Exercises.

(13.) Divide 34-82725 by 39.5275; 897.25 by .98725; 42.9365 by 387.25; and 3.1415926 by 180, true to seven decimal places in the quotient.

(14.) Divide .86253 by 65.125; 9.387625 by 13-425; -8376 by .03; and I by 3.1415926, true to six decimal places in the quotient.

(15.) Divide 11.684626 by 3.1482; 9.587625 by .003525; 73.4286 by 173-25, and 8.475 by 3.83, true to two decimal places in the quotient.

PROMISCUOUS EXERCISES IN DECIMALS.

5

2

43

15 17

(16.) Reduce to decimals, ; ; ; ; ; and 1; extending the answer to six places of decimals where it is interminate.

(17.) When the radius of a circle is 1, its semicircumference is 3.14159265, and the semicircumference is also divided into 180°, it is required to find the degrees in an arc equal in length to the radius.

(18.) The length of an arc equal in length to the radius is 57° 2957795; express the length of the radius in seconds.

(19.) The circumference of a circle when the diameter is 1 is expressed by the fraction nearly; express this fraction as a decimal.

(20.) There are two thermometers in common use, Fahrenheit's and the Centigrade, 180° of Fahrenheit's equal 100° of the Centigrade; express by decimals a degree of Fahrenheit in terms of the degree of the Centigrade, and a degree of the Centigrade in terms of a degree of Fahrenheit.

(21.) A sidereal day is 23 h. 56 m. 4.09 s.; express this as a decimal of a common day—that is, of 24 hours-and carry the result to nine decimal places.

(22.) A nautical mile is 6082.66 feet, and an imperial mile is 5280 feet; express each of these miles as decimals of the other.

Also, find how near the results are to the decimal values of 38 and 39.

(23.) A mètre is 39-37079 English inches, a kilomètre is 1000 mètres; express as decimals of each other, a kilomètre and an English mile.

(24.) Divide the sum of .075 and .0075 by the difference of .75 and .075.

(25.) Multiply the sum of 7.345, 8.375, and .055, by the difference of 12.395 and 11.344.

(26.) If the length of the year be taken at 365 days instead of 365.2422419 days, its true value, what will the error amount to in four centuries?

(27.) Multiply .0235 by 8.08; divide .0625 by .25; and find the value of .8435416 of £5.

(28.) If 90 degrees correspond to 100 French grades, how many degrees and grades are there in the sum of 42.45 degrees and 42.45 grades?

(29.) In the Centigrade thermometer the freezing point is zero, and the boiling point is 100°; in Fahrenheit's the freezing point is 32°, and the boiling point is 212°; what degree C. corresponds to 68° F.? And what degree F. corresponds to 42.25° C. ?

CHAPTER II.

ON ALGEBRAIC AND ANALYTICAL SYMBOLS.

16. Whatever can be increased or diminished by addition, subtraction, multiplication, or division, is called quantity.

17. When quantities are considered generally or algebraically, they are represented by the letters of the alphabet; the letters a, b, c, &c. at the beginning of the alphabet, are generally used to represent quantities whose values are already known, and the letters x, y, z, &c. towards the end of the alphabet, are generally used to represent quantities whose numerical values are unknown, and which it is the object of the calculator to determine.

18. Besides the letters which indicate quantities, another set of very important symbols is used to indicate operations to be performed upon them; these symbols of operation are chiefly the following:

+(called plus) indicates that the quantity after it is to be added to that before it; thus a+b shews that b is to be added to a, and is read a plus b. If in the above the value of a was 7, and that of b was 3, a + b would be 7+ 3 = 10.

19. =

20.

(read is equal to) means that the quantities between which it stands are of equal values; and quantities so connected are said to form an Equation. It is not a sign of operation. (read minus) indicates that the quantity after it is to be subtracted from that before it; thus, a-b means that the value of b is to be taken away from a. Using the same values as in (18), a-b=7-3, which=4; therefore, for these values, a-b=4.

21. ~ means the difference of the quantities between which it stands, without reference to which is the greater; thus, a~b means either ab, or b-a; hence a ~b=b~ a.

22. x means into, or multiplied by, and indicates that the quantities between which it stands are to be multiplied together; thus, ab means a taken b times, or b taken a times. This operation is, however, sometimes denoted by a point placed between the letters to be multiplied together, as a.b; and still more frequently by the letters being placed close together, as the letters of the same word; wherefore, a × b=a·b=ab. If the same numerical values of a and b be taken as in (18), then a × b= a·b=ab = 7 × 3 = 21.

23. When a letter is to be multiplied by itself, instead of repeating the letter, it is customary to write it only once, with a small figure after it to indicate the number of times it enters as a factor; thus, aa is written a2, and read a square; so also aaa is written a3, and is read a cube or a the third power. In the same manner aaaaa = a3, a3 × aa = a1; and the superscribed small figure is called an exponent.

24. means by or divided by, and always indicates that the quantity before it is to be divided by the numerical value of the letter or number after it; thus, a÷b directs us to divide the value of a by the numerical value of b; but this operation is more frequently denoted by writing the quantity to be divided above a line and the divisor below it; thus, a ÷ b = and taking again the values as in (18), a÷b=2=1=2}.

3

=

α

b'

25. (), {}, [] enclosing, or drawn over several quantities, denotes that collectively they are to be considered as one quantity, and the enclosing symbols are called brackets.

26. (called the radical sign), when by itself, denotes the square root; thus, √36 denotes that a number is to be found that, when multiplied by itself, will produce 36, this number is evidently 6, for 6 x 6 = 36; and therefore √36-6.

When a 3 is placed in the radical sign, as 3/a, it denotes the cube root; when a 4 is placed in the radical sign, as a, it

denotes the fourth root; and generally

of the quantity over which it is placed.

indicates the nth root

The same thing is also very frequently indicated by means of fractional exponents; thus aa1, 3⁄4ã=a3, √ã=at, and a

27. ±(read plus or minus) indicates that the quantity placed after it is first to be added to that before it, and then subtracted from that before it, thus giving two values of the same quantity; thus, x = a+c, means that c is to be added to a, which gives one value of x, and then subtracted from it, which gives another value. If in the above a = 12, c=5, then x=a+c, means first x = 12 +5, or x=17; and also x = 12 57, wherefore x has two values, the one being 17 and the other 7.

28. When a number stands before a letter, or combination of letters, it is called a coefficient, and indicates the number of times the quantity before which it stands is to be taken; thus, 7a means that a is to be taken 7 times, and 13(abc + d) means that the product of a into b into c increased by d is to be taken 13 times.

29. In applying the principles of notation stated above, the following AXIOMS will be assumed as true :

(a.) Quantities that are equal to the same quantity are equal to each other.

(b.) If equal quantities are added to equal quantities, the sums are equal.

(c) If equal quantities are added to unequal quantities, the sums are unequal; the greater sum being that which contains the greater of the unequal quantities.

(d.) If equal quantities are taken from equal quantities, the remainders are equal.

(e.) If equal quantities are taken from unequal quantities, the remainders are unequal; the greater remainder being that which remains of the greater of the unequal quantities.

(f) If unequal quantities are taken from equal quantities, the remainders are unequal; the greater remainder being the excess above the less of the two unequal quantities.