Encyclopaedia Britannica [1, 1 ed.]

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ENCYCLOPAEDIA VOLUME

the

BRITANNICA. FIRST.

ENCYCLOPEDIA O R,

Britannica:

A

DICTIONARY C> F

ARTS

and

SCIENCES,

COMPILED UPON A NEW PLAN. IN

WHICH

The different SCIENCES and ARTS are digefted into diftindt Treatifes or Syftems; AND

The various TECHNICAL TERMS, &C. are explained as they occur in the order of the Alphabet. ILLUSTRATED WITH ONE HUNDRED AND SIXTY COPPERPLATES.

By a

SOCIETY

IN

of GENTLEMEN in

THREE

SCOTLAND.

VOLUMES.

V O L.

I.

LONDON: Printed forJoHNDoNALDsoN, Corner of Arundel Street in the Strand, M. DCC.LXXIIL

PREFACE.

T

HE method of conveying knowledge by alphabetical arrangement, has of late years become fo univerfal, that Didlionaries of almoft every branch of literature have been publifhed, and their number ftill continues to increafe. The utility of this method is indeed obvious; and experience has given it the {lamp of approbation. AMONG the various publications of this clafs, thofe of the greateft importance are General Didlionaries of Arts and Sciences: But it is to be regretted, that thefe, of all other kinds of Dictionaries, have hitherto fallen fhortefl of their purpofe, and yielded the lead fatisfaCtion. This is not owing, as fome have imagined, to the nature of the alphabetical plan, as being incompatible with an objeCl fo complex and extenhve as a digeil of the whole body of Arts and Sciences; but to an unaccountable inattention to the proper method of making that digeft, and to the no lefs unaccountable attachment which Compilers feem to have to the prepofterous method of dealing out the Sciences in fragments, according to the various technical terms belonging to each: A method repugnant to the very idea of fcience, which is a connected feries of conclufions deduced from felf-evident or previoufly difeovered principles; which principles, with the relations of the different parts of fcience, cannot be comprehended by the learner, unlefs. they are laid before him in one uninterrupted chain. Dictionaries of Language, of Trade, of Political Geography, Biographical Dictionaries, Claffical Dictionaries, and others of the like nature, whofe fubjeCts comprife a multitude of particulars unconnected with each other, among which there is no reafon for precedence in order, better than what each derives from the letters by which it is expreffed, can fcarce be improperly executed as to the general arrangement; and therefore, if each article be accurate in itfelf, the whole will generally give all the fatisfaCtion expeCted from fuch books. But the fy Hematic nature of the Sciences will not admit of their being difmembered, and having their parts fubjeCted to fuch fortuitous diftributions: Yet they have fuffered this violence in all the Dictionaries hitherto pubhihed; and hence the diffatisfaCtion and complaints of all thofe who have had occahon to confult them.'

IN £ Work of this kind, the Sciences ought to be exhibited entire, or they are exhibited to little purpofe. The abfurdity and inefficacy of the contrary method, which has hitherto obtained, will be evident from an example or two. Suppofe, then, you want to obtain fome knowledge of AGRICULTURE: You reafonably expeCl to be gratified by confulting one or other of thefe Dictionaries ; as in all their Prefaces or Introductions the reader is taught to believe, that they contain the whole circle of Science and Literature, laid down in the molt diflinCt, and explained in the molt familiar, manner. Well, how are you

vi

PREFACE.

you to proceed? The fcience is fcattered through the alphabet under a multitude of words, as Vegetation, Soil, Manure, Tillage, Fallowing, Plough, Drain, Sowing, Marie, Chalk, Clay, Loam, Sand, Inclofure, Hedge, Ditch, Wheat, Barley, Harrow, Seed, Root, ’ • . . y—..^V- . . and x=y—18. whence “ Examp. difference of their fquares, being given, to find the “ quantities.” Suppofe them to be x and y, their fum y—18=^- and ^=90 . . x—y—18=72. /, and the difference of their fquares 2=3300 are to be refolved by Dirett. 3d, as follows. 20^=3300 2 o y= i%- =i65...x=300—y=i35. 2>'—J+3Z Z=20—12=8 . 7+2 z=8 Direct. IV. “ When in one of the given equa“ tions the unknown quantity is of one dimenfion, and 36—3>'—6z=24—2y—2z “ in the other of a higher dimenfion; you mufl find a I2=H-4Z “ value of the unknown quantity from that equation whence y— ^(_ 12-—4Z 8-2z ... . 2d ill value value “ where it is of one dimenfion, and then raife that va“ lue to the power of the unknown quantity in the other “ equation; and by comparing it, fo involved, with the 82Z—12 2Z—128=44Z “ value you deduce from that other equation, you fhall “ obtain an equation that will have only one unknown and z=2 “ quantity, and its powers.” j(=8—2z)=4 That is, when you have two equations of different x (=12—y—z)=6. dimenfions, if you cannot reduce the higher to the fame dimenfion with the lower, you mufl raife the lower to This method is general, and will extend to all equashe fame dtmenfion with the higher. tions that involve 3 unknown quantities: but there are often

A. R B E A L G 93 often eafier and (hotter methods to deduce an equation THEOREM I. involving one unknown quantity only ; which will be beft learned by pra&ice. Suppofe that two equations are given, involving two unknown quantities, as, Examp. VII. rx+j—a C ax+hyzzc Supp. \x+z=zb l dx+ey=f ty+z=c x—a—y then fhail y—af-—dc ~—fu» a—y->rz~b y+z=c where the numerator is the difference of the products of the oppofite coefficients in the orders in whichis not a-to-\-2Z—b-\-c found, and the denominator is the difference of the pro2Z—b+c—a dudts of the oppofite coefficients taken from the orders f —a that involve the two unknown quantities. For, from the firft equation, it is plain, c—by that z 2-f-«2 will be prime to and As furds may be confidered as powers with fradtional “ they are reduced to others of the fame vaconfequantly will be a fraftion in its leaft terms, exponents, lue,that {hall have the fame radical fign, by reducing and can never be equal to an integer number. There- ““ thefe fractional exponents to fractions having the fame fore the fquare of the mixt number is ftill a mixt “ value and a common denominator.” Thus, s/a'zza’1 number, and never an integer. In the fame manner, the / 1 i,, an( cube, biquadrate, or any power of a mixt numbe.r, is V ‘ — ”> l ff— ~~fH, ~= —> and therefore a and ftill a mixt number, and never an integer. It follows from this, that the fquare root of an integer mujl be an a/a, reduced, to the fame radical fign, become 4/^S integer or an incotnmenfirabie. Suppofe that the inteIf you are to reduce 4/3 and if 2 ta ger propofed is B, and that the fquare root of it is lefs and than tf-j-i, but greater than a, then it muft be an in- the fame denominator, confider, 4/3 as equal to fx, the commenfurable; for if it is a commenfurable, let it be 4/2 as equal to 2T, whofe indices reduced to a common where — reprefents any fraction reduced to its denominator, you have 3^ =3^ and 2T —2s, and conleaft terms ; it would follow, that fquared would fequently'4/3=4/33=4/2 7, and 4/2 =4/2* =4/4 ; fo give an integer number B, the contrary of which we that the propofed furds 4/3 and 4/2 are reduced to other have demonftrated. It follows from the laft article, that the fquare roots equal furds 4/27 and 4/4, having a common radical of all numbers but of i, 4, 9, 16, 25, 36, 49, 64, 81, fign.Surds of the fatne rational quantity are multiplied by foo, 121, 144, &c. (which are the fquares of the integer numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, ro, 11, 12, &c.) adding their exponents, and divided by fubtratting them ^ are inconmenfurables: after the fame manner, the cube 3 1i yf1 ~a 3+2 roots of all numbers but of the cubes of 1, 2, 3, 4, 5, thus * a y.\/a—al 6 x=.as, -=z\/a5 •, and 6< 7, 8, 9, See. are incomtnenfurables; and quantities that are to one another in the proportion of fuch num- V^_^_ «_x —a15 —fa'1 bers muft alfo have their fquare roots or cube roots in\/a a* commenfurable. The roots of fuch numbers being incommenfurable are expreffed therefore by placing the proper radical fign o- ma/a'X.f» a—a~-~ mArn ; n—m ; 4/2Y.i * 3/2— 8, Va a/ 10, &c. exprefs numbers ineommenfurabie with unit. 6 4/2 6 Thefe numbers, though they are incommenfurable them- 4/32; I— =4/2. felves with unit, are commenfurable in power with it, becaufe their powers are integers,- that is, multiples If the4/2fards are of different rational quantities, as of unit. They may alfo be commenfurable fometimes with one another, as the 4/ 8, and the 4/ 2, becaufe * and 4/63, and have the fame fign, “ multiply they are to one another as 2 to x : 'And when they have “4/tfthefe rational quantities into one another, or divide. Vop. I. No. 5. 3 f Bb “ them’

G B E R them by one another, andfet the common radical7 Ggn dudls. But when any compound furd is propofed, there c another compound furd which multiplied into it gives over their prodntt o quotient.' Thus, yV* Xv ^5 “ ita rational produtt. Thus, s/ b multiplied by ifa h —'/ gives a—b, and “thethepropofed inveftfgation Va “ which multiplied into furdofwillthatgivefurda */(1*1 -v/^x-Zs^w; m - /«' b a =V “ rational produft,” is made eafy by the following thejWa « ' orems. If the furds have not the fame radical fign, “ reduce them to fuch as lhail have the fame radical fign, and THEOREM I. Generally, ncontinuif you multiply am — bm by z.1 n—xmfyn a a proceed as before,” */aY.ifb zed till the terms be in number equal to the pron 3 • Z 2 XZ4 =2^xZ=2^xZ -Za x4*= duct n mffiallnbe a —; for a — -\-a —See.. . bn~m 6 6 Z8Xi6=Zi28. If the furds have any rational coeffi- Xa>n—hm n cients,x their product or quotient muft be prefixed; thus, an+an-rmb”i+ann—*‘>‘b*m+ —*”•**&, &c. n —a"—mbm—a —*mbxm—aan—imb^>n t Sec.—b 2Z? 5Z6 =ioZi8. The powers of furds are found as the powers of other quantities, “ by multiplying their exponents by the 2in“ dex of the power required;” thus the fquare of y' is II. n 1 THEOREM n 3J t T I2 — nm,—an—See. mulan—m—ar — r»bm-\-a m =2 =Z4; the cube of Z5 =5 =Z JOr you need only, in involving furds, “ raife the quan- tiplied by a n-\-b gives a“z+zb i which is demonllrated “ tity under the radical fign to the power required, con“ tinuing the fame radical fign ; unlefs the index of that as the other. Here the fign of b» is pofitive, when — “ power is equal to the name of the furd, or a multiple odd number. “ of it, and in that cafe the power of the furd becomes is anWhen furd is propofed, “ fuppofe the “ rational.” Evolution is performed “ by dividing the “ index ofanyeachbinomial number equal to m, and let n be the “ fradtion which is the exponent of the furd by the “ leaft integer 1number that is 1meafured by w, then lhail “ name of the root required.” Thus the fquare root “ a'—ms+rjjn-— mbm-^an—^mb >n} Sec. give a compound 3 4 is.3Ztf* or Za 6 4. “ furd, which multiplied into the propofed furd am^fzbm of Za “ will give a rational product.” Thus to find the furd an< 30 e The furd-v/a™* — > f fi^ manner, if a which by \/a—^/b, will give a rational quanpower of any quantity of the fame name with the furd tity. multiplied the leaft number which is meadivides the' quantity under themradical fign without a re- fured byHerey is unit; and let »=i , then TftisdlT mainder, as here a™ divides a x, and 25 the fquare of n in, t —' b r™) &c. _—a1 ^ +«° T T 5 divides 75 the quantity under the fign in Z?5 without b"t-\-a a remainder, then place the root of that power rationally b 3 +Z' =v/a +Z^ t'Zi% which multiplied by »/a— before the fign, and the quotient under the fign, and thus -f-Z^3 — Thus, Z75:=5Z3; Z48=Z3Xi6~4Z3 > Z8i=: ay-\-by, gives a rational product. Here a and , Z 27X3=3^/ 3. and an—m—an—*™bm-\-an—imb*m) Sec. —a^ T-—^ When furds by the lafi: article are reduced to their bl+a^bi—aZ-SbZzzaZ—a'bl+aibl—b^ife— lead: expreffions, if they have the fame irrational part, they are added or fubtra&ed, “ by adding or fubtradHng “ their rational coefficients, and prefixing the fum or “ difference to the common irrational part.” Thus, THEOREM III. Z7J+Z i Z8i+Z24=3Z3 Let amc±zbl be multiplied by an—m—^an—‘lml1l-!ran— 3 nt 3 48=5Z3+4Z3-9Z3 3 n +j2Z3=sZ3Compound furds are fuch as confilt of two or more f n~Ambil, and the product lhall give an^czhm-l . joined together. The fimple furds are commenfurable b't^za in power, and by being multiplied into themfelves give ‘ therefore n muft be taken the leaft integer that ffiall at length rational quantities; yet compound furds multiplied into themfelves commonly give Itill irrational pro- “ give — alfo an integer. Dem.

98

L G b B R A. 99 mb * lz+zan—*mi>51 .... arifing ’will be that of the propofed quantity divided by the binomial furd, exprejfed in its leaf terms. Thus, Xamdtzbl .... &c. 77^- = ^^ = *^; x/6 »f 42 + \/i8 /->>■'—j—/7^—— ^tllh I/) &C. V 7 — V' 3~ 4 -j-gti—mhl. & r±zbm When the fquare root of a furd is required, it may be found nearly by extrading the root of a rational quantity that approximates to its value. Thus to find the fquare root of 3~|-2V'2, we firft calculate y'a^i, 41421; * * * zi=im an and therefore 3+24/2=5, 82842, whofe root is found nl n The fign of is pofitive only when m is an odd num- to be nearly 2, 41421 : So that 4/3 + 2 4/2 is nearly ber, and the binomial propofed is am+bK 2,41421. But fometimes we maybe able to exprefs If any binomial furd is propofed whofe two numbers the roots of furds exa&ly by other furds ; as in this exhave different indices, let thefe be m and /, and take n ample the fquare root of 3+24/2 isT+4/2^ forT+Vlx equal to the leaft integer number that is meafure4 by m 1 + 4/2=1+24/ 2+2=3+2 4/ 2. to know when and how this may be found, and by ; and an~~mz^ian~-'tmbl-\-a>i—'imb'~h^:a>l~im let Tnus order fuppofe that x-\-y is a binomial furd, whofe fquare b &c. (hall give a compound furd, which multi- will be x*+>’*+2xy -• If x andy are quadratic furds then plied by the propofed a^ztzbl fliall give a rational x3+>'2 will be rational, and2 2xyz irrational; fo that 2xy ftiall always be lefs than x -\-y , becaufe the difference produft. Thus \/ a — a/ b being given, fuppofe 2 2 is x +j —2xy—x—y which is always pofitive. Sup/=y, and =1, therefore you have «=3, and pofe that a propofed furd confiding of a rational part 2A, n z n! llJ A ,r, ll 1 1 T and an irrational part B, coincides this, then x + 2 an—m-\.2 — mbl-\-an—' b‘ b+a t-a”— ' b - r8cc. —cr l -j- ^ =A and xy B : Therefore bywith what was faid of o 0 quations. Chap. 12th, a^'b^+at-h^+al ~ ' ^bl+a b^=a^+a b\ a1 / 2 ~\-a\;b^-\-ab-\-a^ b2, "h v2=A—x2=—and therefore, :b^-\-b^—*y a> ~T X'v/^"hv a 4x 2 4 ab-\~^/aXy'^47l*V,bs ^a^x/a~{ a2 y.\^b aX\/b2 4" Ax —x< = ?* and x —Ax’i+ % ab-\-b\/aX>s/b-\-bXi 7 r >/T , which multiplied by the y'*— j gives aP—bm«i =:=*23—b2. from whence we have x A+4/A =B x/b 2 By thefe theorems any binomial furd whatfoever be- A—4/A —B Therefore when a quantity partly ing given, you may find a furd which multiplied by it lhall give a rational product. rational, partly irrational, is propofed to have its root Suppofe that a binomial furd was to be divided by an- extracted, call the rational part A, the irrational B, and thefquare of the great ef member of the root Jhallbe other, as v'ao-f-v'is, by y'j —^3, the quotient may + 4/12 But it may be exan( tlle uare tfie e er art be expreffed by 4/20 * ft °f ^ JJ P V 5—V preffed in a more Ample form 3 by multiplying both nume- be — • And as often as the fquare root of rator and denominator by that furd •which, multiplied 2 2 into the denominator, gives a rational produd: Thus A —B can be extracted, the fquare root of the propofed binomial furd may be expreffed itfelf as a bi4/20+4/ 12 _ 4/20 + 4/12 4/ 5 + 4/3 _ nomial furd. For example, if 3 2+ 24/2 is propo^5—4/3 v/5—\/3 V 5 +4/3 fed, then A=3, 6 = 24/2 and A —B2=9 —8=1. 4/100 + 2 4/60 +6 _ 16 T2 4/60-=8+24/15* B 2 5“ when any quantity is divided by a bino- Therefore x* = In general, 2 ~~ ^ _ 2, and Jj =1 mial furd, as amz±=b’, where m and / reprefent any = 1. Therefore x+7=i +4/2. fractions whatfoever, take n the leaf integer number that is me afared by m11andm ultiply1 both numerator To find the fquare root of—1+4/—8, fuppofe A = and denominator by a — -j-a” “"’^b +an — 3mb2', &c. —1, B=a/—8, fo that A2—B*=9 and ^K/AT— and the denominator of the produd will become rational, and equal to an —- b ni ; then divide all the members of --*+3— = j, and the numerator by this rational quantity, and the quote therefore the root required is 1+4/—2. But Dem. afn—mz£ian—1

A

5

BRA. But though X and are not quadratic fards or ropts be — —- , if H of integers, if they are the roots of like furds, as if. they, “f< t, fo lhall the root required are equal to \/mv z and where m and « are the c root of A=tB can be extraAed.v'Ql Examp. I. Thus to find the cube root of v/968 integers, then h^rn-rn'A*/ z and -f o — K/ m n z ; -f-25, have A*—B*=343-, whofe divifors are 7, 7, 7, ■j , _ A + — *** --. whencewe«=7, A*—B*=wand Q=i. Further, that / 1 is, v 968+25 is a little mpre than 56, whofe nearelt - a= m\fz, = A—V’A — cube root is 4. Wherefore r—4. Again, dividing ^968 and >:-H ■/. a^/ha^/z. The part A here by its greateft rational divifor, werhave A\/ Qz=22\/2,n eaiily diftir/guHhes itfelf from B by its being greater. and the radical part and ~b T~ r ~2^~ > i If x 2and jy are equal to and thenx*-}- neareft integers, is 2 =/. And iaftly,% 0 ts~2\/2 > =wv'z+»a//+2 V -/r:t. So that if z or / the 1 1 / be not multiples, one of the other, or of fome number */f 1' '—n—\, and v/Qj= v 1 =1. Whence 2-v/2+t that meafares them both by a fquare number, then v/iil is the root, whofe cube, upon trial, I find to be ^968 + 25. A itfelf be a binomial. Let X"\-y~\~z exprefs any trinomial furd, its fquare xJ-{~ II. zTo find the cube root of 68—v'4374,' y?-{~2*+2xy+2xz4'2^z may be fuppofed equal to A+B weExamp. as before. But rather multiply any two radicals as 2xy have A*'—B =250, whofe divifors are 5, 5, 5, 02. by 2xz, andz divide by the third ayz which gives the Thence «=jX2=xo, and Qt=4, and V'A+BXV'^. * quotient 2* rational, and double the fquare of the furd V'68+^4774X2 is nearly 7=7- again Ay/Q^ c r 68X ; x required. The fame rule ferves when there are four r 7 quantities -\-z1 -f-i1 +2xy-\-2xs-\-2xz-{-2yz-j-2yfi‘ :'zr, multiply sxr by zxs, and the produdt divided y/4=X36X-v/i, that is, . and ±~T, or - -t-T 2 by 2sy gives 2x° a rational quotient, half the fquare of is nearly //=4, —n—^/S, *c =4=/. 6 } Therefore 2X. In like manner 2xyX2yz~^yzxz1 which divided by and y^Q^v^4—-y^2, whence the root to be tried i» 2xz another member gives 2y*, a rational quote, the half of the fquare of 2y. In the fame manner z and s 4 — y/6 may be found; and theira fum x+y+Z+^> the fquare root of the feptinomial x +d2+2S~t'J'2,4*25y'+2.vr-}-2xz+ V* 2yz-\-2ys, difeovered. XIV. Of the Genesis ««£/ResoluFor example, to find the fquare root of 10+V/24+ Chap. 1 o ft. of Equat i ONS in general; and the 4/40+^60; I try which I find to be tnumber of Roots an Equation of any Degree ^/i6=4, the half of the fquare root of the double of may have. which, viz. 4:Xv'3=V2, is one=member of the fquare After the fame manner, as the higher powers are root required; next the half of the produced by the multiplication of the lower powers of fquare root of the double of which is 4/3 another mem- the fame root, equations of fuperior orders are generated by the multiplication of equations of inferior orders ber of the rootr required; laltly, =10, which involving the fame unknown quantity. And “ an equaany dimenfion may be confidered as produced gives f° the third member of the root required : “ tiontheof multiplication of as many Ample equations as from which we conclude,/ that/ the ,fquare root of 10+ ““ by ■*/24+4/40-\-6o is v 2+v 3'fV 5 ; and trying, you it has dimenfions, or of any other equations v/hatif the fum of their dimenfions is equal to the find it fucceeds, fince multiplied by itfelf it gives the pro- ““ foever, dimenfion of that equation.” Thus, any cubic equapofed quadrinomial. , For extra&ing the higher roots of a binomial, whofe tion may be conceived as generated by the multiplication of three fimple equations, or of one quadratic and one two members being fquared are commenfurable numbers, fimple equation. A biquadratic is generated by the there is the following of four firii[>le equations, or of t; and “ the fquare of the fum of any quantities is and mayroots be extended to all kinds of equations. “ always equal to the fum of their fqirares added to In quadratic equations, the two roots are either both “ double the produdts that can.be made by multiplying pofitive, as in this1 “ any two of them,” therefore p'—B+ig,1 andZJconfe, (x—aXx—5—) x —ax-i-ab—o, quently 3 — 2y. For example, a-\rb-\-c\ —a rb' -^\r — bx cl-)r2abr2ac-\-2bc; p1=.B-\r2q. An&u-\-b-\-c-{-dV where there are two changes of the figns: Or they are both negative, as in this —a'1 -^b1l -\-cz-Yd1 ■\-2Y.ab-\-ac-\-aci-\-bc-)rbd-\-cd, that is, again, p' =B-\-2q, or B—p71—2q. And fo for any other number of quantities. In general therefore, “ B the (r+ix«+i=)«-Hh,^+i,j=0i “ fum of the fquares of the roots may always be found where there is not any change of the figns : Or there is “ by fubtradting 2y from p1 the quantities p and q one pofitive and one negative, as in being always known, fince they are the coefficients in the (mXx+A=) J _ propofed equation. “ The fum of the cubes of the roots of any equation is equal to y>3—S/y+S'S or to' Bp—-/’y+gr.” For where there is necefiarily one change of the figns; beB—qY.p gives always the excefs of the fum of the cubes caufe the firft term is pofitive, and the laft negative, and of any quantities above the triple fum of the products there can be but one change whether the 2d term be 4that canz be zmadez by multiplying any three of them. or —. the rule given in the laft paragraph extends Thus, a -\-b -\-c —ab—ac—(~B — qXp)— Therefore quadratic equations. «3+^3+c3—%abc. Therefore if the fum of the cubes to Inall cubic equations, may be, is called C, then ffiall% B—qXp—C—jr, and C—Bp—qp i°. ,A11 pofitive, as thein roots this, x—aXx—bXx—£-=o, +3'= (becaufe B—p —2y)=/>3—B/y+Br. the figns are alternately -f- and —, as appears After the fame manner, if D be the fum of the 4th infromwhich and there are three changes of the figns. powers of the roots, you will find that D~pC—qB-\-pr 2°.theThetable; —qr, and if E be the fum of the 5th powers, then ffiall x-\~aXx-j~bXxroots s may be all negative, as in the equation E-pD—qC-\-rB—ps^r^t. And after the fame manner the figns. Or,■ rc-—c> where there can be no change of the fum of any powers of the roots may be found; the 0 progreffion of thefe expreffions of the fum of the powers 3 . There m^y be two pofitiye roots and one negative, being obvious. as in the equation x—aXx—lXx-^-c=o • which gives As for the figns of the terms of the equation produx3'—a ^ 1 +ab~) ced, it appears, from infpe&ion, that the figns of all the —b V x —ac v x-^-abc—o. terms in any equation in the table are alternately + and 4-^3 —bej —: thefe equations are generated by multiplying continually x—a, x—b, x—c, x—d. See. by one another. Here there mufl be two changes of the figns; becaufe if The firft term is always fome pure power of x, and is is greater than c, the fecond term mufi be negative, pofitive; the fecond is a power of x multiplied by the its coefficient being —a—b-{-c. quantities—a, —b, •—c, &c. And fince thefe are all if a-^b is lefs than c, then the third term mufl negative, that term mull therefore be negative. The be And negative, its coefficient -rah—ac~—bc(ab-—cXa-\-b) * third term has the products of any two of thefe quantiin that cafe negative. And there-cannot poffibly be ties (—a, —b, —c, Sec.) for its coefficient; which pro- being three changes of the figns,. the firft and laft terms having duAs are all .pofitive, becaufe —X— gives +. For the fign. like reafon, the next coefficient, confiding of alh the the40fame may be one pofitive root and two negative, produAs made by multiplying any three of thefe quanti- as in .theThere equation x+uXx-f Z’Xx—c=o, which gives ties muft be negative, and the next pofitive. So that *34-0 x1—ac 4-*0v x-—alc—o. the coefficients, in this cafe, will be pofitive and negative by turns. But, “ in this cafe the roots are all pofitive fince x—a, x—b, xzzc, x—d, x~e. See. are the aflumed fimple equations. It is plain then, that “ when all the “ roots are pofitive, the figns are alternately + and —.” But if the roots are all negative, then x-J-aXx-HX * Becaiife the re{Jangle aXb is lefs than the fquare a-fbXal-b, and therefore rsuch lefs than afbXe.

B R A. ic4 A li O E where there muft: be. always one change of the figns, “ another that {hall have its roots greater or lefs than roots of the propofed equation by fome given diffince the firff term is pofitive and the laft negative. And ““ the ference.” there can be but one change of the figns, lince if the 2d Let the equation propofed be the cubic x3—px1Jrqx term is negative, or a-^-b lefs than c, the third muft be —r=o. And let it be required to transform it into anonegative alfo, fo that there will be but one change of the ther equation roots fhall be lefs than the roots of figns. Or, if the fecond term is affirmative, whatever this equation bywhofe given difference (e), that is, fupthe third term is, there will be but one change of the pofe_>=x—e, andfome x=>+c ; then, inftead of figns. It appears therefore, in general, that in cubic e- x and its powers,confequently fubftitute j-\-e and its powers, and quations, there are as many affirmative roofs as there are there will arile this new equation. changes of the figns of the terms of the equation. There are feveral confe&aries of what has been already demonftrated, that are of ufe in difcovering the —py '—ipey—pe1 { roofs of equations. But before we proceed to that, it + qy+q? ^ ’ will be convenient to explain fome transformations of equations, by which they may often be rendered more roots are lefs than the roots of the preceding efirnple, and the inveftigation of their roots more eafy. whofe quation by the difference (