Bibl. Utenhor.

Octavo r?. ÎÎ5.76

In

B O O K J containing The Fundamental Principles of this Art.

Together with

AU the Praólical Rules of

containing,

A great Variety of PROBLEMS, ,

In the moft important

Branches of the MATHEMATICS.

Vtit qukquam in univerfa Mathefi ita difficile aut arduum Bccurrere poffie,. que nan inoffenfi pede per hanc methodum penetrare Uceat,

Schoot. Pref, to Des Cartes.

LONDON:

Printed for J. Nourse, in the Strand ; Bookfeller in Ordinary to His MAJESTY.

THE

ThE fubje£i of the following book is Algebra, « fcience of univerfal ufe in the^ Mathematics. Its buftnefs and ufe is to folve difficult problemSy to find out rules and theorems in any particular branch of fcience -, to difeover the properties of fueb quantities as are concerned in any fub-' jell uoe have a mind to confider. It properly follows thefe two fundamental branches. Arithmetic and Geometry, Z»«/ is vajtly fuperior in nature to both, as it can folve quefiions quite beyond the reach of either of them.

This m an art truly fublime, and of an unlimited extent ; for if the conditions of a problem be never fo complex, and though the quantities concerned are never fo much entangled with one another, yet the Ægebraifi can find means to diffiolve and feparate them •, or if they be ever fo remote, his art can furnifh him with methods to bring them together and compare them. It is true, he is often obliged io traverfe many roundabout ways, to get the relation of the quantities con-cerned ; yet by certain rules he can purfue the computa- ' tion of his problem through all thefe intricate turnings and windings ; and by hisfkill and fagacity can hunt

through all thefe labyrinths, HU he arrives fafely

A 2

îv The PREFACE.

at the end of the chacs^ viz. the folution of the problem.

The extent of this curious art is fo great that it has gained the title of Univerfal Mathematics ; and is called by hü ay of eminence^ The Great Art-, and has been efieemed the very apex of human reafon. It is alfo called SpzcAOMi. Arithmetic, Univerfal Arithmetic, The^ Analytic Art, The Art of Refolution and Equation ; with a view to fome or other of its properties or operations.

The nature of this excellent art is fuch, that it may be applied to any fubjett, provided the principles of that fubjvit, it is applied to, be underftood. Its great beauty is, that it deals in generals. For whjlfl other branches go no farther than their own particular fubjeU, and can only find folutions in particular cafes ; this art finds out general folutions, general rules, general theorems, and general methods.

This noble fcience has alfo this peculiar property, that it not only invefligates rules in all the other parts of the Mathematics ; but by the mofl fubtle art and invention, it finds out its awn rules, models them according to any form, and varies them at pleafure, fo as to anfwer any end propofed. It would be in vain io attempt to enumerate all the ufes of this admirable art.

By making ufe of letters inflead of numbers, it has one great advantage above arithmetic, viz. that in the feveral operations of arithmetic, the numbers are lofl or fwallowed up, and changed into others : but here they are preferved dijiinlt, vifible, and unchanged, By which means general rules are dtawn from particular folutions, to anfwer all cafes of like nature.

Br

The PREFACE. nbsp;nbsp;nbsp;nbsp;V

Ey help of algebraic charaSlers, geometrical de-monfirations are often rendered more fhort, compendious, and clear. So that by this means voe avoid the tedioujnefs of a long verbal procefs, vohich othervcife •we fhould neceffarily be involved in ; and vohicb never fails to darken and obfcure the fubjebt.

It is highly probable the cmcients made ufe of feme fort of analyfis, whereby they found out their noble theories. For it is hardly pojfible fo many fine theorems in Geometry, fhould be groped out or fumbled on, without feme fuch method. But as it was then only in its infancy, it muß have been far ßort of the perfeblion we have it in at prefent.

As to the Reader's qualifications, it is abfolutely ne-ceffary that he underßand Arithmetic and Geometry, as the keys to all the refl. And it is alfo neceffary that he underfland the principles of every branch of fcience, to which he would apply algebraic calculations ; other-wife it would be in vain to attempt the folution of any problems therein, by the help of Algebra.

Then as to the method I have followed, it is this. 1 have gathered together the mofl valuable rules and precepts, which lie flattered up and down in all the befl books of Algebra ; and what was deficient, 1 have fupplied as well as 1 could. ‘Then I have thrown all thefl precepts and rules of working, into fo many problems ; which 1 have reduced into as fhort a compafs, and exprejfed in as plain terms as poflible, fl as thty may be clear and intelligible. And the method I have taken 1 fuppoje will appear to be very fimple and tafy, and will readily be apprehended by fuch people as have found confufion and difficulty in other methods. I believe I have omitted nothing that is fundamental-, and if any thing of lefs moment is paffied by, it is either becaufe

' Ji Thé preface.

t ” fupplied by fame cthef

r^ 1 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;problems are

in jucb order, that the eaßefl appear firfl, and lead on to the harder, 'which follow in due courfe afterwards : thefe make up the firfl book. And the fécond book contains the application of Algebra to all forts of problems, of-which there is great variety, and many of them perfe£ily new-, others that are not fo, have

generally new folutions to them. So I hope I have delivered both the principles and the praSlice at large, and yet have not clogged the Reader with any fuper* fiuity.

W. Emerfon«

THE

CONTENTS.

Page

DEfinitions nbsp;nbsp;nbsp;nbsp;- nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;———

Charaélers -------

Notation — ■ ■■— ---- ----

Axioms --- --- ----- ——

BOOK I.

The fundamental Principles.

Seft. I. The Operations in Integers -------—

Seél. II. Fraélions nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;----

Seft. III. Surds ---- ------

Seél. IV. Of managing Equations

Seót. V. Subftitution, Extermination, i^c,

Seft. VI. Infinite Series

Seól. VII. Several fundamental Problems

Prob. LXV. To find two quantities whofc fum and difference is given

Prob. LXVI. To find the lead common dividend 198 Prob. LXVII. To find the difference of the fquares, having the fum and difference given

Prob. LXVIII. Two quantities given, to find the fquare of the fum

Prob. LXIX. Two quantities given to find the fquare of the difference

Prob. LXX. Given the fum and difference to find the reélangle

prob. LXXI. Given the n‘lgt; power of a binomial, to find the difference between the fquare of the fum of the odd terms, and the fquare of the fum of the even terms nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;203

Prob. LXXII. To find any root of a binomial furd 204

Prob. LXXin. To explain the properties of o, and infinity

Prob. LXXIV. To find the value of a fraélion, when both numerator and denominator are o

Prob. LXXV. To find whole numbers anfwering the equation ax-^z.by c

Prob. LXXVI. To find a number that being divided by given numbers, will leave given remainders 219

Prob

vüi nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;The C o N T E N T s.

Prob. LXXVn. To find the limits of an equation containing feveral unknown quantities nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;pag. 225

Prob. LXXVIll. To find the limits in two fuch equations nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;228

Prob. LXXIX. The inveftjgation of the rule of alligation nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;_ nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;231

Prob. LXXX. Inveftigation of the rule of falfe 234 Prob. LXXXI. Inveftigationof the rule of exchangeißö Prob. LXXXn. To find rational fquares, cubes,b’r.237 Prob. LXXXIII. To find the maxima and minima of quantities nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;241

Prob. LXXXIV. To turn numbers into logarithmic feries nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;244

Prob. LXXXV. To turn logarithms into numerical feries nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;247

Prob. LXXXVI. To demonftrate a propofition fyn-thetically, from the analytical folution 250 Seä, VIII. The refolution of equations, and extraction of their roots in numbers nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;251

Scél. IX, The geometrical conftruöion of equations 297 Sedl. X. To inveftigate a problem algebraically 316

BOOK IL

The Solution of particular Proikmj,

Page

Seö. I. Numerical Problems

Sed. II. Problems concerning intereft and annuities 348 Sed. III. Problems in arithmetical and geometrical pro-greffion

Sed. IV. Unlimited problems

Sed. V. Problems for finding rational fquares, cubes,

.

Sed. VI. Geometrical problems

Sed. VII. Problems in Plain Trigonometry

Sed. VIII. Problems in Spherical Trigonometry

Sed. IX. Problems of the Loci

Sed. X. Mechanical problems

Sed. XI. Phyfical Problems

Sed. XII. Problems concerning feries

Sed. XIII. Problems concerning exponential quantities 497

Sed. XIV, Problems of Maxima and Minima 507

ALGEBRA.

DEFINITIONS.

I, A'L G E B R A is a general method of com--/3- puting Problems, by help of the letters of the alphabet, and other charaders. It is of the fame nature as Arithmetic, but more general, and therefore it is called Univerfal Arithmetic^ as likewife the Analytic Art. The peculiar pradice of this method is, to allume the quantity fought as if it was known, and proceeding to work by the rules of this art, till at laft the quantity fought, or fome powers thereof, is found equal to fome given quantity, and confequcntly itfelf becomes known.

• 2. Like quantities, arc thofe that confid of the fame letters; as a, ^a, —^a. Alfo bb, •^bb, — i\bb-, alfo labc, i^abc, —abc-,

• 3. Unlike quantities, are thofe çonfiifing of different letters, or of the fame letters, difl'crently repeated. As«, b,zc,—2^. Alfo «, 2««,—ßäaa.

• 4. Given quantities, are thofe whofe values are known.

• 5. Unknown quantities, are thofe whofe values are not known.

• 6. Simple quantities, are thofe confiding of one term only ; as ^b, i^dcc, amp;c.

• 7. Compound quantities, arc thofe confillint: of fe-veral terms, as a-^fb, 2a—?c, a '’b—^d,

Pc/iiR-e

• 2 nbsp;nbsp;nbsp;nbsp;DEFINITIONS.

• 8. Pofitize quantities, are thofe to be added.

• 9. l^egative quantities, are thofe to be fub-tracted.

• IO. Like figns, are either all , or all —, (See the Charadcrs.)

• II. Unhke ßgns are and —.

• 12. The Coefficient, is the number prefixed to any letter or letters in any term. As 3 is the coefficient of 3/7t2. If no number be prefixed, then i muft be underftood, -is a a fignifies iaa.

• 13. y/ Binomial quantity, is one confifting of two terms, as 2 a nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;A Trinomial of 3 terms, as

a b — L. K ^ladrinomial of four, nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;A

liefidual is a binomial, where one of the quantities is negative.

• ) 4. Pozver of a quantity, is its fquare cube, bi-quadrate, Ißc.

An Equation, is the mutual comparing of one thing with another, by the fign of equality put between them.

• 16. A dependent Equation, is an equation which may be deduced from fome others.

• 17. An independent Equation, is one that cannot, by any means, be produced from the others.

• 18. Pure Equation, is an equation containing but one power of the unknown quantity; as a Jimple Equation, a pure Quadratic, a pure Cubic, amp;c.

• 19. An affielled Equation, is that which contains fevcral powers of the unknown quantity ; and is denominated according to the higheft power in it; as an affepled êffiadratic ; an affebled Cubic ; an affebled fourth Power, amp;c. Thus a fimple equation contains only the fimple quantity itfelf. A quadratic, a quantity of 2 dimegt;.fions ; a cubic, a quantity of 3 dimenfions ; a biquadratic, of 4 dimen-fions, ^c

• 20. Index or Exponent, is the number fet over a letter fhewrng what power it is : as a' ; here 3 fliews

DEFINITIONS. 3 fhews it is the third power; or that is equivalent to a a a. And thus 0 is the fame as lt;3 a «ß ; the fame as rz a zr lt;3 0, amp;-c, the index always Ihewing hov/ oft the letter is repeated.

2 1' FraSiion, confifts of two quantities placed one above another, v/ith a line between them, a

as the upper {a) is called the numerator^ the lower (A) the denominator.

A Surd, is a quantity that has not a proper root, as fquare root of a a}, cube root of

amp;c. roots of compound quantities that contain other furds are called, Univerjal Surds.

23. A rational quantity, is a quantity that has no radical figru

Char allers ufed in Algebra.

-F more, to be added, being the fign of addition, This is called an affirmative fign. 1 hus a b fignifies b added to a.

— lefs, abating, the fign of fubtraélion. '1'his is alfo called a negative fign. Thus a — b, fignifies Z» fubtraéted from a.

Thefe figns always affeól the quantity following ; and are always lo be interpreted in a contra,ry fignification. If fignifies upward, forward, gain^ increafe, above, before, addition, Szc. then — is to be interpreted downward, backward, lofs, decreafe, below, behind, fubtraAion, Zzc. And if 4- be be underftood of thefe, the/i — is to be interpreted of the contrary.

B 2 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;co dif-

• 4 nbsp;nbsp;nbsp;nbsp;C H A RACIER S.

co dijference-, a fignifies the tliffer-ence between a and Z».

X r4ultiplied by ; as fl X Z», fignifies a multiplied by b. Likewife fl b^ fignifies fl multiplied by b. All letters joined together fignifies a multiplication. For brevity’s fiike points are often ufed inftead of x, as n --. -----, fig-

23

n—I nbsp;n—2

nifies « X--X----• 2 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;3

• -4- divided by, as a — b, fignifies fl divided by b, and fignifies the fame.

• z=. eq^ual (o, as a -p Z» =z 2d, fignifies a and Z- equal to 21/.

C“ greater than, as fl tr is a greater than Z».

c_ leffer than, as a c_ b, is fl lefs than a root, as s/a, is fquare root of a.

cube root of fl. ^a, fourth root of a, Stc. it is called a Radical Sign.

involved to, as 2, involved to the fquare ; nbsp;nbsp;nbsp;nbsp;3 involved to the cube.

Uc.

Im extraSied. luj 2, fquare root. Im 9, cube _________ root, fÿc.

a b c, a line, or vinculum, drawn over feveral quantities a, b, c, denotes them to be efteemed a compound quantity.

E X P L A N A T I O N.

flfl —Zi^-I- c d, fignifies Z» fubtracled from fl A, and 2, cd added.

a a b b' — cd — dd, fignifies, that c r — dd is fubtraded from' a a — hb.

a a 2 a~b

EXPLANATION

—----- ----- 5

aa 2aâ V) rr — s s, fignifies the difference between 2 0 and rr — j j.

abc c fignifies the produft of a and and c c

a-{-bY.aa^ fignifies the fum of a 4- b multiplied by a a. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;. nbsp;nbsp;nbsp;nbsp;'

fignifies the produd of jn^Q is to be added to a.

—■‘t.

a a — 2 ab , fignifies the fquare of rhp mm pound quantity 0 « — 2 z:

y/ b b cc fignifies the fquare root of b b -p c r,

\/ 2 a b — c Cy fignifies the cube root of 20^__cc

'^ZZb'' fignifies 0 a divided by a ■— b.

J tx— a~a of di-vided by ä' — a a.

a^b^ fignifies a a a x b b^ or the cube of « mqi. liplied by the fquare of iJ.

“^ax — x: X \/ß a X, fignifies the fquare root of sax multiplied by 3 æ x— x x ; and fo of others.

Quantities that have no fign prefixed, muff be underftood to have the fign ; leading quantities feldom have the figns put down, when they are affirmative.

If A B and C D be two lines ; then A B x C D in a geometrical fenfe, fignifies the rectangle made by the lines AB and CD.

., - A B nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;.

C^’ nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;B has to

CD.

NOTATION.

I- In the computation of problems, putthefirft letters of the alphabet, b, c, d, J, ƒ, h, for known quantities, and the laft letters of the alphabet for unknown ones. Yet fome put vowels for

B 3 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;unknown

• 6 nbsp;nbsp;nbsp;AXIOMS.

unknown quantities, and the reft of the alphabet for known ones.

• 2. For general forms, put the capitals A, B, C, D, for the general quantities.

• 3. Or in univerfal forms, let the quantities be denoted by the Greek capitals, r, 7, ©, n, S, Y, lt;î), y, fi, and indices, coefficients, by the fmall letters, nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;0, nbsp;nbsp;nbsp;nbsp;/.i. v, -n-, t, e.

• 4. In cafe of neceffity, make ufe of any other fort of letters, or of any charaóters, that have names, as b , V, lt;î,Q, ?, ïi, ï, SI, t, yi, amp;CC.

AXIOMS.

• I. If equal quantities be added to equal quantities, the fums will be equal.

• 2. If equal quantities be taken from equal quantities, the remainders will be equal.

• 3. If equal quantities be multiplied by equal quantities, the produóls will be equal.

• 4. If equal quantities be divided by equal quantics, the quotients will be equal.

• 5. The equal powers or roots of equal quantities, are equal.

• 6. If to or from equal quantities, unequal ones be added or fubtraóltd ; the fums or remainders will be unequal.

• 7. If equal quantities be multiplied or divided by unequal quantities ; the produdls or quotients ■will be unequal.

• 8. Quantities feverally equal to’ a third, are equal to one another.

• 9. The whole is equal to all the parts taken together.

• 10. If a quantity be addedi, and the fame quantity fubtraded, they deftroy one another, and are both reduced to nothing.

BOOK

BOOK I.

The fundamental Principles of Algebra.

S E C T. I.

‘The primary Operatiom of /ligebra in Integers.

PROBLEM 1.

add feveral Quantities together.

I RULE.

IF the quantities are like and have like figns -, add all the coefficients together, for the coefficient to that quantity, and prefix the fame fign.

Ex, z.

to nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;— 2O2Xxjy’

add 4- nbsp;x^abb nbsp;nbsp;nbsp;nbsp;— nbsp;nbsp;nbsp;x x

nbsp;nbsp;'^abb nbsp;nbsp;nbsp;nbsp;— nbsp;ly X X

4- abb nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;----

--—■ 3?4y

Sum 4- \r,c,abb

2 R U I. E. -

If like quantities with unlike figns •, add all the affirmative coefficients, into one fum ; and all the B 4 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nega-

8 nbsp;nbsp;nbsp;nbsp;nbsp;ADDITION. B. I.

nc2;ative ones into another ; fiibttaél the leHer funi from the greater, and to the difference prefix the fign of the greater, with the proper quantity.

Ex.

— 1. a a •— 9 b c d 4- nbsp;nbsp;nbsp;nbsp;4“ 2e

4- 7 0 i? —20 bed — d d -56 4-3^r^î4- A- d.

Sum 4- 8 ^7 —25 bed nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;-{-ye 3 R U L E.

Set down all the unlike quantities with their proper ligns.

E.X. 6.

4- 2 rt

4- 3 b — c

______d_^

Sum 2^24-3^ —

Ex. 7.

13 a a — 2 a -- 4 Æ 2 «

— Q. dd 4quot; 6 d

Sum 4- I3«a — ^ab b C — 2 d d 6 d.

Ex. 8.

2 f e 4- 3

— ^ee 4- 5 e ƒ 4- 2 ƒƒ — 11

4- 6 ƒ g — ef ff— 2

Sum 4- 5 nbsp;nbsp;nbsp; T ef i-ff 4- 3

The reafon of this rule is evident for like figns ; and in unlike figns, it follows from the nature of affirmative and negative quantities, that the difference ought to be taken, to make up the total. As if a man owes 10/. then 10 I. ought to be deduft-ed from his ftock to find his real worth.

Cor. I. lichen fever al quantities are to be added together, it is the fame thing, in whatever order they are placed.

Thus a 4- — c — a — c b — — c a h — b 4- agt;— c, amp;c. for all thefe are the fame.

Cor. 2. lienee thefum of any number of affirmative quantities, is affirmative ; and the fan of any ntimb.er of negative quantities, is negative.

PROB L E M II.

'To fubtrail quantities from one another.

RULE.

Change the figns of all the quantities to be fub-traóled ; and then add them all together by Prob. I. and their fum will be the remainder fought.

Ex.

-2O 6

or — J 4

Ex. 2.

from 6 a — 3 x -j- 6j — y

take 8« 4j; 6j-)-5

Kem. '— Ï a — / a’ o — 12

£x. 3.

from a b a b

take a — b nbsp;nbsp;lt;— æ

Rem. Ï b nbsp;nbsp;nbsp;nbsp;1 a

Ex. 4.

from 004-20^-}-^^

take 4-40^

Rem. a a — a b b b

Ex.

, from a a — b b

take c c — dd

Rem. ÔÆ — bb—cc-{-dd

Ex. 6.

from 3 aa — la c d — d d — ff

take — Q.a a — ß a — a,b — 2 dd

Rem. 4* 3 quot;k c d ab -f-dd ~~'Jf

Cor.

Sea. 1. MULTIPLICATION. it

Cor. I. Hence, To fubtra^ one quantity from ano-is the Jame thing as to add them together, tvben ^11 the fights of the fubtrahcnd are changed, a b -■ ■ a “I“ b.

For it is the fame thing to fubtraa —, as to add -, and to add —, as to fubtraa d-. For fuppofe a man to owe lo/; becaufe it is a debt it, ntuil be writ —to/, therefore if any body would take away this —lo, it is the fame thing as if he lt;nbsp;added lo to his ftock : but before it is difcharg-ed, this —lo is the fame, as io dtduaed out of his ftock.

PROBLEM III.

T0 multiply one quantity by another.

R U L E.

Multiply every particular term (or fimple quantity) of the multiplier, into every term of the multiplicand, one after another -, fo that the coefftcients be multiplied into the coefiicienis -, and the letters into the letters, by placing them all together, like letters in a word. And prefix -p to produas of like figns, and — to unlike ones. The fum of all is the produft fought.

Ek. I.

— a nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;c

— b — Ï b ßd

ab nbsp;nbsp;nbsp;—(y ab nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;io, c d

Ex. 2.

• lt;3 4- nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;zr 3

rt -p nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;a — b

• a a -p ab nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;a a -p a b

• -P ab-\-bb — ab — b b

aa z ab-\-fb a a — b b nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;Ex,

» MULTIPLICATION. B. I.

3.

• 3 a — 2 b a ‘\-b

\^a a — iC) ab

• 4 - ab — '?,bb

1500 2 Zgt; — 'i, b b

Ex. 4.

a a a b — b b

a—b

«5 O, b —ab b — a ab — ab b b^

a^ nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;—ï a bb nbsp;nbsp;b^

Ex. 5.

ab — c -i~ r s

5^—7^_____ ________

rab — 15rfz/ ^rrs — quot;jabd 21 cdd — rsd.

Ex. 6.

2^ a a — 20^ 5

ö^3 2z2^ — 3

3 lt;3 — ïb a' 4- ß a a

4- 6ba'gt; — A^aabb 4- ^^ab — gaa — Gab — ïg

30 4- i^ba^—^bbaa—i^aa 4- iG ab— 15

£x. 2'

a a bb

c c — d d

c c a a 4- c cbb — d da a — d db b

c c aa — dda a 4* c c b b — dd b b

. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;Ex.

Seót. I. MULTIPLICATION. 13

Ex. 8.

a’

a'-

_

That every term in the multiplicand muft be multiplied by every term in the multiplier, is thus made evident. Let a 4- b be multiplied by t 4- ; it is plain, a-\-b muft be taken fo often as there are fuppofed units in c and t/, that is, as often as there are units in c, and alfoas oft as there are units in d. Therefore the produét vdll be a-\~b x f4quot; a-^-by.d. But for the fame reafon a by^c — ac-\-bc, alfo a-^-bxd — ad-^bd. Whence the pro-duél will be ac-\-bc^ad-{-b-d-, that is, the fum of all the products of every term multiplied by every term.

That like figns give 4-, and unlike figns—, in the produft, will appear thus.

Cafe I. Let 4- a be multiplied by 4- b. Then, fince this multiplication fuppofes, that 4- is to be fo often added together as there are units in 4-^1 and the fum of any number of affirmatives is affirmative, therefore the whole fum is affirmative, that is flX4-Zgt;zz4-d^.

Cafe 2. Let 4- a be multiplied by ■—b. Now frnce this implies that 4- a is to be as often fub-tracted as there are units in b ; and the fum of any number of negatives, is negative, therefore that whole fum, is negative, that is, 4- X — b

■— ab.

Cafe 3. Let — ß be multiplied by 4quot; b. It is plain here, that — a is to be fo often raken as there are units in b -, and the fum of any number of negatives being negative, therefore the whole fum is negativej that is, — a x 4- b — ab.

14 MULTIPLICATION. B. I.

Otkerwife, Let d — a be multiplied by b -, then f^CaJe i.) the product will be bd together with — a X but b d is too big, as being the produél of d by b, inltead of lt;ƒ — a by Z-{d—a being lefs than therefore b d, being too much, the produól —• a x b muft be fub-traétcd ; thatis, the true product will \yç.db — ab-^ and confequently — ab — — «x ^.

Cafe 4. Let —a be multiplied by—b. Here — a is to be fubtradled as often as there are units in b : but fubtraóling negatives is the fame as adding affirmatives {Cor. 1.. Prob. 2.) ; confequfently the produét is 4- a b.

Or thus. Since « — a — q., therefore ß — a x — b — o, becaufe o multiplied by any thing produces o ; therefore fince -y a — ax — b — o -, and the firft term of the product is —ab {Cafe zf, thcrefo^ the laft term of the produól muft be •4- ab, to make the fum o, or ■— ab-\-ab — Q\ that is, — a x — b z=: ab.

Otherwife. Let —a be multiplied by —b. Then {Cafe z.} the product will be —bd together with —ax — b-, but —bd the quantity to be lubtradled is too big, being the produét of d by —b, inftead of — a by —b, {d — a being lefs than df, therefore the quantity — b d to be fubtraóted being too much, fomething muft be reftored, that is —r- a x — b muft be added ; and the true produét will be — b d -f a b ; and therefore nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;— ax — b.

Cor. I. /ƒ federal guar.tities are to be multiplied together ; it is the fame thing in whatever order it be done. Thus ab c — a c bvz. c a b —bca, iicz. for all thefe are equal.

Cor. 2. powers of the fame quantity are multiplied together, by adding their ind:ces. ‘Thus 2-!-^ a^X a' w: a a\

Cor.

Scâ. I. MULTIPLIC.ATION. 15

Cor. 3. Any odd number of —, multiplied together produce — i 4«^/ any even number of —, pro-duce •

SCHOLIUM.

In the multiplication of compound quantities, it is the beft way to fet them down in order, according to the dimenfions of fome of the quantities. And in multiplying them, begin at the left hand, and multiply from the left hand towards the right, the way we write, which is contrary to the way we multiply numbers. But this will be moft expeditious, and the feveral produéts will by this means be fo ranged under one another, that like quantities will fall in the fame places, which is the eafieft way for adding them up together.

In many cafes, the multiplication of compound quantities is only to be performed by writing their fums, each under a vinculum, and'putting the fign (x) of multiplication between. As if the fquare of a a — x x was to be multiplied by a g —bh, and that by a c b d, it may be written thus, ax — XX Xag —-bhy;.ac-\-bd.

PROBLEM IV.

To divide one quantity by another.

I RULE.

In fimple quantities, which will divide without a remainder; divide the number by the number, and put the anfw’er in the quotient. Then throw out all the letter's in the dividend which are found in the divifor, and place the remaining letters in the quotient. And like figns produce -f-, and unlike figns —, in the quotient.

Ex.

Ex. 7.

— 8 X x) — 16 a;’ ( 2 Af — 16 X’

2 R U L E.

In compound quantities, range the terms of the divifor and dividend, according to the dimenfions of fome letter. Then, by Rule i, divide the fird term of the dividend by the firft term of the divifor, placing the refult- in the quotient. Multiply the whole divifor by the quotient, and fubtraèt it from the dividend, to which bring down the next term of the dividend, call this the Dividual.

Divide the firft term of the dividual by the firft term of the divifor -, then multiply and fubtradl as before, and repeat the fame procefs till all the quantities be brought down. This is in efFeót the very fame rule as is ufed in arithmetic.

Ex. S.

a} ab -p ac— a {b-\-c — 1 the quotient a b

-p a c

-p a c

— a

— a

Q

Ex. 9.

2 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;zb a a — 3 f ß ß (a

zb a a — 3 cßß

C nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;Ex.

Ex. 10.

a c A- i c

a d -\- b d a d -\- b d

Ex. II.

— 4)7’—~ 47 12 (j—3

_____— 47

— 377 nbsp;nbsp;nbsp;

y-3yy o

jEgt;f. 12.

aa o,b

— ab — b b

— ab — bb

o

jEjf. 13.

ÿa—b) nbsp;nbsp;—12^îæ—baa'irXQab-^ibb {aa—404-2^

3Æ* nbsp;nbsp;nbsp;nbsp;—baa

—itaa nbsp;nbsp;nbsp; ioß^

—12«« nbsp;nbsp;nbsp;nbsp; ir^b

6ab—ïbb 4- 6«^—ïbb

o

3 RULE.

When the divifor does not exaélly divide the dividend ; place the dividend over the divifor, in form

20 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;DIVISION. B. I.

This and fuch like examples will be better under-ftood after the next fection.

Ex. 'i8.

19.

—ïb'-} nbsp;nbsp;nbsp;nbsp;nbsp;Ç--~

That like figns give , and unlike figns — in the quotient, will appear thus. The divifor multiplied by the quotient mull produce the dividend. Therefore, r. When both are , the quotient is -h, becaufe then X I. ) ( mull produce -J- in the dividend. 2. —)—(- - 2. When they are both —, the 3. -b)—(— quotient is again, becaufe -f-x— 4. —) (— mull produce — in the dividend.

Again, 3. When the divifor is -f-and the dividend —, the quotient is —, becaufe — X muft produce — in the dividend. 4. Laftly, If the divifor is —, and the dividend -b, the quotient will be —, becaufe — x — produces in the dividend.

Cor. I. One power of a, quantity, is divided by another fewer thereof-, by fubtracting the index of the divifor, from the index of the dividend. ‘thus, 0’ _3—3 nbsp;nbsp;, A nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;4^^“^ nbsp;nbsp;4

——0, nbsp;nbsp;-a,-. Aad nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;~——--—~n:

, nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;Cor.,

Seft. I. INVOLUTION.

Cor. 2. Hence any power of a quantity may be taken out of the denominator and put into the numerator, and the contrary ; by changing the fign of the index, eri nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;cib~^ ,

~. And —- z= ba^.

20^- nbsp;nbsp;nbsp;nbsp;nbsp;2 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;Ö—3

Cor. ß, IJence — divided by , or divided h '—gt; give the fame quotient, viz. —, Heat is,

a a

~~b- ~~b' ,

PROBLEM V.

‘To involve a quantity to any power.

I R U L E.

Multiply the quantity fo often into itfelf as the index denotes. And where the root is , all the powers are . And where the root is —, all the odd powers are —, apd all the even powers -p. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;'

Ex. I.

a'- root '

0 fquare a® cubé

4th power amp;c.

root fquare cube 4th power.

Ex.

il INVOLUTION. B. L ! £ä'. 3.

Involve a-\-i to the cube or ■povier,

a I

quot;I“ b

cc ci —d b a b b b

fquare a a 4- 2ab 4- b b

a b

a^-i-zaab-{-abb

4- aab-}-2abb-^bi cube «’4-3'ïö^4-3«5^4-j5’

2 RULE.

Multiply the index of the quantity, by the index of the power, and make the figns as in Rule i.

4.

Ex. 5;

•-a root

fquare nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;or

cube nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;or aZZZZ^®

m power orÔ^^”’

3 rule.

3 rule.

In a binomial. The power will confift of i term more than the index of the power. The hio helt power of both is the index of the given powerband the index of the leading quantity continually de-creafes by i in every term, andin the following quantity, the indices of the terms are o i 2° 3, 4, nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;_ nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;’ ’ ’

Then for finding the unciæ or coefficients. The firft is always i -, the fécond, the index of the power. And in general, if the coefficient of any term be multiplied by the index of the leading quantity, and divided by the number of terms to that place -, it gives the coefficient of the next following term.

Laftly, When both terms of the root are all the terms of the power will be ; but if the fe-. cond term be , then all the odd terms will be ^4-, and all the even terms —.

Ex. 6.

Involve a-}-e io the power.

The feveral terms without the coefficients will be ß’, a^e, a'ee, a^e\ ae^, -, and thç coefficients i nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;^*^^3 10x2 5x1

’ nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;2 ’ 3 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;-5-’

that IS, I , 5, ÏO , IO , nbsp;nbsp;2

And therefore the 5th power is' 5 nbsp;nbsp; 10 nbsp;nbsp;nbsp; 10 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;ae^ e^.

Esc. y.

Inth, a—x SI, tie ^tb fawtr.

the root is nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;4X'

that is, nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;M

^'4 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;4 R U L E.

24 INVOLUTION. B. I.

4 R U L E.

In trinominals, quadrinomials, ^c. Let one letter remain, and put another letter for the reft of the quantities ; then involve this binomial by Rule 3 -, then inftead of the powers of the afliim-ed letter, find (by Rule 3.) the powers of the compound quantity it reprefents, which put in its ftead.

Ex. 8.

Involve nbsp;nbsp;nbsp;nbsp;—x to the third power.

Put e for I—X, then the cube of e e is ß’ 3 nbsp;nbsp;nbsp;nbsp;nbsp; 3 nbsp;nbsp;nbsp;nbsp; ^’ (Rule 3), that is,

^ _3ßßX^—x gzî X nbsp;nbsp;nbsp;nbsp;-Çb^x. But (Rule 3.)

2^.v-{-.ïy, and b—x^—b^—^^bx-^-^^x^ h—jf’—zz’ 3ßß5—3ß^?A; ^abb nbsp;nbsp;(gt;abx -{-^(ixx-^rb'^—^bbx-^-'^bxX'—X’.

Cor. 1. The nf^ pozoer of a-^-e, that is., n n nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;t

ß e -a ;;^ e ^j x — aquot;''quot;'ee-A^nx

2

’f f” , nbsp;nbsp;n—t n—2 n—; »—4

^’ «x-—x—X—- a

• nbsp;nbsp;. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;^3 nbsp;nbsp;nbsp;4-

This c'-fie is proved by involving as far ^s you \\ 1 , or the feveral powers will always agree with the rule. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;°

or. 2. xillpowers of an affinnative quantity, are affama t.,e. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;powers of a negative quan

ti.}, are negative ; and all even powers affirms.tive. • nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;index of the power of any quantity,

ef the power, and index of the quantity. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;■'

___ nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;faßor, multiplied together, tip' —a X b^“

P R O-

problem VI.

To extra£l tie root of any q^uantily.

Evolution is juft the reverfe to involution ; and is performed as follows.

I R U L E.

For fl m pie quantities-, extract the root of the coefficient for the numerical part, and divide the index of the letter or letters, by the index of the power, gives the index of the root.

Ex. I.

The cube root of a? is or a. ,4 the fquare rcot of is Qr gaa. the /([uare root of 2a'^b^i3 or flS’v/z . 9 the cube root of —izsS’ is* —or ■—55?.

2 R U L E.

For the fquare root, of a compound quantity; range the terms according to the dimenfions of fome letter. Then find the root of the firft term (i Rule), and fet it ia the quotient: fubtraól its fquare, and bring down the next term, which divide by double the quotient, and fet the anfwer in the quotient. Multiply the divifor and quotient by this laft quotient, which fubtraól from the dividual, proceed thus, iuft as in common arithmetic.

Ex.

26 EVOLUTION. B. I.

2.

ExtraEî thefquare root cf nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;—2ax nbsp;

4^x-l-xx.

—lax—(04-2^—x root

*04-2^) 4zzä4quot;4^^

—-2ax—^x 4-a'j# —2ax—43x4-ä'j^

Ex.

Extras the fquare root of ææ—6»^ 4-2 24 4-9»»— €»24-2Z.

4-22 —6nz K “ 4. X

0« nbsp;nbsp;• 4-22

-3,77=6» r“

2'* 1 nbsp;nbsp;nbsp;nbsp;1 nbsp;nbsp;nbsp;nbsp;. O'---6»Z

nbsp;nbsp; 2^ XX

—6n ^nn

4-22 —6nz

4- zz

Ex^

Seft. 1. EVOLUTION.

jEx. 4’

Extras the fquare root of aa-\-xx.

2a -P **

aaJ ,

. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;4-XX4-—

^aa

XX Âr \ O — ** »«4

a Sa’' 4aa — x^ x^ X* 4aa 8a ~^64a^

8lt;jlt;- 64a'’ 3 rule.

In higher powers. Find the root of the firft member, which place in the quotient : fubtraft its power, the remainder is the refidual. Involve this ' root to the next lower power, and multiply it by the index of the given power, for a divifor ; by this divide the firft term of the refidual, the quotient is the next term of the root. Then involve the whole root as before, and fubtraél ; and repeat the operation, till all the terms of the root be had.

£x. 5.

ExtraSî the cube root of nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;—^.ox^ gSx—64.

4-6x5—^ox^ g6x—6^ (Ä-X4-2X—4 root.

jf«

3x*) 6x! ( -F 2x

X‘-F6x5 4-i2y44-gy;—

3X*) O nbsp;12X4(—4

4-6x5—4Oxgt;4-q6x—64.~xx 4- 2X—4^. o

Ex.

•8 EVOLUTION. B. I.

Ex. 6,

Extratî ilie root cj 160^—

—2i6«Zgt;5 4*8i^ .

J 6a^—^i^6a^b 21 (gt;aabb—11 ótfiJ’ 81 b^^iia—ifb root, 24«’} o —lt;^6a''b —4^

—lt;^6a^b

16« —^(ya^b-gt;^z\iaabb—216ab^-j-S i b^ o

4 R u L E.

The roofs of compound quantities, may fome* times be difcovered thus. Extraél the roots out of all the fimple powers or terms in it ; then conneél thefe roots by the ligns or —, as you judge will beft anfwer. Involve this compound root to the proper power ; then if it be fame. with the gi-ven quantity, you have got thç root. If it only differs in the figns, change fome of them, till its power agrees with the given one throughout. j

Ex. 7.

Eo extract the cube root of lt;3’—6«’^-lquot; i 2«^’»—8^’.

Here the root of is «, and the root of —8^’ is • lb. Then a—2b is the root, for its cube is a’—6«'Z' i2a^’— Sb\ as required.

jEx. 8.

Extract the 4th root of i6a'*^—Q6a’x' 2i6a‘:*’ —2i6ax''4-8ix .

and 81X , are la and 7 hererore if aa-p^x be made the root and in

volved.

7

Seót. I. evolution. 29 volved, it is i6a^ g6a^x 2i6aaxx 2 i6ax^ 8i5f4, which differs in the figns, from the quantity given. 1 herefore make 2û—3^ the root, which being involved fucceeds ; the power beino-1 ba^—gba^x 21—216z?x’ 4- 81

5 R u L E.

When the quantity given has not fuch a root as is required, fet it down in form of a furd.

Ex. 9.

Ex, IO.

Ehe cube root of a^—ba-b i2abb-irW, is v/ a'—6«=Zgt; i2ûZ’Z'4-8ii)i.

Ex, It.

Wbat is the t^tb root of the root is

even root, of an

affirmative quantity, may be either -{-or __

For the fq^are root of „..y be \r-~a,

-. is X? n nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;4th power of.

quot; IS 4-a , as well as of

J me jign, as the quantity itfcf

tor „ cubed ,s u:, a„a ,„ted is

Cor.

30 evolution; b, I.

Cor. 3. The fquare root, or any even root, of ' negative quantity, is impoftble.

For neither öx «, nor —ayfr-a, can pro, duce —aa.

Cor. 4. 27»^ wh j-oot of a produ^, is equal to the root of each of the faSlors, multiplied togethen

” nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;n

SECT,

3»

SECT. II.

0/ F R A C T I o N s. •

Th E operations of algebraic fractions are ex^ aélly the fame as thole of vulgar fractions in arithmetic -, therefore he that has made himfelf mafter of vulgar fraélions, will eafily underftand how to manage all forts of algebraic fractions, as in the following problems.

PROBLEM VII.

To reduce a given quantity to a fraHion of a given denominator.

rule.

Multiply that quantity by the given denominator, and under the produél write the fame denominator.

Ex, I.

Let a-\-b bave the denominator x.

ax-\-bx .

------- nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;, anfwer.

X

Ex. 2.

Let xse-^yy bave the denominator r.

—nxi

- ' nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;, anfwer.

I

Ex. 3.

Let have the denominator b—c.

Cor.

31 FRACTIONS. E. I.

Cor. The value of a fraSlion is not altered, by multiplying both numerator and denominator by the rab rabd ab fame quantity. Thus nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;.

rc red c

PROBLEM VIII.

To reduce a mixed number to a fraction.

R U L E.

Multiply the integral part by the denominator of the fraftion, and to the produél add the numerator, under which write the common denominator.

Ex. I.

T nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;7 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;7T71 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;b ,

Let a — ~ given. 1 hen —— is the fraftion required.

Ea'. 2.

„ nbsp;nbsp;. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;aa — ax

Suppoje a—X

X ax—xx-1-öß—ax nbsp;nbsp;nbsp;aa—xx .

Here ——...... , or -----— is that re»

X

quired.

PROBLEM IX.

To reduce an improper frabiion to a vohole or mixed number.

R U L E.

Divide the numerator by the denominator, as far as you can, gives the integra' pait -, and place the remainder over the denominator for the trac-tional part.

Ex.

Se^- fractions. Ex. I. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;'

Ghamp;n nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;igt;}ai—aa lt;nbsp;nbsp;nbsp;nbsp;a a

b nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;4^ V nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;anfwer.

■—aa Ex. 2,

a—X

quot;—x'jaa-^xx f nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;2XX

od—ax V^ '’^ g__J anlwer.

lt;jx w a.v—XX

4-2XX

PROBLEM X.

To find the greateß common dwifor^ for the terms of afraSlion, or for any Koo quantities,

RULE.

■ The quantities being ranged accoidino- to the dimenfiotis of Come letter divide the greater by the leffer, and the laft divifor by the laft re-’ mainder, and fo on continually till nothing remain ; then the laft divifor is that required. But obferve, t firft to throw out of each divifor, all the fimple di-vifors, (or others) that will divide it ; and then proceed. The fimple divifors are had by infpeótion.

Ex. I.

cd-\-dd nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;. nbsp;nbsp;nbsp;. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;. ,

Let ------—be the framon tropoied.

aac-{-aaa nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;■' nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;c c j

cd dd} aac-\-aad (

or c-\-d } aac aad (^aa aac-{-aad

o

, nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;D nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;There-

34 fractions. b. I. Therefore c-\-d is the greateft common divifor

aac aaa aa

’ Ex. 2.

y __alb aa^iab^-bb

aa4'2ab-4-bb} a'^—^ibb nbsp;nbsp;nbsp;nbsp;nbsp;{a

-^ogt;-4-'i-aab-\-abb ■—zaab—zabb remainder.

20^^) aa^zab^bb {

or a-4-b ) aa.\-iab^bb {a-4-b

-f- üb

lt;{b-\-bb '

nb-\-bb

Q

Therefore ig the greateft common divifor.

3gt;

a‘'—b^} a^—bbai (^a —b^a

rem.—bba^-4-b^a') nbsp;b^{

or 4lt;J ——b^(^aa-4-ib

—bbaa

-{-bbaa—b^ , ■^bbaa—b^

Q

the common divifor is __bb.

PRO-

Sea. II. FRACTIONS. 35

PROBLEM XI.

‘I’o reduce afraSlion to its lowefi terms,

RULE.

Find the greateft common meafure (Prob. X\ by which divide both numerator and denominator of the fraftion ; the quotients will be the numerator and denominator of the fraótion required.

Ex. 1.

cc-}-dd

7a'^aad f^ofo/eà.

the greateft common divifor is r-f d. Therefore d

- — the fraftion required.

Ex. 2.

Here a b is the greateft common divifor : then

aa—ab

~'„ r-c the fraótion fought.

E.i(. z.

Suppofe nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;7

the greateft Co

f aa-^bb , ^^^'^bba^ - -71— the fraftion required.

PRO-

36 FRACTIONS. B. I.

PROBLEM Xn.

To reduce fraSlions of different denominators, to fractions of the fame value, having a common denominator.

I RULE.

Multiply each numerator, into all the other denominators, for a new numerator ; then multiply all the denominators together for a common denominator.

Ex, I.

r . nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;t

Let nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;be given.

, r , nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;ab-\-bb

thele become 5-, —z—.

be' be nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;. ■ f

Ex. z.

1,1 ~2i fropofed. G

they become

^dg bdg bdg

Ï RULE.

Divide the denominators by their greateft common divifor, then multiply both numerator and denominator of each fraftion, by all the other quotients, which will produce as many new fraélions.

3-

ibb ' zb' b'

'i-bc

^bb ' i^bb' nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;fractions required.

or --

ibb' zbb' zffb'

En.

38 F R A C T I Ö N S. . B. 1 Ex. 4.

fl a ro a—

a—i add b —

ab—bb — caa

fum 04.^ ------.

problem XIV.

To fublra^ one frabiion from another.

RULE.

Reduce them to a common denominator ; then fubtraél the numerators : and under the difference, write the common denominator.

Ex. I.

fl ^ f., c hrom -J- fubtrabî -j.

a—bc

—— = difference.

Ex. 2.

__ ab-\-bb à’ ~bd~

_.aad

ab-\-bb—^aad

---- z: remainder.

3-

take

^^d

reduced ————i2flr .

remainder nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;—Èbc4-1

Ex,

5elt;a. II. FRACTIONS.

• Ex. 4. _ aa nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;aac

brom a—“ or a--

, nbsp;nbsp;, nbsp;nbsp;nbsp;a—b nbsp;nbsp;nbsp;nbsp;, . ab—bb

take b --, or P 4---j—

c nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;bc

—ab-\-bb

ditîerence a—b ------j------

PROBLEM XV.

T0 multiply frabîions.

1 \J -L E.

In fraélions, multiply the numerators together for a new numerator ; and multiply the denominators together for a new denominator.

Ex. i:

Multiply

, nbsp;nbsp;nbsp;nbsp;nbsp;■ a-Xf

then -7—, or

Ex.

Multiply ~ by b a-\-b ab-\-bb •

Ex. 3.

1.' 1 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;, aa-\-bb

iy

, nbsp;nbsp;nbsp;aa—bb aa-^bb nbsp;nbsp;nbsp;nbsp;a^—b*

—k--X -T',— = nbsp;nbsp;nbsp;,-T—, produft.

Zgt; f nbsp;nbsp;nbsp;nbsp;nbsp;bbc-\-bcc

D 4 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;2 R U L E.

40 FRACTIONS.. B. 1.

2 RULE.

When the numerator of one, and denominator of the ocher, can be divided by fome common di-vifor, take the quotients inftead thereof.

Ex. 4.

Ex. 5.

7 aa-iriab-^-bb dd eJ-M ■ ‘'y

1 J d ad-\-bd reduced ^337 X f = ~^2_d, '» pro^uâ:.

• 3 rule.

If a fraólion is to be multiplied by an integer, which happens to be the fame with the denominator ; take the numerator for the produót.

Ex. 6.

, , aa—ïbb

Multiply ——— by a—i, quotient 0«—zbb.

• 4 RULE.

When a fraftion is to be multiplied by an integer j multiply the numerator by the integer.

Ex. ’J.

...r , ■ 1 nbsp;nbsp;nbsp;nbsp;aa.^^bb ,

Multiply — by XX.

aaxx-{- ^bbxx aa-\.Qi,f,

icd quot;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;P‘'0^-

Ex.

Ex. 8.

2a--2X ,

—P— by a x ïaa—ixx

----- = produa.

bb—be product

Schol. 5y this rule., a compound fragten may be reduced to a fimple one.

PROBLEM XVL

T’a divide one frail ion by another.

1 RULE.

In fraftions, multiply the denominator of the divifor by the numerator of the dividend, for a new numerator -, alfo multiply the numerator of the divifor into the denominator of the dividend, for a new denominator.

Ex. I.

Divide by c \ a /ad , tJtVï; the quotient.

42 FRACTIONS. B. I,

Ex. 2.

divide

O'—O nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;d •

\o-\-bf aa—ib a—b)~T nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;quotient

• 2 rule.

If the fraólions have a common denominator ; take the numerator of the'dividend, for a nume-. rator ; and the numerator of the divifor, for thé denominator.

Ex. 3.

aa—bb nbsp;nbsp;zab—bb

Dmde nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;bj '

« «

a a—bb

1“'“'

Ex. 4.

O(t-^ïab -^-bb

Let ---—-J--- dtwde —~

c—d

, nbsp;nbsp;nbsp;nbsp;nbsp;agt;—abb

aa^i.ab-\-bb = quotient.

aa — ab

~a.4-b~ ~ quotient reduced.

• 3 RULE.

When fraótions are to be divided by integers j multiply the denominators of the fraélions, by fuch integers.

Ex.

, nbsp;nbsp;o—E , •

' Eivtde ---- by d.

c

o—b

quotient is •

Exi

Selt;a. II. F R A C T I o N S4 43

Ex. 6.

, aa— zib a-Yb divide--;—.

a—b aa—ibb nbsp;nbsp;nbsp;nbsp;aa— zbb

a—by,a-\-b nbsp;nbsp;nbsp;nbsp;aa—bb^ quotient.

• 4 R U L E.

When the two numerators, or the two denominators, can be divided by fome common divifor i throw out fuch divifor, and proceed by Rule i.

Ex. 7.

/T—b nbsp;nbsp;nbsp;_ nbsp;nbsp;aa—bb

Eet divide '—, cd nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;c-{-d ’

, nbsp;nbsp;nbsp;, I A facd-\-bcd

Ex. 8.

ôÆ-l-ô^ nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;a*'—b^

Let -----■T' divide--

a—b nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;aa—zab-{-bb

, nbsp;nbsp;, A al—a?-b-\-abb—bi f ai—aab-{-abb—bgt;

reduced —)--------7----- I--------

1 ' nbsp;nbsp;nbsp;nbsp;nbsp;a—b nbsp;nbsp;nbsp;nbsp;' nbsp;nbsp;nbsp;aa—ab

the quotient.

, nbsp;nbsp;nbsp;nbsp;nbsp;. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;bb

ti^t IS, the quotient nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;.

From hence may be deduced the following corollaries.

Cor. I. value of any fractional quantity is not at all changed., by changing all the ftgns of both nu-

merator and denominator. Thus

—c nbsp;nbsp;nbsp;c—r ’

Cor. 2. T'A« value of any compound fraClional •iuantity, is equal io the fum of all the particular fimple

44 fractions. b. I. fifnple fra£îions, that compofe it. ’Thus rx zcx—ixrz _ nbsp;rx ^cx nrx

'ir—2x nbsp;nbsp;nbsp;nbsp;nbsp;3r—2.x jr—2;f nbsp;^r—2x'

.V ^fraaion be multiplied by any given guantity ■ st is the fame thing whether the numerator be multiplied by that quantity, or the denominator di-‘J , « dab dabd dab

■v,d,db,„.

Cor. 4. T’he prcduSî of two frabiions, is equal to Ue frabîton, that has the produbl of the numerators for the numerator ; and the produbt of the denominators for its denominator.

—c nbsp;nbsp;ar—ac

bx-\-xx’

5' V frablion is to be divided by fom^ guantity ; it is the fame thing whether the numerator be divided by it, or the denominator multiplied, r. 2flz nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;2az nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;2ar nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;2a

For — -dr r —--. And — — r — ~

X nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;rx nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;X nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;X'

Cor. 6. If any fort of quantity is to be divided by a frablion -, it is the fame thing, as to multiply the faid guantity, by the frablion inverted. Thus r s nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;a b

ab ~ ~—ab-je.-. And ~ -i-— or g c a r ar t ~ ~^~b ~

PROBLEM XVII.

To involve frablional quantities.

It. \I I. F.

Involve the numerator into itfelf, for a new numerator ; and the denominator into itfelf for a new denominator i each as often as the index of the power.

Sea.li. fractioj^s.

ÄX. 4.

What is the cube root of

' the root is

Va^—h^'

Lx.

What is the cube root of

the root is nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;d^ j ^„bd—d^

zacjj

Lx. 6.

ILhat is the ^th root of ___!x^—y^

8lt;jjf J.—Sx'jyj 4-_y**

the root is \/

8 ax^—-8 x’'yy -{-J’* ’

or Î/ — ~ ------- * 8«âÏ—ÿT^ïZj^

power or root of a fraSiion, is equal ^th nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;of ^^0 numerator^ divided

y power or root of the denominator.

cr.

48

SECT. III.

Of SURDS.

p U R D 5 are fuch quantities as have not a pro-per root. Simple Surds are thofe which confift but of one term. Compound Sards are thofe which confift of feveral fimple ones. And tZ«/-•uerfal Surds are thofe confifting of feveral terms under any radical fign.

Surds are faid to be commenfurable, when they are as one number to another -, and incommenfura-bloy when their proportion cannot be expreffed in numbers.

PROBLEM XIX.

To deftgnate or exprefs the roots of q^uantities by frabiienal indices.

1 RULE.

Divide the index of the quantity by the number cxprefiing the root ; the quotient is the index of the root required.

Ex. I.

Let the quantity a be propofed.

then v/æ —=: nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;See.

Ex.

Let ^ab’’ be propofed.

’^^3“!'' =

Ex,

3‘

Let ei^ ie ^iven.

then s/a^— nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;-s or 4, ^a^~=.a^^

Ex. 4.

Ze/ aa—xx be frofüfed. _____________ ________________1.

\/aa—XX — aa—xx , nbsp;nbsp;‘

y™ quot;nbsp;» nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;“’Jquot; amp;'c

Ex.

Let — be riven.

X *

then J-^ nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;_ _L —

X

X nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;■

Ex, (t.

TbEc^ ___I- nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;

then ~^2=L. y-k _ nbsp;nbsp;nbsp;nbsp;nbsp;, ècc,

~ a'rbc-^ abbc^ - n^b^c

Ex. 'J.

-J. Cl à

‘^yaa

b^ ~ bi' bi nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;bi^- quot;“Æ’

E

Seft, III. nbsp;nbsp;nbsp;nbsp;nbsp;SURDS.

13.

ret nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;aab .

“’ x’ i iquot; 7^ nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;•

they are rra^ X(I ä~\ rrd^

now” to explain this ; let there be a rank of /ƒ’ nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;aaaaa^ amp;!.c. the

lame „.11 (by Def. ao.) be dinoted .,

' nbsp;nbsp;nbsp;1 tr’, amp;c. Now tliefe quantities, o, a‘, n^,

c. are in geometrical progrefTion, and their indices. in arithmetic progreffion, as is plain. Now iince I, ß, a‘^, nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nr»

tionals, therefore’ thcfe win nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;Ptop“''-

proportionals. * nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;«“metrical

1gt; \/a, v/lt;l‘. \/llt;, nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;^a‘, ^/a’,

, nbsp;nbsp;nbsp;.

that IS I, a, aa, a\ ,/a\ «♦, See. and I,

a'^', See.

and I, 1^^^,

quot;gt;/aT, aa^ Sec. and fo on.

of analogy, the indices of all thefe, are alfo in arithmetic progreffion.

Take anyone of thefe feries as 1,

a,^ Sec. thefe will be equivalent to i,

5 a J ÓCG.

Suppofe now the feiies i, nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;con-

to the denomiantor ; and the indices, which con-tinually decreafu will then btcnmp t- ' i will Hand thus ; nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;quot;quot;sai've, and

Powers

Seel. III. SURDS.

or thus, a nbsp;nbsp;nbsp;nbsp;a a \ a nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;a\ a\

amp;c. and the fame is equally true of any of the other ferfes.

Cor. I. The powers of any quantity are a Cet of geometrical proporiionals from i ; and their indices, a fet of arithmetic proportionals from o.

thus, powers nbsp;nbsp;i, n, a^^ ei'^ ar, f increafing.

indices o, i, 2, 3, 4, f. increafing.

alfo, powers i, nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;deercaf.

indices o, —i, —2, —3, —4, nbsp;nbsp;nbsp;decreaf,

Cor. 2. Hence the double, triple, quadruple, the index of any quantity, is the index of. the fqiic.re, cube, biquadrate, icc. of that quantity.

Cor. 3. Hence alfo, the index of the produH of any two-powers (whole or frailed) of any quantity, is equal to the fum of the indices of thefe pozvers. And therefore to multiply any two powers together, is to add their indices. Thus aryc.a'—a^ a^yf —0“’, amp;:c.

Cor. 3. index of the quotient of two powers, dividing one another, is equal to the index of the dividend — the index of the divifor •, whatever the indices be. And therefore, to divide by powers, is to Jubtraii their indices. Thus — ' — a^ , and

~ = a nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;=a\ ^c.

Cor. 4. Any power is taken out of the denominator, and put into the numerator, by changing the fign of the index : and the contrary. Thus *

*

J

7? = nbsp;nbsp;nbsp;nbsp;=

-S'' nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;a--b'‘ '

E 3

54 nbsp;nbsp;nbsp;nbsp;nbsp;. S U R D S. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;B- B

Cor. 5. In fraliional indices., tie numerator figt;ecsJS the power, and the denominator the root.

Schol. In all the following problems^ it will be the beft way to reduce the furds to fractional indices.

PROBLEM XX.

To reduce a rational quantity to the form of a furd.

rule.

Multiply the index of the quantity, by the index of the furd root given -, to which fet the radical fign, or index of the furd.

Ex. I.

Reduce 6 to the form of

Here or 6^ = 36, and nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;is that

required.

Ex. 2.

Reduce a to the form of

Here zza\ and

Ex. 3.

Reduce a-\-b to the form of ,yi)c

Anfw. ^a-[-b , or -----,—~ aa-)^2ab-\-bb .

4. a

Vf/c form of ,yd.

Anf /-quot;I • bbe ’’ of the form y/pi^

problem XXI.

Ta reduce quantities of different indexes, to other equal ones, that fhall have a common index given.

R U L E.

Divide the indexes of the quantities by the given index ; the quotients will be the new indexes for thefe quantittes. Over thefe quantities with their new indices, place the index given.

Ex. I.

Reduce 12* and 7^ to the common index

— ) — f — firft index. 7

\ then 12^ and 7^ quot;quot;nbsp;are

)4 f- fécond index \ quantities required. 36^3

Ex. 2.

Reduce æ’ and to the common index |. (0 firft index. 7 ,

3 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;I then a'E, and are-the

’ f 9 r . ] nbsp;nbsp;nbsp;( furds.

• — / ~k~ fee. index. \

• gt; 1. Ï

PROBLEM XXn.

To reduce quantities of different indites, to others equal io them, that (hall have the leaf common index.

RULE.

Reduce the indices of the given quantities, to a common denominator, in the leaft terms. Then involve each quantity to the power of its numerator i and take the root denoted by the common denominator.

E 4 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;Ex.

5^ nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;SURDS. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;B. f.

Ex. I. Reduce ar.d to the leaß common index.

J nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;2

— and -gt;■■ are nbsp;nbsp;- nbsp;nbsp;and nbsp;—. nbsp;nbsp;Therefore

4 nbsp;nbsp;nbsp;nbsp;nbsp;t) nbsp;nbsp;nbsp;nbsp;nbsp;,12 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;12

and — c''^.

or b^ and becomeand or bbb'i ‘ ^ and quot;TtJ ' Ex. 2.

Eet bquot;^ and äi)\^ be giuen.

■— and — are reduced to and —•. / y

Therefore b'^ and 7? become and ~dc\ or Ißi and or and

3-_______ nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;3_______________ Let a-Eb., and \/aa-^xx be propofei.

Thcfe are lt;7 ^^ and aaxx'^. The indices are reduced to and Therefore the furds

become nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;and aa—xx nbsp;nbsp;; oz

ß’-j-3Äß^ 3i3Z’^ 7’'^, and 7 —or --__ nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;6---- - ■ ,, v/lt;2’ 3Äiz/’4-3aZ’Zgt;4-^’, and ^a*—za'xx-j-x^

PROBLEM XXIIL Lo reduce furds to their moßßmple ternis^

V. \] L E.

Divide by tlie greatcit power contained in it, and fet the root before the furd containing the remaining ■ quantities. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;Ex.

JLx. I.

Reduce y/ jS to aßmpler form. \/48=y/3Xi6 3= 4v/3 the lord required.

£;r. I.

Ret \/häfttübc be propofed.

6400 — 8ö. Then s/bj,aabc — ^a^/bc.

3-

Reduce a^x—

Here y/aa — a., and the furd becomes ay.ax-^xx^'

or «y/öX—XX .

The M i.

! aa—iLa-\-i,bb nbsp;nbsp;nbsp;nbsp;a- zb

•J nbsp;nbsp;nbsp;nbsp;7c =

, nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;a—ïb

becomes x y/^^,

PRO-

PROBLEM XXIV..

l'a find whether two furds are commenfiurable cr not.

RULE.

Reduce them to the leall common index, and the quantities to a common denominator, if fractions, except when like terms are commenfurable.

Then divide them by the greatell common divifor, (or by fuch a one as will give one quotient rational ;) then if both quotients be rational, the furds are commenfurable ; otherwife not.

I.

* 8 and ^8 be propofied.

Thefe are v/2x9 and 2x4. Divide by 2, and the quotients are ^/q, and ^4. ; that is, 3 and 2 ; therefore they are commenfurable.

Ky. 2.

Let the fiurds be \/^ and \/T2l. ^5'

Thefe are and nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;Divide by 2,

and the quotients are 5/25 and 5/36, that is 5 and 6 ; and the furds become —v/z and -^y/2 5 and are therefore commenfurable, bcinv^ as 6 to nbsp;—.

5

3-Let v/qS and ^/S be prop of cd.

Divide by 8, the quotients .are ^6 and i therefore they are incontmenfurable.

Ex.

Ex, 4.

vide by and the quotients are 1 and c5 that is, Z-è and cc -, therefore the furds arc commenfurable.

______5-

Suppofe \/a^-Jf-aabb and \/aabb-\-b\

Thefe are aay.aa-^-bb,, and nbsp;nbsp;nbsp;bby_aa -f bb.

Therefore dividing by aa^^-bb., the quotients are \/aa,, and \/bb,, or a and Zgt;, and therefore they are commenfurable.

Ex. G, Let nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;and nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;.i^^n

I4Zgt; nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;«Z» given.

Divide the de-nominators by 2, then they are reduced to v'yÄ .^^b nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;are therefore

incommenfurable.

PROBLEM XXV.

7'0 add furd quantities together.

RULE.

Reduce quantities with unlike indexes, to thofe of like indexes.

Alfo reduce fractions to a common denominator, or elfe to others that have rational denominators (or numerators).

Then reduce the quantities to the fimplcft terms (Prob. 2 3.) ■ This being done ; if the furd part be the fame in all, annex it to the fum of the rational parrs, with the fign (x) of multiplication.

If the furd part is not the fame in all, the quan. tides can only be added by the figns 4- and __.

Ex. I.’

Add s/G to 2^6.

The fum isT^a x v/6 or

Ex. 2.

Add ^8 to \/

^8=2^2, and v^5O=5v/2, and the fum r=2-l-5Xv/2 = 7v/2

Ex.

--- nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;3 _______

Add V §00 to nbsp;nbsp;108.

33 nbsp;nbsp;3

\/5Oo = V/4X125 = nbsp;nbsp;nbsp;nbsp;nbsp;And

v/108 y/=^3v/4- Therefore the fum = 5 3 Xx/4 =

Ex. 4..

Add nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;to ^A^aabi.

yhzy ai-e reduce_dj2_4r’^^v/3^ and ab^

And the fum nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;x s/3^.

Ex.

Seft. Iir. SURDS. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;amp;i

£x. 5.

_ Given and y/a^.

4^ — 4« nbsp;~ 4«ï — 16a« * — \/16aa —

2lt;y(ia. A.rïd nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;a^aa. And their fum

----4__'

— ^ 2X\/aa ~a-\-2 Xs/d-

Ex. 6.

Thefe reduced to a common denominator, be-come

o

that is, 6v/~ and 5^/ L, nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;fu^ is

Or thus^ •

’,1

ta •ï.'i'

Thefe become \/-L nbsp;nbsp;,

4 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;jcrf

and

s U R D s.

and 4vZ—[_ that is nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;and — v/— •

4X27 nbsp;nbsp;nbsp;4

whofe fun,îs 1 -3 nbsp;nbsp;4

Or thust

4 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;108

n 4Z ' nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;/ 64 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;/I nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;Ä ’ y T

and V— — nbsp;nbsp;nbsp;nbsp;nbsp;— 4 V ——- —

io8 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;4x27 —

, And the fum — —gt;/—. 3

£»•. 8.

g.

Add s^ccddaa—ccddxx^2 to \^d^aa—d^xxlt;^ï-

They are reduced to nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;— xxVz, and

ddy/ag—xx^z, and the fum is

lt;■34-33 X nbsp;nbsp;aci—xxlt;AZ.

Ex. IQ.

3

To aa — nbsp;nbsp;nbsp;4.

Add ^aa 4- ly/a

Surn 3\/ûô — v/^’d-t/i 2 4-2v/lt;7—\/l'

PRO-

Seel. III. SURDS. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;63

problem XXVI.

To fubtra^ furd g^uantitiei.

RULE.

Reduce, as in the laft rule; then fubtraól the rational quantities, and annex the difference to the common furd, with the fign ( x ) of multiplication.

Examples.

__£2_Subtraél y/G from 2^/6, the remainder is 2—I X nbsp;nbsp;= ^6.

• 2. v/50-VS = 5^^ — z,/! - 0^2. 33

• 3. v/'5oo-v/.o8 = 5s/4-gt;-3gt;4 zx 2gt;4.

• 4. = 4öÄv^3^ — ab^

— ^a—ab % s/^b.

5- v/lt;— v/4a_= nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;_

^4 nbsp;nbsp;nbsp;nbsp;nbsp;'SURDS. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;B. I.

PROBLEM XXVII.

2ö multiply fur ds.

I RULE.

Surds by furds ; if they have not the fame index already, reduce them to the fame -, then multiply the quantities under the common index.

Ex. I.

Multiply y/5 by y/3- nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;,

the produól x/r^

Ex.

Ex. 3.

Multiply

Reduced to^ and ; the produél

——\/aabbd^.

Ex. 4.

Multiply by a^, nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;1

Produd nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;= a''^.

i RULE.

A furd by a rational quantity ; conned them with the fign ( X ) of multiplication ; or elfe reduce the rational quantity to the form of that furd, and multiply by Rule i.

Ex. s-

Multiply \/4a—by ïa.

The produd is nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;—3x.

Or 2a — nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;then the produd zx

\/ i6a‘—i2aax.

3 R U L, E.

When rational quantifies are annexed to furds ; multiply the rational by the rational, and the furd by thç furd.

Ex. 6,

Multiply nbsp;nbsp;nbsp;\/a—x by t—d x/e.x.

The produd-- x/~^xxax

—nd___ a^'iX'—axx.

F

Ex. J.

. amp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;----- ^^ax'^

Multiply ~-^igt;yax by b—xV

Here ~i^\/ax — — y. ax — ■

And nbsp;—x^/ nbsp;nbsp;nbsp;zz b—x x

~~b^^\'

a .—. J . nbsp;nbsp;nbsp;nbsp;,---- , aaxquot; s

X ô’x’V X into Z»—X X zz

■ nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;-1 ■» nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;6

ab—ax nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;ab-—axy/a'gt;x9 _

b bb - i, Ib ~

6

ab—ax^ f a'^x’' nbsp;nbsp;, nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;, „

—]gt;— ~bb' produit.

8.

Multiply (i-\-^b—d

aa-i-a^/â—ad

prùduft aa—ad—b-\-dy/b

Ex. 2-

Multiply 2a—^a^yd h ^'^—ïc^yd

6ac—gac\/ d

—A.ac\/d-}-bacs/dd

product 6lt;?f—i'^ac\/d-\-6acd

Ex. IO.

Multiply \/a—y/b—^\ h s/a-\-Vb—'y/'i

produdh u nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;—V^â-

a\/^—3» or \/4a—^ 4-^/3.

Schol. If impoffible or imaginary roofs be multiplied together, they always produce —, other-wife a real product would be railed from impofli-ble fadhors, which is abfurd. Thus, s/—«Xv/—b — \/—ab., and .ç/—a X —\/—b — '——^b, nbsp;nbsp;nbsp;nbsp;nbsp;Alfo nbsp;—a X \/—a — — a ,

and v/—a x —..y—a — -j- u, ôcc.

PROBLEM XXVin.

’To divide furds.

I R U I, E.

the fame fimpie quantity; fubtradh their indices from each other.

Ex. 1.

Divide by quotient

Ex. 'i.

1 I

T^ivide a'‘ by ’

I r nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;w—'1

quotient ß” — a”’'”

2 RULE.

2 RULE.

If they be different quantities ; reduce them to the fame index, if they are not fo already. Then divide the quantities under the common index.

Ex. 3.

Divide by 5} gt;5 the quotient.

Eat. 4.

Dm* nbsp;nbsp;nbsp;nbsp;nbsp;iy

c ' ZC ^ab\'\aaà( .qatf

Ex, 5.

Divide ^aabbd' by t^d. 6 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;s ____

V« — \/d\ di}aabbd' {-^aabb —\/ab t quot.

3 RULE.

If rational quantities are annexed ; divide rational quatities by rational quantities, and furds by furds.

Ex. 6.

Divide “tyi6a’—i2aa.x by 2a.

quotient —t laax

gt;/ 4«—jjv.

Ex.

Ex. 7-

, ac—ad -----— 7 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;„ ~'y

Divide ---T—\/aax-—axx by rV

1.0

ïb ' zb \ '

Then X = quotient.

£x. 8.

». 1 ,. ßb “quot;Oix Divide -—I— \.b

Then the quotient

ÛX /ß’ ax , a ß /

-- nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;= -b^ -b

Ex. g.

aa^^ad'-^^bquot;^d^^b {^a^^^^b'~'d aa—a\/b nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;quotient.

-\-a-yb

-\-a\/b—b

O O —ad

—ad-iç-dy^b

70' nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;i

S U R Ù S. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;B.j/ ä

a-^b-^abbc {ab—b^bc

a'^b-^-a'-bs/bc

—O' b\/ bc—abbc

—a’-bs/bc—abbc

• nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;I

. O

t .

Ex. 11.

Divide s/aa—b ^^ h

yaa—b-ir^^ f

aa—a\/b—

a^/b—^g—b ^^

\ nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;b——'^ y/g

O

Ex. 12.

'D^'^'îde \/ ; .^/hbca-k-\/aab—bc—-^abc

\/ : \/oc 4-\/O •

\/^(-\-\/a}^l)l}ra _j_ ^aab—bc—•■\/abc{..yba—.^bc \/bbca-\-^aab ’ nbsp;nbsp;nbsp;’ nbsp;nbsp;■''

O — bc—^abc ^bc—gt;yabc

Q

4 R U L E.

When the quantities will not divide, fet them down in form of a fraftion.

Ex.

Ex. 13.

Divide : bed -{-^abb : by gt;yab—\/e^c.

problem XX1Xgt;

’Eo involve furd quantities to any power.

I RULE.

Multiply the index of the quantity, by the index of the power to be raifed.

Ex. I.

Let ^2 /ȣ cubed.

\/'2’ =*2^’. Then 2’^^ 3 qj. 2* is the cube, that is 2’ or ^8 the cube of ^2.

2.

What is the fquare of -^^bcc.

quot;is/bec = 3 X Its fquare =

9 X bcc^ bbc*' — lt;^c\/bbc.

3-

What is the cube of a\/a—x.

ci\/a — x — a'y.a—-x ■ cubed it is that is, the cube nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;-b ^cix''—x\

2 rule.

If quantities are to be involved to a power denoted by the index of the furd root j take away the radical fign.

Ex. s. .âfüb

7c h’^^red.

Its fquare is

cc

Ex. 6.

mat 13 the cube of V^a^-b^ ^^b^abT. af'—bf-{-'ib\/ahb.

3 rule.

Compound furds are involved as integers, obferv-ing the rule of multiplication of furds.

Ex. 7.

3 \/5 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;'^^^4

9 3\/5^

3gt;X 5~l~5

the fquare 144-6^/5 Ex-

Ex. 8.

Let a—be cubed.

ca—as/b

—a^yb-\-b

aa—za^yb-xb a—^b

a'—iaas/b-\-ab

— aa ^b-\-ïab—b.yb

the cube —'3,aa\/bj^ab—b^b,

PROBLEM XXX.

quot;To extrait any root of a furd.

R U L E.

Divide the index of the quantity or quantifies, by the index of the root to be extrafted.

Ex. I.

Extrabi the fquare root of a'.

J 1__

The root

Ex. 1.

Extrabi the cube root of ab'’. The root is a^b^^ _

Ex.

mat i, the

The root it

»

Ex.

74 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;S U R D S. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;B. L

*

Ex. 4.

T’Ebat !s tbe cube vcot of \/aa—xx.

The root is aa—— aa—xx^zz^aa^—xx.

• 2 R U L E.

When the index of the root to be extraded, is the fame as the index of the power of that quantity ; take away that index, and the quantity itfelf is the root.

5.

TEbat is the fquare root of 3‘Æ*.

Anfw. ^a, the root.

Ex. 6.

TEbat is tbe cube root of ßax—

Anfw. nbsp;nbsp;nbsp;—^xx, the root,

• 3 rule.

Compound furds are extrafted as integers, due regard being had to the operations of fimpie furds. AVhen no fuch root can be found, prefix the radical iign.

Ex. 7.

For the f^uare root of aa—

aa—4«y/^- -4^ {a—iy/b aa

la—ts/b} o —4ö’^^-i-4iJ

—4a^Z'-|-4^

Ex.

Ex. 8. nbsp;nbsp;nbsp;nbsp;nbsp;______

What is the cube root of aa—Vax—xx.

-

Anfw. v/ax—^^ax—xx, the root.

PROBLEM XXXI.

7ö change a binomial furd quantity into another, R U L E.

This reduaion is performed by an equal involution, and evolution. Involve the binomial to the power denoted by the furd or furds, then fet the radical fign of the fame root before it.

Ex. I.

’To transform 2 4-^/3 to another.

Its fquare. 44-34.4^3-74.4^3

the fquare root, v/74-4\/3'

Ex. Ï.

Reduce nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;a univerfal furd.

Its fquare 2 4-3 2^/6 nbsp;nbsp;nbsp;5 4-2v/6

the root s//;4-2lt;/6.

Ex. 3.

\/a X be given to reduce.

The fquare a ^x~^.^ax the root

Ex. 4.

3

be given.

»/abb^b

^-Vis/aab-y 3

SUR D S.

Cor. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;5 and in ge

neral a~ b~ ~ VTX ^vIquot;.

problem xxxil

^0 extrait the fyuare root of a binomial (or rtfiàual) Jurdt A B, or A B j oy (rtnomialj Sxc.

I rule, /or binomials. lÉLThen v/Â B = -ir'_____ and^Â=5 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;- v/4z:2

ï

1? -r /:^ï±p zA—D nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;,

For if V nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;— be involved by

Prob. 29. it will produce A 4- s/ A A — D IX that Is A B, as it ought. And \/-

.A—D s/---- will allo produce A

I.

To extrabl the root of ^-{-.^20

Here Ar:;, Brzy/zo, and 5/' \/2^ = D.

Then the fquare root of

Ex.

Sea. III. SURDS., nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;77

Ex. 2.

=; nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;the root.

3-

To extraSi the root of

y/ A A — B B = s/ïÇf r: D=.5. And the

^9 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;'gt;9

root = -r y/— that is, 2 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;2 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;*

Ex. /if.

What is the fquare root of 6—2^/5.

Here v^A — B B =: v/36—20 = D ±: 4. ft J yA-bß nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;-A—D

And v/~~~ — and v/=: i.

And the root = nbsp;nbsp;—i.

Ex. 5.

ExtraSt the root of ।

\/ KX — B ’B = v/iTz: D = 4. And \/2i 4 nbsp;nbsp;A—D _ s/i\—4

2

/2 r— 4

2

Extrait the root of aa-^-ix^aa—xx.

Here A = aa^ B ~ 2x nbsp;nbsp;nbsp;—xx. Then

AA—BB — \/aa—4(3*;if’4-4x “ aa—2xx ~D.

• 1 hen ' nbsp;nbsp;nbsp;nbsp;nbsp;— aa—xx^ and nbsp;---- =: xx^ and

the root —x-^-^ax—xx.

Ex, J.

U^at is the root of 64.^8—.,

Then

\/AA—BB=D =

=^8.

2

And the root zz v : 3 \/^ — I \/2. (fee Ex. 2.) ; therefore the root — i ./2—

• 2 RULE, y^r trinomials^ Zee.

For trinomial, quadrinomial furds, (sfc. divide half the prcduól ot any two radicals by a third, gives the fquare of one radical part of the root. This repeated with different quantities, will give the fqiiarcs of all the parts of the root, to be con-nefted by and —. But if any quantity occur oftener than once ; it inuft be taken but once.

For. if A' jj-i-z be any trinomial furd, its fquare will be A’ y 2'- 2 ■-^2xz-{-7yx-, then if half the produ61 of any .«o reólangles as 2xyx2xz (or 2xy^') be divided by fome third 2yz, the quotient quot;i-x'-yz

~2ÿz “ nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;fquare of one of

the parts ; and the like for the reft.

Ex.

Ex. 8.

To extraEl the jquare root of 6 4-^8-—^12—VZ24.

Here

2^/24

and nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;-

2v/8 nbsp;nbsp;nbsp;-

Ex. g. «, To find the fquare root of J2 \/ —\/ 484-^80—^244-^40—^60. „ nbsp;nbsp;nbsp;\/32X4-8 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;.

2^/80’ “ nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;produces no-

V A • nbsp;nbsp;\/22y48

thing. Again, —2.--- nbsp;=4/16=4. nbsp;And

2v/^4

\/4OX6o _ nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;^^2 4/4.0

quot;2^/ 7 -^^5 = 5 i and quot;nbsp;=v/4=2 i

J \/48X24 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;v^'J2Xî^o

See. therefore the parts of the root are ^^4, v^5, \/3» v^4ï amp;c. and the root 24-4/z—*/2

v^5 gt;nbsp;fo*quot; being fquared it quantity given.

Cor. 1. /k biywmials, if Y) be a rational quantity, the root will confili of two furds , and the farts of each under the radical ftgn will conßfi of a rational quantity (D), and a furd {ex').

Cor. 2. If both A D rational, the root will confiji either of the two fnrds, or elfe of a rational fart and afurd-, which is the only cafe that is ufeful in this exttaflion.

P R O-

PROBLEM XXXIII.

*ïo extras any root (f) of a binomial Jurd A B, or A—B.

RULE.

Let A A—EBrzD, take Q^fuch, that QD—bS the leaft integer power. Let quot;'^A B x v^Q—r, the neareft integer number.

Reduce Av^Q to the fimpleft form p^/s. n

Let nbsp;— t, the neareft integer.

2v/j

, nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;ts/s \/Its—n

Then the root =: ---------------, jf

v/Q, extrafted.

Note, 4- is for the binomial A B, and __ for the refidual A—B.

£x. I.

What is the cube root of ^9684-25.

Here D = 343 = 7x7x7- nbsp;nbsp;QX7’=«’, and

0 = 1, «=7. Then v^A Exx/Q^ ^^5^ = r = 4. Ay/Q, = nbsp;nbsp;nbsp;nbsp;— nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;and

7,.

—/I- -t-2. And 2v/2

Z^/=2v/2, Vtii—nzz.y/'i—7 = 1. \/Qj=i.

And the root —------ = Z\/2 4-1 i which

I

fucceeds.

Ex.

®-v'4374;

- «Î nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;5’X2’=4D

-~p\/s., and ^s — r.

for its cube is 68—2^y/().

Ex. 3.

Extraa the ^th root of

Here D = 3, n—'^^ QjzS t, r=^,

ac nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;1 o nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;5 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;V V

v/Q ——\/9' And the root to be tried

Scuoj^ivM.

If the quantity be a fraction or has a common divifor, extraft the root of the dcncminator or of that common divifor, I’cparately. They that would fee the demonftration of this rule, mav confult Gravefande’s or Mac Laurin's .Algebra. For as it feldom happens that -fuch quantities have a proper root; it is not worth while fpending any more time about them.

G nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;PRO-

PROBLEM XXXTV.

A compound furd being given, conßfling of iiao, three, \ or more terms, which are furd fquare roots : to find fuch a multipber or multipliers, by which multipl fing the given furd ; theprodubl will be rational.

R U' L E.

Change the fign of one of the terms in a binomial, or trinomial, or the figns of two terms in a quadrinomial j and by this multiply the given furd.

Ex. I.

£(?ƒ a ^3 le given^ Multiply by a—

produól aa—3.

• nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;Ex. 2'.

Given y/ —y/x.

Multiply by \/5 \/x produfl: nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;—x nbsp;nbsp;rational.

3.

V'5^K/i-v'2 Ic

Multiply by nbsp;5 ^3 ^/2

5 v/i2—\/io v/i5-l-3--v/6

---- --2

product 64-2^/15

multiply by —6 2v^i5 produd nbsp;nbsp;nbsp;nbsp;nbsp;60—36=: 2^---

S3 £x. 4.

There, is given nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp; v/lt;i

Multiply by •v/^ v/^ v/f— produa 0

Qr nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;ƒ nbsp;nbsp;0^ 2 de.

multiply by nbsp;nbsp;nbsp;nbsp;f-\-2^ab—2^dc

produa nbsp;nbsp;nbsp;nbsp;nbsp;ff-{■ y/-\-i^ab—^dc

S 4f\/ab multiply by j-

produa g — 1 (iffab

In this procefs f is put for the rational part «4-^—and for ƒƒ 4.4öZgt;—44c.

Cor. A binomial becomes rational after one operation, a trinomial after two, and a quadrinomial after three, iic.^

PROBLEM XXXV.

A binomial beifig given, conjifing of one or two furds, whofe index or root is any power of 2 ; to find a multiplier or multipliers that fhall make it rational.

R U L E.

Multiply It by its correfponding refidual (that is when one fign is changed) ; and repeat the fame operation, as long as there are furds.

Ex. I.

' Let nbsp;nbsp;a—be given.

Multiply by

produa nbsp;nbsp;nbsp;0—b nbsp;nbsp;nbsp;nbsp;nbsp;rational.

G 2

SURDS. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;B. 1

Ex. 2. 4- nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;4-

■L«/ v/5 \/3 propofed, * *

Multiply by v/5 —

I produéh

multiply by 5/5 v^3

2 produéh — 3=2, rational.

3’

S

there be given y/b.

Z

-Multiply by \/a—y/b

4

• 1 produól nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;\/a —\/b

4

multiply by \/a\/b

• 2 produit nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;— .yb

mult. by nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;\/a \/b

• 3 pfoduél nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;a — b rational.

Zx. 4. I 4 Zi?/ nbsp;nbsp;nbsp;nbsp;nbsp;Ö \/b be given.

Multiplier nbsp;nbsp;nbsp;nbsp;a — ./b

I prod. nbsp;nbsp;nbsp;nbsp;0« —\/b

mult. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp; y/b

2 prod. — b

Cor. quot;The nnmber of operaiionSy is eqtiaî to the pDV}Cf of 2 in the index.

P R O B'

PROBLEM XXXVI.

Any binomial furd being given, to find a multiplier •vohich ßoall produce a raiion..l produbi.

RULE.

If the furds have not the fame index, reduce them to the fame, (Prob. 21.)

Take the two quantities (throwing away the radical fign or index) -, change the fign of one of them. That done, involve thefe to the next inferior power denoted by the index of the root (Prob. 5. Rule 3), but leave out the uncise or coefficients : then place the common radical fign before each quantity, but after its fign. And this will be your multiplier.

Shorter thus,

« »

Binomial v/B.

« nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;« nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;n

Multiplier nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp; v/A”--B -p yAquot;-3B‘

»

^/Aquot;-4B’ -P amp;c. nbsp;nbsp;nbsp;nbsp;,

The upper figns muft be taken with the upper, and the lower with the lower ; and the feries continued to n terms.

Ex. I.

3

be given.

Multiplier — .^7X3

— Vl'X.l'if.S, — v/7X3X3

____ V/7X3X3 3

7 3 = .0, rational.-------

G 3

SURDS,

£x. 2. s nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;3

a—be propofed.

Multiplier lt;?a4-^v/2 produél nbsp;nbsp;nbsp;nbsp;a'^—2.

3-3

TjÉt \/a b bc propofed.

Mult. produôl a

Ex, 4.

£«/

4

reduced nbsp;nbsp;nbsp;nbsp;nbsp;4- ^^9, given.

^4- nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;■¹¹

Multiplier v/5’— v/5^X9 s/sX9^'^\/^ produâ. 5~-9 — —■4;

Or thus^

¹

* Surd 5/9 v/5.

produél 9 nbsp;nbsp;nbsp;——4.

Ex. Z'. 4 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;4

Let nbsp;nbsp;nbsp;nbsp;nbsp;'— ^yb' ie given.

Multiplier ^^5 4- nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;4- \/a^b^ 4- ^b’).

Or

----4quot; aab^ 4“ b^a^lb-]-ib^b produéT;;^;;:^—---

Ex. 6.

\/a — y/b be propofed.

reduced to '✓

put jf—a’, y — bb.

Surd 5/x _

mult, 4- '‘y/x^y^ 4-6

_____ nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;\/4“ \/y^,

produéTx—y - a,^bb.'~

problem XXXVII.

y? fraSlion being given vühoje denominator is a compound furd i to reduce it to another whofe denominator is rational.

RULE.

Find fuch a multiplier (by Prob. 34, 35,01 36), as will make the denominator rational. By this multiply both numerator and denominator.

Ex. I.

0

1«

3Xv/54-5Ä nbsp;nbsp;nbsp;_

v^5-V2Xvquot;5W2 ■quot;

^,/5

G 4 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;I-v.

SURDS. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;B. I.

Ex. 2.

Let there be s^men -y/1 nbsp;nbsp;nbsp;nbsp;3 '

Makiply both terms by v/z-v/t. the fraffion becomes

7—3=4 ’

3-

Sufpofe 3—z

Multiply by 3 v/2, then nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;is the

fraélion required. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;'

Ex,

' T nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;—^‘\/bc

ö v/Z-c propofed.

Multiply by a—then is the fradion fought.

5-

Let

5-.y^ nbsp;nbsp;nbsp;be,given.

Multiply .by 5 ^3 .

1 Gy/b 4- 3 y/gfl 2 v/3Zlt; ‘2-5~3—ï'2-

Ex. 6. '

. r ’o

Suppofe Ï

Multiply by v/7‘ v/7X5 \/5\ and the frac-^

tion becomes I£Z4942ogt;35 tov/2c _

, nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;3 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;,

5v/49 5v/35 5v/25.

Multiply by 5/5’—s/5*.3 gt;5.3‘—5/3S And the fraction is

Or thus^

Multiply the terms of the fradion -7 nbsp;—5^—

\/5 ^^3

by v/5—\/3gt; and it becomes

again multiply the terms of the laft fraction by y^5 v/3» and it becomes

5’^—5’3’ 3'5*—3^ . .

■ nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;lt;nbsp;/0.0.

Ex. 8. g 'Lit —,---—— be the fraßion.

Multiply by \/3 v/2—i, and the fraaion will ' 8v/3d-8\/2—8 nbsp;nbsp;nbsp;4\/:î 4v/2—4

be nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;= riV6 •

multiply by —2 ^/6, and it becomes

• — 8v/ 3—8\/ 2-1-8 4'\/ ’84-4^/ J2—4\/6

• '

— 4 2^/184-2^/12——4v/3—4v/2 rx 4 6^2 4v/3 ““2^6—ät\/3

= 4 2v/2

SECT.

90

SECT. IV.

Several Methods of managing Equations.

An Equation is the mutual comparing of two equal quantities, by the help of this character the part on the lefthand is called the firft fide of the equation -, that on the right, the fécond fide. And the fingle quantities are called terms of the equation.

An equation is either two ranks of quantifies equal to one another, and feparated by this mark ( —); or one rank equal to nothing. And they are to be confidered either, as the laft conclufion to which we come in the folution of a problem ; or as the means whereby we come to it. In the firlt cafe, the equation is compofed of only ons unknown quantity mixed with known ones, and may be called the final equation. But thofe of the laft fort involve feveral unknown quantities} and therefore they are to be fo managed and reduced, that out of all the reft there may emerge a new equation, with only one unknown quantity, which is that we feek. And diis is to be made as fimple as it can, in order to find the value of the unknown quantity.

An equation is named according to the dimerr-fion of the higheft power of the unknown quantity in it. A ftm-ple equation is that which contains only the quantity itfelf ; as a~b—c. A quadratic equation, is when the higheft power is a fquare, as aa—ba—d. A cubic equation, when the higheft power is a cube, as a^^ba'’—ca—d. A fourth power when the higheft power is Inch, as

—^a'-\-azz.d, amp;c.

PRO-

Seót. IV. Managing EQUATIONS. 91

PROBLEM XXXVin.

To turn proportional quantities into equations ; and equations into proportions.

In the folution of problems, it often happens, that we have feveral quantities in geometrical proportion, which are to be reduced into an equation ; which will be done thus ;

RULE.

Multiply the extremes together for one fide of the equation, and the two means for the other fide ; or the fquare of the mean, when there arc but 3 terms.

On the contrary in a given equation, divide each fide into two faélors ; and make the two factors of one fide the two means -, and the two factors of the other fide, the extreams.

Ex. 1.

Jf a-.b ; : c 4-ƒ :d. Then adzzbc-^lf.

Ex. 2.

Let a-i^b K—b ; : ^^âa^xx : A = —^\lt;aa—xxl

Ex. 3.

V lt;td zz ic-Ybf. Then a : b : : c f : d.

Ex.

Manuging E QJJ A TIO N S. B, I.

4. ar-\-br nbsp;nbsp;nbsp;nbsp;ca—cb nbsp;_____

— = —2 Vaa—xx.

Then nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;: r

•gt; nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;u

r c ___ or a b : a-\-b : : — : nbsp;nbsp;nbsp;nbsp;aa—xx.

Ex. 5.

Eet bc-\-bä — da—eg.

Then i : nbsp;nbsp;nbsp;: c i/ : da—eg.

or b : \/da—eg ; : \/da—: c-j-d.

PROBLEM XXXIX. reduce an equation.

Wlien a queftion is brought to an equation, the • unknown quantities are generally mixed and entangled with the known ones ; and therefore the equation mud be fo ordered that the unknown quantity may Hand clear, on the firft fide of the equation ; and the known ones on the fécond fide. Which is done thus :

I RULE.

When any quantity is on both fide the equation, throw it out of both.

Ex. J.

]f o^x-Eiib — 4c—d -f- ^b.

Throw out 6b. Then 2X—4C—d-E^b.

2 RULE.

hen the known and unknown quantities are both on one fide ; tranfpofe any of them to the contrary fide, and change its fign. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;£x.

Sect. IV. ' Managing EQUATIONS. 93

Ex. 2.

If 5 jf 3^ = rx M.

Then 5 X =: rx 4- bd —

And 5 X — rx — bd — '^b.

For to tranfpofe a quantity with a contrary fign, ÎS the fame thing as to add it, or elfe to fubtraft it from both fides ; therefore the quantities on each fide, remain ftill equal, by Axiom i. and 2.

3 RULE.

If there be fraftions in the equation,* multiply both fides by the denominators.

• 3 - nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;ur .. .

aa nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;dx nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;; 1

Suppofe -J nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;,

hd'if

Multiply by a a cb — f multiply by æ, a’ q- bca — bfa — bdx.

This procefs is plain from Axiom 3.

• 4 RULE.

When any quantity is multiplied into both fides of the equation, or into the higheft term of the unknown quantity j divide the whole equation thereby.

Ex. 4.

If jba^ bcaa — bcda..

Divide by ba, jaa -{■ ca — cd.

J* ■J K

divide by 7, q

7

The truth of this appears by Axiom 4.

5 RULE.

Managing EQU ATIO NS. B.

• 5 R U L E.

If the unknown quantity is^affeéled with a furd t tranfpofe the reft of the terms ; then involve each fide according to the index of the furd.

5.

If v/aa—ba -Oc c zz. d. Then y/aa—ba — d—c. fquared aa — ba =; dd—2dc-{-cc.

This procefs is plain from Axiom 5.'

• 6 RULE.

When the fide containing the unknown quantity is a pure power ; or if being adfefted, it has a ra* tional root : then extract fuch root on both fidcs of the equation.

Ex. 6.

If 0’ zzbi bbc.

Cube roof. 0 =:

Ex. •].

If ** 6x 9 —tab.

Square root nbsp;nbsp;* 3 =

and nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;X =: ±\/2ob—3.

Schol. All thefe rules are to be ufed promifcu* oufiy, as one has occafion for them i till the equation be duly cleared.

PROBLEM XL.

To explain the nature and origin of adfebled equations,

I. Any adfeded equation may be confidered a» made up of as many fimple equations, as the di-menfiefl

Scft. IV. o/ EQUATIONS. 95 menfion of higheft power is. Suppofe

amp;c:c. then a—azzo, a—b—o^ xgt; c—o. AncMf all thcfe be multiplied together, then X a x a—b x x—c =.0 -, that is,

—abc—Of a cubic equation,

■—b -\-ac

whole roots are a, b, c.

In like manner, x—a y, x—b y, x—c x x—d—Ot produces a biquadratic equation,

*♦—a nbsp;4-a^ X'- — abc x abcd~o.,

—b nbsp; af nbsp;—abd

c nbsp;nbsp;-^-bc nbsp;—acd

“•~d nbsp;nbsp;-{-da nbsp;—bed

-{■db

-{-de

whofe roots are a, b^ d.

Thefe two equations may be written or denoted thus, X’ — px* nbsp;nbsp;— r zz Q and

x jx* — rx -f- s zz o. And any fuch • e-quation being found in the folution of a problem j the bufinefs is then to refolve it into its original compounding fimple equations, and fo to find the roots a, b., c, ècc. For each of thefe fimple equations gives one value of x, or one root. And if any one of thefe values of x be fubftituted in the equation inftead of x, all the terms of the equa^ tion will vanifh__and be zzo. For fince

X—a X X'—b y. X—C2_^S' nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;It is plain, when

one of the faftors x—a is zzo, the whole pro-duft will be zzO. And of confequence there arc three roots in the cubic equation', and four in the biquadratic ; and in general there are as many roots, as is the dimenfion of the higheft pov/cr in it, and no more.

2. If

$6 Nature of EQUATIONS. B. 1.

• 2. If it happen that the roots a, b, c, ècc, be equal to one another, then x—a will be =0, or X—a nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;amp;C. and ä'—a is had by evolu

tion, fince the given equation is generated by involution.

• 3. That there are no more roots than thefe is plain i for if you put any quantity, as ƒ for x, which is equal to none of the roots b, amp;c. Then fince neither ƒ—f—b^ nor ƒ—c, amp;cc, is o, their produft cannot vanifh or be rzo, but muft be fome real produél j and therefore f is no root of the equation.

• 4. Since the fquare root of a negative quantity is impoflible ; therefore if we have fuch an equation as this, XX q- aa—o, or xx — — aa, then which are two impoflible roots of that equation. So that a quadratic equation has either two impoflible roots or none. And therefore in any equation, there is always an even number of impoflible roots ; fince each quadratic that goes to the compounding it, muft have either two or none. Therefore no equation can have an odd number of impoflible roots. Hence therefore -the number of real roots in a cubic equation, will cither be one or three -, in a biquadratic, four, two, or none. In a fifth power, 5, 3 or i ; iàc.

Ç,. From the foregoing equations it is plain, that the coefficient of the firft term (or that ol the high-eft power) is i. The coefficient of the fécond term (or next higheft power}, is the fum of all the roots, a, b^ c, amp;c. with their figns changed. The coefficient of the third term, the fum of the produds of every two of the roots. ï'he coefficient of the fourth term, the fum of the produds of every three of them, with contrary figns, iâc. The odd terms having always the fame fign, and the even terms a contrary one. And the

abfoluie

Sea. IV. Nature c/ E Q U A TIO N S. abfolute number is always the produA of all the roots together.

• 6. Hence it follows, that when the fum of all the negative roots is equal to the fum of all the affirmative, the fécond term vanifhes, and the contrary. And if all the negative reélangles be equal to all the affirmative ones, the third term vaniffies. And if all the negative folids be equal to all the affirmative ones, the fourth term vanifhes, out of the equation -, and fo forward.

• 7. But the roots of equations may bè either -p' or —, yet fcill the fame rules hold good. For let the fign of any of them as c be changed into — c that is, let jf rrzo ; then in the cubic equation the fécond term will be ■—a—b-^c ; that is, the fum of the roots with a contrary fign ; the third term will be ab — ac — be , that is, the fum of the produfts of all the roots j and fo of the reft.

• 8. Hence alfo in every equation cleared of fractions and furds, each of the roots, each of the rectangles of any two of the roots, each of the folids under any three of them, each of the produfts of any four of the faid roots, ^c. are all of them juft divifors of the laft term or abfolute number. Therefore when no fuch divifor can be found, it is evident there is no root, no reftangle of roots, no foiid of roots, ^äc, but v. hat is furd. For in the cubic equation, a^ b, c, and ab, ac, be are all of them divifors of the laft term abc : and fo of higher powers.

• 9. In any equation, change the figns of all the fitft ; then let the coefficients of the firft, fécond, third, i^c. terms be i, p, q, r, s t, v, Sic, refpedively. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;’

H

Then

9^ Nature of E QJJ A TIO N S. B. I.

Then obferving the figns, we (hall have

f rz furn of the roots, nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;Sic.

^A 2ÿ — fum of rhe fquares of the roots ècc.

;gt;B jA 3r = the fum of their cubes —a^ b^ amp;I.C.

/C^ÿB4-rA 4f zz the fum of the biquadrates»

Where A, B, C, t?f. are the firft, fécond, third» èfr. terms.

For ^zzö ^4-c amp;c. =:A.

Alfo /A or nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;z: a^ b^ c^ zab

2«f 2f4/=:B—2^. Therefore Bzz/gt;A 2j, amp;c.

To go through the calculations of the reft would be tedious, and of little ufe.

10. In equations of the third and fourth power, we find, when the roots are all affirmative, the figns are and — alternately ; fo that there are as many changes of the figns as is the index of the power, or as the number of roots. But if the roots are all negative, the figns are all throughout, there being no changes of the figns. Whence in thefe cafes, there are as many affirmative roots, as changes of the figns in all the terms, from to —, and from — to . And the fame rule holds in general, that is, there are as many affirmative roots in any equation as there are changes of the figns. But the equation is fuppofed to be compleat, that is to want no terms, and to have ‘ numeral coefficients. And likewife the number of negative roots is known thus -, as often as two of the figns , or two of the figns — ftand next one another, fo often there is a negative root. It would be needlefs to trouble the leader with the proof of thelc things ; fince it can only be done tn particular cafes, and not in a general way-

And

Sea. IV. Naiure of E QU AT ION S. nbsp;nbsp;99

And befides when impoffible roots happen to lie hid in the equation, they caufe the rule to fail.

• II. When the roots are all affirmative, the terms of the equation are alternately and — through the equation ; but when the roots are all negative, the figns are all ; and therefore, as by changing the figns of the roots, the figns of the alternate terms are changed ; fo on the contrary, changing the figns of the alternate terms, changes the figns of all the roots. And this holds in general, as will be evident by producing two equations from the fame roots, with contrary figns.

• 12. Since any adfeéted equation, as —px*-\-qx—r=.o, is made up of fimple equations, fuch as X—a—Oy X—^=0, amp;c. Therefore if one root as a be known, the whole equation may be exaft-ly divided by x—a ; and fo reduced to a lower dimenfion. Alfo when all the roots a, c are found out,' then will the continual produft of X—a, X—i, X—c, exadly produce the fame equation. It is no wonder that an equation has fevcral roots ; becaufe in fuch cafes, there are more folu-tions to a problem than one. So that in one cafe of it, X is =.a, in another cafe in a third

See. and they are all comprehended in the general equation. And hence though there be lèverai roots in an equation, yet only one of them will anfwer one cafe, or the particular queftion propofed. '

• 12. That any root fubftituted for x in the given equation, will make the whole equation to vaniffi, by deftroying all the terms, is proved thus. Let the equation be,

*’ — ax’’ -E abx — aie ~o.

— h -E ac — c be

H 2 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;And

i-

loo Nature of EQUATIONS. B. 1. And let the roots be a, b, c, as before. Then fiibditute any one, as lt;7, inftead of x, and the equation will become

0’— baa — abc—o, —baa caa —caa abc

Where the terms manifeilly deftroy one another. And the fame will happen, by fubflituting b or c, for X.

• 13. If the laft term of an equation vanifhes fas a b c^ nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;12), then one root will be o-, for then

the whole equation may be divided by the un-knt'wn quantity x or x—o. If the two lall terms vanifli {abx acx-\-bcx, and —abc)^ then two roots aie —o; if the three lalt terms vanilh, then three roots will be o ; ^c.

And (vn the contrary, if one, two, or three roots, Uc. be —o, the hit term, the two lall, or the three lail terms, àfc. ill vanifli out of the equation, and the remaining part of the equation will contain the reft of the roots. Thus in the equation, 7\rt. 12. if the re 's b, c be —o ; there remains only x'— «q-iÿ-i- nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;or ä—a—Q,

an equation containing tlr remaining root 0.

• 14. Änd in any power )f a binomial, if each term be multiplied by rhe index of the unknown quantity therein -, it will tlureby be reduced to the next inferior power. I'o prove this, we mull ob-ferve, that the coefficients of a binomial, are the very fame, whether you reckon forward from the beginning, or backward from the end ; that is, the hril and lall are the fame -, the lecond and laft but one -, the third and lall but two, yt. her the coefficients of any power of x-\-b^ are the fame as of Z’q-x. In the quad.atic xx-{-ïbx-\-bb, the

Sed. IV. Nature of EQUATIONS, loi the coefficients kire i, 2, i. In the cubic nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;they are t, 3, i. In the

fourth power they are i, 4, 6, 4, i. In the fifth power, i, 5, jo, 10, 5, I ; and fo on.

Therefore, let any power of x-\-b be denoted thus, x’-}-/;*-’'— -------4-

»■«—i.n—2 nbsp;, nbsp;nbsp;nbsp;nbsp;nbsp;n.n—i

• -*---- x'b'lt;-ï —-— nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;

nxb«-^ 4- nbsp;nbsp;; n being the index of the power,

and let m be that of the next inferior power, or m~n—1. Now let each term be multiplied by the index of x in each term ; that is, by «, «—2, amp;Ç. and we ffiall have

-- nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;1 . H— 'J

—1 . x”“'/’ 4--' x'^—'^bb

_____ 2

n . n—I . nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;_

------ x'b’‘—i 4- n . n—I . x^'b”-^

• gt;1x1”—^ 4- o. And dividing all by «x, it becomes

«—I . x’—'^b 4-

. «—2 . n—■? ' nbsp;nbsp;nbsp;—-----x^-^b\ nbsp;nbsp;

--

—I . n—2 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;----

~--x'-b^'-i 4- n—1 .xb’^—'- 4- nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;-, that

IS, reftoring w, x^ m .’'‘—^b 4- nbsp;nbsp;nbsp;nbsp;nbsp;—^-x”—^bb 4-

---- nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;2

1 . ƒ---X x'':-îb\ Ac.....4---------

which is nianiithly the /«'* power of x4-^.

IT 3 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;15- Aho

102 Nature of EQ^UATIONS. B. I.

• 15. Alfo if the equation refulting from the Jaft operation be taken,' and its feveral terms again multiplied by the index of x in each term ; it will be reduced to the next power below that, and fo on for more operations. And therefore after each operation one root will be deftroyedj or fo many roots will be deftroyed as there are operations, and the reft will remain.

• 16. And further: If there be feveral equal roots of one fort, and alfo feveral equal ones of another fort, in any equation. And if the terms of that equation be multiplied by the feveral indexes of the unknown quantity in each term ; an equation will arife wherein one of the equal roots of each fort will be deftroyed. And in general, whatever roots there be in any equation, if the terms be refpec-tively multiplied by the indexes of the unknown quantity therein, an equation will come out where' in one root of every fort will be deftroyed, whether there be equal roots, or al! different. But thefe things being of little confequence, I thall not de* tain the reader any longer about them.

• 17. As impoffible roots are fuch as .ire produced from the fquare roots of negative quantities ; fo impoffible equations are thofe produced from impoffible roots j as this equation —^a^^aa-^- loa -1-22—0, which is produced from thefe two, aa-i-2a-l-2=o, and aa—6a-l-11 —o -, the former produced from a-f-i-l-v/—i, and 0-I-1——i; and the latter from a — 3 4- nbsp;nbsp;— 2 , and

fl—3—y/—2. I'helefort of equations have roots that are barely impoffible.

Likewife, there are equations that are doubly impoffible, or impoffible equations of the fécond degree. And thefe are produced from equations involving two degrees of impoffibility, as this fl*-I-4«’ Saa-1-80-1-5=0, which is produced from the

Scét. IV. î^ature of EQUATIONS. 103 the equations, aa 4- 2a 2 — i =0, and «0 204-2—y/—i~o. Such as thefe cannot be reduced into rational quadratics, as the other may.

PROBLEM XLI.

^0 increase or dimini/h the rooti of an e^uation^ by any given quantity.

RULE.

For the unknown letter fubftitute a new letter, — the given increment, or 4- the given decrement. And fubftitute the powers thereof, in the equation, inftcad of the powers of the unknown letter.

Ex. I.

Let nbsp;—px'^-}-qx-i—r~Q, be given and let the

robots be lefenedby the quantity e.

Suppofe yzzx—e^ or jf=:74-^. Then

= J’4-30'quot;

—px'- — —py^ nbsp;nbsp;'^p^—pe'’

4-î* = nbsp;nbsp;nbsp;nbsp;nbsp;nbsp; nbsp;nbsp; je

—r =.

is the equation required.

£y. 2.

Intreafe the roots by 4, «ƒ this equation a’4-a’—xolt;»-f-8:z:o.

Suppofe «4-4=^» or az=.e—4,

Then nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;—i2e*4-48f—64

4-«* = ee r— 8^4-16 ioazz .—10^-1-40

4- 8 :z: nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;4. 8

— life - -30? * :zö, the equation required; reduced, «»—,1^4.20=0, a quadratic.

H 4 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;Cor.

104 Nature of EQUATIONS. B I.

Cor. I. The lafl term of the transformed equation, is the very fame as the equation given, having e in the place of r (in Ex. i.)

Cor. 2. hî^hen the laß term vanifhes, the number effumed —4, Ex. 2.) /j one of the roots in the equation propofed.

Sihol By this rule, all the roots of an equation may be made affirmative -, by increafing them by a proper quantity. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;I

PROBLEM XLII.

‘To multiply or divide the roots of an equation, by a given number or quantity. •

RULE.

Affiime a nfew letter; and divide or multiply it by the given number; and fubftitute its powers in the equation, inllead of the unknown quantity;

Multiply by 3, /Z’z'j equation y'--^_y— o.

3 ~

Suppofe y—— z, then fubAituting

Z' 4- . 1'46 we have------ y——7- —i -

2’—I2J—146—0.

Ex. Z.^ - -, nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;: J

Divide by \/2, the equation'

Let X—j’^3, which put for x, we have or 3y—27 1=0.

Cor. Ry this ru’e, fractions or funds may be taken out of an equation ; by dividing the new letter by the common denominator ; or by multiplying the new letter by the fund quantity. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;P R O-

Sea. IV. feature of EQUATIONS. 105

PROBLEM XLIII,

• To change the roots of an equation into their reciprocals.

RULE.

In the given equation, inftead of the root, fub-Ritute a unit divided by fomc other letter.

Example.

‘ Let 3^’—2y-l-i—o, be given.

Put 7 = -r, then — I =0.

reduced 3—2z* z’ — o. or nbsp;nbsp;nbsp;nbsp;z5—2z‘ 3 = o.

^chol. this rule the greateft root is changed into the leaft, and the leaft into the greateft, ^c,

PROBLEM XLIV.

To compleat a deficient equation.

Kn equation is compleat, when it has all its terms, or thofe containing all the powers of the unknown quantity -, and deficient, when any power is wanting.

RULE.

Increafe or diminifh the roots of the equation, by fome given quantity (by Prob. 41).

Example.

Suppofe nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;—5—o, deficient.

Let e iz=.a, then

~ e’-hsef seq-i

4-23 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp; 2e4-2

—5 =_______

5^—2=0, compleat.

Schol.

Schel. An equation may be rendered com pleat, by multiplying by the fame letter with fome quantity added, as «4-1 ; but then it raifes the equation a degree higher.

PROBLEM XLV.

ie'prefs an equation to a lower dimenfion one of its roots being given.

Ï RULE.

Put the equation ~o, and divide it by the unknown quantity — the root given.

Example.

Given atq-Æ’—rorf-j-8r:o, one root a——4. ö 4=^o) ö’ ö*——3Æ-I-2—o the a’ -b4a* nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;equation req.

—-3a’—toa

——12«

-|-2a-|-8

-p 2a 4“ 8

o

2 R U L E.

Put a new letter added to chat root, equal to the unknown quanrity ; and fuhliitute that and its powers in the equation.

Example.

Let nbsp;nbsp;nbsp;nbsp;nbsp;—io«4-8=o, be given., and a—’—iti

Put nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;Then

«’ nbsp;nbsp;nbsp;f’--i2f‘4-48/?—64

nbsp;— 4- ee ^8^4-16

’~-io«z:: nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;__io?4-40

=___8

z; e'^—iif*4quot;3^^quot;b° e^—iie 4- 30 =0.

Sea. IV.' EQUATIONS. fö/

problem XLVI.

To find how many roots are affirmative^ and how many negative^ in a given equation.

RULE.

Range the terms of the equation according to the dimenfions of the unknown quantity. And if the equation is not compleat, make it fo by Prob, 44,

Then obferve how often follows —, or — follows , that is, how many changes of the figns there are j and there are fo many affirmative roots in the equation.

Alfo, as often as two like figns ftand together, fo often there is a negative root.

. Given a*—X’—19XX 49X—30=0:

Here the figns are 4- — — —» and there are three changes ; from the firft to the fécond, from the third to the fourth, and from the fourth to the fifth term : therefore there are three affirmative roots. Alfo, in the fécond and third terms, two negatives ftand together, and in none elfe, confequently there is one negative root.

2.

' Sttpfo/e x 5X’—7Jf*—29x4-30=0.

The figns are 4- 4- —- — roots nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;neg. af. neg. af.

So there are two affirmative, and two negative roots.

Ex.'

xos ƒgt; nbsp;nbsp;nbsp;‘■l'ranfmutation of

. ' £ nbsp;nbsp;nbsp;nbsp;nbsp;£x, g., nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;, ,

Let^ the^ equation be^ a'^—ya ózzo. This equation being defedlive is to be complcated.

* gt;---7ö- .6—o.

mult, by a i' hz ‘o. r nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;-------—---

*—?«' ..

V = Ui 'A

«♦ a’ — 7a’ — « -I- 6—0.

‘ So there are two affirmative, and two negative roots in this laft equation,' and one of the negative roots being —1, (by the multiplication of a i zzo,) therefore, the given equation contains two affirmative roots, and one negative. ~ nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;‘

The reafon of this rule appears from Art. 10* Prob. 40.

.0'

5 ,___ Scholium.

• This rule does not'hold good, if there be- im- ' poffiblc roots in the equation -, except fo far as thefe! impoffible roots may be taken for ambiguous ones, that is, for either affirmative or negative roots. As in- the-equation x’—óx' igv—rowhich ’ by this rule gives three affirmative rootsj'but,in ’ reality it has but one root, which is 2, the reft are imaginary.

There are alfo fome rules whereby to judge how many impoffible roots' are in an equation,, but they are fo very tedious, and of fo little ufc, that I ftifill not trouble the reader .with them. See Vniver/al Arithmetic^

ScÄ. IV. E Q^U A T I O N S. '109 PROBLEM XLVII.

To change the affirmative roots into negatives^ and the negatives into affirmatives.

RULE.

Place cyphers for the deficient terms, if there be any ; then change the figns of all the even terms, that is, of the fécond, fourth, fixth, amp;c. terms of the equation.

Ex. I.

Given .x'4-8^ 24—o.

That is, »’ 04-8x4-24—o. transformed x’—o 8x—243:0.

In the given equation xzz—2, in the transformed equation »zz 2.

Ex. z.

Suppofe »*—4%’—i9X‘ io6x—i2Oz:o. transformed X 4X’—19»*—iu6x—120—0. In the former equation the roots are 2, 3, 4 and —5 -, and in the latter 5, ——3, and —4.

The reafon of this procefs is plain from Art 11. Prob. 40. and may be demonftrated thus. In the given equation, we have x for the root. Now fuppofe —X to be a root. Let this be fubfiituted in the given equation, and it produces —x'—8» 24—o, that is, x’ Sx'—24—0, as in Exam. i. And » 4X’—19»‘—io6x—iz.o—o, as in Exam. 2. For it is plain, all the odd powers of x will now be negative, which before were affirmative, the rert remaining as before. Whence the figns of all the odd powers will be changed, according to the rule.

S E C T.

.ÏIÓ

s E C T. V.^

Ranging the term j quot;working by general for mi j Jùbflitution and reflitution j taking aquot;way any term of an equation j extermination of w known quantities -, the defignation of quantities by letters ; regiflering theßeps.

PROBLEM XLVIII.

2c the terms of an equation, or difpofe of them in the beß manner for any operation.

RULE.

Th IS is done by placing thefe terms foremoft that contain the higheft power of the unknown quantity j and in the following places, thofe of lefs dimenfions ; fo that the powers in the fcveral terms may continually decreafe from the higheft, according to the fériés of the natural numbers. But in many cafes, the contrary method is to be followed, and the loweft power taken firft.

Ex. r:

Let az^ 2*——bquot;' o^ab^ —q.

Place it thus, 2 nbsp;nbsp;nbsp;nbsp;* * 4- nbsp;nbsp;nbsp;—q,

-^b nbsp;nbsp;nbsp;nbsp;^b\

Ex.

Suppofe Jf -4-ajf’ .\‘bx’’^^bx^ •{’CX—dx-’-^b^ ranged x 4-«A;’-t-Zgt;x* f3f-|-ö^’=o.

PRO-

Sea.V. .GENERAL FORMS.

PROBLEM XLIX.

To by a general form.

RULE.

Write down each letter or quantity in the general form, and after it (with the fign xx), each letter it reprefents in that particular cafe i which will give feveral equations.

Then caft your eye over the general form, and obferve the general quantities therein, and look for them on the firft; fide of the equations ; and what you find them equal to, on the right hand, write down, inftead of them, each one by one, till you have gone through the general form i and you will have the folution.

When the quantities are many, it will be the beft way to write down the general form firft, and the particular one under it, each quantity under its correfpondent j then it will appear by infpetlion what letters to fubftitute.

Ex. I.

l'a involve aa—xx to the ^th power.

This is to be done by the general torm in Cor. x.

Prob. 5. therefore we have

I* nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;a “ aa

e~—XX n — s i

Whence ö-1-e — aa—xx —aa -p ^y.aa X

GENERAL FORMS. ßj.

Ex. 2.

ExtraSl thefquare root of 28—300.

This is to be done by the form in i Rule, Prob. 32. Here A=28, ^—^^00, D=\/784—^^^^22, »A D _ ^28 22 _ nbsp;nbsp;nbsp;yA—D

2 nbsp;nbsp;~ nbsp;nbsp;nbsp;nbsp;2 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;2 nbsp;nbsp;nbsp;=

.28--22 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;,__

V—Therefore s/A—B = 5—^/3 the root required. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;’

Ex. 3.

^0 find a quantityi by 'which if y/2—^6 bemul-tipliedy the produbl will be rational.

This is to be done by Prob. 36.

Here »=15, A=2, Bzz6. 5 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;3

And the multiplier \/16 1^8x6 ^4x36 4-5

V^2X2i6 v^I296.

mult.v/i6 v^8x6 v/4X36 v/2X2i64-^i296 by v/ 2 —^6

.5 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;S nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;S nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;5

v/32 v^96 v/8x36 v/4X2io y/2592 555

—^96—\/8x36—x/4X2i6—

2—6 = —4. produót.

PROBLEME.

Jhorten the work by fubflitution and reflitution.

In any operation, when the quantities grow very numerous, or very much compounded, it wiH make the work very tedious ; and therefore it cught to be made fliorter as follows.

RULE.

amp;a.V. SUBSTITUTION, ù?c. 113

RULE.

Aflbme a new letter to reprefent or ftand for any humbef of given quantities 5 and likewife fome different letter to ftand for the coefficient of any power of the unknown quantity ; do fo for as many of the coefficients as are compounded. Likewife, put letters for the numbers concerned ; then work with thefe inftead of the original quantities, which will make the work eafier. And this is called Suifii-tuiion.

When the operation is over, each number or compound quantity mull be reftored again inftead of its letter j and this is called Reftitution.

Ex. I.’

Eet aa-\-ba—ca-^da—de.

Put s—b—c-\-d.. Then the equation becomes aa :a—de.

Ex. z.

f ----- -

a—zx'AVaa—xx

Put c—d—p. Then

-------- ,

—Z.X, ^aa—XX —

pxx-{-cx

multiply by pxx.^cx. Then

ß/gt;Ä'X—zpX'^ acx—2£xxx \/aii—xx = bx.

Put ap—ze—q. Then qxx—zpxgt; -{-acx nbsp;nbsp;nbsp;nbsp;aa—xx ~ bx.

acx qxx—2px Vaa—xx — bx.

Iquared nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;âa—xx—bbxx, Zzz.

where the values of p, be reftored.

I

PRO-

n4 SUBSTITUTION, ^c, -BJ.

PROBLEM LI.

take away the fécond term of an equation.

RULE.

Divide the coefficient of the fécond term by th® index of the higheft power ; annex the quotient, with its fign changed, to fome new letter, which fubftitute for the root, in the given equation.

Ex. I.

Suppofe a^-^aa—ioj4-^=o.

Put f--- a. Then 3

al -3 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;. !

Z I

lO

—icfl= nbsp;nbsp;nbsp;nbsp;nbsp;--IO « 4--

3

8 — nbsp;nbsp;nbsp; 8

I II

o ? — IO—II—»

3 tion required.

Ex. 2.

£f/3^j3 4-(j —o, be given'. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;'

Let y=:x4— —x4-2lt;i« •^4

then = x 8aA:’ 24a*x* 3'ï^’* ’oa*

' —807’= —8axi—48a* X*—^6a^x—640

a^ nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;___~P

o c:z* * — 24a‘x‘—64^’^—47quot;*=O’

Schol, Hence by this and the 43d problem, an equation may be found, which wants the laft te^m

Sea. V. EXTERMINATION. 115 but one. For if the fécond term be taken away by this problem, and the equation transformed by Prob. 43, you will have the equation required.

PROBLEM LIT.

7c take away any term out of an equation.

RULE.

Take a new letter for the root, to which add an unknown quantity ; and fubftitute this fum and the powers thereof, into the given equation. Then any term put equal to nothing, will determine the va-; lue of that aflumed unknown quantity.

Ex. I.

Suppofe nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;-J-^x^—gx — 2 =0.'

Put y-^-e — X.

Then iv* —

3*’ zz “-3^1’ —oyye ^eyye'^—^e’ f 3*^ ~ nbsp;nbsp;nbsp;nbsp;nbsp;nbsp; 3^7 nbsp;nbsp;nbsp;nbsp;nbsp;4quot;3^’*z—O'

—By —5« \

Then, if the fécond term is to be taken away* make jy’iszo or 4f=3 ; therefore czz—.

4

Ex. z.

7he fame fuppofed \ to take away the third term.

Here we fhall have nbsp;nbsp;nbsp;nbsp;nbsp;—9/lt;^ 3gt;:y~o; re-

* —°’ refolving of which qua-o-ivpe nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;gives the value of e. Then

vanilh. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;lo that the third term may

I '2

EXTERMINATION. B. I.

£x. 3,

7 he fame thing fill fuppofed-, to take away the fourth or fifth term.

For the fourth term, 4e’—9f* 6f—S'zzo, a cubic equation whole root is e-, and makes the fourth term vanifli.

For the fifth term, e^——5lt;? 2~o, a fourth power whofe root is e. Then y e—x, which fubftituted in the equation, makes the laft term vanifii.

Cor. T. Hence the third, fourth, fifth, amp;e. term, VIay he taken out of the equation ; by refolving a quadratic, cubic, fourth power, ice. equation.

Cor. 2. Hence if the laft term of an equation {ai e^—3i?'q-3f«—5^4-2) be —o, then one root {x} is —Q ; for then x—o, or x will divide the equation. If two of the laft terms be ~o, two values of the root will be =0, and fo on. But if the laft term does not vanifh, there is no root —o.

Schol. After the fame rule any term may be made equal to any given quantity j by putting the faid term equal to that quantity.

PROBLEM LIII.

T'ö exterminate a fmgle letter, or a quantity ef one dh menfion, out of fever al equations.

I RULE.

Seek the value of the quantity to be expelled, in two equations -, and put thefe values equal to one another.

Ex.

• I .

Ltt daquot;^by

• . and 2x y f lt;ixler:m;iate y. ■

By tranfpofing b, a-\-x—b—-\ and by tranfpo* fing 2x, j’:z3Zi—2x. Therefore «4-a— b~ ^b__ And by reduftion ‘^x—^b—and à'—l.'rZ--

3

Ex. 2, Let ax—iby—abl and xy - bb, î nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;y.

Here æx—ab — zby.^ andj=:--- zb '

air _ r c ax~algt; bb Allo J' = Y ; therefore nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;and re-

ducing XX—bx — ~.

• 2 RULE.

Find, by reduction, the value of one unknown quantity, in one equation ; and fubftitute that value for it, in all the other equations. Proceed thus with another unknown quantity,

£x. 3.

Zz/ æ4-a:=:2^—y~t and ^ax-yx=.d nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;gt;

By the firfl; equation yzzzb—a—x, put this value in the fécond equation ; then

3flx nbsp;nbsp;XX zb—a—X = d., that is, 30X—-

■j-xx—dy or 4Z7X—zbx-\-xx=.d.

Ex,

'118 EXTERMINATION. B. T.’

Ex. 4.

Suppoje gt;x-\-y-^z-=za

3j zz .v 22gt; to expunge z andy^ az — xy. j

By the firft equation, zzz^—x—y.

By the fécond equation, 35—*4-2«—2X—2j',