This appears from the property of the absolute term, and from this obvious maxim, that if a number of quantities be multiplied toge

298+12ƒ+6ƒ3+ƒ3 -2x=-4-2f

-5-5

23—2x−5 =−1÷10ƒ+6ƒ2+ƒ3=0;

ƒ3+ƒ2+10ƒ—1 = 0,

which is still a cubic; but since, from the nature of our substitution, must be less than 1, the cube of it may be omitted as inconsiderable, and this will give

ther, and the sign of an odd num-therefore, to find ƒ we have the ber of them be changed, the sign equation of the product is changed. For when a positive number is substi tuted for 2, the result is the absolute term of an equation, whose roots are less than the roots of the given equation by that quantity. If the result has the same sign as the given absolute term, then from the property of this term, either none, or an even number only, have had their signs changed by the transformation; but if the result has an opposite sign to that of the given absolute term, the sign of an odd number of the positive roots must have been changed.

In the first case, then, the quantity substituted must have been either greater than each of an even number of the positive roots of the given equation, or less than any of them; in the second case, it must have been greater than each of an odd number of the positive roots. An odd number of the positive roots must, therefore, lie between them, when they give results with opposite signs. The

same observation is to be extended to the substitution of negative quantities and the negative roots. Let it, for example, be proposed to find an approximate value of x in the equation

23-2x-5=0.

Here we easily see, that one root is between 2 and 3; for these numbers being substituted for a, will give, the one a positive, and the other a negative result. The root, therefore, is greater than 2 and less than 3, and we may, therefore, write a=2+f, or a=3-f; and by finding f, or an approximate value of it, in either case, will obviously also give an approximate value of x, by adding the value of f to 2, in the first case, and subtracting it from 3 in the second case.

Suppose a 2+, and subtituting this value for a in the proposed equation, we have,

36

–5+√31
6

;

which value off, being added to 2, will give an approximate value of x.

But generally, when ƒ is less than unity, all the higher powers of f may be omitted, without any very sensible error; which being done in the present instance, will give 10f-1=0, or f=0·1, and therefore r=21 nearly.

Now as f=0·1 nearly,

'let

f=1+g, and substituting this value for ƒ in the preceding equation, we have

fs 0·001 0·03g + 0·3 g2 + g3 6/2006 + 1·2g+ 6g 10f1. +10g

in general, is by the following for- root, we are rather under than mula, first published by Barlow: over the real state of convergency; Let xn+axn-1+ bxn―2+cxn-3+ for generally, if the first supposi &c.=w, by any equation; tion be a simple integer, the first and xn+axin-1+bxin-2+cx/n-8+ approximation may be obtained to &c. v, al being an approximate value of 2; then

(w-v)2x!

(n 1)w + (n + 1) xn+ (n-1)ux-1+(n−3)bxn−x+ &c.;

(w-v)2x

three places, which, though it be not always correct in the last place, yet by employing it as a new approximation, the next value of a will, in most cases, be found true to six or seven fignres, which is a degree of convergency that (n−1) v + (n+1) + cannot be obtained by any other (n−1)ax/n—1+(n−3)bxn−2+ &c. rule that we are acquainted with, The first formula being applica-at the same time that the operation ble to the cases in which is is much more simple than by any greater than unity, and the second other rule whatever. when is less than unity.

or, x=x2+

These formulæ are general for equations of all dimensions, but when reduced to particular cases, they become much more simple: thus, reducing them both into one, we have for

(1.) Equations of the third Degree,
(w-v)xl
x=x+

w or v+2x13+ax/2

An example will sufficiently illustrate the preceding remarks. Example. Find the value of x in the cubic equation

x3+3x2+3x= 130; Assume 4, then by formula (1) x'3= 64 2x'3 128 3.x'= 48 ax^2 W =130

3x'

= 12

48

306 deno minator

306) 24 (4.08 nearly. Therefore now r=4.08.

Hence, adopting this new value,

we have

bxs- dx

3x'2

3.x == 12.24

315-77) 39372 ('001247. Whence, 4'08-0012474078753 Answer.

And in the same way we might find the particular formula answering to an equation of any other degree. Application of the preceding For-w-v= mula.-In order to this it may be observed, that a is an approximate value of x, found first by trial, as near the true root as possible, which in all cases ought to be true to the nearest integer. Then substituting this value of x', a nearer approximate value of will be obtained, which will in all cases double the number of figures. Then, considering this as a new value of x', another still nearer value of a will be determined; and

so on.

In stating the degree of approximation to be that of doubling the number of figures in the assumed

This root is true in the seventh figure; and, in the same manner, the root of any other equation may be determined.

APSES, in Astronomy, are the two points in the orbits of the pla nets where they are at their greatest and less distance from the sun or earth; the former being called the higher apsis, and the latter the lower apsis: but the higher apsis is more commonly called the

pole are measured by arcs of the meridian, and the longitude by an arch of a parallel circle.

ARCH of Progression, or Direction, is an arch of the ecliptic, which a planet seems to pass over, when its motion is direct, or according to the order of the signs.

ARCH of Retrogradation, is an arch of the ecliptic described when a planet is retrograde, or moves contrary to the order of the signs. ARCH of Position, or Angle of Position, is the same with the horary angle.

nate, the abscissa, and a = the thickness at the keystone; hence, when a, h, and r, are any given numbers, a table is formed for the corresponding values of x and y, by means of which the curve is constructed for any particular occasion.

And in a similar manner, if the curve of the intrados, and the depth of the key-stone be given, the equation of the extrados may be computed. Thus, for example, in the case where the intrados is a circle, the equation, readily deduced from the above construction, is

y√ (a2 + b2-y3)

√(32—62)

height above the centre = y. It will be found that a line at right angles to the perpendicular, and meeting it at the point where √ (a + m) — n falls, will be an assymptote to the extrados. Hence the extrados, in the case of a circular arch, is a curve of the fourth order, very much resembling the conchoid of Nico. medes, and that it coincides very nearly with the curve, in which a road is usually carried over a bridge. This holds good, of whatever portion of the circle the arch is supposed to consist.

ARCH of Vision, is the sun's depth below the horizon at which a planet or star, before hid in his rays, begins to appear. This arch is different for different planets; being for where a=the height; and (putting Mercury 10°, Venus 5°, Mars 114°, n=half the span, and m=the height Jupiter 10°, Saturn 11; a star of of the keystone)(a+m)TM —n2= b; the 1st magnitude 12°, 2d magnitude the distance of the point from the 13°, &c. This angle is not, how-centre of the arch, and its ever, constant in all cases for the same planet, but varies a little with the latitude and declination, &c. With respect to Venus, it is sometimes reduced to 0, as she is at times visible when the sun is some degrees above the horizon. ARCH of Equilibrium, in the Theory of Bridges, is that which is in equilibrio in all its parts, and therefore equally strong through. out, having no tendency to break in one part more than another. It is not of any determinate curve, but varies according to the figure of the extrados; every different extrados requiring a particular intrados, so that the thickness in every part may be proportional to the pressure.If the arch were equally thick throughout, the catenary curve would be the arch of equilibration; but as this can selAREA, in Geometry, is the superdom or never happen, it is a mis-ficial measure or surface of any taken idea to suppose this curve figure. The areas of similar plane the best in all cases. It therefore figures are to each other as the appears, that when the upper side of the wall is a straight horizontal square of their like sides, or other lineal dimensions. line, the equation of the curve is thus expressed:

log.

a+x+ √(2ax+x2)

a

ARCTIC Circle, in Astronomy, a small circle of the sphere parallel to the equator, and distant 23° 28/ from the arctic or northern pole.

ARCTIC Pole, the northern pole of the world.

AREOMETER, an instrument for measuring the density or gravity of fluids. It is now commonly made of glass; consisting of a round hollow ball, which terminates in a long slender neck, hermetically sealed at top; there being first as the much mercury put into it, as will height of the arch, y the ordi-serve to balance or keep it swim

y=hX

a+r+√(2ar+r2)

log.

a

where h half the span,

ming in an erect position. The of the different nations among stem or neck is divided into de- whom it has been used. Its origin, grees or parts, which are num-like that of all sciences, is involv. bered, to show the specific gravity by the depth of its descent.

ARGUMENT, in Astronomy, is in general a quantity upon which another quantity or equation depends; or it is an arch, whereby we seek another unknown arch, bearing some proportion to the first: hence

ed in obscurity; nor is it fully ascertained to whom we are indebted for the improved system now in use.

Arithmetic is divided into various kinds, according to the nature of the numbers that form the subject of it; but the most simple form, and that which is the root or founARGUMENT of Inclination, or dation of all the others, is the arithARGUMENT of Latitude, of any metic of abstract or simple numplanet, is an arch of a planet's bers. A number is called abstract orbit, intercepted between the as- when it merely answers to the cending node and the place of the question, "how many?" and has planet from the sun, numbered no allusion to the value of the said according to the succession of the many, and no reference to things signs. of any kind, but which admits of Menstrual ARGUMENT of Lati-having the name and value of any tude, is the distance of the moon's kind of things whatever applied true place from the sun's true to it. In this sense, one is the place; by which is found the smallest and simplest number that quantity of the real obscuration in can be mentioned; and the other eclipses. numbers proceed by constant_adAnnual ARGUMENT of the Moon's ditions of one, without limit. This Apogee, or simply Annual Argu-being the case, the first thing rement, is the distance of the sun's quisite for the formation of an arithplace from the place of the moon's metical language, is the invention apogee; that is, the arc of the of a limited number of original ecliptic comprised between those names, which shall comprehend a two places. great and almost endless variety of ARGUMENT of the Parallax, de-numbers. For this purpose, what notes the effect it produces on an observation, and which serves for determining the true quantity of the horizontal parallax.

is usually called a "scale of numbers," becomes necessary, and the limits of this scale are the powers of any number, m which is taken ARGUMENT of the Equation of as the modulus or root. Thus mo, the Centre, is the anomaly, or dis-m1, m2, m3, m1, will form five ranks tance, from the apogee or aphelion; or places in the scale, any number because this equation is calculated whatever being taken for m; and in an elliptic orbit for every de- the scale will receive its name acgree of anomaly, and varies accord-cording to the number so taken : ing to the variation of the ano-thus,

maly.

ARIES, or the Ram, in Astronomy, the first of the old 12 signs of the zodiac: it is marked r, in imitation of a ram's head. The sun enters this sign generally about the 20th of March.

If m 2, the scale is binary.
If m 3, the scale is ternary.
It m7, the scale is senary.
It m=10, the scale is denary.
lim 12, the scale is duode

nary.

When the lower or any of the ARITHMETIC, that part of ma-intermediate places is blank, it is thematical science which treats of supplied by a character (0) having the nature and properties of num-no separate value, and another bers, the representing of them by power of the modulus is given by proper symbols, and the application adding this character. of them to the business of calcula- Whatever may be the value of tion. This science has undergonem, mo is always 1. Hence, the various improvements, and has par- first five places in each of the taken of the genius and language above scales, expressed in com

52