EQUATIONS. art by=c2 Ex. 2. Let { Let rty=a, and 2.1 +2y = 2 a; this To find r and y. latter equation being deducible from the formr-ny=ds From the first, mar + mby = mc mer, it involves no different supposition, nor from the other, mar — nay = ad requires any thing more for its truth, than by subtraction, mby tnay=mc - ad that r ty=a should be a just equation. mc-ad therefore, y = mbt na PROBLEMS WHICH PRODUCE SIMPLE Again, nar tnby =nc mb r--nby =bd From certain quantities which are known, by addition, na + mbox=nctbd to investigate others which have a given relatherefore, x = tion to them, is the business of Algebra. When a question is proposed to be resolved, we must first consider fully its meaning and If there be three independent simple equa- conditions. Then substituting for such untions, and three unknown quantities, reduce known quantities as appear most convenient, two of the equations to one, containing only we must proceed as if they were already detwo of the unknown quantities, by the pre- termined, and we wished to try whether they ceding rules; then reduce the third equation would answer all the proposed conditions or and either of the former to one, containing the same two unknown quantities; and from not, till as many independent equatious arise as we have assumed unknown quantities, the two equations thus obtained, the unknown which will always be the case if the question quantities which they involve may be found. be properly limited; and by the solution of The third quantity may be found by sub these equations, the quantities sought will be stituting their values in any of the proposed determined. equations. 2r + 3y + 4z= 16 Prob. 1. To divide a line of 15 inches into Ex. Let 3r+ 2y — 5z = 8 ? To find r, two such parts, that one may be three-fourths ( 5 1-6 +5 = = 6. y, and z. of the other. Let 41 = one part, From the 2 first equat. fir +9y + 12x= 48 then 3 r - the other. 6r+ 4y— 102 = 16 7x = 15, by the question, by subtr. 5y +222 = 32 15 from the 1st and 3rd, 10 r + 15 y + 20%= 80 - 12 y +62 = 12 60 by subtr. 27 y + 142 = 68 one part, 7 and 5y +222=32 45 3 the other. Prob. 2. If A can perform a piece of work z=1 in 8 days, and B in 10 days, in what time 5y +222 = 32 will they finish it together? that is, 5 y + 22 = 32 Let I be the time required. 5y = 32 — 22 = 10 1 In one day, A performs 8 part of the work; y=== 2x + 3y + 4z=16 therefore in x days, he performs ä parts of that is, 2.r+6+4 = 16 2.x=16-6-456 it; and in the same time, B performs x=3. parts of it; and calling the work 1, The same method may be applied to any number of simple equations. + 1. That the unknown quantities may have 10x + 8 x = 80 definite values, there must be as many inde 18 r = 80 pendent equations as unknown quantities. 80 Thus, if x +y = a, r=a-y; and as 18 = 4 9 days suming y at pleasure, we obtain a value of x, such, that rty=a. Prob. 3. A and B play at bowls, and A These equations must also be independent, bets B three shillings to two upon every that is, not deducible one from another. game; after a certain number of games it VOL 1 I 7 10 r 4 r = 8 3r= =677 10 8 18 appears, that A has won three shillings; but the signs of all the terms, that the equation had he ventured to bet five shillings to two, may be reduced to this form, x = px= and lost one game more out of the same num 9. Then add to both sides the square of ber, he wonld have lost thirty shillings: how half the co-efficient of the first power of the many games did they play? unkwown quantity, by which means the first Letx{b s be the number of games A side of the equation is made a complete won, square, and the other consists of known quany the number B won, tities; and by extracting the square root then 2 x is what A won of B, on both sides, a simple equation is obtained, and 3 y what B won of A. from which the value of the unkwown quan2 x - 3y=3, by the question; tity may be found. y = 13 found to be IV 9+ ? Ex. 1. Let x + px=q; now, we know y +1.5, B would win, that x +px + + is the square of a + 5y + 5 - - 2x +2 = 30, by the question. add therefore, to to both sides, and we have or 5 y =2x = 30-5-2= 23 therefore, 5 y 27 =- 23 x+pa+b=a+; then by extractand 2x – 3y = 3 by addition, 5 y − 3y = 26 ing the square root on both sides, 2y = 26 x+=+Ve+ p2 and by transp 2x=3+3y=3 +39 = 42 x = 21 pa x+y=34, the num. of games. In the same manner, if x2 - px=9, x is ON QUADRATIC EQUATIONS. p? When the terms of an equation involve the square of an unknown quantity, but the first power does not appear, the value of the Ex. 2. Let 22 12 x + 35 = 0; to find ... square is obtained by the preceding rules ; By transposition, x? - 12 r = -35, and by extracting the square root on both adding the square of 6 to both sides of the sides, the quantity itself is found. equation, 2 - 12x + 36 = 36 — 35 = 1; Ex. 1. Let 5.32 - 45 = 0; to find x. then extracting the square root on both sides, By trans. 5.ro = 45 6=+1 2 = 9 x = 6+1=7 or 5; either of which, therefore, x=vq=+3. substituted for x in the original equation, The signs + and are both prefixed to answers the condition, that is, makes the the root, because the square root of a quan whole equal to nothing. tity may be either positive or negative. The sign of x may also be negative; but still x Ex. 3. Let x tv5x+10=8; to find z. will be either equal to + 3 or - - 3. By transposition,5x+10=8. squar, both sides, 5 x +10=64 — 16x + xr? Ex. 2. Let a x?=bcd; to find x. a? -- 21 x =1064 =- 54 bcd 441 441 compl. the sq. 32 54 Ocd 441 441 4 If both the first and second powers of the 21 15 unknown quantity be found in an equation: extracting the square root, s 2 Arrange the terms according to the dimen 21 – 15 sions of the unknown quantity, beginning with = 3 or 18. 2 the highest, and transpose the known quantities to the other side ; then, if the square but on trial it appears, that 18 does not an By this process two values of x are found, of the unknown quantity be affected with a co-efficient, divide all the terms by this co swer the condition of the equation, if we supefficient, and if its sign be negative, change pose that ✓57+ 10 represents the posi 21 + a 225 4 tive square root of 5 x + 10. The reason is, by extracting the sq. roots, x+y=+11 that 5r+ 10 is the square of — V5r+10 and xy=3 as well as of tv5x + 10; thus by squar therefore, 2.r=# 14 ing both sides of the equation v5.0 + 10 x=7, or -7 - 4. =8 — 1, a new condition is introduced, and and y = 4, or a new value of the unknown quantity corres PROBLEMS PRODUCING QUADRATIC ponding to it, which had no place before. EQUATIONS. Here, 18 is the value which corresponds to the supposition that r— ~57 + 10 = 8. Prob. 1. To divide a line of 20 inches into Every equation, where the unknowu quan two such parts, that the rectangle under the tity is found in two terms, and its index in #hole and one part, may be equal to the one is twice as great as in the other, may be square of the other. resolved in the same manner. Let .y be the greater part, then will 20 be the less, and x = 20 - 8.20 = 400 — 20 x by the, Ex. 4. Let z + 4z3= 21 question. z+421 +4=21+4= 25 22 + 20 x = 400 qua + 20 r + 100 = 400 + 100 = 500 x+10=500 V500 10. therefore = 9, or 49. Prob. 2. To find two numbers, whose sum, Ex, 5. Let y* -- 6 ya - 27 = 0. product, and the sum of whose squares, are * 6 y2 equal to each other. y* -6y? +9= 27+9= 36 Let x + y and r · y be the numbers; ya - 3=+6 their sum is 2 .z their product .za - 2 the sum of their sqs. 2 x2 + 2 y? gd + y2 therefore, 3x = 2x2 q? 3 you try'= 2 2x ya 27 or 3 = -2 -12 = 27 or i Ex. 6. Let it +ry+=0 27 3 2 y=+ 53 xy When there are more equations and unknown quantities than one, a single equation, 3-V-3 involving only one of the unknown quantities, inay sometimes be obtained by the rules laid Since the square of every quantity is posidown for the solution of simple equations ; and tive, a negative quantity has no square root; one of the unknown quantities being discover- the conclusion therefore shews that there are ed, the others may be obtained by substituting no such numbers as the question supposes. its value in the preceding equations. See BINOMIAL THEOREM; EQUATIONS, nature of ; Series, SURDS, &c. &c. Ex 7. Let <x+y=65}To find & and y. The first and principal applications of alge Alcebra, application of lo geometry. Pry= 28 From the second equation, 2 xy=56 bra were to arithmetical questions and and adding this to the first, rä+lryty=121 computations, as being the first and most subtract. it from the same,xe - 2xy + y=9 useful science in all the concerns of human life. Afterwards algebra was applied to tity may be derived by the ordinary megeometry, and all the other sciences in their thods of reduction of equations, when only turn. The application of algebra to geome one unknown quantity is in the notation; or try, is of two kinds ; that which regards the till as many equations are obtained as there plane or common geometry, and that which are unknown letters in the notation. respects the higher geometry, or the nature For example: suppose it were required of curve lines. to inscribe a square iu a given triangle. Let The first of these, or the application of ABC, (Plate Miscellanies, fig. 1.) be the algebra to common geometry, is concern given triangle ; and feign DEFG to be the ed in the algebraical solution of geome required square ; also draw the perpendicutrical problems, and finding out theorems in Jar BP of the triangle, which will be given, geometrical figures, by means of algebraic as well as all the sides of it. Then, considercal investigations or demonstrations. This ing that the triangles BAC, BEF are simikind of application has been made from the lar, it will be proper to make the notation time of the most early writers on algebra, as follows, viz. making the base AC=b, 'as Diophantus, Cardan, &c. &c. down to the perpendicular BP=p, and the side of the present times. Some of the best pre- the square DE or EF=r. Hence then cepts and exercises of this kind of applica- BQ= BP-ED=p-x; consequently, tion are to be met with in Sir I. Newton's by the proportionality of the parts of those “ Universal Arithmetic,” and in Thomas two similar triangles, viz. BP: AC:: BQ: Simpson's “ Algebra and Select Exercises." EF, it is p:6::p-x:r; then, multiply Geometrical Problems are commonly resolv extremes and means, &c. there arises pr= ed more directly and easily by algebra, than bp by the geometrical analysis, especially by bp-bx, or bx+pr=bp, and r= +p young beginners; but then the synthesis, or the side of the square sought; that is, a construction and demonstration, is most fourth proportional to the base and perpenelegant as deduced from the latter method. dicular, and the sum of the two, taking this Now it commonly happens that the alge sum for the first term, or AC + BP:BP :: braical solution succeeds best in such pro- AC:EF. blems as respect the sides and other lines in The other branch of the application of geometrical figures ; and, on the contrary, algebra to geometry, was introduced by those problems in which angles are con- Descartes, in his Geometry, which is the cerned, are best effected by the geometri- new or higher geometry, and respects the cal analysis. Sir Isaac Newton gives these, nature and properties of curve lines. In among many other remarks on this branch. this branch, the nature of the curve is exHaving any problem proposed, compare to. pressed or denoted by an algebraic equagether the quantities concerned in it; and, tion, which is thus derived : A line is conmaking no difference between the known ceived to be drawn, as the diameter or some and unknown quantities, consider how they other principal line about the curve; and depend, or are related to, one another; upon any indefinite points of this line other that we may perceive what quantities, if lines are erected perpendicularly, which they are assumed, will, by proceeding syn are called ordinates, whilst the parts of the thetically, give the rest, and that in the first line cut off by them are called abscissimplest manner. And in this comparison, ses. Then, calling any absciss x, and its the geometrical figure is to be feigned and corresponding ordinate y, by means of the constructed at random, as if all the parts known nature, or relations, of the other were actually known or given, and any lines in the curve, an equation is derived, other lines drawn that may appear to con- involving r and y, with other given quanti. duce to the easier and simpler solution of ties in it. Hence, as x and y are common the problem. Having considered the me to every point in the primary line, that equathod of computation, and drawn out the tion, so derived, will belong to every posischeme, names are then to be given to the tion or value of the absciss and ordinate, quantities entering into the computation, and so is properly considered as expressing that is, to some few of them, both known the nature of the curve in all points of it; and unknown, from which the rest may most and is commonly called the equation of the naturally and simply be derived or expressed, by means of the geometrical proper In this way it is found that any curve line ties of figures, till an equation be obtained, has a peculiar form of equation belonging by which the value of the unknown quan- to it, and which is different from that of curve. every other curve, either as to the number 2ub, that is, the square of a +b is equal to of the terms, the powers of the unknown let. a? + b? + 2 ab, as derived from a geome. ters r and y, or the signs or co-efficients of trical figure or construction. And in this the terms of the equation. Thus, if the very manner it was, that the Arabians, and curve line HK, (fig. 2.) be a circle, of which the early European writers on algebra, deHI is part of the diameter, and IK a per- rived and demonstrated the common rule pendicular Grdinate: then put HI=r, IK for resolving compound quadratic equations. =y, and p= the diameter of the circle, And thus also, in a similar way, it was, that the equation of the circle will be po- r= Tartalea and Cardan derived and demon. y?. But if HK be an ellipse, an hyperbola, strated all the rules for the resolution of cuor parabola, the equation of the curve will bic equations, using cubes and parallelopipebe different, and for all the four curves, will dons instead of squares and rectangles. be respectively as follows : viz. Many other instances might be given of the use and application of geometry in For the circle............p.r- x = y', algebra. For the ellipse ........pr :-=y, ALGOL, the name of a fixed star of the third magnitude in the constellation PerFor the hyperbola ......pr. -.pr+x=y, seus, otherwise called Medusa's heud. This star has been subject to singular variations, For the parabola .. px =y?; appearing at different times of different magnitudes, from the fourth to the second, where t is the transverse axis, and p its pa. which is its usual appearance. These varirameter. And in like manner for other ations have been noticed with great accurves. curacy and the period of their return is This way of expressing the nature of determined to be 2d 20h 48' 56'. The curve lines, by algebraic equations, has cause of this variation, Mr. Goodricke, who given occasion to the greatest improvement has attended closely to the subject, conjecand extension of the geometry of curve tures, may be either owing to the inlines; for thus, all the properties of alge- terposition of a large body revolving round braic equations, and their roots, are trans. Algol, or to some motion of its own, in ferred and added to the curve lines, whose consequence of which, part of its body, abscisses and ordinates have similar proper covered with spots or some such like matties. Indeed the benefit of this sort of ter, is periodically turned towards the application is mutual and reciprocal, the earth. known properties of equations being trans ALGORITHM, an Arabic term, not ferred to the curves they represent; and, unfrequently used to denote the practical on the contrary, the known properties of rules of algebra, and sometimes for the curves transferred to their representative practice of common arithmetic; in which equations. last sense it coincides with logistica numeBesides the use and application of the ralis, or the art of numbering truly and higher geometry, namely of curve lines, to readily. detecting the nature and roots of equations, ALIEN, in law, a person born in a and to the finding the values of those roots strange country, not within the king's alleby the geometrical construction of curve giance, in contradistinction from a denizen lines, even common geometry may be made or natural subject. subservient to the purposes of algebra. An alien is incapable of inheriting lands Thus, to take a very plain and simple in- in England, till naturalized by an act of stance, if it were required to square the parliament. No alien is entitled to vote in binomial a +b; (fig. 3.) by forming a the choice of members of parliament, has square, as in the figure, whose side is a right to enjoy offices, or can be returned equal to a +b, or the two lines or parts on any jury, unless where an alien is party added together denoted by the letters a and in a cause; and then the inquest of jurors b; and then drawing two lines parallel to shall be one half denizens and the other the sides, from the points where the two aliens. parts join, it will be immediately evident Every alien neglecting the king's procla. that the whole square of the compound mation directing him to depart from the quantity a +b is equal to the squares of realm within a limited time, shall, on con. both the parts, together with two rectan- viction, for the first offence be imprisoned gles under the two parts, or a' and b2 and for any time not exceeding one month, and |