THE POPULAK EDUCATOR 



you may observe) the commands of your father. 20. I feared I was 

 displeasing you. 21. Take care to improve the morals and to exercise 

 the body of the boy. 22. I feared that an enemy was injuring me. 

 23. The boy feared that his mother was silent. 24. I took care to 

 improve (that I might improve) the morals and to exercise the body 

 of the boy. 25. I took care that you should improve the morals and 

 (that you should) exercise the body of the boy. 26. I took care that 

 the teacher should improve the morals and exercise the body of the 

 boy. 27. I fear that you will (may) not come. 28. The husband fears 

 -that his wife will (may) die. 29. The teacher feared that the scholar 

 would not obey his words. 30. The bad boy fears that the teacher 

 will come. 



EXEECISB 93. ENGLISH-LATIN. 



1. Ille me monebat. 2. HH regem monebant. 3. Ego vos monerem. 

 -4. Vos me moneretis. 5. Illi puerum monuerunt. 6. Tu mulierem 

 znonebas. 7. Ego prseceptorem monebo. 8. Tace. 9. Tacete. 10. 

 Tacento. 11. Mulier repente tacuit. 12. Cura ut emendes. 13. Cura 

 tit civium mores emendes. 14. Timeo ne tibi displiceat. 15. Pueri 

 timebant ne patri displicerent. 16. Omnibus placet. 17. Bonus malis 

 displicebit. 18. Cur taces? 19. Metuunt ne Caesar patriam vincat. 

 .20. Bonae sorores timent ut fratres valeant. 21. Valesne ? 22. Timeout 

 valeas. 23. Si corpus exercueris valebis. 24. Mater timet ut mild 

 aditus in ccelum patent. 



LESSONS IN ALGEBRA. I. 



DEFINITIONS. 



.ART. 1. ALGEBRA is a general method of solving problems, 

 and of investigating the relations of quantities by means of 

 letters and signs. 



The following will afford illustrations of this method of 

 arriving at the solutions of problems by the use of signs and 

 letters instead of figures as in arithmetic. 



PEOBLBM I. Suppose that a man divided 72 pounds among his 

 ithree sons in the following manner : To A he gave a certain 

 number of pounds ; to B he gave three times as many as to A ; 

 and to C he gave the remainder, which was half as many pounds 

 -.as A and B received. How many pounds did the donor give to 

 ach? 



To solve this problem arithmetically, the pupil would rea- 

 son thus : A had a certain part, that is one sliare ; B received 

 three times as much, or three shares ; but C had half as much 

 .as A and B ; hence he must have received two shares. By 

 .adding their respective shares, the sura is six sliares, which, by 

 ihe conditions of the question, is equal to 72 pounds. If, then, 

 <> shares are equal to 72 pounds, 1 share is equal to \ of 72, 

 namely, 12 pounds, which is A's share. B had three times as 

 many, namely, 36 pounds; and C half as many pounds as both, 

 namely, 24 pounds. 



Now, to solve the same problem by algebra, he would use 

 letters and signs, thus . 



Let x represent A's share ; then by the conditions, 

 x multiplied by 3, or x X 3 (when X , the sign of multipli- 

 cation, is used instead of the words "multiplied by"), will repre- 

 sent B's share, and 



4r, the sum of the shares of A and B divided by 2, or 4o; -h 2 

 (when -T-, the sign of division, is used instead of the words 

 " divided by "), will represent C's sliare. 



Now, a; X 3 may be written 3x, and 4-x -r- 2 may be written 

 2a;; so then adding together the several shares of A, B, and C, 

 namely, x, 3x, and 2x, and putting +, the sign of addition, 

 between them, we get K + 3x + 2x, which is equal to 6x ; or 

 using =, the sign of equality, for the words "is equal to," we 

 get x + 3x + 2 = 6x. Then 6x = 72, for the whole is 'equal 

 to all its parts; and la? = 12 pounds, A's share; 3x = 36 

 pounds, B's share ; and 2x = 24 pounds, C's share. 



Proof. Add together the number of pounds received by each, 

 and the sum will be equal to 72 pounds, the amount divided 

 between A, B, and C. 



In this algebraic solution it will be observed : First, that we 

 Tepresent the number of pounds which A received by x. Second, 

 to obtain B's share, we must multiply A's share by 3. This 

 multiplication is represented by two lines crossing each other like 

 a capital X. Tliird, to find C's share, we must take half the 

 sum of A's and B's share. This division is denoted by a line 

 between two dots. Fourth, the addition of their respective 

 shares is denoted by another cross formed by an horizontal and 

 .a perpendicular line. Take another example : 



PROBLEM II. A boy wishes to lay out 96 pence for peaches 

 end oranges, and wants to get an equal number o? each. He 



finds that he must give 2 pence for a peach, and 4 pence for 

 an orange. How many can he buy of each ? 



Let x denote the number of each. Now, since the price of one 

 peach is 2 pence, the price of x peaches will be x X 2 pence, or 

 2x pence. For the same reason, a; X 4, or 4# pence, will denote 

 the price of x oranges. Then will 2x + 4, or 6x, be equal to 

 96 pence by the conditions of that question, and Ix or x (for 

 when 1 is the co-efficient of a number [See Art. 16 below] it is 

 always understood, and never expressed) is equal to of 96 

 pence, namely, 16 pence, and 16 is therefore the number he 

 bought of each. 



2. Quantities in algebra are generally expressed by letters, as 

 in the preceding problems. Thus b may be put for 2 or 15, or 

 any other number which we may wish to express. It must not be 

 inferred, however, that the letter used has no determinate value. 

 Its value is fixed for the occasion or problem on which it is 

 employed, and remains unaltered throughout the solution of 

 that problem. But on a different occasion, or in another problem, 

 the same letter may be put for any other number. Thus, in 

 Problem I., x was put for A's share of the money. Its value 

 was 12 pounds, and remained fixed through the operation. In 

 Problem II., x was put for the number of each kind of fruit. 

 Its value was 16, and it remained so throughout the whole of 

 the calculation. 



3. By the term quantity, we mean anything that can be mul~ 

 tiplied, divided, or measured. Thus, length, weight, time, number, 

 etc., are called quantities. 



4. The first letters of the alphabet, a, b, c, etc., are used to 

 express known quantities ; and the last letters, z, y, x, etc., 

 those which are unknown. 



5. Known quantities are those whose values are given, or 

 may be easily inferred from the conditions of the problem 

 under consideration. 



6. Unknown quantities are those whose values are not given, 

 but required. 



7. Sometimes, however, the given quantities, instead of being 

 expressed by letters, are given in figures. 



8. Besides letters and figures, it will also be seen that we use 

 certain signs or characters in algebra to indicate the relations of 

 the quantities, or the operations which are to be performed with 

 them, instead of writing out these relations and operations in 

 words. Among these are the signs of addition (+), subtraction 

 ( ), equality (=), etc. 



9. Addition is represented by two lines (+), one horizontal, 

 the other perpendicular, forming a cross, which is called plus. 

 It signifies "more," or "added to." Thus a + b signifies that 

 b is to be added to a. It is read a plus b, or a added to b, or a 

 and b. 



10. Subtraction is represented by a short horizontal line ( ) 

 which is called minus. Thus, a b signifies that b is to be 

 " subtracted " from a ; and the expression (see Art. 22 below) 

 is read a, minus b, or a less b. 



11. The sign + is prefixed to quantities which are considered 

 as positive or affirmative ; and the sign to those which are 

 supposed to be negative. For the nature of this distinction, see 

 Articles 36 and 37. 



12. The sign is generally omitted before the first or leading 

 quantity, unless it is negative ; then it must always be written. 

 When no sign is prefixed to a quantity, + is always understood. 

 Thus a + b is the same as + a + b. 



13. Sometimes both + and (the latter being put under 

 the former, +.) are prefixed to the same letter. The sign is then 

 said to be ambiguous. Thus a +_ b signifies, that in certain 

 cases, comprehended in a general solution, b is to be added to a, 

 and in other cases subtracted from it. 



Observation. When all the signs are plus, or all minus, they 

 are said to be alike; when some are plus and others minus, they 

 are called unlike. 



14. The equality of two quantities, or sets of quantities, is 

 expressed by two parallel lines, =. Thus a + b = d signifies 

 that a and b together are equal to d. So 8 + 4 = 16 4 = 10 

 + 2=7+2 + 3. 



15. When the first of the two quantities compared is greater 

 than the other, the character => is placed between them. Thus 

 a > b signifies that a is greater than b. 



If the first is less than the other, the character =: is used ; as 

 a < b, namely, a is less than b. In both cases, the quantity 

 towards which the character opens is greater than the other. 



