number, the remainder will be the square of the left-hand digit. What is the number? Ans. 93.

72. A, travelling to London, overtook at the 50th milestone a flock of sheep, proceeding at the rate of 3 miles in 2 hours; and 2 hours afterwards met a waggon moving at the rate of 9 miles in 4 hours. B, travelling at the same rate, overtook the sheep at the 45th milestone, and met the waggon 40 minutes before he came to the 31st milestone. Where would B be when A reached London ? x = distance between them, y = rate of

10y

3

3242 + 27

Ans. x=25, y=9.

LITERAL ANALYSIS.

WHEN the known quantities are expressed in numbers, these numbers disappear during the progress of the operation, and the answer, when obtained, does not exhibit the process by which it has been deduced from the assumed data. This is the mode of solution given in the preceding parts of this work, and it was necessary for beginners; but it does not exhibit sufficiently the true difference between arithmetic and algebra, but rather confounds them. The essential character of algebra, taken in its most extensive meaning, is, that the results of its operations do not give the particular values of the quantity or quantities sought; they only represent the operations which ought to be made upon the given quantities, for obtaining the values of those sought, according to the conditions of the problem; so that the principal object of algebra is the investigation of theorems and the exhibition of rules for the arithmetical or geometrical solution of problems. For accomplishing these purposes, it is necessary to represent the known quantities by letters, as well as the unknown ones. The former are represented by the first letters of the alphabet, a, b, c, &c. and the unknown ones by the last letters, x, y, z, &c. The question is translated into equations, and these equations are resolved by the preceding rules; and then the values of the unknown quantities will be expressed in a general way, from their relations to those which are given in the question. Consequently, if this general expression be transferred from algebraical characters into common language, it will give a general rule for the solution of all questions of the same kind. But the expressions will answer the same purpose as accurately in algebraical characters, and then they are called Theorems, or Formulæ.

..1. Given the sum s, and the difference d, of two quantities

2

x and y: to find the quantities. x+y=s, and x-y=d: by adding these equations we get 2x=s+d, whence x="+d; and by subtracting the equations we get 2y=s-d, and These values, expressed in common language, give the following rules, viz.

y=

2

d

To find the greater, add the difference to the sum, and divide by 2.

To find the less, subtract the difference from the sum, and divide by 2.

2. Given the sum s, of two quantities x and y, and the difference of their squares D, to find the quantities. x+y=s, and x2 y2D; and dividing the latter by the

D

8 D

y==; whence, as before, x = + and

8

28

3. As exercises, the student may investigate the following, viz. Of two quantities, their sum, difference, product, quotient, sum and difference of their squares, any two being given, to find all the rest. The operations will be similar to those used in the two last questions; and the results, except for the sum and difference of their squares, are given in the following Table, in which x and y are the quantities, s = their sum, d = their difference, p= their product, q = their quotient, Z= the sum of their squares, and D= the difference of their squares.

The use of this table is plain. Suppose the sum of two numbers to be 277, and their difference to be 115; then the = = 196.

greater number is (" + d) = (277+115) =

392

Suppose again the difference of two numbers to be 10, and their product 119.

The greater number is d+(d2+4p)1

10+/576

2

10+24

=17.

2

Suppose the sum of their squares to be 250, and the difference of their squares to be 88.

250+88

The greater number is (Z+ D) + = (250 +88) &

The less is (2D)
(Z = D) = (250 — 88 ) } = √/81 = 9.

= √169

4. Given the sum s, of the products of two quantities, by known multipliers m and n, and also the sum of their products c, by other known multipliers p and q, to find the quantities.

Here mx+ny=s, and px+qy = c; and multiplying the former equation by p, and the latter by m, they become pmx+pny =ps, and mpx+mqy = mc; and subtracting, we get npy-mqy = ps · me; and dividing by np-mg, we

; and in the same way we find x =

98-nc np-mq

5. Given the sum s, of the quotients of two quantities by known divisors m and n, and also the sum c, of their quotients by other known divisors p and q, to find the quantities.

x y

= S, and + =c, whence nx+my = mns,

P

and qx+py=pqc; which, resolved as the last, gives

6. Given the values m and n, of two ingredients, to find the quantities which must be taken of each, to form a given quantity a, of a compound of a given value e.

7. Given the times m and n, in which two agents could produce the same effect separately, to find the time in which they could do it jointly.

8. Given the times m, n, and r, in which three agents can perform the same work separately; to find the time in which they can do it jointly.

9. Given the times m, n, and r, in which every two of three agents can perform the same work; to find the time x, in which they can do it jointly, and also the times y, z, and v, in which each of them can do it separately.

10. Given the specific gravities m and n, of two ingredients,

and the quantity a, of the mixture, with its specific gravity r; to find the quantities of the ingredients.

11. Given the first distance d, of two moving bodies, and their velocities m and n; to find the time of their conjunction. Ans. x= M-N

d

12. Given the sum 2s, of two numbers, and also the sum of their squares, of their cubes, of their fourth, or of their fifth powers, &c.; to find the numbers.

NOTE.-If their difference be 2x, the numbers will be s+x and s―x; and then the sum of their squares will be 2s2+2x2, the sum of their cubes 2s3+6sx2, the sum of their fourth powers 2s1 + 12s2x2+2x4, and the sum of their fifth powers 285 +20s3x2 + 10sx4, all of which are of the quadratic or simple form, and may be resolved as before; but the sums of the higher powers exceed the quadratic.

Letz sum of their squares, c = sum of their cubes, q = sum of their fourth powers, and p = sum of their fifth powers; 2 3 3 ) = ( — 3 s 2 ± √ √ } q+8s1

then x=

= (3 — 20 2 ) 3 = (3

=(−8°±/fo + 54

68

3.

283

13. To find two numbers of which the product is given p, and also the product P, of the sums when each is increased by a given number (a and b).

14. To find two numbers such, that their sum, their product, and the difference of their squares, shall be all equal.

15. Given the sum a, of two numbers, and the sum of square roots b; to find the numbers.

their

16. Given the excess of the product of two numbers above their sum a, and also the sum of their squares b; to find the numbers.

Ans. Let m√(2a+b+1);