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man.

SECTION XVI.
Resuaż op EXAMPLES.

COMPARISOX-EXCORE, &c.
Avez vous autant de livres anglais, Have you as many English boobs, as
que de livres italiens ?
Italian books 1

The superlative absolute is formed by placing très, fort, or
J'en ai tout autant
I have just as many.

bien, cery, before the adjecuire ( 14, 11). J'ai autant de ceux-ci que de ceux. I have as many of these as of those. Ces chandeliers sont très utiles. These candlesticks are very weful. là.

Notre tailleur est bien obligeant. One tailor is very obliging. Il est aussi heureux que vou. He is as happy as you.

2. The superlative relative is formed by adding the article Avez vous plus d'assiettes que de Hare you more plates than dishes 1

le, la, les, to a comparative (f 14 (9)). plats ? J'ai plus de ceux-ci que de ceux-là. ' I have more of these than of those. Votre neuve est le plus savant de tous. Your nephero is the most learned of all. Est-il plus complaisant que ses Is he more obliging than his brothers 3. Encore is used in French in the sense of more, some more, frères ?

any more, still used affirmatively and interrogatively, but not Le Français a-t-il moins de légumes Has the Frenchman server vegetables negatirely. que de fruits ? than fruits i

Arez vous encore du café? Have you any more coffee! Il a moins de livres que de manu- He has fewer books than manuscripts.

J'ai encore du café.

I are saure (or some more) coffee. ecrits.

J'en ai encore.

I luce some more, or soane left. Il n'a pas autant de ceux-ci que de He has not so many of these as of ceux-là? those,

4. Ne-plus is used in the sense of not any more, and no En a-t-il moins que votre frère ? Has he less (of tiem) than your more, or none left.

Je n'ai plus de livres.

I have no more books.
Il en a tout autant.
He has quite as many.

JC A'ai plus de chocolat. I have no chocolate ieft
EXERCISE 29.

6. Ne-guère means but little, but fer.

Je n'ai guère d'amis.
Bleu, -e, blue.
Manuscrit, mn.

I have but few friends.
Fer, m. iron.

Je n'en ai guère.

I have but fero-rat little. Courage, m. courage. Fromage, m. cheese. script. Davantage,* more. Hollandais, m. Dutch Marécbal, m. blacksmith. 6. The pronouns moi, toi, lui, eux, are used instead of the Drap, in, cloth.

Modestie, f. modesty. nominative pronouns je, tu, il, ils, after the que of a comEnnemi, m. enemy. Italien, -ne, Malian. Soie, f. silk,

parison, and when the verb is understood. Espagnol, -e, Spaniard. Jardin, m. garden.

Tres, very.

Vous êtes plus heureux que moi. You are happier than I. Estampe, f. engraving. Manteau, m. cloak. Verre, m. glass.

Vous avez plus de mérite que lui. You have more merit than he, 1. Etes vous aussi content que votre frère ? 2. Je suis aussi content que votre frère. 3. Votre père a-t-il autant de courage

Résumé op EXAMPLES. que de modestie? 4. Il a moins de modestie que de courage. Votre marchand est bien obligeant. | Your merchant is very obliging. 6. Le libraire a-t-il autant de manuscrits que d'estampes ? 6. Voilà le meilleur de ces garçons. That is the best of those boys. Il a plus de celles-ci que de ceux-là. 7. A-t-il autant d'amis Nous avons encore des amis. We hare some more (or still) friends. que d'ennemis ? 8. Il a plus de ceux-ci que de ceux-là. 9. Vous avez encore du crédit. You have still (or yet) credit. A-t-il autant de pain que de fromage? 10. Il a tout autant Avez vous encore une piastre? Have you a dollar left: de celui-ci que de celui-là. 11. Le maréchal a-t-il plus de Le maçon a-t-il encore des briques ? Has the mason more bricks I chevaux que votre frère. 12. Il en a plus que mon père et nl n'en a plus.

He has no more-he has none left.

He has no more bricks. plus que mon frère. 13. N'avez vous pas froid ? 14. Non, il n'a plus de briques. Monsieur, je n'ai pas froid, j'ai très chaud. 15. Avez vous il n'en a plus guère.

He has but few. deux manteaux de drap. 16. J'en ai un de drap et un de Je n'ai guère de livres.

He has buit few left.

I hace but few books. velours bleu. 17. N'avez vous pas plus de verres que d'assi. Avez vous plus de courage que lui ? Hare you more courage than he! ettes ? 18. Nous en avons davantage.* 19. Le maréchal a-t-il II a moins de courage que moi. He has less courage than 1! plus de fer que d'acier ? 20. Il n'a pas autant de celui-ci que combien de piastres avez vous en- How many dollars have you still, of de celui-la. 21. Il a moins de celui-ci que de celui-la? 22. core ?

hare you left? Les Hollandais ont ils de beaux jardins ? 23. Leurs jardins

EXERCISE 31.
sont très beaux. 24. Les jardins des Italiens sont plus beaux
que ceux des Espagnols.

Correct, -, correct. Neveu, m. nephew. Sæur, f. sister,
Crédit, m. credit. Nièce, f. niece,

Salade, f. salad.
EXERCISE 30.

Beaucoup, much. Nouvelles, f. news. Tante, f. aunt. 1. Are you more attentive than your sister ? 2. I am not Boyer, Boyer. Quel, which, which one. Tous, all. Bo attentive as your brother. 3. Have you more courage than Dictionnaire, m. diction- Savant, -e, learned. Ville, 1. town, city. my brother ? 4. I have quite as much. 6. Has the black

ary. smith as much money as iron? 6. He has more of the latter

1. Votre dictionnaire est il très correct? 2. Il est plus cor. than of the former (Sect. 8, R. 6). 7. Has he more modesty rect que celui de Boyer. 3. Votre dictionnaire est le plus than the Spaniard ? 8. He has more. 9. He has more than correct de tous. 4. Quel est le meilleur de ces jardins ? 5. your friend's sister. 10. Are you not cold, Siri 11. No, Sir, Celui-ci est le meilleur de tous les jardins de la ville. 6. Avez but I am afraid and sleepy. 12. Has the Dutchman more

vous encore de l'argent ! 7. Je n'ai plus d'argent, mais j'ai cheese than the Italian? 13. He has more cheese and more

encore du crédit? 8. Avons nous encore de la salade? 9. money; 14. Have you as much English silk as Italian silk ? Nous n'en avons plus. 10. Nous n'avons plus de viande. 16. I have more of this than of that. 16. Who has more 11. Qui en a encore? 12. Mes frères et mes sæurs en ont friends than the Spaniard ? 17. Your friend has more.

18. encore.

13. En avez vous encore beaucoup: 14. Je n'en ai Has the Spaniard as much of your money as of his ? 19. He plus guère. 15. Votre tante a-t-elle plus de robes que votre has less of mine than of his. 20. Have we more silk cloaks nièce ? 16. Elle n'en a pas beaucoup. 17. Votre neveu est il than cloth cloaks? 21. We have more of these than of those. plus savant que votre nièce ? 18. Il n'est pas aussi savant 22. Have you good cloaks? 23. Yes, Sir, I have good cloaks; qu'elle. 19. Elle est plus savante que lui. 20. Avez vous good hats, and good leather shoes. 24. Have you more plates encore froid? 21. Je n'ai plus froid, j'ai bien chaud. 22. ikan dishes ? 25. I have not more plates than dishes ; but I N'avez vous plus de nouvelles ?. 23. Je n'en ai plus. 24. En have

more glasses than plates. 26. Are you not very cold 7 avez vous beaucoup ? 25. Je n'en ai guère. 27. No, Sir, I am neither cold nor warm. 28. Has your car.

EXERCISE 32. penter wood ? 29. Yes, Sir, he has wood, money, cheese, and meat. 30. Who has more money than the carpenter ? 31. is not very correct. 3. Has your father more courage than he ?

Has your brother a very good dictionary: 2. His dictionary The Duitser. has more. 32. Who has more engravings 4. He has much more courage than your nephew. 6. Hare tinan booho : 33. The bookseller has more of these than of your brothers credit? 6. They have but little credit, but they chose. 34. Are you as attentive as your friend ? 35. I am have money. 7. Is your aunt obliging? 8. My aunt is very more attentive than my friend.

obliging? 9. Have you still books, pens, and paper ? 10. i Davantage means more. It can never be placed before a noun ; it may have no more books, but I have still good

pens and excellent be used instead of plus, at the end of a sentence.

English paper. 11. Who has still paper: 12. I have no more,

119

tato.

but my brother has some more. 13. Have you any news,

EXERCISE 33. Sir? 14. No, Madam, I have none to-day. 15. Have you as Abricot, m. apricot. Fleur, f. flower. Pomme, f. apple. much wood as my brother's son? 16. I have more than you Anana, m. pineapple. Légume, m. vegetable. Pomme-de-terre, f. pso or he. 17. Are you still wrong? 18. No, Sir, I am no longer Beurre, m. butter. Magasin, m. warehouse. (plus) wrong, I am right. 19. Are your sisters still hungry? Cerise, f. cherry. Oncle, m. uncle. Prune, f. plum.

Poire, f. pear. 20. They are neither hungry nor thirsty, but they are still Epicier, m. grocer.

Sucre, m. sugar.

Thé, m. tea. sleepy. 21. Is your niece as learned as he? 22. She is more Etranger, -e, foreign. Poivre, m. pepper.

Jardin, m. garden. learned than he and (que) his aunt. 23. Have you no news, Sir? 24. No, Madam, I have no more news. 25. Who has 1. Combien de pommes-de-terre votre frère a-t-il ? 2. Il news ? 26. I have no more. 27. Have you them all? 28. n'en a pas beaucoup. 3. L'épicier a-t-il beaucoup de sucre Yes, Sir, I have them all. 29. Has your aunt much of it left? dans son magasin : 4. Il n'en a guère, mais il a beaucoup de 30. She has but little more

of it? 31. Has your brother any beurre et de poivre. 5. Votre jardinier a-t-il beaucoup de

33. He has two cerises ? 6. Il a plus de cerises que de prunes. 7. Les prunes more English horses ? 32. He has no more.

8. Les cerises sont more. 34. Have you a handsome French shawl left? 35. I sont elles meilleures que les cerises

meilleures que les prunes. have no more French shawls, but I have an English one.

9. Avez vous quelques poires

mûres ? 10. Nous en avons quelques unes, nous avons aussi SECTION XVII.

beaucoup d'ananas et d'abricots. ii. Votre oncle a-t-il quel1. The adverbs of quantity, combien, how much, how many bon et de beau. 13. Il a de beaux légumes et de belles

fleurs.

que chose de bon dans son jardin ? 12. Il a quelque chose de trop, too much, too many ; beaucoup, much, many ; assez, enough ; meaning no, when coming before a noun or an adjective, are frère et celles de votre jardinier. 18. N'avez vous pas aussi pcu, little

, few; guère, but little, few ; and the word pas, 14. Avez vous des fleurs étrangères ? 15. J'en ai quelques followed by the preposition de.

les miennes? 19. Non, Monsieur, je ne les ai pas. 20. Que Combien de fleurs avez vous ? How many flowers have you ?

en a beaucoup? 21. Personne n'en a beaucoup. 22. J'en ai J'ai beaucoup de fleurs. I have many flowers.

quelques unes. You have too much leisure.

23. Avez vous assez de thé? 24. J'en al Vous avez trop de loisir.

assez. 25. J'en ai plus que lui. Your sister has time enough. Votre scur a assez de temps. 2. The adverb bien, used in the sense of beaucoup (much,

EXERCISE 34, many), is followed by the preposition de, joined to or blended

1. Has your gardener many vegetables ? 2. Yes, Sir, he has with the article le, la, les Sect. 4].

many. 3. How many gardens has he? 4. He has several Vous avez bien de la complaisance. You have much kindness

gardens and several houses. 6. Have you many books ? 6. Elle a bien des amis.

She has many friends.

† have but few, but my friend has many. 7. What coat has 3. Quelque chose, something, anything (Sect. 5, 6], and rien, your brother ? ' 8. He has a good cloth coat. 9. Has your nothing, not anything, take de before an adjective.

uncle many peaches ?_10. He has but few peaches, but he has Votre ami a quelque chose d'agré. Your friend has something pleasant. many cherries. 11. How many plums has the tailor: 12. able.

The tailor has no plums, he has cloth and silk. 13. What silk Avez vous quelque chose de bon ? Have you anything good ?

has your friend the merchant? 14. He has a great deal (bearJe n'ai rien de bon. I have nothing (not anything) good,

coup) of silk, and a great deal of money. 15. Has the gar4. Quel, m., quelle, f., quels, m. p., quelles, f.p., are used dener anything good in (dans) his garden? 16. He has many interrogatively for which or what before a noun.

pineapples. 17. Has he more vegetables than fruit? 18. He Quelle serviette avez vous ? What or which napkin have you ! has more of this than of those. 19. Has your uncle many Quelles bourses votre ami a-t-il ? What purses has your friend?

pears and cherries ?

20. He has a few, and he has many 5. Que is used for what before a verb,

apples and plums. 21. Have you a few? 22. I have still Qu'avez vous ?

What is the matter with your many, but my brother has no more. 23. Which peaches has 6. Lequel, m., laquelle, f., lesquels, m.p., lesquelles, F. p., he? 24. He has large (grosses) peaches. 25. Which (ones) are used absolutely for the word which, not followed by a noun, have you? 26. I have the best peaches. 27. Has the mer.

chant anything good in his warehouse: 28. He has nothing and equivalent to which one, which ones.

good in his warehouse, but he has something good in his Lequel votre fils a-t-il ?

Which (one) has you son?

garden. 29. How many potatoes has the foreigner? 30. He Lesquelles avons nous ?

Which (ones) have we? 7. Quelques is used before a plural noun for a few, some; good vegetables. 33. Is he right or wrong?. 34. He is right,

has not many. 31. Has he good vegetables ? 32. He has quelques uns, m., quelques unes, f., are used absolutely, with

but you are wrong:

35. He has neither this book nor that, the same meaning Plusieurs means several, and is invariable. he has the bookseller's. Le Danois a-t-il quelques pommes ?

Has the Dane a few apples ?
Il en a quelques unes.

He has a few.
Il en a plusieurs.
He has several.

LESSONS IN BOTANY.-No. IV.
Resume Op EXAMPLES.

THE CORN PLANTS-Continued.
Combien de poires avez vous ? How many pears have you?

In our last lesson, it was shown that corn plants are cultivatca
Nous avons beaucoup de poires.
We have many pears.

grasses; we have now to consider a few more examples.
We have many of them).
Nous en avons beaucoup.

Maize, or Indian corn, was cultivated in America before the
Nous avons assez de cerises. We have cherries enough.
Nous n'en avons pas assez. We have not enough (of them).

discovery of that country by Columbus, and indeed from time
Vous n'avez guère de pêches.
You have but few peaches.

immemorial. It is a plant of much larger growth, in the Votre jardinier a bien des pêches. Your gardener has many peaches.

leaves, the ear, and the grain, than any other sort of corn. N'avez vous pas de pêches ? Have you no peachesi

See fig. 1, next page.
J'ai beaucoup de pêches et d'abri- I have many peaches and apricots.

Indian corn is also known by the name of Turkey-corn; this

name being given to it from the circumstance that maize is Le boucher a-t-il quelque chose de Has the butcher anything good. cultivated in that country. It is the largest and handsomest

of all the grasses cultivated for food. When growing luxuIl a quelque chose de bon et de He has something good and bad.

riantly, it attains a height of from five to six feet; while its mauvais.

broad leaves, springing from its straight thick stem, and its Il n'a rien de bon. He has not anything (nothing) good.

elegant spike of flowers at the summit, present a form which Quelles poires f. avez vous ? What or which pears have you :

is rarely surpassed. Next to rice it supplies food to the Nous avons celles de votre scur ? We have your sister's.

It forms the staple crop Quel habit m. avez vous ?

greatest number of the human race.
Which or what coat have you !

in North America, where the farmers make it answer a great
We have the tailor's.
Nous avons celui du tailleur.
Qu'avez vous de bon ?
What hare you good!

number of purposes, besides supplying their families with
Lequel avez vous ?
Which (one) have you ?

bread. Maize is also very extensively cultivated in Mexico : Lesquels votre frère a-t-il? Which (ones) has your brother 1 and from the genial nature of the climate, and the general fer

cots.

bon.

tivity of the soil, the returns yielded to the farmer are most stalk is of such great height, as to present the appearance of a abundant.

tree in miniature.

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countries it is sown on lands which are absolutely flooded for namely, 144 and 64; find their difference, and the result is also
weeks together; and when the grain has germinated, the 80. These results will take place with any two numbers; a fact,
water is drawn off. The fields are left dry for a month or more, of the truth of which you may be satisfied, by trying as many pairs
till the young plants are three or four inches high, when the of numbers as you please. This invariable result is expressed in
plantations are again flooded, and left in that state for two or general terms, in the words of the following theorem :-
three weeks. This soaking destroys the weeds which have 1. The product of the sum and difference of any two numbers is
sprung up with the rice. After this, the plants are left dry till equal to the difference of their squares.
the ears are formed, (see fig. 3), being repeatedly hoed between

Again, take any two numbers, say 12 and 8 as before, and find and kept clear of weeds ; the watering is again removed, and the their sum and difference, namely, 20 and 4; then find the squares flood is left on the ground till the grain is ripe. “The verdure of these numbers, namely, 400 and 16. Next, find the squares of of the young plant,” says Heber, when visiting Ceylon, " is the two assumed numbers 12 and 8, and twice their product, particularly fine; and the fields are really a beautiful sight, namely, 144, 64, and 192. Find the sum of these two squares, when surrounded by, and contrasted with, the magnificent namely, 208'; then, add to this, the double product. 192, and the mountain-scenery.'

sum is 400; subtract from it the same product, and the remainder Rye is used by several nations of Europe as bread-corn. It or difference is 16. The same results will take place with any is not so much liked by birds as many other plants, and of this other two numbers you choose to try. Hence, we deduce the some farmers take advantage. They sow a narrow border of two following general theorems :rye round their fields of wheat and other grain ; and when 2. The square of the sum of any two numbers is equal to the sum thus fenced they are not attacked by poultry, nor even by the of their squares, increased by twice their product. wild birds. As these seldom alight in the centre of corn-fields, 3. The square of the difference of any two numbers is equal to but confine their depredations to the outer boundary, they the sum of their squares, diminished by twice their product. visit the rye, and finding what they do not like they proceed Our object in presenting these theorems, as well as those ap. no further. The bread made on the continent from rye, is very pended to the rule of subtraction, is to lead the student by degrees black, and as leaven is used instead of yeast it is sour, and to to the consideration of some general rules, which are equally true a stranger accustomed to better food extremely unpalatable. of all numbers, and are not confined to particular instances such That which is sold in London by some bakers as rye bread is, as those by which they were illustrated. By help of a few new on the contrary, well-flavoured and very good ; similar, in fact, definitions, these theorems may be made a means of introduction to to brown wheaten-bread.

the universal language of algebra. In this science, the letters of In addition to the corn-plants already mentioned, there are the alphabet are employed to represent numbers, not fixed numbers, others belonging to the large family of the grasses, which but any numbers whatever ; and therefore, all theorems which can might be employed as food, and which are only neglected be demonstrated by means of letters, must be considered as univerfrom the smallness of their seeds. None are unwholesome in sally true, and equally applicable to all numbers. In addition to their natural state except darnel, a common weed in many the definitions, and explanation of signs already given in No. III., parts of England. This is one of the vegetable products page 36, we may add the following relating to universal arithmetic. reserved by Providence for other purposes.

i. When numbers are represented by letters, they are usually How many plants, we call these weeds,

called quantities ; when a particular value is given to them in any Against our wishes grow,

problem or question, they are then called known quantities ; when And scatter wide their various seeds

their value is not giveni, but required in any problem or question, With all the winds that blow.

they are then called un known quantities. The former are denoted Man grumbles when he sees them rise,

by the initial (beginning) letters of the alphabet, as a, b, c, &c. ; To foul his husbandry ;

and the latter by the terminal (ending) letters of the alphabet, as Kind providence this way supplies His lesser family.

2, y, x, &c. In expressing general theorems, however, either of

these kinds of letters may be employed. Scattered and small, they 'scape our eye,

2. The arithmetical signs are used to denote that the arithmetical But are not wasted there ; Safe they in clefts and furrows lie,

operations are to be performed upon the numbers represented by The little birds find where.

the letters. Thus the expression atb=c, means that the sum of It is worthy also of remark that the tall sugar-canes and the two numbers represented by a and 6 is equal to the number regigantic bamboos of tropical climates are only grasses on a of the two numbers represented by a and 6 is equal to the num.

presented by c. The expression - b=C, means that the difference larger scale, agreeing with our own in every essential particu- ber represented by c. The expression axb=c, means that the lar, and differing mostly in size. They afford to the Indian sa vage almost all he wants, except the food which he derives product of the two numbers represented by a and 6 is equal

to the from his rice or his maize. ** With their lightest shoots

number represented by c. Very often, and mark this particularly, he makes his arrows; the fibres of the wood form bow very often the sign X is omitted ; that is, ab=e means the same as strings; and from the larger stems he fabricates a bow; a long atb=c. The expressions a+b=c, or = C, or a : b=c, all and slender shoot affords him a lance-shaft, and he finds its hardened point a natural head for the weapon. With the

mean the same thing, --namely, that the number represented by a hardened stems he builds the walls and roof of his house ; its divided by the number represented by b, gives for a quotient the leaves afford him an impenetrable thatch ; split into narrow

number represented by c. strips, it gives him the material for weaving his floor-mats,

3. When numbers or quantities are enclosed in a parenthesis, thus and other articles of domestic convenience ; its fibre furnishes (a+b), (2—6), &c., the expression signifies that the number or him with twine, and its leaves provide him with paper, when quantities so included are to be treated as a simple number or he becomes sensible of the utility of such a material. Would quantity, or rather that the operation of the signs applied to them he commit himself to the waves, the stems form the hull of are to be performed, and that the result is to be treated, as it is in his boat, which by a few skins stretched over it is rendered reality, like a simple a umber or quantity. Thus, the expression water-tight; they also give him masts, and their slips of wood (a+b)xc=d, means that the sum of the quantities a and b is to be become cordage or are woven into sails. In China, India, and multiplied by the quant ity c, and that the product is equal to the Japan, bamboos are used for a great number of useful pur- means that the sum of the quantities a and b, is to be multiplied

In like manner the expression (a+b)x(c+d)=e, poses.

by the sum of the quan tities c and d, and that the product of these

sums is equal to e. LESSONS IN ARITHMETIC.-No. VIII.

4. Applying these symbols of numbers and of operation to the In our last number, we intended to annex to the Rule of Multis general theorems in No. V, page 67, they will stand thus :plication, some theorems of frequent occurrence, and of considerable

(1) (a+b)+(a−b)=2a use in practice. These we now proceed to give. Take any two numbers, say 12 and 8, and find their sum and difference, namely, In these expressions a cienotes the greater number and the less, and 20 and 4 ; then, multiply these quantities together, and the product reasoning upon them generally we shall be convinced of the univer. $ 80. Nest, find the squares of the two numbers 12 and 8, sality of their application. Thus, in the rem (1), we are to add a to

a

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to 3-6; that is, we are first, to add 6 to a; second, to substract complete product of a by a that is wanted, but the product of the from a; and third, to add the results. Now, we know that if we difference between a and b by a; here, the general principle is add a to a, we shall have twice a, or 2a ; but if we add something, that the product of the difference of any two numbers by a there viz., b to the one a, and take away the same thing, viz., 6 from number is equal to the difference of the products of those i the other a, we shall still have twice a or 2a ; because what we add numbers by the third number ;* but this product a mesto to the one we take away from the other, and therefore the sum will diminished by the product of the same difference by 6, as it is neither be increased nor diminished, that is, the sum will still be product of the difference (a - b) by the difference of a and , the same as if we had only added the two together.

is required. Now, if we multiply the difference (a−b) by ül 5. Again, in theorem (2), we are to subtract a- from a+b; quantity o of the difference (a-6), we shall have the produce that is, we are first, to add to a ; second, to subtract 6 from a ; ab - b*, as above; and subtracting this product from the forex and third, to subtract the latter result from the former. Now, we product a? - ab, we have a+b: - 2ab; for, if we subtract know that if we subtract a from a, we shall have nothing left; but, which makes a --- ab - ab, or a.- 2ab (since ab is to be se> if we add something, viz., b, to the one a, and take away the same tracted twice), we have subtracted too much by ba; thing, viz., b, from the other a, we shall have twice b, or 26 left; therefore add 63 to the result in order to obtain the true remur because, what we add to the one, we take away from the other; viz., atb.- 2ab. and therefore, in taking the latter a away from the former a, we 10. Another theorem of great importance, is necessary : take away too much by the quantity b, and therefore we must add given here, as it is an extension v horems (2) and (3) of this b to the former b, in order to obtain the full remainder. 7, to the combination of seva rás urnbeers by the signs of edés

6. As we have defined the square of a number in our last lesson, or subtraction : viz.we may now inform the student how it is denoted, or marked on The square of the sum of several numbers, is equal to term the number itself. As the same factor occurs twice in the product of the squares of those numbers, and twice the product of the called the square, this occurrence is indicated by placing the num- number by all that follow it; tưice the product of the second ca ber 2 on the right of the number or quantity, in a smaller cha- by all that follow it; and so on, to the last number. Thus

, 3 racter, and on a higher level than the latter ; thus 62 denotes the the several numbers, 4, 6, and 10; their sum is 20, and its & square of 6, or 36; and a denotes the square of a, or axa. In is 400. Now, this square of 400, is equal to the sum of the sun like manner, (10+2)2 denotes the square of 10+2, or of 12; and of the numbers 4, 6, and 10: viz. 16, 36, and 100; with twie (a+b) denotes the square of a +b, or of the sum of the quantities product of 4 by 6, and by 10, viz. 48 and 80; and twice the a and'b. Again, (4+5+6) denotes the square of 4+5+6, or of duct of 6 by 10, viz. 120 ; for 16+36+100448+80+120– 15; and (a+b+c)2 denotes the square of a+b+c, or of the sum Expressed symbolically, this operation stands as follow: ita of a, b, and c.

10) = 4 + 6+ + 10 + (2X4X6)+(2X4X10)+(2x6x: 7. Applying the symbols of numbers and of operation to the gene- or, 20-400=16+36+100+48+80+120. ral theorems, placed near the beginning of this lesson, they will Applying to this theorem, the symbols of numbers and of open stand thus :

tion, it will, in the case of four quantities, a, b, c, d, stand the (1) (axb)x(a−b)= Q2 — 62

(a+b+c+d)?=a+to+da+2ab+2ac+-2ad12be **** (2) (a+b)=a +62+2ab.

2cd. The operation indicated here, is that the sum (a+b+c+ (3) (a-Va=a? +62-2ab.

be multiplied by itself; now, if we multiply (a+b+ct In these expressions, still a denotes the greater number, and 6 the the first part, we have a tab+ac+ad; if we multiply by in less, and we may now reason upon them in a general way, as we second part, we have ab+8+be+bd; by c, the third part, did on the expressions, taken from the theorems in page 67. Thus, have ac+be+r+cd; and by d, the fourth part, we have als in theorem (1) we are to multiply the sum (a+b) of any two quan cd+d. But, as the whole product is the sum of all these para tities a and b, by their difference (a-b), and find the product;

we have first, a +b+r+d?, the sum of the squares of the per now, if we multiply the sum (a+b) by one of the quantities a, of then, we have abtab, or 2ab ; actac, or 2ac ; adtad, o the difference (a−b), we shall have the product aa-ab; for axa= bc+bc, or 2bc; ba+od, or 2bd ; cd tcd, or 2cd; whence

, we have a a’, and b Xa=ab; and adding these products, we have a tab, second, 2ab+2ac+2ad+2bc+26d+2cd, the sum of the producer by the general principle of Rule 4 in Multiplication. But this of twice each part by all that follow it. Hence, the com product by a must be diminished by the product of the same sum square of, (a+b+c+d) is a +62+e+a+2ab+-2ac420 by b, as it is the product of the sum (a+b) by the difference of a +od+2cd; and the truth of the principle is manifest. and b, that is required. Now, if we multiply the sum (a+b) The rule for squaring a quantity consisting of several part by the quantity 6 of the difference (a−b), we shall have the produced from the preceding principle is simply this : Square the duct ab +62; for axb=-ab, and bxb=b; and adding these pro-term, and multiply all the terms that come after it by turite ducts, we have ab +62, by the general principle above mentioned. term; do the same with the second term; and so on, to ti á Subtracting this product, therefore, from the former product, we term. The sum of all these products is the answer. have a--; inasmuch as ab-ab=0); the principle of this operation being that the product of the sum of any two numbers by

QUESTIONS ON THE PRECEDING LESSON. their difference, is equal to the difference of the products of the sum by each of the numbers. This principle is a natural conse- is meant by a known quantity? How is it represented ?

1. What is meant by a quantity? How is it represented quence of that given in Rule 4 of Multiplication, when applied to meant by an unknown quantity: How is it represented?.FI a multiplier divided into two parts.

the sum of two numbers or quantities is added to their difiers 8. In like manner, in theorem (2), we are to multiply the sum what is the result?. When the difference of two number (a+b) of any two quantities a and b, by the sum (a+b) of the same quantities is taken from their

sam, what is their result?" quantities, and find the product ; now, if we multiply the sum together,

what is their product? What is the square of the (a+b) by one of the quantities a of the sum (a+b), we shall have of two numbers or quantities equal to? What is the square the product atab, as before ; but this

product" by a must be difference of two numbers or quantities equal to? Whaine increased by the product of the same sum by b, as it is the product square of the sum of several numbers or quantities equal to? of the sum (a+b), by the sum of a and b that is required. Now, “Add the quantities x+y and 1-y together. if we multiply the sum (a+b) by the quantity b of the sum (a+b), Subtract the quantity 2-y from the quantity +4. we shall have the product ab +62, as before ; and adding this pro- Multiply aty by z-y, and tell the product. duct to the former product, we have a2+62+2ab, for abtab Find the squares of ty and of y. =2ab.

Find the squares of ctyto+m, and of a tbtctate 9. Lastly, in theorem (3) we are to multiply the difference (a-6)

What are the sum and difference of mand ? of any two quantities a and b, by the difference (a - b) of the same

What is the product

of the sum and difference of m anda! quantities, and find the product. Now, if we multiply the differ

What are squares of the sum and difference of m anda?

What are the factors of p2-92 ence (a - b) by one of the quantities a of the difference (a - b), we shall have the product që-ab; for, ax=a?, and b Xa= ab; and

What are the factors of p2++2pq and of patagapa! subtracting the latter product from the former we have a? - ab; that this subtraction is necessary, is obvious, because it is not the paragraph.

• This principle is merely a generalisation of that gives in the presin

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