::=1. progressions are produced. But, in most cases, the relation of the E
. 2. Let
. terms of a series is not constant, as it is in those that arise by division. Yet their inf, = S. Then relation often varies according to a cer. 1.1.1 tain law, which is sometimes obvious on 5tittat, &c. in int.
tát, &c. in inf. = inspection, and sometimes it is found by dividing the successive terms one by an. by subtraction, -to totta other, &c. Thus, in the series
8 , 16 , 128 , 1+ žr+ ir + 33 33 + 315 x4, +, &c. in inf. = 7 &c by dividing the 2d term by the 1st, the 3d by the 2d, the 4th by the 3d, and of 1.372.4" 3.5" 4.6 Tiu
trat, &c. in inf. so on, the quotients will be
3 2 4 6 8
3 +, 5-7, 7-X, ğ x, &c.; and, therefore, the terms may be conti. Ex. 3. Let -2 +3 +34 +, &c. nuel indefinitely, by the successive mul- in inf. = S, tiplication by these fractions. Also in the following series 1+ āx + 40**+ 12823 + 1152 x4, s. &c. by dividing the adjacent terms suc. cessively by each other, the series of by subtraction, quotients is
1.2.3 2.3.4 3.4.5 1 9 25 49 5 t, 20 %, 42 %, 7 x, &c.; or 1.1 3 3 5.5 7.7 2.3*,4 57, 6.7 it', 8. 9 *, &c.
3.4 T3 15 +, &c. in and, therefore, the terms of the series may be continued by the multiplication of these fractions.
SERIES, summation of. We have be. Ex. 4. Let mtmIrti I 2,+ fore seen the method of determining the sums of quantities in arithmetical and...
+mone=S, geometrical progression, but when the m + n-1.min terms increase or decrease, according to an 1 . 1 ether laws, different artifices must be men mtrtm+2, + m + 35 tis.
of patent sistema +, ke in inf.
(ton terms) = 1
subplumose ; receptacle chaffy. There are four species.
SERPHIUM, in botany, a genus of the by subtraction,
Syngenesia Polygamia Segregata class m.m + mtr m + 2r and order. Natural order of Compositæ
Nucamentaceæ Corymbiferæ, Jussieu. + &c. (to n terins) +.
Essential character: calyx imbricate; n+ nr
corolla one-petalled, regular: seed one, hence, t
t, &c. oblong, below the corolla. There are m.m + go mtr.m + 2r
four species, all natives of the Cape of Good Hope.
SERPENTINE, in mineralogy, a spe.
cies of the Talc genus, diviiled by Werand +
t, &c. ner into the common and precious: the m.m + rm + rim + 2r
common is chiefly green, though pass(to n terms) = - 1
ing into various other colours, which
are seldom unitorm. There are gene
in 1 If n be increased without limit,
rally several colours together, and these are arranged in striped, dotted, and
clouded delineations. It occurs masvanishes, and the sum of the series is
sive; insernally it is faintiy glimmering, mr.
which passes into dull, when there are If m=r= 1, we have to
no foreign parucies to give a slight degree of lustre. It is soft, not very brit. tle, and frangible. Feels a little greasy, not very heavy. It is infusible before the blow.pipe without addition. It con
sists of Similar to the method of subtraction is the following, given by De Moivre.
Magnesia - - - - - 23 "Assume a series, whose terms con.
Silica . . . - - - 45 verge to o, involving the powers of an in.
Alumina . . . determinate quantity, r ; call the sum of
Iron . . . the series S, and multiply both sides of
Water - - - .. the equation by a binomial, trinomial, &c. which involves the powers of r, and inva. riable co-efficients; then, if x be so assumed, that the binomial, trinomial, &c. may vanish, and some of the first terms be It is one of the primitive rocks; is transposed, the sim of the remaining se- found in many parts of Germany, lialy, ries is equal to the terms so transposed.” Siberia, in this country, Scotlandi, and the x ri xi .
Shetland islands. It iakes a good polish, Let 1 3ta t,&c.in inf=S.
and is turned into vessels and ornaments Multiplying both sides by r—1, we have of a great variety of shapes. In Upper
Saxony, several hundred people are em
ployei in quarrying, cutting, turning, 24
and polishing the serpentine, which oc. curs in that neighbourhood, and the ar.
ticles into which it is manufactured are . c3
carried all over Germany. The precious or -1+
is found in Silesia.
SERPICULA, in botany, a genus of the
Monoecia Tetrandria class alid order. then,-1+ toatat, &c.= 0; Natural order of Inundatæ. Onagræ. Jus.
sieu. Essential character: male, calyx 1,75+75-zāt, &c. in inf. = 1. four-toot
four-toothed; corolla four-petalled ; fe
male, calyx four-parted; pericarpium SERIOLA, in botany, a genus of the nut tumentose. There are two species, Syngenesia Polygamia Equalis class and viz. S verticillata and S. repens. order. Natural order of Compositæ Se. SERPULA, in natural history, a genus miflosculosæ. Cichoracex, Jussieu. Es- of the Vermes Testacea class and order: gential character : calyx simple; pappus animal a tercbella : shell univalve, gene
raliy adhering to other substances: often neck: the colour is variable, and separaied internally by divisions at uncer- changes, according to the season, age, tain distances. About fifty species have or mode of living, and frequently vabeen enumerated.
nishes, or turns to another in the dead SERPENTES, in natural history, an body: tongue filiform, bifid ; skin reti. order of the Amphibia, containing seven culate.” The distinction between the genera, viz.
poisonous and innoxious serpents is on
ly to be known by an accurate examiAchrochordus Cæcilia
nation of their teeth ; those which are Ampbisbæna Coluber
poisonous being always tubular, and calAnguis
Crotalus culated for the injection of the poisonous Boa
Auid, from a peculiar reservoir commu
nicating with the fang on each side of Serpents are distinguished as footless the head. These teeth or fangs are amphibia : their eggs are connected in a situated in the upper jaw: they are chain: penis frequently double: they frequently accompanied by smaller fangs, breathe ihrough the mouth. The amphi- seemingly intended to supply the place bia were divided by Linnæus into four of the others, if lost by age or accident. orders; viz. Reptilia, Serpentes, Mean. The fangs are situated in a peculiar tes, and Nantes. Of the Meantes or gli bone, so articulated with the rest of ders, which were characterized # breath. the jaw as to elevate or depress them ing by means of gills and lungs together, at the pleasure of the animal: in a quifeet branchiated, and furnished with escent state, they are recumbent, with claws, there was but a single genus, riz. their points directed inwards or backthe siren: this has since been classed wards; but when the animal is inclinwith the reptiles. See Reptilia and ed to use them as weapons of offence, SIREN.
their position is altered by the peculiar The nantes, or swimming amphibia, mechanism of the bone in which they characterized by their having fins, and are rooted, and they become almost per. by breathing by means of lateral gills, pendicular. were afterwards distributed into the or. Serpents in cold and temperate cliders of fishes denominated branchiostigi, mates conceal themselves, during winand chondiropterygii, which have since ter,in cavities, beneath the surface of the been ranked by Dr. Shaw, and others, ground, or in any other convenient places under the general term cartilaginous of retirement, where they become nearfishes. See CHRONDAOPTERIGIOUS. ly or wholly in a state of torpidity. Some
We have thought it right to give this serpents are viviparous, as the rattle. account of the changes in the Linnæan snake, the viper, &c. : while the innoxisystem, which we have generally adopt ous species are oviparous, depositing, ed, having omitted any mention of the as we have observed, their eggs in a facts under the former articles. “Ser kind of chain in any warm and close pents," says the translator of Gmelin, situation, where they are afterwards is are cast naked upon the earth, without batched. The broad undivided laminæ Jimbs, exposed to every injury, but fre. on the bellies of serpents are termed quently armed with a poison the most scuta, and the smaller or divided ones deadly' and horrible: this is contained in beneath the tail are called subcaudal tubular fangs resembling teeth, placed scales, and from these different kinds of without the upper jaw, protruded or re- laminæ the Linnæan genera are charactracted at pleasure, and surrounded terized, with a glandular vesicle, by which this SERRATULA, in botany, saw-wort, a fatal fluid is secreted : but lest this tribe genus of the Syngenesia Polygamia £quishould too much encroach upon the lis class and order. Natural order of Comlimits of other animals, the benevolent positæ Capitatæ. Cinarocephalæ, Jussieu. Author of nature has armed about a Essential character: calyx, subcylindrififth part only in this dreadful manner, cal, imbricate, awnless. There are twenand has ordained that all should cast ty species. their skins, in order to inspire a neces. SERROPALPUS, in natural history, a sary suspicion of the whole. The jaws genus of insects of the order Coleoptera: are dilatable, and not articulate, and the antennæ setaceons; four feelers, unequal; csophagis so lax, that they can swal. the anterior ones longer, deeply serrate, low without any mastication, an ani. composed of four joints, the last joint mal twice as thrice as large as the very large, truncate, compressed, patelli.