Lessons in Algebra.

(For Pupil Teachers in their Fourth Year.)

BY THE EDITOR.

7. A collection of letters and signs used algebraically is called an algebraical expression. The parts of an expression which are connected with one another by the (+) or (-) signs are called the terms of that expression. When an expression consists of only one term it is said to be simple; as ab, a2b2, xyz; when it consists of two terms, as a + b, n − y, or 2ab – b2c, it is called a binomial expression; of three terms, as a+b-c, xy + y2-z, or 4ab+b2 - 1, it is called a trinomial. In general, however, all expressions which are composed of more than two terms are called multinomial expressions.

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8. When terms have their letters and indices alike,—that is, when they only differ in their coefficients and their signs, they are called similar or like terms; thus, 6ab and -4ab, n2y and 4n2y, 4a2bc and a2bc are pairs of like or similar terms.

CHAPTER II.-ADDITION AND SUBTRACTION.

9. In algebra, as in some other sciences, the signs (+) and (-) are used not merely to indicate that quantities are to be added or subtracted, but also, and generally, that these quantities are directly opposite in character. In Trigonometry, for instance, lines which extend in certain directions are termed plus or positive lines, and are distinguished by the sign (+); those which extend in an opposite direction are termed minus or negative lines, and are distinguished by the sign (-). In some thermometers all degrees counted upwards from a certain point are sometimes termed positive degrees; those which are counted downwards from the same point are distinguished, on the other hand, by the minus sign (-). Similarly, in Algebra, the student will be helped, especially in this chapter, if he consider all positive quantities to represent-say-gain, and all negative quantities loss; for instance, suppose we wish to collect, or add as it is called, the following terms, 4d-3d+ 2d-d+5d, here, the positive terms together equal 11d, and the negative terms - 4d; if we now consider the positive quantities to represent gain, and the negative loss, we see at once that the result of collecting a

gain of 11d and a loss of 4d, would equal a gain of 5d, that is, would equal 5d, whatever d may mean. Again, collect or add 3d 3d + d - 6d + 3d; here the plus, or gain, amounts to 7d, and the minus, or loss, to 9d; therefore, since the loss is 2d more than the gain, the result is a loss of 2d; that is, the whole line equals - 2d. Again, collect 4d - 2d + 3d - 5d; here the positive quantities, or gain, amount to 7d, the negative, or loss, amounts to the same; the result is therefore neither gain nor loss, that is, the whole line equals 0.

10. In all examples in addition and subtraction, we arrange the like terms in columns with their respective signs, and for addition collect each column as we did the lines above and the

line of figures on p. 25. Having arranged the columns, the student does not concern himself any more with the letters and indices, but only with the coefficients and signs.

Examples.-Add together 4a - 3b+2c, a + 2b − c, 3a + b − 5c, -7a+2b-c. (2.) 2x2 - 3xy+6, -4x2+ 2xy +4, 3x2+xy − 7, -5x2-3xy+3. (3.) 2ab+3ab2a2b2 - abc, -3a2b + 4a2b2 — 4abc, ab2+a2b2+6abc, - 6a2b-4ab2 – 3a2b2.

The student should, of course, verify the answers. the asterisks indicate that the positive and negative are equal and therefore neutralise one another.

EXERCISE III.

In No. 3 quantities

Find the sum of—1. 6a+3b-4c-3, 2a-2b+3c+5, 9a 7b+3c-6, -4a+2b+3c − 7.

2. 2a-3a2b+2b2-4, - 6a+2a2b-b2+6, 2a + 4a2b-b2 2, − – 4u+a2b+5b2+7, a 3a2b+262-5.

3. 14x3-21x2y-3xy2+y3, - 6x3+17x2y+5xy2-3y3, -3x2+ 4x2y-xy2+5y3, x3+6x2y+11xу2-4y3.

4. 3x2+2xy-3y2, -5x2-3xy - 2y2, 7x2+xy+4y2, −6x2+ 5xy + 3y2, 6x2+5xy - 2y2.

5.

· 3a3+5a2b+6ab2 – 5b3,

6a3-2a2b+3ab2 + b3, 3a3 + 5a2b-5ab2+3b3, - a3 + a2b-ab2+3b3, 4a3-3a2b+ab2 - 2b3.

6. 4x3+2xyz-3x2y+2z3, 4x2y – 3z3 + xyz − 4x3, 7xyz - 4x3 + x2y-323, 7x3- xyz + 2x2y - z3, -x3-9xyz - 4x2y +223.

7. 3a3-4a2b-5ab2 - 5b3, - 7a3 + 3ab2 + b3, 6a2b+3ab2, 2a3 + 4a2b3ab2+4b3, -2a3-8a2b+2ab2 - 6b3.

8. y3 +23 — 2.xy, x3 + y3 − 3xy, − 3x3 + 2xy − y3 +z3, −z3+5xy -3x3, 5x3 y3-23+14xy.

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(2.)

Answers. (1.) 13a-4b+5c-11. (3.) 6x3+6x2y+12xy2-y3. (4.) 5x2+10xy. 4ab2. (6.) 2x3- 31⁄23. (7.) - 4a3 – 2a2b - 6b3.

(1.)

7a+6b+5c

4a+2b+ c

3a +46 + 4c

− a + a2b+7b2+2. (5.) 3a3+6a2b+ (8.) 16xy.

The student will find no difficulty with the first of these examples. If 4a's be taken from 7a's, the remainder, of course, equals 3a's; and so on for the other two terms. The second example, however, is not quite so easy; in the first column the lower term is precisely like the upper, there is therefore no remainder after subtracting the lower from the upper. The two terms of the second column, too, are precisely alike in every respect, there is therefore no remainder after subtraction. In the third column, if a loss of 3d be taken from a loss of 7d, there would remain a loss of 4d only. A and B engage in a transaction by which they incur a joint loss of £7, if B now take as his share of the loss a loss of £3, there would remain to A a loss of £4 only. Now, carefully compare the above examples with those given below, and notice that the former are

like the latter, except that in the latter every sign in the lower lines has been changed, the answers are, however, alike in both

sets, but in the second set the answers were obtained by adding the two lines together instead of by subtraction. Thus, whenever the student cannot easily subtract he must adopt the following rule: -CHANGE THE LOWER SIGN AND ADD.* He should not re-write them as we did above, merely consider the lower sign to be changed. For instance, in example (1) we may say minus 4a added to 7a, minus 2b added to 6b, minus c added to 5c, and the answer in each case will be precisely the same as if we subtracted in the usual way.

EXERCISE IV.

1. From 8a+ 96+10c take 5a +7b+8c. 2. From 7x-2y - 3z take 3x - 2y - 3z.

3. From 3a2 - 2ab+b2 take a2 - ab+b2.

4. From 3x+5x2y2 – 7y3 take 2x3 – 6x2y2+4y3.

5. From a3 + 4a2b – 3ab2 – 63 take 4a3 + 4a2b+2ab2 – b3.

6. From x3 + y2 – 23 + 3 take · x3 +4y3 + 3z3 + 2.

7. From 3x2y-2xy2+y3 - 6 take 8x2y - 6xy2+7y3. 8. From 2a3 - 7a2b2 - b3 take 6a3 + a2b2 + 2abc.

9. From 7x2y - 3xy2+ y3 − x3 take 2x3 + y3 – 6. 10. From 2a2 - 3ab - 3 take 6a2 + b2 + 3.

Answers.—(1.) 3a+2b+2c. (2.) 4x. (3.) 2a2 - ab. (4.) 3+11x2y2-11y3. (5.) -3a3-5ab2. (6.) 2x-3y3-423 +1. (7.) - 5x2y + 4xy2 - 6y3 - 6. (8.) — 4a3 – 8a2b2 – b3 - 2abc. (9.) 7x2y-3xy2-3x+6. (10.) -4a2 - 3ab-b2-6.

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ERRATUM.-On page 27, January Number. Example 4; for 4√d read 4√α.

Latin Lessons for Private Students.

CHAPTER I.-Spread of the Latin Language-Latin the Foundation of Modern French, Italian, Spanish, and Portuguese-Proportion of Latin Words in our own Language—The Latin Alphabet-Pronunciation— Quantity-Accent-Parts of Speech.

1. THE Latin language was originally the language of Latium, a country on the west of Italy, stretching some 60 or 70 miles

* In a future lesson we shall prove that this rule is applicable to all cases of subtraction; at present we must ask the student to take it for granted, as he could not understand the proof.

south from the River Tiber. Rome was the chief city of Latium, hence, when the Romans by their conquests spread themselves over the rest of Italy, the Latin, or, as it might be called, the Roman language, became the language of nearly the whole of that country. With the Romans, too, Latin spread over the greater part of France, Italy, Spain, and Portugal, and, as a rule, became the language of the educated classes of these countries. It thus became the basis of the languages from which were afterwards developed modern French, Italian, Spanish, and Portu guese.

A very large proportion-about one-fourth-of the words of our own language, too, is derived either direct from the Latin, or indirectly through the French.

2. The Latin alphabet consists of 25 letters, being the same as the English,* except that it has now.' The letters y and z, however, do not occur in old Latin, but were introduced in Greek words considerably later.

3. For all the ordinary purposes of students using these lessons, Latin words may be pronounced as if they were English words of the same spelling, except in a very few instances. These instances will be dealt with as they occur.

4. The student will remember that in English words the vowel is sometimes long and sometimes short. In a somewhat similar manner some Latin syllables are long, while others, though spelt with the same vowels, are short. Thus the letter 'u' in the English word duke and the Latin word duco, I lead, is long; while the same letter in ductile and the Latin root "duc" is short. Again, the 'o' in the English word donation and the Latin "donum," a gift, is long; while the same letter in "domineering" and dominus, a lord, is short. This characteristic property of a syllable to be pronounced long or short is called its QUANTITY.

When a syllable is sometimes long and sometimes short its Quantity is said to be Doubtful.

In the strictly grammatical portions of these lessons all long Latin syllables will have the mark (-) placed over the vowelthus, duco, I lead; dōnum, a gift. All short syllables the mark (~)-thus, duc, dominus. Doubtful syllables are sometimes marked thus (), as duplex, i.e., duplex or duplex, double. The Quantity of diphthongs will not be raarked, however, as they are

*The forms of the letters of our alphabet are derived as a whole from the Latin; but the sounds from the Anglo-Saxon.