M

ALGEBRA.

been greater.thaa each of ail odd number of the positive roots, I.to odd number of the positive roots, therefore, must lit*-between |them when they give results with opposite sign-. The same ob-servation is to he extended to the substitution of negative quan-tities and the negative roots.

from this lemma, bv means of trials, it will not be difficult tofind the nearest integer in a root of a given numeral equation.

Let the equation be .c 1 —-2x—5=0.

1. In, this case a root is between 2 and 3; for these numbersbeing inserted for .r, the one gives a positive and the other a negative,result. Hither the number above the root, or that below it, maybe assumed as the lirst value; only it will be more convenient totake that which appears to he nearest to the root.

2. Suppose a=2+f, and substitute this value of a in theequation.

x 3 = 8 + 12/+6/*+/»

— 2x= —4 —2 f

i 3 —2r — ;> = — Y +767'+ 6 t 2 +/ 3 = o.

If/be less than unit, its powers./'- and /* may he neglected inthis first approximation, and Uf = 1, or/= l).1 nearly ; thereforex — 2.1 nearly.

3. As/— (i. 1 nearly, let /= . 1 + g, and insert this value of/ in tlie preceding equation.

,/ 3 = 0.001 -j- 0.03g + 0.3++ g 3

6D = 0.05 -4- 1.2g + 6+'

10/= 1 +10g

/ 3 + 5/- + 10/— 1 = 0.061 + 11.23 g + 0.3m! + + = 0, andneglecting g- and g 3 as verv small, 0.61 -j- U.23g = 0, or g =— 0.061

—.= — .0054, hence/'=s 0.1 -(- g = .0946 nearlv, and x

11.23

= 2.0946 nearlv.

4. 'I'his operation may be continued to any length, as by sup-posing g — — .0054 + It, and so on, and the value of .1 =2. 09435147 nearly.

By the first operation a nearer value of v may be foundfilms: since /= . I nearly, and — 1 -J- 10/'+ 6/’- +/ 3 = 0, /=

1 1

-, that is, /=-= .094 true to the last

lO+tj/'+Z 2 10+ .6+ .01

figure, anil x = 2.094.

In the same manner may the root of a pure equation be found,and this gives an easy method of approximating to the roots ofnumbers which are not perfect powers.

This rule is applicable to numeral equations of every order;and, by assuming a general equation, general rules may be de-duced tor approximating to the roots of any proposed equation.By a similar method we may approximate to the roots of literalequations, which will be expressed bv infinite series.

But perhaps the easiest rule is that of double position, or trialand error, much recommended bv Dr. Hutton.llute 1. Hind, by trial, two numbers near the true root, use themseparately instead of the unknown quantity, and mark theerrors arising from each.

2. Multiply their difference by the least error: then divide theproduct by the difference of the errors, when they are alike,but by the sum when they are unlike.

3. Add this quotient to the number which gave the least error, ifthat number be too little, but subtract if it be too great; theresult will give the true root nearly.

4. ’I ake this and the nearest of- the two former, or anv tint isnearer than it, and by proceeding as above, the root will befound more exactly ; and by repeating the work it may hefound as near as you please. Hack new operation generallydoubles the number of true figures.

Ante. it is best to use two assumed number* differing bv t, inthe hist figure on the right hand, because then the multiplier, Hide2, is only 1.

Hr. To find the value of ,r in the equation ,r> — 15.D + 63r— 40. Here it is soon found that .r is verv little above I. As-sume therefore 1.0 and 1.1, then by the first a 3 — 15i 2 + 63.rbecomes 1 — 15 + 63 = 49, too little by 1 ; error — 1. Bv thesecond it is 1.331 — 18.15 + 69.3 = 52.481, too much by 2.431 ;

1 X .1

error + 2.481, and —--— = .03 nearly, the correction ; then

1+2.481

1.03 = root nearly. Again assume 1.03 and 1.02 ; the first resultis 1.092727— 15.913.) + 64.89 = 50.069227 ; hence the error is+ .009227. By the second we have 1.061208*— 15.606+64.26

.01 X .069227

= 49.715208; the error is — .2S4792 and -^ =

.354019 '■

.0019555, ami 1.03 — .0019555 = 1.02804 = x nearly.

This rule may be applied at once with nearly the same case t<*an unreduced equation ronl,lining studs and compound quan-tities; and hktwise succeed very well in exponential equations, orstub as have the unknown quantity in the exponent oi the power.

Application of Ai.gebka to (If.omf.tkv.

(35.) A line whether known or unknown, may be representedby a.single letter: a rectangle by the product of the two lettersrepresenting its sides: and a rectangular paratlclopipcd by theproduct of three letters: two of which represent the sides of anyof its rectangular bases, and the third the altitude.

These are the most simple expressions of geometrical magni-tudes ; and any other having a known proportion to them, mayin like manner be expressed algebraically. Conversely, the geo-metrical magnitudes, represented by such algebraical quanlilio,may be found; only the algebraical dimensions above the third,not having any corresponding geometrical dimensions,. must beexpressed bv proportionals. '1 bus, if the algebraical equationii 1 -+ * = t- 1 — </', is to be expressed geometrically, a, b. r, andd, being supposed to represent straight lines; let a: l>: e-.J-.g, mcontinued proportion, then «'■ a : g, and <r‘ : a 1 + 6*: : a : a+ a ; then let a: c: I :: k : /, and a': c' also, let c: d:m:n : p, and c*: d 1 : : c : p, or c 1 : c' — if 1 :: c : c — p. By combiningthe two former proportions, id : a' + b ': i: a +:;, and combiningthe latter with this last found, c‘ — J’: a’ + b ‘:; c — p X l:c Xa +,g; therefore c — p x l — c X a +/;, and c : c — p : : l:a T~/-

It any known line is assumed as 1, as ils powers do not appear,tlie terms of an equation, including any of them, may be of verydifferent dimensions; and before it can be properly expressedbv geometrical magnitudes, the deficient dimensions must besupplied by powers of the 1. When an equation iias been deriv-ed from geometrical regulations, the line denoting 1 is known ;and when an assumed equation is to be expressed by the relationsof geometrical magnitudes, the 1 is to he assumed.

In this manner may any single power be expressed hyaline.If it is.r 5 , then to 1," i tind four quantities in continued propor-tion ; so that 1 : i : ;/i : u: p: <], am! 1 : <] : : l 5 : a 5 , or q = x"; andso of others.

The opposite position of straight lines -may be expressed bythe sic ns + and —.

Thu>, let a point A be given in the line AP, any segment APP A M P B

taken to the right hand being considered as a positive, a segmentAp to the left is properly represented by a negative quantity.Ifu and b represent two lines; and if, upon’ the line All Irom thepoint A, AP be taken towards the right equal too, it i-»«y be ex-pressed by a ; then PM taken to the lelt and equal to b, will I)*'properly represented by — b, for AM is equal to a — b. Ifii = b, then M will fall upon A, and a — b = 0. By the samenotation, if b is greater than a, M will fall to the lelt of A ; andin this ease, if 2« = h, and if Pp he taken equal to b, thena — b = — a will represent Ap, which is equal to u, and situatedto the left of A.

Df.monstkatiov of TltKOllF.MS.

(36.) All propositions in which the proportions of magnitudesonlv are employed, and all propositions expressing the relationsof the segments of 'a straight line, of tiieir squares, rectangles,cubes, and p..rallelopipcds, are demonstrated algebraically withgreat ea*e.

This is particularly the case in those propositions, winch maybe geometrically deduced without any contraction of the squares,

rectangles,