Page:Cyclopaedia, Chambers - Supplement, Volume 1.djvu/668

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mild nature, others painful, malignant, and meaning to be cancerous. Surgeons diftinguifh them alfo into the larger and the fmaller, and the harder and fofter ; they are fometrmes fupported by a broad, fometimes by a narrow root inele excrefccncesnot only deform the mouth, but hurt the fpeech and maftication, and are always beft fpecdily extirpated. When they grow to a fmall root, they are beft extirpated by ligature, or tying a thread firmly about the root ; but when the root is broad, they are beft eaten off by caultics, as the oil of tartar, fpirit of fal armorliac, and the like, and when thefe mild ones prove ineffeaual. It is beft not to have re- courfc to ftronger, but to raife the tubercle with the hook or pliers, and carefully extirpate it with the knife, not cutting off the gums, fo as to uncover the roots of the teeth, which would be probably attended with a caries of the bone. T he blood (hould be permitted to flow from thefe for fome time, but if the haimorrhage continues too long, the patient mud wafh his mouth with hot red wine, with allum diftolved in it, repeating it till it ceafes. And afterwards tinaure of myrrh, and honey of rofes, fhould be ufed till the parts are pcrfea y well; and, if the tubercles fhould fprout up again, they fhould be taken down in time by the fame methods. Heijler\ bur- gery, p. 462. See alfo Schelhammer de Epulide.

EPULONES, in antiquity, a clafs, or order of priefts, who had the management of feafts, games, and feftal faenfices, under their infpeaion. Pitifc. Lex. Ant. in voc.

EQUANT Circle, in the old aftronomy, a circle defenbed on the center of the Equant. Its principal ufe is to find the va- riation of the firft inequality.

EQUATED Bodies. On Gunter's k&ot there are fometimes placed two lines, anfwering one another, and called the lines of equated Indies : They lie between the lines of lohds and iuperficies, and arc noted with the letters D, I, C, S, O, T, for Dodecahedron, Icofihedron, Cube, Sphere, Oaahedron, and Tetrahedron.

The ufe of thefe lines are, I. When the diameter of the fphere is given, to find the fides of the five regular bodies, feverally equal to that fphere. 2. From the fide of any of the bodies being given, to find the diameter of the fphere, and the fides of the other bodies, which fhall be equal feverally to the firft body given.

If the fphere be firft given, take its diameter, and apply it over in the feaor in the points S, S : If any of the bodies he firft given, apply the fide of it over in its proper points : bo the parallels taken from between the points of the other bo- dies, fhall be the fides of thofe bodies, equal feverally to the firft body given.

EQUATION (6>/.)— Cai/iEquATlON. The fecond terms of a cubic Equalim being taken away, they may be all reduced to this form : xs -f a x + h = o. Where

This rule is commonly afcribed to Cardan, and from him has been called Cardan's Rule ; but it is faid by fome % that Tartalea was the inventor, and, by others ", bcipio Ferreus, to whom, it is faid, Cardan himfclf afcribes the invention.— [» Lagtii Elem. d'Arithmet. & d'AIgebr. p. 479. b Wolf. Elem. Mathef. Tom. 1. p. 336. Edit. 2. and Saunderfon's Algebra, p. 702.] When, in a cubic Equation, xi — ax-\-l = c, a is negative,
 * = ]/—.yb+ v/iW-KV 3 + v / — 'J ~ s/'* bl > + i'-l a ' i -

the expreffion y' i u b -f- ^ 7 a'^ will be transformed into s/iob — 7li ai, which becomes impoffible, or imaginary, wneni' 7 a J is greater than %hb; for */$bb — 1 ' 7 a 3 will then be the fquare root of a negative quantity, which is im- poffible. And yet, in this cafe, the root .v may be a real quantity. But algehrifts have not yet been able to find a real expreffion of its value. This cafe is called the irreducible, ireduclible, or impracticable cafe. See Irreducible. The irreducible cafe may be folved by the trifeaion of an arc, for which the reader may confult Saunderfon's algebra, p. 713.

This method requires a table of fines, and if fuch be not at hand, we may have recourfe to Dr. Halley's univerfal me- thod of extracting tile roots of Equations, in the philofophi- cal tranfaaions, or in Lowthorp's abridgment, Vol. r. C. 1. $. xx. Vid. infra.

There are feveral other methods, of extraaing the roots of cubic Equations, extant in books of Algebra. Mr. Cotes obferves, in his logometria, p. 29, that the folu- tion of all cubic Equations depends either upon the trifeaion of a ratio, or of an angle. See this method explained and demonftrated in Saunderfon's Algebra, p. 718. Biquadratic Equation. — Des Cartes gave a method of re- ducing biquadratic Equations to cubic. This method is ex- plained in moft treatifes of algebra, fince his time ; and very clearly by the marquis de l'Holbital in his conic lections, Art. 356. See alfo Maclaurin's Algebra, p. 228- feq. Hence a biquadratic Equation being propofed, it may be folved, by reducing it firft to a cubic Equation ; and then finding the roots of this new Equation, by the trifeaion of an angle, or of a ratio. Thefe roots, fo found, having a given


 * I

relation to the roofs of the propofed biquadratic, thefe will alfo be given. Equations of higher Degrees. — We have no univerfal rule to exprefs, algebraically, the roots of Equations, higher than biquadratic. But fuch Equations may be folved university in numbers, by Dr. Halley's method of approximation. This method proceeds by afluming the root defired, nearly, true to one or two places ; which may be done by a geome- trical conilrucYion, or by a few trials ; and then c;rrcctmg the afiumption, by comparing the difference between the true root 2nd the aflumed, by means of a new Equation, whole root is that difference, and which he ihews how to form front the Equation propofed, by ftibftitutk-n of the value of the root fought, partly in known, and partly in unknown terms.

The Doctor's method will eafily be understood by an exam- ple. See p. 9.

Dr. Brook Taylor found that this method was capable of a greater degree of generality, and that it was applicable, not only to Equations of the common form, that is, to fuch as confiit of terms, wheFein the powers of the root are pofitive and integral, without any radical fign, but alfo to all expref- fions in general, wherein any thing is propofed as given, which by any known method might be computed j if, vice verfa, the root were confidered as given. Such are all radi- cal expreiTions of binomials, trinomials, or of any other no- rmal, which may be computed by the root given, at leaft, by logarithms, whatever be the index of the power of the no- rmal ; as likewife expreiTions of logarithms, of arches by the fines or tangents, of arcs of curves by the abfciiHe, or any other fluents, or roots of fluxional Equations. See Phil. Tranf. N°. 352. Jws's Abridg. Vol. 4. c. 1. §. xvii. The method of finding the roots of Equations by approxima- tion, extends to all kinds of Equations^ and though it be not accurately, yet it is practically true j as it gives the value fought, to a very great degree of ex£C~tnefs, nay, to any affignablc degree, if any one will take the pains of computa- tion.

The method of Sir Ifaac Newton, and of Mr. Raphfon, is, in effect, the fame as that mentioned in the Cyclo- paedia, under the head Approximation. See further in Newt. Meth, of Flux, and Mr. Colfon's Comment, p. 186. But Dr. Halley's Method is more expeditious in practice. See Phil. Tranf. N°. 210. or Lowthorp's Abridg. Vol. 1. c. 1. §. xx. The principles of thefe methods may alfo be found in Mr. Maclaurin's algebra, part. 2. c. 9. or in Mr. Simpfon's algebra, p. 147, feq. who gives feveral for- mulae for the approximating to the roots of Equations. This gentleman has alfo another method for this purpofe, which we {hall mention further on.

Mr. Daniel Bernoulli, in the Afta Petrspalitana, Tom. 3. p. 92. feq. has given a new and ingenious method of ap- proximating to the roots of Equations, without any previous trials. The method is deduced from the nature of recurring feries. The book wherein it is publifhed being in few peo- ple's hands, we mall here give a full account of this me- thod. Let the propofed Equation be put into this form:

1 = a x -f- bx x -f- cx> -f- e ^+-}-, &c. Then form a feries, beginning with as many arbitrary terms as the Equation has dimensions, but with this condition, that if A, B, C, D, E, feffl denote the terms following each other in direct order, then muff E — a D + bC-\-cB -- e A +> fcfa an ^ l £t tnere De two proximate terms, M, N, in this feries fufficiently continued ; then will the antecedent M, divided by the confequent N, be nearly equal to the required root.

For inftance, let 1 = — ix + 5^* — 4*3 ~f- **. Form a fe- ries, beginning with four arbitrary numbers, 1. 1 . 1 . 1 (as the propofed Equation is of four dimenuons) and let a new term always be formed from the double of the laft preceding, taken negatively, more the quintuple of the penultimate, lei's the quadruple of the antepenultimate, more the penante- penultimate. This feries will be 1. 1 . 1, 1. o. 2. — 7 . 25 . — 93 • 34 1 • — 12 54> ^ c - &n ^ the approximated root of the Equation propofed will be nearly = — AV*- Again, let 1 — x-\- 2xx+ 3*3 -f 4*4 + 5* s - Form a feries, beginning with five arbitrary terms, 1. 1 . 1 - 1 . 1 . 15 . 29 . 71 . 183 . 477 . 1239 • 3*7*5 ^ c - Here ^~A or A will be nearly equal to one of the roots of the Equation. The root here found is the lcaft of the foots, without attend- ing to the figns ; that is, the root fo determined is that which is lead diftant from nothing.

To find the greateft root of a propofed Equation, let it be difpofed in the following manner :

x'" = ax n -— l + bx tn — z -f- cx K — 3 -}- d.

Then form a feries, beginning with as many arbitrary terms as there are dimenlions in the Equation, fo that if A, B, C, D, E, £sfr. denote fo many contiguous terms directly follow- ing each other in the feries, E muft be every where = flD + bC -f- cB -f- dA, Ufe. Laftly, let there be two proximate teroiSj M and N, in this feries, fufficiently continued, then

will