From eq. 1. x=22-y-z; substitute this value of x in the second and third, and (44-2y-2z-3y+5z=40, or) 5y-3 z =4; also (66-3y-3z+4y-2z3=-100, or) 2z+3z-y= 166; let now the value of y (=32+) in the last but one be sub 3z+4 stituted in the last, and it becomes (2 z3 +3z 10z3+122=834. Now it appears from trial, that z is greater than 4, but less than 5; let these two numbers therefore be substituted for z, then Equation. 146+476 For a nearer approximation. Let 4.2 and 4.3 be put for z, and 1st Supp. 2nd Supp. Wherefore z=(4.3-.022874) 4.277126, very nearly. Whence y=( 14.356599, nearly. 3 z+4 5 =) 3.366275, and x=(22-y-z=) 2. Given z-x-10, xy+xz=900, and xyz=3000, to find x, y, and z. From eq. 1. z=10+x; substitute this value for z in the 900-10x--- x2 second, and it becomes xy +10x+x2=900, and y= x ; write this value for y, and 10+x for z in the third, and it will become (9000+800 x—20 x2-x3=3000, or) x3 +20 x2 — 800 x= Here by trials x is found to be greater than 23, but less than 24; then using these two numbers as suppositions, and proceeding as before, x=23.923443456, y=3.696558933, and z= 33.923443456, nearly. 3. Given x2+y=157, and y2-x=6, to find x and y. Ans. x=12.34, y=4.321. 4. Given x+xy=80, and x2y-y2=495, to find x and y. Ans. x=8, y=9. 5. Given + y2=12, and x2+y=8, to find x and y. 6. Given x+yz=20, y+xz=22, and z+xy=28, to find x, y, and z. 67. Dr. HUTTON'S RULE for extracting the roots of numbers by approximation. RULE I. Let N the number of which any root is required to be extracted, the index of the proposed root, r=the n number found by trials, which is nearly equal to the root, namely, TN nearly, and let x=the root, or x"=N exactly. n+1.N+n-1.Î1 II. Then will x: Xr, nearly'. n+1.r"+n—1.N The rule is thus demonstrated; let N=the given number, the root of which it is proposed to evolve; n = the index of the root, r=the nearest ra tional root, v=the difference between rand the exact root, x=r+v=the exact root; then since Mi=r+v, we shall have N=r+v]"=r"+nr¬¬1v + n 2 -yn➡ 2y2+,&c. (Vol. I. P. 3. Art. 54.) and by transposition and division, &c. in which, rejecting smallness, v may be considered as = on account of its But from the first equation, Nrnrn- III. To find a nearer value, let this value of x be substituted for r in the above theorem, and the result will approach nearer the root than the former. IV. In like manner, by continually substituting the last value of x for r, the root may be found to any degree of exactness. EXAMPLES.-1. Let x=19 be given, to find the value of x. Here N=19, n=4, and the nearest whole number to the fourth root of 19 is 2; let therefore r=2, then will r"=16, and x= n+1.N+n—1.r 5 x 19+3 x 16 5 × 16+3x 19 n+1.r"—n—1.N xr= 286 ·x2=) =2.08, nearly. 137 To repeat the process for a nearer approximation. Let r=2.08, then r" (2.08+) 18.71773696; these numbers being substituted in the theorem, we shall have x= 5x19+3 x 18.71773696 5 × 18.71773696+3 × 19 ×2.08=) 151.15321088 x 2.08= 2.0877975, extremely near; and if a nearer value of x be required, this number must be substituted for r, and repeat the operation. Ans. x 7.999, &c. Ans. x 6.019014897. Ans. x=3.9999, &c. 2. Given 3=510, to find x. 5. Required the sixth root of 21035.8? Ans. x=5.254037. 6. Extract the sixth root of 272. 68. PROBLEMS PRODUCING EQUATIONS OF THREE OR MORE DIMENSIONS. 1. What number is that, which being subtracted from twice its cube, the remainder is 679? Ans..7. 2. What number is that, which if its square be subtracted from its cube, the remainder will exceed ten times the given number by 1100? Ans. 11. r+ 7, which is the rule. This is the n+ ].r"+n—1.N n+ 1.r1+n—IN investigation of the rule in Vol. I. page 260: the theorem was first given by Dr. Hutton, in the first Volume of his Mathematical Tracts; it includes all the rational formulæ of Halley and De Lagni, and is perhaps more convenient for memory and operation than any other rule that has been discovered. 3. What number is that, which being added to its square, the sum will be less by 56 than 1 its cube? Ans. 8. 4. There is a number, thrice the square of which exceeds twice the cube by .972; required the number? Ans. 9 10 5. If to a number its square and cube be added, four times 43 54 the sum will equal of the fourth power; required the number? Ans. 6. 6. If the sum of the cube and square of a number be multiplied by ten times that number, the product shall exceed twice the sum of the first, second, third, and fourth powers by 180; what is the number? Ans. 2. 7. Required two numbers, of which the product multiplied by the greater produces 18, and their difference multiplied by the less, 2? Ans. 3 and 2. 8. The days being 16 hours long, a person who was asked the time of day, replied, "If to the cube of the hours passed since sun-rise you add 40, and from the square of the hours to come before sun-set you subtract 40, the results will be equal:" required the hour of the day? Ans. 8 in the morning. 9. To find two mean proportionals between 1 and 2. Ans. 1.25992, and 1.5874. 10. The ages of a man and his wife are such, that the sum of their square roots is 11, and the difference of their cubes 31031; what are their ages? Ans. 36 and 25. 11. If the cube root of a father's age be added to the square root of his son's, the sum will be 8; and if twice the cube root of half the son's age be added to the square root of the father's, the sum will be 12; what is the age of each? Ans. the father's 64, the son's 16. 13. There are in a statuary's shop three cubical blocks of marble, the side of the second exceeds that of the first by 3 inches; and the side of the third exceeds that of the second by 2 inches; moreover, the solid content of all the three together is 1136 cubic inches; required the side of each? Ans. 4, 7, and 9 inches. PART VI. ALGEBRA. THE INDETERMINATE ANALYSIS.. 1. A PROBLEM is said to be indeterminate, or unlimited, when the number of unknown quantities to be found is greater than the number of conditions, or equations proposed ". • For some account of the subject, see the note on Diophantine problems. b If the number of quæsita exceed the number of data, the problem is unlimited. If the quæsita be equal in number to the data, the problem is limited. If the data exceed the quæsita, the excess is either deducible from the other conditions, or inconsistent with them; in the former case the excess is redundant, and therefore unnecessary; in the latter it renders the problem absurd, and its solution impossible. To give an example of each. 1. Let x+y=6 be given, to find the values of x and y. Here we have but one condition proposed, and two quantities required to be found, the problem is therefore unlimited; for (admitting whole numbers only) r may = 1, then y=5; if x=2, then y=4; if x=3, then y=3; if x=4, then y=2; if x=5, then y=1. 2. Let x+y=6, and x-y=4, be given. Here we have two conditions proposed, and two quantities to be found, whence the problem is limited; (see Vol. I. P. 3. Art. 89.) for x=5, y=1: and no other numbers can possibly be found, that will fulfil the conditions. 3. Let x+y=6, x—y=4, and xy=5, be given. Here is a redundancy, three conditions are laid down, and but two quantities to be found. By the preceding example x=5, y=1; wherefore ay=5 × 1 = 5, or the latter condition (ry=5) is deducible from the two former. 4. Let x+y=6,x−y=4, and xy=12, be given. Here is not only a redundancy, but an inconsistency; for the greatest product that can possibly be made of any two parts of 6, is 9, that is, xy=9; it cannot then be divided into two parts, x and y, so that ry=12; wherefore the latter condition is inconsistent with the two former, and renders the problem impossible. There is a mistake in the appendix to Ludlam's Rudiments, 5th edit. p. 338. Art. 107. by which the subject is altogether perverted. |