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want.

You go to sleep easily. southern margin of the volcano. These changes in the form | Vous vous endormez facilement.

I auake very early. and position of cones of eruption give to Vesuvius, at different Je m'éveille de très bonne heure. epocns, a different appearance. In this eruption of October, Pourquoi vous approchez vous du Why do you come near the fire

feu ? 1822, in twenty-four hours after the falling in of the great cone

Je m'en approche parceque j'ai I come near it because I am cold. of cinders just mentioned, and when the small but numerous

froid. streams of lava had flowed off, then a fiery eruption of ashes Nous nous éloignons du feu. We go from the fire. commenced, which continued without intermission for twelve Nous nous en éloignons.

We go from it. days, and covered the sides of the mountain.

Nous nous approchons de notre We go near our father. These different measurements of Vesuvius suggest grounds for père. a very bold theory in geology. How is it that the north margin Nous nous approchons de lui. We go near him. of the volcano, that called Rocca del Palo, maintains such a

EXERCISE 75. uniformity of height while the other is lowered? The probable cause is that the north margin is in the process of being now Aussi, also.

Encre, f. ink.

Ordinairement, generaised up gradually by the upward tendencies of subterranean Aussitôt-que, as soon Fenêtre, f. window. rally. forces. Between the years 1816 and 1822 we are sure that

Feu, m. fire.

Plume, f. pen. that margin was from 3,970 feet to 4,022. When it was mea- Canif, m. penknife. Fourchette, f. fork. Pourquoi, why. sured, thirty or forty years before, the height was from 3,875 Demoiselle, young lady. Heure, f. hour, o'clock. Prêt-er, 1. to lend. feet to 3,894. How is this? Future investigations will, per- Domestique, m. ser- Moins, less, before. Quart, m. quarter, haps, decide how much of this difference is due to errors in

Obligé, e, obliged. Taill-er, 1, to mend. measurement, and how much to the actual rise of the mountain

1. Pouvez vous vous passer d'encre? 2. Nous pouvons nous by the expansion of heat from below. "If the lava þeds of en passer, nous n'avons rien à écrire. 3. Vous servez vous de Rocca del Palo,” says A. Von Humboldt," really become votre plume: 4. Je ne m'en sers pas; en avez vous besoin : higher we must assume them to be upheaved from below by 5. Ne voulez vous pas vous approcher du feu? 6. Je vous volcanic forces."

suis bien obligé, je n'ai pas froid. 7. Pourquoi ces demoiselies s'eloignent elles de la fenêtre: 8. Elles s'en éloignent

parcequ'il y fait trop froid. 9. Ces enfants ne s'adressent ils LESSONS IN FRENCH.—No. XVIII. pas à vous ? 10. Ils s'adressent à moi et à mon frère. 11. A By Professor Louis FASQUELLE, LL.D.

quelle heure vous éveillez vous le matin? 12. Je m'éveille

ordinairement à six heures moins un quart. 13. Vous levez SECTION XXXVIII.

vous aussitôt que vous vous éveillez? 14. Je me lève aussitôt 1. The reflective verb, se passer, is used idiomatically in the que je m'éveille. 15. De quels livres vous servez vous ? 16. sense of to do without. It is followed by the preposition de, Je me sers des miens et des vôtres. 17. Ne vous servez vous when it comes before a noun or a verb :

pas de ceux de votre frère ? 18. Je m'en sers aussi. 19. Les Vous passez vous de ce livre ? Do you do without that book?

plumes dont (Sect. 31, R. 8} vous vous servez sont elles bonnes ? Je ne puis m'en passer. I cannot do without it.

20. Pourquoi votre ami s'éloigne-t-il du feu: 21. Il s'en 2. Se servir (2 ir. see $ 62], to use, also requires the prepo

éloigne parcequ'il a trop chaud. 22. Pourquoi votre domes

tique s'en approche-t-il? 23. Il s'en approche pour se chauffer, sition de before it object :

24. Vous ennuyez vous ici: 25. Je ne m'ennuie pas.
Je me sers de votre canif. I use your penknife.
Je me m'en sera pas.
I do not use it.

EXERCISE 76.
3. The second example of the two rules above, shows that,
when the object of those verbs is a thing, it is represented in

1. Will you lend me your penknife ? 2. I cannot do withthe sentence by the pronoun en :

out it, I want it to mend my pen. 3. Do you want to use my

book? 4. I want to use it, will you lend it to me? 5. What Je m'en sers, je m'en passe. I use it, I do without it.

knife does your brother use? 6. He uses my father's knife 4. The pronoun* used as indirect object of a reflective verb, and my brother's fork. 7. Will you not draw near the fire ? if representing a person, follows the verb [$ 100 (4)] :

8. We are much obliged to you, we are warm. 9. Is that Je puis me passer de lui. I can do without him.

young lady warm enough? [Sect. 34, 3.] 10. She is very cold. Je m'adresse à vous et à elle. I apply to you and to her. 11. Tell her (dites lui) to come near the fire ? 12. Why do you 5. S'endormir (2 ir. see § 62]; to fall asleep, and s'éveiller,

go from the fire? 13. We are too warm.

14. Does your

brother leave the window? 15. He leaves the window beto awake, are also reflective.

cause he is cold. 16. To whom does that gentleman apply i Je m'endors aussitôt que je me I fall asleep as soon as I go to bed.

17. He applies to me and to my brother. 18. Why does he couche.

not apply to me? 19. Because he is ashamed to speak to Je m'éveille à six heures du matin. I arcake at six o'clock in the

morning.

you. 20. Do you awake early every morning? 21. I awake

early, when I go to bed early. 22. Why do you go to sleep? 6. S'approcher, to come near, to approach ; s'éloigner, to draw 23. I go to sleep because I am tired. 24. Are you afraid to back, to leare, take the preposition de betore a noun. Their

go near your father? 25. I am not afraid to approach him. object, when a pronoun, is subject to Rules 3 and 4 above :

26. Can you do without us : 27. We cannot do without you, Votre fils s'approche-t-il du feu ? Does your son draw near the fire?

but we can do without your brother. 28. Do you want my Il ne s'en approche pas. He does not come near i.

brother's horse: 29. No, Sir, we can do without it. 30. Do Il s'éloigne de moi et de vous. He goes from me and from you. you intend to do without money? 31. You know very well

that we cannot do without it. 32. Is your brother weary of Résumé of EXAMPLES.

being here ? 33. He is not weary of being here. 34. Come Vous servez vous de ce couteau ? Do you use that knife !

near the fire, my child. Je ne m'en sers pas, il ne coupe pas. I do not use it, it does not cut. De quels couteaux vous servez vous? What knives do you use 1

Section XXXIX. Nous nous servons de couteaux We use steel Inires.

1. The verb aller (1 ir. § 62), conjugated reflectively, and d'acier. Pouvez vous vous passer d'argent ? Can you do without money ?

preceded by the word en, i. e. s'en aller, corresponds to the Nous ne pouvons nous en passer. IVe cannot do without it.

English expressions to go away, to leave :Vous passez vous de votre maitre ? Do you do without your

teacher ? 2. INDICATIVE PRESENT OF THE VERB s’EN ALLER, To Go Nous nous passons de lui. We do without linn.

AWAY.
Vous adressez vous à ces messieurs ? Do you apply to those gentlemen ?
Nous nous adressons à eux et à We apply to them and to you.

Je in'en vais, I go away;

Nous nous en al- We go arody:
Tut t'en vas,

Thou art going lons,
away:

Vous vous en allez, You are going The role does not apply to the reflective pronoun, which is sometimes Il s'en van He goes away;

away; ou indirect objecte

Ils s'en yont. They go anda.

vous.

nous ?

3. THE SAME TENSE CONJUGATED INTERROGATIVELY, 5. Will you make haste to finish your letter? 6. I make Est-ce que je m'en Do I go away? Nous en allons Do we go away ?

haste to finish it. 7. Does the gardener get angry with his nis?

trother? 8. He gets angry against him when he does not T'en vas tu ? Art thou going Vous en allez Do you go away? make haste. 9. Make haste, my friend, it is ten o'clock. 10. away! vous ?

Why do you not make haste: 11. I like to play, but I do not S'en va-t-il ? Is he going away! S'en vont ils ? Are they going like to study. 12. Do you like to stay at my house : 13. I

away? like to stay there. 14. Are you rejoiced at the arrival of your 4. Se fâcher, to be or became angry, requires the preposition mother? 15. I rejoice at it. . 16. Is not your brother wrong contre or de before the noun or pronoun following it :

to go away so soon?_17. He is right to go away, he has much Se fache-t-il contre votre frère ? Does he become angry against your tunes ?

to do at home. 18. Do you rejoice at other people's misfor

19. I do not rejoice at them. 20. I rejoice at your brother! Il se fâche contre lui.

success. He is angry with him.

21. Does not your brother draw near the fire? 22. Vous vous fâchez d'un rien.

23. Does that young You get angry at nothing.

He goes from the fire, he is too warm.

lady get angry with you? 24. She gets angry at trifles (de 6. Se réjouir, to rejoice, is followed by the preposition de :- rien). 25. Do you like to be in Paris ? 26. I like to be there.

Je me réjouis de votre bonheur. I rejoice at your happiness. 27. Can you do without me to-day? 28. We cannot do with6. Se plaire [4 ir. see § 62,] to take pleasure, to delight in any you want your penknife ? 30. I want to use it. 31. Make

out you-make haste to finish your work (ouvrage). 29. Do thing, to like to be in a place, takes à before its object :

haste to rise, it is six o'clock. 32. Is it fine weather ? 33. No, Je me plais à la campagne. I like to be in the country.

Sir, it rains. 34. Is your father well this morning? 35. Yes, Je me plais à étudier, à lire. I take pleasure in studying, in reading. Sir, he is very well.

7. Se dépecher, se hâter, to make haste, take de before their object: Dépêchez vous de finir vos leçons. Make haste to finish your lessons.

LESSONS IN ARCHITECTURE.—No. II. Pourquoi ne vous dépêchez vous pas ? Why do you not make haste?

AFTER the rough sketch of the origin of architecture in our last, RESUME OF EXAMPLES.

we must notice in proper order that system of construction,

the monuments of which cover a great part of the old world. Le marchand s'en va-t-il aujour- | Does the merchant go away to-day? This system had its origin among the Shemitic tribes, which d'hui ?

at the commencement of civilisation peopled the fairest part of Nous nous en allons demain. We are going away to-morrow. Je m'en vais quand je suis fatigué. | I go away when I am tired.

the globe. This early system, noted for the rudeness of its Pourquoi vous fâchez vous contre

form, its stability without mortar, and the great size and irreguWhy do you get angry with him? lui ?

larity of its materials, is attributed to the Pelasgians, a people Il se plait à jouer, il n'étudie ja- He takes pleasure in playing, he originally from Upper Asia, who, according to Herodotus, mais.

never studies.

spread themselves over Phænicia and Asia Minor, and colonised Vous plaisez vous chez vos parents? Do you like to be at your relations ? Greece and Italy. Examples of this style of architecture, De quoi vous réjouissez vous ? At what do you rejoice!

called Pelasgic, are found extending from the borders of Persia Nous nous réjouissons de votre We rejoice at your success.

and Armenia to the western limits of Asia. Crossing the Medisuccès.

terranean, it spread over Greece, where the most remarkable Nous nous en réjouissons. We rejoice at it.

monuments described by ancient authors, from the age of Pourquoi vous dépêchez vous ? Why do you make haste !

Hesiod and Homer, are traced, according to tradition, as far Nous nous dépêchons d'écrire. We make haste to write. Nous nous plaisons en Angleterre. We like to be in England.

back as eighteen centuries before our era. This was the style

of construction used in the heroic times of ancient Greece; Nous ne nous plaisons pas à Paris. We do not like to be in Paris, Nous ne nous y plaisons pas. We do not like to be there.

and at a later period it was employed on certain important Vous plaisez vous à New York ? Do you like to be in New York ?

occasions. Nous nous y plaisons. We like to be there.

The migrations of the Pelasgi carried this system into Italy,

and we meet it at every step, particularly in the central counEXERCISE 77.

tries. Examples are also to be seen in nearly all the western Ambassadeur, m. am- Jouer, 1. to play. Prochain, e, next. islands of the Mediterranean, in the Balearic Isles, and some tossador.

Malheur, m. misfortune. Retourn-er, 1. to return. even on the coasts of France and Spain. In fine, by a remarkArrivée, f, arrival. Mieux, better.

Semaine, f. week. able coincidence, travellers who have drawn and described the Autri, m. others. Midi, m. noon.

Tante, f. aunt.

monuments of Palenque and Papantla, cities of Mexico deCour-ir, 2 ir. to run. Parceque, because. Turc, turque, Turkish.

stroyed long ago, and grown over by forests, exhibit construcJamais, rever.

tions similar to those of the Pelasgi. The gigantic remains 1. Vous en allez vous bientôt ? 2. Je m'en vais la semaine of the Pelasgic monuments, to this day subjected to examinaprochaine. 3. Pourquoi vous en allez vous ? 4. Parceque je tion by travellers, bear traces of different modes of building. ne me plais pas ici. 5. Vous plaisez vous mieux chez votre | Those which seem to be the most ancient are composed tante qu'ici : 6. Je m'y plais mieux. 7. N'avez vous pas of blocks of stone, or rather of rocks, so rude and so immense tort de vous en aller si tốt:* 8. J'ai raison de m'en aller. 9. that Pausanias, in speaking of the walls of Tyrins, built thirtyNe vous rejouissez vous pas des malheurs d'autrui ? 10. Nous six centuries ago, describes them thus :-* These walls are ne nous en réjouissons point. 11. Cet homme se fâche-t- constructed of unhewn stones, and are all of such dimensions il contre le jardinier: 12. Il se fâche contre lui parce qu'il ne that a yoke of oxen could not shake the smallest of them. veut pas se dépêcher. 13. Se fâche-t-il bien souvent? 14. Il The interstices are filled up with smaller stones, which serve to se fâche à tout moment, il se fâche d'un rien. 15. Ne vous unite the larger ones." These walls present the same appeardépéchez vous jamais ? 16. Je me dépêche toujours quand ance now which they did in the days of Homer and of Palj'ai quelque chose à faire. 17. Ne vous plaisez vous pas à sanias. They are about 25 feet thick, and about 43 feet in courir et à jouer? 18. Je me plais à jouer et mon frère se plait height. Two temples, close to each other, in the island of à lire. 19. Vous réjouissez vous de l'arrivée de l'ambassadeur Gozo, near Malta, are analogous in their construction to the Turc: 20. Je m'en réjouis. 21. Ne vous plaisez vous pas en walls of Tyrins. They are built of immense blocks of stone, Amérique : 22. Je m'y plais beaucoup mieux qu'en France. forming a sort of artificial hill, in which are placed the naves 23. Votre écolier ne se plait il pas chez vous. 24. Il se plait and arches of the temples; but some of the rocks bear traces chez moi, mais il désire retourner chez son père. 25. Dépê- of masonry. chez vous, il est déjà midi.

It has been proved, by careful examination, that these edi.

fices were dedicated to the gods of Asia. To conclude ; the EXERCISE 78.

walls of Tarragona, on the east coast of Spain, are constructed, 1. At what hour does your friend go away? 2. He goes like the preceding, of immense rocks in their natural state. away every morning at nine o'clock. 3. Do you go away The application of instruments to building, at a later period, with (@tec) him 4. I go away with him when I have time. I caused the edifices of the Pelasgians to sosume another form. The stones taken from quarries, were cut into irregular polygons, the union of these materials architectural forms, without going and placed one upon another in such a manner as to make the through that initiatory process which characterises the origin different faces of the geometrical figures which they employed, of all human inventions. Yet the plains of Chaldea soon ex. coincide, the salient angles filling up the re-entrant angles hibited constructions which had a great influence over primi. formed by two adjoining stones. (Fig. 5.) This was the tive art in the east, and formed the basis of a system which Fig. 5.

ordinary manner of building un- extended its branches even to the west. The want of stones der this system of construction. It in Mesopotamia soon taught the inhabitants to mould bricks, is met with from the lake Van, on and their most ancient temple mentioned in the Bible, called the frontiers of Armenia, to the the tower of Babel, was an immense pyramid built of bricks west of Italy, Sardinia, and the piled on one another, and forming, according to report, eight Balearic Isles; and it is found stories or rows, gradually receding from each other. At the

in temples and in tombs, in pub- top of this building they sacrificed to Baal; at a later period Pelasgic wall.

lic and private buildings, and in the Chaldean kings placed his statue there, when their artists innumerable military constructions. At last, a third method had made some progress in the art of sculpture. It is probable presents itself in the walls of these early buildings ; namely, that this pyramidal-formed temple owed its origin to their rethat in which the stones are fashioned in the square form ; and membrance of the practices of those Caucasian countries the buildings themselves, assuming the same form, exhibit a whence the Shemitic tribes derived their origin. Herodotus greater degree of civilisation ; and the invention and application gave a glimpse of the truth, when he said that the Scythians of more exact instruments. The walls of the ancient Mycenæ made their temples or altars with a great quantity of wood were built in this manner. (Fig. 6.)

heaped in the form of a pyramid. However the case may be, Fig. 6.

this very simple form, which appears to have come naturally to the mind of those men who were the first to raise large constructions, spread itself over all Asia ; the ancient pagodas of India are built in this form ; the most ancient monuments of Lower Egypt and Ethiopia, where the Shemitic tribes settled in Africa, are all of them pyramids. (Fig. 7.) In Asia,

Fig. 7.

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Walls of Mycenæ, and gate of Lions.

Pyramids of Memphis. The continued and progressive order of these Pelasgic con- whole cities, Ecbatana for example, presented numerous constructions, is one of the most interesting facts in the history centric enclosures rising one above another in such a way as of the art of building-particularly when we refer them to an

to exhibit the pyramidal form. The celebrated gardens of antiquity which goes back to the heroic time of Greece. Babylon, formed of numerous terraces, one above another, had Doubtless the gradual improvement which is to be seen in the also the same configuration. In short, this must be considered walls constructed by this original people, does not reveal all the as the progress of architecture, when we see that the most revolutions of this art in early antiquity; but it enables us to ancientreligious edifices of the Mexicans are immense pyramids, perceive the progress of the greater part of the civilised world, a simple at first like those of Chaldea, and of Lower and Upper progress which it must necessarily follow, because it is the Egypt; but at a later period, ornamented with sculpture like nature of all human inventions to pass from early and rude the pagodas of India. Ancient public buildings were also attempts, to successive periods of improvement and perfection. found in Mexico of a pyramidal form. It is evident that these The Pelasgic monuments, sketched and studied at the present first regular constructions were thus generally established ; day, extend over a zone, which, comprising the breadth of and the greater part of the primitive world adopted them, with Western Asia, stretches over Greece and Central Italy; and the exception of those countries where great political events this is not the whole of the ancient world, as we have already interrupted the first movements of civilisation and suspended said, in which early monuments composed of rocks in their the march of the arts ; with the exception also of those whose natural state, have been seen by ancients and moderns ; but inhabitants, less endowed by nature, necessarily remained they have been discovered in all the northern countries, and in the rear of civilisation, and only received a movement of in Africa, from Egypt to the neighbourhood of Carthage ; and this kind from their neighbours, or from an invasion of some we have reason to believe that in these countries, to the primitive constructions, a second period succeeded, more refined in its productions, and forming a step from the first attempts, to the more perfect examples, of which we behold the numerous ruins in India, in Central Asia, in the valley of the Nile, and in the oases of the desert. These monuments of transition, so to speak, have disappeared under early and actual civilisation, and have even escaped the investigation of travellers. FIRST REGULAR CONSTRUCTIONS, PYRAMIDS, &c.

21.991 The Pelasgi, proceeding from the Asiatic plateaus (table

671) lands), directed their steps towards the west ; other Shemitic tribes marched towards the south and east, and peopled India, Persia, Assyria, and Arabia, as well as Ethiopia and Egypt. The art of these tribes, like that of the western branch, passed through a rude and primitive state, as we have shownthrough the BBTH-EL style, or constructions in unhewn stones.

The Palace of Luxor, in Egypt. It cannot be supposed that these tribes were more privileged people, more advanced in civilisation. The first builders than others, and were able, without previous attempts, to hew worthy of the name from their ability to mould bricks, and stones regularly, to mould and cement bricks, and to give to hew stones to raise their gigantic monuments, were compelled

Fig 8.

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3
8

24

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.

to follow the road in which they were placed. The want of utility, and cannot be depended upon as a satisfactory check upon
experience, the absence of instruments and machines prevented the operations. One thing, however, we will admit—that where
them from raising, at first, great edifices with vertical fagades the application of this method indicates an error, there must be an
or fronts, such as they were enabled to construct at a later error; but the evil is, that it does not indicate the existence of
period. To form large foundations, and to raise above them many errors when they really have been made. The misplacement
materials with gradual and numerous recesses such as would of a number, or even a figure, in a sum in addition, or in a product
prevent the fall of the upper parts of the building, was the in multiplication, is a very serious error, and might lead to very
first law of construction and of statics to which they were injurious consequences in business, and yet the proof by casting out
obliged to submit. This is so true that after having made their the nines would not detect such an error! And it is on this ground
great steps in the art of building, the Indians, the Chaldeans, that we object to its practical utility.
the Ethiopians, and the Egyptians, and become able builders, EXAMPLE.—Multiply 75432 by 476, and prove the operation by
they still continued in the path of which the pyramid was casting out the nines.
starting-point, by raising their edifices in such a manner as to

Multiplicand 75432
give to their façades a great inclination in order to obtain Factors

Multiplier 476
greater stability; a wise system, which was adopted by the
Btruscans when they left Asia, where these principles were

452592
long established. They were also spread over a part of Italy,

528024 and traces of them are found at Norchia. The same ideas ex.

301728 erted their influence over the early edifices of the Greeks, and

Product 35905632

.6
they are found in a modified form among the finest specimens
of their later architecture. They are recognised, for instance,

METHOD OF PROOF.
in the Parthenon, where the inclination of the jambs of the By casting the nines out of the multiplicand, as shown at page
doors and windows still exists. Mexico also bears witness to 266, we obtain the remainder 3 ; by casting the nines out of the
this, as may be seen in our remarks on the first regular con- multiplier, we obtain the remainder 8; and by multiplying these
structions of that country.

remainders together, we obtain the product 24. By casting the QUESTIONS ON THE PRECEDING LESSONS.

nines out of this product, we obtain the remainder 6. Now, by Among what early tribes had the Pelasgi style of architecture its casting the pines out of the product of the factors in the above origin?

operation, we obtain also the remainder 6. Hence, it is presumed Who were the Pelasgi; and how far back are their monuments that the operation is right. traced ?

If the remainder, arising from casting the nines out of either of In what countries are the Pelasgic monuments found; and in the factors, or of both, should be 0, the remainder arising from what parts of America were similar constructions discovered ? casting the nines out of the product should also be 0; because the How many centuries ago were the walls of Tyrins built, and what product of 0 by any figure or number is 0. was the size of the stones? Where are examples of temples 'built The proof of division by casting out the nines, proceeds on the like these walls to be found ? Where did the square form of stones first appear, and where did tient is one of the factors, the divisor is the other factor, and the

same principle as that referred to in multiplication ; for the quobrick-building first commence ?

dividend is their product. To this principle must be added the very simple one, that if the same thing be added to equals, the

wholes are equal. Accordingly, when there happens to be a reLESSONS IN ARITHMETIC.-No. XIV. mainder over in the division of one number by another, the

remainder arising from the casting out of the nines from this reThe proof of multiplication, by casting out the nines, proceeds on mainder must be added to the product of the remainders of both the the following principle: that if two factors and their product by factors, in order to get the true remainders of the nines; because divided by any number, the product of the remainders of the factors, the remainder of the division is included in the dividend, or else wlien divided by that number, will give the same remainder as the there would have been no remainder at all. The rule for proving product of the factors. Thus, if 24 and 20 be the factors, 480 is division by casting out the nines, then, is the following :-Cast their product; and if each of these factors be divided by a num- the nines out of the divisor and the quotient, and multiply these ber, say 7 for instance, the remainders are 3 and 6, and their pro- remainders together ; cast the nines out of this product, and note duct 18, divided by 7, gives the remainder 4; but the product 480 the remainder ; cast the nines, also, out of the remainder of the divided by 7, gives the same remainder 4, as we have said. Com- division, and note this remainder : add it to the former remainder bining this principle with what was explained in our last lesson on noted, and cast the nines out of their sum ; the remainder of this the subject of finding the remainder of a number divided by 9, the casting out should be the same as that arising from casting the rule for proving multiplication, " by casting out the nines,” is as nines out of the dividend, if the work be right; this rectitude or follows:--Cast the nines out of the factors, i.e., the multiplicand accuracy being subject to the same limitations and restrictions, as and the multiplier, and multiply the two remainders together, cast those above explained under the rule for proving multiplication. the nines out of this product, and also out of the product of the EXAMPLE. - Divide 7543210 by 476, and prove the operation factors; if the remainders be the same in both cases, it may be by casting out the nines. presumed that the work is right. The accuracy of the operation,

Divisor. Dividend. Quotient. as we said in our last number, cannot be depended on, for the rea

476) 7543210 (15847 sons there stated, and others that might be added : thus, the

476 arrangement of the partial products might be all wrong by the dis

2783
placement of the products, of the units, or of the tens, or of the

2380
hundreds, &c., i.e., of any one by itself or of all of them to-
gether; and yet the casting out of the nines would not indicate the

4032
fact, but would give the same result as if they were all arranged in

3808 a manner perfectly correct. Hence the propriety of giving other

2241 methods of proof, as we have done in past numbers, ---methods which

1904 are sure to detect the slightest inaccuracy, and put the calculator

3370
fully on his guard. For doing this, we have been found fault with

3332
by a number of individuals, who fancy that if they know any thing at
all they must know arithmetic, and that they can teach us, forsooth,

38 remainder
better than we can teach them! But true knowledge is always hum-

Divisor 476.

8 ble, and ready to acknowledge its own deficiencies or mistakes ; it is

Quotient 15847

7 also anxious to gain useful information from every quarter, and more

Product 56
willing to learn than to find fault. We hope that such persons will take

Remainder 38
these hints in good part. We assert, without fear of their displea-
sure, that the method of proving multiplication, or, indeed, any

Sum
role in arithmetic, by “casting out the nines," is of no practical

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METHOD OF PROOF.

The following is a method of interesting young pupils in Casting the nines out of the divisor gives remainder 8, and out of the irksome rule of addition. Tell the members of a the quotient gives remainder 7. The product of these remainders class to put down on their slates a sum in addition, a certain 34765 is 56, and casting the nines out of it gives remainder 2; also, cast- number of lines deep, and a certain number of figures broad, 27184 ing the nines out of the remainder of the division gives remainder as in the margin; and to leave room enough below for you 36958 2, and the sum of these remainders is 4. Lastly, casting the nines to add as many lines. This being done, you proceed to 47366 out of the dividend gives the same remainder 4 ; hence it is pre- add as many lines to each in the following manner :-Looksumed that the work is right; but, as we have said before, this is ing at the first line, write down rapidly the difference between only a presumption.

each figure in it and 9; do the same The principle of most of these operations was very well ex. with the second line, then with the third,

and so on.

34765 plained by some of our correspondents, among whom we must

You then tell them to add up

numbers

27184 mention the following :-Robert Boland, Great Shelford, Cam- all the lines, and find the amount of the

put down 36958

by the bridgeshire; J. L. N., Dublin (who applied it most ingeniously to whole. But before they do this, you may

47366 other scales of notation); Charles Currie, Glasgow (who added just as well tell them the answer or sum

pupils. some curious and ingenious rules for dividing by a number consist - which it will make. You do this by

65234 numbers ing of nines, extracted from old authors) ; P. G. Anderson, Bir- giving them the product of a number

72815 mingham ; Joseph Webster, Bramley (who applied the proof of consisting of as many nines as there are

63041

by the casting out the nines to involution and evolution, and gave rules figures in the breadth, by the figure which

52633 teacher. for the same); J. Carey, Clapham ; and others. From the review indicates the number of lines in the ori. of all the correspondence we have received, we are convinced that ginal depth of the sum before you added 399996 Sum. many of the observations made on the properties of the number 9, your own lines to it. This operation especially in their application to the methods of proving arithmeti- will be seen at once in the example placed in the margin. cal operations, are more curious than useful ; and that their place

In this operation it is to be observed, that, may be fairly taken by something of much greater value.

however different the numbers may be which

99999 We were reminded of a curious property of 9 and its multiples are put down to be added, it is plain that,

4 by a correspondent, Joseph Bowman, Preston,-namely, that the by the addition of your figures to them, number 12345679 being made the multiplicand, and the multi- they must all come to the same sum. 399996 Answer. ples of 9 as far as 81 the multipliers, the products will consist of Here you multiply 99999, the number numbers all composed of the same figure, and that figure will be composed of as many nines as there are the number of times 9 which constitutes the multiple employed; figures in the breadth, by 4, the number of lines in the original thus :

breadth, that is, the depth of the numbers put down by the pupils. 12345679 12345679

12345679

The reason of this process is so obvious, that we shall leave it for 9 18

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the consideration of those who may be interested in the subject,

satisfied that they are sure to make the discovery for themselves. lllllllll 98765432

86419753 12345679

24691358 222222222 333333333

FRENCH EXTRACTS. The young learner can perform the remainder of the operations with 35, 45, 54, &c., for himself. The reason of these curious Having been frequently requested by our correspondents to reresults will be seen at once by separating the multipliers into their commend to the students of French, some book for reading, in factors, and performing the operation as follows:

order to extend their knowledge of that language, and to give them 9 12345679 12345679

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an interest in some useful study besides, we have selected for their 2

9
9

special use, a collection of the most valuable MAXIMS and MORAL

SENTIMENTS wbich the French language affords. This collection 111111111 il1111111

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is arranged alphabetically, and will be continued in several num. 3

bers of the Popular Educator, until complete. When this is 222222222 333333333

done, our readers will be in possession of as fine a body of social Although we have seen these curious results many years ago, we

morality and experience of the world, as the language of any country give our correspondent full credit for his individuul rediscovery.

can afford. The names of the authors are added to the sentiments Another correspondent, J. L. N. (Dublin), suggests the following or maxims, to give them zest and authority ; those of La Rochecurious operation in subtraction as an amusement for the junior foucauld, La Bruyère, Bacon (our own Francis), Pascal, Montaigne, classes in a school. Tell the members of a class to put down upon of the wisdom of their sayings, and the excellence of their style.

Fenelon, Montesquieu, and many others, will be a sufficient guarantee their slates individually any number they please, as in the margin ; then, to put down under this number, the same 543268 figures written in any other order whatever, taking care that 438625

PENSEES MORALES ET MAXIMES. its first figure, or the figure of highest value, shall be less 104643 than the one above it ; next, to subtract this number, or row of figures, from the one above, and mark out one of the figures tirent quelque avantage, ni de si heureux que les imprudents ne

Il n'y a point d'accidents si malheureux dont les habiles gens ne of the remainder. Lastly, tell them to add together all the rest of the puissent tourner à leur préjudice.-La Rochefoucauld. figures, and be prepared to tell you the result. Then, beginning at one end of the class, ask the number required, subtract it in your mind from the next multiple or 9 above it, and tell the re louer

de bon cæur.-La Rochefoucauld.

C'est, en quelque sorte, participer à une bonne action que de la mainder aloud ; this remainder will be the number which each individual marked out, Thus ; if in the above example 6 were marked

Il faut faire comme les autres : Maxime suspecte, qui signifie out, the sum of the rest of the figures would be 12; now the next presque toujours : il faut mal faire, dès qu'on l'étend au-delà de

ces choses purement extérieures, qui n'ont point de suite, qui multiple of 9 above 12, is 18 ; and subtracting 12 from 18, you dépendent de l'usage, de la mode et des bienséances.-La Bruyère. have 6, the number marked out. The principle of this apparent trick is fully explained in our last lesson on Arithmetic. telling them to cast the nines out of the rest of the figures, and tell défauts, et que la considération est la seule indemnité de la

Puisque l'âge diminue nos agréments en nous laissant nos you the remainder, the process would be easier, for you would only vieillesse, tâchons de devenir plus respectables à mesure que nous have to subtract it from 9, and this remainder would be the figure devenons moins aimables.- Levis. marked out. But this process woula more readily reveal your secret. It is to be noted, that when the sum of the rest of the y manquons souvent d'expérience malgré le nombre des années

Nous arrivons tout nouveaux aux divers ages de la vie, et nous figures is itself 9, or a multiple of 9, the figure marked out may be La Rochefoucauld. either 9 or 0. This renders two cases dubious, and diminishes the

AMBITION. completeness of the artifice. Pupils rejoice to find their master in L'esclave n'a qu'un maitre, l'anıbitieux en a autant qu'il y a de a dilemma.

gens utiles à sa fortune.-La Bregine

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ACCIDENT.

ACTION.

AGE.

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