









4  is the same as  2×2,  6  is the same as  2×3, 
8  _ _ _  2×2×2,  9  _ _ _  3×3, 
10  _ _ _  2×5,  12  _ _ _  2×3×2, 
14  _ _ _  2×7,  15  _ _ _  3×5, 
16  _ _ _  2×2×2×2,  and so on. 

12  is the dividend, 
3  is the divisor, and 
4  is the quotient. 
6)  34  (5 
30  
4 
9)  41  (4 
36  
5 
30xy divided by +6y gives 5x, and 
54abc divided by 9b gives +6ac; 












1  2  3  4  5  6  7  8  9  10 
1  1  1  1  1  1  1  1  1  1 
2  3  2  5  2  7  2  3  2  
4  3  4  9  5  
6  8  10  
1  2  2  3  2  4  2  4  3  4 
P.  P.  P.  P.  P. 
11  12  13  14  15  16  17  18  19  20 
1  1  1  1  1  1  1  1  1  1 
11  2  13  2  3  2  17  2  19  2 
3  7  5  4  3  4  
4  14  15  8  6  5  
6  16  9  10  
12  18  20  
2  6  2  4  4  5  2  6  2  6 
P.  P.  P.  P. 













2 times, or twice [1/2] makes [2/2], or 1 integer; 2 times, or twice [1/3] makes [2/3]; and 3 times, or thrice [1/6] makes [3/6]; or [1/2] 4 times [5/12] makes [20/12], or 1[8/12], or 1[2/3].But, instead of this rule, we may use that of dividing the denominator by the given integer, which is preferable, when it can be done, because it shortens the operation. Let it be required, for example, to multiply [8/9] by 3; if we multiply the numerator by the given integer we obtain [24/9], which product we must reduce to [8/3]. But if we do not change the numerator, and divide the denominator by the integer, we find immediately [8/3], or 2[2/3], for the given product; and, in the same manner, [13/24], multiplied by 6 gives [13/4], or 3[1/4]. 102. In general, therefore, the product of multiplication of a fraction [a/b] by c is [ac/b]; and here it may be remarked, when the integer is exactly equal to the denominator, that the product must be equal to the numerator.

[12/25] divided by 2 gives [6/25]; [12/25] divided by 3 gives [4/25]; and [12/25] divided by 4 gives [3/25]; &c.104. This rule may be easily practised, provided the numerator be divisible by the number proposed; but very often it is not: it must therefore be observed, that a fraction may be transformed into an infinite number of other expressions, and in that number there must be some, by which the numerator might be divided by the given integer. If it were required, for example to divide [3/4] by 2, we should change the fraction into [6/8], and then dividing the numerator by 2, we should immediately have [3/8] for the quotient sought. In general, if it be proposed to divide the fraction [a/b] by c, we change it into [ac/bc], and then dividing the numerator ac by c, writing [a/bc] for the quotient sought. 105. When, therefore, a fraction [a/b] is to be divided by an integer c, we have only to multiply the denominator by that number, and leave the numerator as it is. Thus [5/8] divided by 3 gives [5/24], and [9/16] divided by 5 gives [9/80]. This operation becomes easier, when the numerator itself is divisible by the integer, as we have supposed in § 103. For example, [9/16] divided by 3 would give, according to our last rule, [9/48]; but by the first rule, which is applicable here, we obtain [3/16], an expression equivalent to [9/48], but more simple. 106. We shall now be able to understand how one fraction [a/b] may be multiplied by another fraction [c/d]. For this purpose, we have only to consider that [c/d] means that c is divided by d; and on this principle we shall first multiply the fraction [a/b] by c, which produces the result [ac/b]; after which we shall divide by d, which gives [ac/bd]. Hence the following rule for multiplying fractions. Multiply the numerators together for the numerator, and the denominators together for the denominator. Thus [1/2] by 2 gives the product 2, or 1;
Number  1  2  3  4  5  6  7  8  9  10  11  12  13 
Square  1  4  9  16  25  36  49  64  81  100  121  144  169 


Numbers.  3  3[1/4]  3[1/2]  3[3/4]  4 
Squares.  9  10[9/16]  12[1/4]  14[1/16]  16 



Numbers.  1  2  3  4  5  6  7  8  9  10 
Cubes.  1  8  27  64  125  216  343  512  729  1000 


Powers  Of the number 2  Of the number 3 
1st  2  3 
2nd  4  9 
3rd  8  27 
4th  16  81 
5th  32  243 
6th  64  729 
7th  128  2187 
8th  256  6561 
9th  512  19683 
10th  1024  59049 
11th  2048  177147 
12th  4096  531441 
13th  8192  1594323 
14th  16384  4782969 
15th  32768  14348907 
16th  65536  43046721 
17th  131072  129140163 
18th  262144  387420489 
1st  2nd  3rd  4th  5th  6th  
10  100  1000  10000  100000  1000000  &c. 
1st  2nd  3rd  4th  5th  6th  
a,  aa,  aaa,  aaaa,  aaaaa,  aaaaaa,  &c. 
[1/aaaaaa]  [1/aaaaa]  [1/aaaa]  [1/aaa]  [1/aa]  [1/a]  1  a  
1st.  [1/(a^{6})]  [1/(a^{5})]  [1/(a^{4})]  [1/(a^{3})]  [1/(a^{2})]  [1/(a^{1})]  
2nd.  a^{6}  a^{5}  a^{4}  a^{3}  a^{2}  a^{1}  a^{0}  a^{1} 











log343  =  2.5352941  
log2401  =  3.3803922  } added 
5.9156863  their sum  
log823540  =  5.9156847  nearest tabular log 
16  difference 

