ALGEBRA. 



Or simplifying 



X 



= 10 



1 20 yards in 12 days. 



Example 5. If 16 men earn ^45 in 28 days, 

 how much will 20 men earn in 35 days ? 



= what 1 6 men earn in 28 days. 



I man earns in 28 



i day. 



16 X 28 



45 X 20 X 35 



20 men earn in i 



16 X 28 



X 



20 



35 days. 



45 X 5 X 5 

 i6 



These examples are sufficient to shew how all such 

 questions are wrought, and it may here be re- 

 marked, that questions belonging to such rules as 

 Interest, Percentages, Profit and Loss, Partnership, 

 Discount, Stocks, &c. are all treated in a similar 

 manner. For example, let it be required to find 

 the interest of ^560 for 4$ years, at 3^ per cent. 



Here 



Jt 



100 



3^X560 



100 



5j X 560 X 4j 



100 



interest of ^100 for i year, 

 it i i ii 



M 560 i ii 

 n 560 4 years. 



SQUARE ROOT. 



When we multiply a number by itself, the pro- 

 duct we obtain is said to be the square of the 

 original number, and the original number is said 

 to be the square root of this product. For ex- 

 ample, 15 multiplied by 15 gives 225; then 225 

 is said to be the square of 15, and 15 is the square 

 root of 225. Having given a number, then, it is 

 easy to find its square, for we have only to multipy 

 it by itself. It is not so easy, however, to find its 

 square root, and it is. important to be able to do 

 so. For example, suppose we wish to find the 

 side of a square which contains 625 square feet ; 

 it is here necessary to find the square root of 625. 

 We first set off the figures in pairs (for 

 the square of a single figure never con- 

 sists of more than two places), and we 

 find that the result contains two figures, 

 that is, that it consists of tens and units. 

 Now, by trial, we see that it lies between 

 20 and 30, for 20 multiplied by 20 gives 400, and 

 30 by 30 gives 900. As a first approximation then 

 to the root required, we shall take 20 ; the square 



6-25(20 

 400 



225 



of 20 is 400, this taken from the given number 

 leaves 225. Now, whatever 



40+ 

 5 



5 

 _5 



25 



6-25(20 

 400 



225 

 225 



the remaining part of the 

 root is, it may be shewn 

 that twice its product by 20 

 (the part of the root already 



found) together with its 



square make up 225 ; it, 



therefore, only remains to find a number, that, 

 multiplied by 40, and also by itself, the two pro- 

 ducts together may make 225. It is evident that 

 this number must be 5, for 40 X 5 is 200, and 

 5 X 5 is 25, and these together make 225. The 

 several steps in this process will stand as in the 

 margin. But this in practice is somewhat simpli- 

 fied. After finding that the first part of the root is 

 20, we write it in the result as 2 tens that is, we 

 write the 2 in the second place ; the square of 20 

 is 400, which is written under the given number, 

 but without the zeros that is, the 4 is written 

 under the hundreds, then 40 + 5 is written in the 

 divisor as 45, and the product by 5 is written 

 under the remainder 225. 



5-90-49(243 



6-25(25 * 



4 44 



45 



225 

 225 



190 



176 



483 



1449 

 1449 



Again, to find the square root of 59049. The 

 working will be as is shewn in the above. The 

 method for extracting the cube root is very com- 

 plicated, and the operation is seldom needed in 

 practical arithmetic. (See Chambers's Arithmetic, 

 Educational Course.) 



ALGEBRA. 



It is evident that the value of all methods of 

 computation lies in their brevity. Algebra must 

 be considered as one of the most important de- 

 partments of mathematical science, on account of 

 the extreme rapidity and certainty with which it 

 enables us to determine the most involved and 

 intricate questions. The term algebra is of Arabic 

 origin. It embodies a method of performing cal- 

 culations by means of various signs and abbrevia- 

 tions, which are used instead of words and phrases, 

 so that it may be called the system of symbols. 

 Although it is a science of calculation, yet its 

 operations must not be confounded with those of 

 arithmetic. All calculations in arithmetic refer to 

 some particular individual question, whereas those 

 of algebra refer to a whole class of questions. One 

 great advantage in algebra is, that all the steps of 

 any particular course of reasoning are, by means 

 of symbols, placed at once before the eye ; so that 

 the mind, being unimpeded in its operations, pro- 

 ceeds uninterruptedly from one step of reasoning 

 to another, until the solution of the question is 

 attained. 



SIGNS AND SYMBOLS. 



The sign +> which is called plus, means that 

 the two quantities between which it is placed are 

 to be added. Thus 5 + 3 is read 5 plus 3, and 



605 



