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constructed by the Assyrians, if they did erect any, and those of the other people of Asia. The nations we have named in our preceding observations, were in the height of civilisation, while the Grecian arts were in their cradle; and it is difficult to admit that the Greeks had not learned their first lessons in

LESSONS IN LATIN.-No. XX.
By JOHN R. BEARD, D.D.
ON PARSING.

architecture by the study of the Asiatic or African orders IN the following Latin exercises ascertain, write down, and which we have described. In fact, the most ancient type of imprint on your memory the parts of the several verbs; that the Greek orders, the Doric, particularly at its commencement, is, the mood, tense, person, and number, together with the is nearly the same as that exhibited in the tombs of the exact English meaning; at the same time tell the person, Heptanomis, and which Champollion called Proto-Doric or tense, and mood endings, as well as give the stems. This you primitive Doric. The genius of Greece developed this first should do very completely with each lesson in succession. idea, enriched it with details which the Egyptians had ne- You thus make a commencement in what is called parsing, glected, and formed out of it the first basis of its national archi- that is, telling or assigning the parts (in Latin, pars, a part). tecture. The principal character of the Greek Doric is the Parsing applies to nouns and adjectives, as well as to verbs, nobleness and dignity of the whole order, the severe simplicity indeed, to all parts of speech; it is also concerned with of its details and the moderation of its ornaments. The syntax, or the combination of words into sentences; so that columns have no base; the shaft is ornamented by wide and you cannot parse your lessons completely until they are shallow flutings; the capital is composed of a large moulding terminated. But you have now advanced far enough to begin in the form of a cup, or flat vase, resting upon two or three parsing, and would be rewarded if every day before you little fillets surmounted by a square tablet. The triglyphs, the attempt a new lesson, you were to take "a back lesson," and fluted ornaments at the extremity of the architraves, which parse it carefully; that is, go over again from the first what are seen in the frieze and entablature, belong exclusively to you have done with the strictest regard to the forms and this order; the square spaces or metops between the triglyphs are frequently occupied with sculptures of isolated subjects; I will give you an example of what I mean by parsing:Let us take the short Latin sentencebut the polished freeze, and consequently, the continued subject, are in this order very rare. Still this order does not exclude all decoration; and in buildings of a common character it loses its heaviness, and becomes very elegant; the mouldings then become finer, and some are decorated with various ornaments. See fig. 12.

According to Vitruvius, it was in the temple of Juno, at Argoe, where the Doric order of architecture first rose to a marked eminence, and became the model for the magnificent edifices afterwards erected throughout Greece. It was next employed in the temple of Jupiter Nemeus, at Nemea, between Argos and Corinth; of Jupiter Olympius, at Olympia, in Elis, in a splendid triple portico in the city of Elis; and in three temples of the same city, viz., those of Juno, Minerva, and Dindymene, or Cybele; at Eleusis, in the great temple to Ceres: in the temple of Minerva at Sunium; and in the temple of the same goddess at Athens, called the Parthenon (see fig. 13); in the entrance to the Acropolis, and in other public buildings of great magnitude and splendour at Athens. In many of the Islands of Greece and Magna Græcia, there were also temples of the Doric style of architecture; as that of Apollo, in Delos; of Juno, in Samos; of Jupiter Panellenius, of Agina, and of Silenus, in Sicily; and many others in places of inferior note. Many of these temples were of great magnitude; they were universally of an oblong form; in some the porticoes were only at the end; in others they were extended quite round the interior of the building, some in single, and others in double ranges; some were covered with roofs, others were left partly uncovered, and some were divided by ranges of pillars along the middle of the interior. The superstructure was placed upon a platform composed of three steps, which surrounded the whole building, and upon which the columns were all placed without bases. The number of columns were either six along the ends, and thirteen along the sides, or eight along the ends, and seventeen along the sides. When built upon so large a scale, with the ranges of columns so distinctly isolated, the essential parts of the Doric order produced effects not surpassed for simplicity and majesty; and even the imperfect remains which have escaped the ravages of time and barbarity appear to have far exceeded the expectations of connoisseurs, In the earlier examples of this order the diameters of the Doris columns were very considerable in proportion to their height; for instance, the column of the temple of Bilenus, in Bily, was only five diameters in height; but in course of time, these relative dimensions were changed, and a proportion more adapted to the production of delicate wilket was introduced, The Doric style of architecture was, with very few exceptions, the only one employed in Greece or Kam colonies in Sicily, and Italy, and in Asia Minor, you win the period of the Macedonian conquest. In Asia Minn, and particularly in Ionia, there speedily arose, subsewd to the period, an order of architecture more elegant than By Thune. But our notice of this order must form the subject in mur meat, lesson,

rules.

Tullia patrem amat.

The first thing I have to do is to construe it, or put it into corresponding English words. On looking at it, I see that Tullia is in the nominative case. Consequently, Tullia is the subject, and with it I must begin. But, patrem comes next: am I to take patrem in the second place? This I cannot do; for patrem is in the accusative case, and consequently must be dependent on some verb. The verb is there. Amat then comes after Tullia. Putting the two together I have, Tullia amat. Tullia loves. What does Tullia love? Patrem, her father. The whole then is, Tullia loves her father. Here you see a departure in the English from the Latin idiom. With such deviations you should familiarise your mind by constant and careful observation. The departure here is this, that to make good or idiomatic English, I am obliged to add the pronoun her, "her father," there being in the Latin no word corresponding to her. Do not hence suppose that it would be bad Latin to say Tullia amat patrem suum, her father; but it is not customary to employ the pronoun in such cases, except it is wanted for the sake of emphasis.

Having translated the sentence, I must now parse it. I shall take each word in its grammatical order. Tullia, Tulliae, a noun feminine of the first declension, nominative case, the subject to the verb amat.

The stem is Tullia (thus Tullia, G. Tulliae, the e of the genitive being removed, Tullia remains as the stem).

After giving the parts and relations of a noun as above, you should "go through" or decline the noun. So with all nouns, pronouns, verbs, and adjectives.

Amat, from amo, is a verb transitive of the first conjugation, indicative mood, present tense, third person singular, agreeing with its subject Tullia, according to the rule "a subject must agree with its verb in number and person." "The four chief parts of amo are-amo, amavi, amatum, amare. The stem of amo is am, the stem of the present tense is ama, the personendings are o, as, at, amus, atis, ant. Amao is contracted into amo. Then go through the tense uniting the stem with the person-endings. You would act wisely if, in addition, you made amat the subject of inquiry; thus, what would amat be in the subjunctive mood? In the passive voice? In the subjunctive passive? By what change is amat made plural? What is the corresponding second person singular? Plural? What does amat become in the future tense? In the pluperfect indicative? Go through the imperfect of amo. the perfect subjunctive first person singular; third person plural.

Give

These things may seem minute and troublesome to you: they would, however, be required by any good teacher; and attention to them is, I assure you, requisite to make a sound scholar; it is also requisite for that mental discipline which the study of language may give, and which, in its perfect form, is of very high value.

Another word remains-patrem ; patrem from pater, patris

an imparisyllablic noun, of the masculine gender, the third
declension, consonantal stem patr, in the singular number
accusative case, being the object of the transitive verb amat,
by which it is governed, according to the rule, "transitive
verbs require their object to be in the accusative case."
Observe, that in thus setting before you a specimen of
parsing, I have given you two rules in Syntax; thus-
I. A subject must agree with its verb in number and person.
II. Transitive verbs require their object to be in the accusa-
tive case. Of these rules you will forthwith have need to
make constant application. Commit them to memory, and
repeat them memoriter whenever applied. A verbal and exact
repetition of them, and of all rules, is desirable at first; after-
wards, I wish that you should give the substance rather than
the words of a rule, for if you express its substance you show
that you understand its import.

VOCABULARY.

Comparo 1, I get together, acquire; emigro 1, I go out, quit (E. R. emigration); flo I, I blow; intro 1, I go into, enter (E. R. entrance); judico 1, Ijudge; latro 1, I bark; libero 1, I set free (E. R. liberation); numero 1, I number; observo 1, I keep under my eye, observe; occupo 1, I fall upon, take possession of (E. R. occupation); vigilo 1, I watch, keep awake, guard (E. R. vigilant); ventus i, m. wind; terror, óris, m. terror; timor, óris, m. fear (E. R. timid); narratio, ónis, f. a narrative; interitus, ûs, m. ruin; placidus, a, um, placid, tranquil; ingens, ingentis, very great; vehemens, vehementis, vehement, very strong; jam, adv. already; nuper, adv. lately.

LATIN-ENGLISH.

Ego te laudabam, tu me vituperabas, frater judicabat; ego te laudabo, tu me vituperabis, frater judicabit; ego ambulavi, tu vigilavisti, ventus flavit; ego ambulaveram, tu vigilaveras, ventus flaverat; ego te laudavero, tu me vituperaveris, frater judicaverit. Quum milites urbem intrabant, omnes cives timoris pleni erant; quam in silvâ ambulabamus, vehemens ventus per altas quercus flabat, dum nos placidus somnus recreabat. Vos vigilabatis; quamdiu eris felix, multos numerabis amicos; bonos semper laudabo, improbos semper vituperabo; si acriter pugnabitis, o milites, patriam interitu liberabitis; si virtutem amabis, omnes

boni te amabunt.

Remark that sometimes an abbreviation takes place in the perfect tense, and the tenses formed from the perfect tense. Thus, instead of saying in full, vigilavisti, as above, the Latins shortened the word into vigilasti, leaving out the vi. This process is called syncopation, and verbs thus contracted (drawn together) are said to be syncopated. Other syncopated forms ensue; as laudasti for laudavisti; amasti for amavisti; amasse for amavisse; also in other conjugations, as complesti for complevisti; audieram for audiveram ; audierunt for audiverunt. I resume the exercises; in which instances of syncopation will be found.

option); pecco 1, 1 sin; redămo, I love again; rogo 1, I ask; sano 1, I heal (E. R. sanatory); supero 1, I surpass, overcome; tracto 1, I handle, treat; evenit, it happens; conscientia, ae, f. conscience; uva, ae, f. a grape; adhibeo 2, I apply; adhibere curam, to take care; medicus, i, a physician (E. R. medical); honestas, átis, f. honesty, honour aegrotus, i, a sick man; immortalitas, átis, f. deathlessness; utilitas átis, f. utility; statio, ónis, f. a station, a post; maturus, a, um, ripe mature; immaturus, a, um, unripe; religiose, conscientiously; acer rime, very bravely; vide, see thou; vide ne, see thou do not.

RULE: The conjunctions ut, that, in order that, so that, so as and ne not to, so that not, to prevent, require after them the subjunctive mood.

LATIN-ENGLISH.

Saepe evenit ut utilitas cum honestate certet; vide ne pecces diligenter tractent; ita vivere debemus ut in omni re rectam concontra virtutis praecepta; omnes parentes optant ut filii litteras scientiam servemu3; me redicus omnem curam adhibet ut aegrotum sanet; amo te ut me redămes; dux imperavit ut milites stationes suas servarent; heri ambulabam ut tristem animum exhilarem ; exercitus noster acerrime pugnabat ut urbem ab interitu servaret.

ENGLISH-LATIN.

cepts of virtue? a wise father takes care that his children do not See that your son does not sin; dost thou sin against the presin; the generals take care that the soldiers keep their posts; you do not (ne) sin); good mothers take care that their children obey take care to prevent your children from sinning (literally, that they their commands; thou oughtest to live so as to (so that you may) preserve honour.

RULE: After non dubito, the conjunction quin is used requiring the subjunctive mood; thus non dubito quin, I doubt not but, or that. In the same way, nemo dubitat quin, no one doubts that; quis dubitat quin? who doubts that? dubium non est quin, there is no doubt that.

LATIN-ENGLISH.

bam quin milites nostri hostes superavissent; quin milites nostri Non dubito quin milites nostri superaverint hostes; non dubitasuperavissent hostes non dubitabam; non dubito quin milites nostri hostes superaturi sint; non dubitabam quin milites nostri hos es superaturi essent; non dubitabam quin vos patrium servitute liberaturi essetis; dubium non erat quin exercitus noster omnes labóres et aerumnas facile toleraturus esset; quis dubitat quin Hannibal contra Romanos fortissime pugnaverit? non dubitabitis quin ego vos semper amaverim; quis dubitat quin bonos semper laudaverimus, malos semper vituperaverimus? non est dubium quin semper fidem servaritis (syncop. for servaveritis) nemo dubitabat quin hostes urbem expugnavissent; nulli civium dubium erat quin pro patriâ libertate acerrime pugnavissetis ; quum hostes urbem oppugnabant, non erat dubium quin ingens terror omnium civium animos occupavisset (sync. occupasset).

ENGLISH-LATIN.

No one doubts that you will fight bravely for the liberty of your country; no one doubts that he fought bravely; no one will doubt that he will fight bravely; no one doubted that he had fought bravely; doubt that my father will come; who doubts that I shall conwho doubts that the soldiers will capture the city? There is no doubt that you endeavour (studeo) to preserve honour; I do not scientiously preserve the city?

Quia semper virtutis præcepta observastis (for observavistis) magnam vobis laudem comparastis; cur per totam noctem vigilasti? Præceptores meos semper amavi; nonne amasti tuos? ingens terror omnium civium animos occupabat; narratio quam Acriter contra hostes pugnastis; quum milites urbem intraverant, mihi nuper narraveras, vehementer me delectaverat; quum exercitus hostilis urbem oppugnaverat, nos jam emigraveramus; si animum virtutibus ornaveris (ornaris) semper beatus eris; quum Though non dubito quin, &c., requires the subjunctive mood hostes urbis nostrae agrog devastaverint, urbem ipsam oppug-in Latin, the verb must be Englished by an indicative mood; as may be seen in the English examples just given. In order to make this quite plain, I will give another instance: Non dubito quin bonus sit avunculus tuus.

nabant.

ENGLISH-LATIN.

We praised thee, thou didst blame me, father was judging; thou wilt praise me, he will praise thee, father will judge us; thou hast walked (syncopated form), I have watched, the winds blew; I will walk abroad; thou art watching; the wind was blowing; the soldiers will enter the city; the soldiers were entering the city; the soldiers are entering the city, the soldiers have entered the city; the soldiers had entered the city; a very strong wind blows through the house; dost thou number many soldiers? I have numbered many friends; he has liberated (set free) his country from ruin; hast thou watched all night? Love thy preceptors; let them love their parents; O boys, love virtue. The narrative delighted my brother; the narrative delights the girls; the narrative will delight father and mother; thou hast acquired fame by the narrative of the ruin.

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I doubt not that thy uncle is good.

Here, then, you see the verb which in Latin must be in the subjunctive mood, must stand in the indicative mood in English. Such is a by no means unusual fact. RULE: With the imperative the negative ne is used, and not the negative non, as ne crede, do not believe.

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after verbs denoting motion; the second supine, that in u, is used after certain adjectives.

Remark: The supines are supposed to have been nouns of the fourth declension, that in um, a noun in the accusative case; that in u, a noun in the dative case (u for ui).

LATIN-ENGLISH.

Parentes mei in urbem migraverunt habitatum; exercitus hostilis adventavit agros nostros devastatum; uva immatura est acerba gustatu; aerumnae sunt durae toleratu; sitis difficillima est toleratu; pira sunt dulcia gustatu; luscinia cantans animos nostros delectat; hostes adventant expugnaturi urbem; non est dubitandum de immortalitate animarum; obtemperandum est praeceptis virtutis.

ENGLISH-LATIN.

The soldiers approach to deliver (supine in um) the king; the hos ile army approaches to capture the city; ripe grapes are sweet to the taste (supine in u), unripe grapes are difficult to be endured; they come to seize (fut. perf. act. of occupo) the fields; birds by singing (gerund) delight the mind; we must fight bravely for our native land (patria), the soldiers have approached to fight for their homes and their wives.

LESSONS IN ARITHMETIC.-No. XV.

CONTRACTIONS IN MULTIPLICATION.

ALTHOUGH the general rules previously given be adequate to the solution of all examples that occur in multiplication, yet in many Instances, by the exercise of judgment in applying the preceding principles, the operation may be very much abridged.

factor, and so on till you have multiplied by all the factors. The last product will be the answer required.

The factors into which a number may be resolved, must not be confounded with the parts into which it may be separated. T former have reference to multiplication, the latter to addition. that is, factors must be multipaed together, but parts must be added together to produce the given number. Thus, 56 may resolved into two factors, 8 and 7; it may be separated into m parts, 5 tens or 50, and 6. Now, 8x7=56, and 50+6=56. EXERCISES.

1. What will 24 horses cost, at 74 crowns apiece. 2. What cost 45 hogsheads of tobacco, at 128 crowns a hogs head?

3. What cost 54 acres of land, at 150 crowns per acre? 4. At 118 shillings per week, how much will it cost a family board 49 weeks?

5. If a man travels at the rate of 372 miles a day, how fir vi he travel in 64 days?

6. At 163 crowns per ton, how much will 72 tons of lead o 7. What cost 81 pieces of broadcloth, at 245 shillings apie 8. What cost 84 carriages, at 384 crowns apiece? CASE II.-When the multiplier is 1 with ciphers annered to a It is a fundamental principle of notation, that each removal tí figure one place towards the left, increases its value ten times; 80sequently, annexing a cipher to a number will increase its vale value a hundred times, or multiply it by 100; annexing times, or multiply it by 10; annexing two ciphers will increse ciphers will increase it a thousand times, or multiply it by &c. Thus, 15 with a cipher annexed, becomes 150, and is same as 15x10; 15 with two ciphers annexed, becomes 156, is the same as 15X100; 15 with three ciphers annexed, became

DEFINITION.-Any number which may be produced by multiplying two or more numbers together, is called a Composite Number. Thus, 4, 15, 21, are composite numbers; for 4=2×2; 15=15000, and is the same as 15X1000, &c. 5X3; 21-7X3.

The factors which, being multiplied together, produce a composite number, are sometimes called the component parts of the number. The process of finding the factors of which a given number is composed, is called resolving the number into factors. EXERCISES.

1. Resolve 9, 10, 14, 22, into their factors. 2. What are the factors of 35, 54, 56, 63? 3. What are the factors of 45, 72, 64, 81, 96? Some numbers may be resolved into more than two factors; and also into different sets of factors. Thus, 12 2×2×3; also 124x3=6×2.

4. What are the different factors and sets of factors of 8, 16, 18, 20, 24?

5. What are the different factors and sets of factors of 27, 32, 36, 40, 48?

We have seen that the product of any two numbers is the same, whichever factor is taken for the multiplier, and that the product of any three or more factors will be the same, in whatever order they are multiplied. For, the product of two factors may be considered as one number, and this may be taken either for the multiplicand, or the multiplier. Again, the product of three factors may be considered as one number, and be taken for the multiplicand, or the multiplier, &c. Thus, 24=3X2X2X2-6X2X2= 12×2=6×4=4×2×3=8×3.

CASE I. When the multiplier is a composite number. EXAMPLE 1.-What will 27 bureaus cost, at 31 crowns apiece?

Analysis. Since 27 is three times as much as 9; that is, 27= 9X3, it is manifest that 27 bureaus will cost three times as much as 9 bureaus.

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Crowns 837 cost of 27 bureaus.
EXAMPLE 2.-What will 36 oxen cost, at 43 crowns per head?
Solution.-36-9X4; and 43X9X4=1548 crowns. Ans.
Or, 36=3×3×4; and 43×3×3×4=1548 crowns. Ans.

To multiply by a composite number. Rule:

To multiply by 10, 100, 1000, &c. Rule:

in the multiplier, and the number thus formed will be the pressi Annex as many ciphers to the multiplicand as there are required.

To annex means to place after, or at the right hand,

EXERCISES.

1. What will 10 boxes of lemons cost, at 63 shillings per di Ans. 630 shillings.

2. How many bushels of corn will 465 acres of land produ: at 100 bushels per acre?

3. Allowing 365 days for a year, how many days are ther 1000 years?

4. Multiply 153486 by 10000.

5. Multiply 3120467 by 100000.
6. Multiply 52690078 by 1000000.
7. Multiply 689063457 by 10000000.

8. Multiply 4946030506 by 100000000,
9. Multiply 87831206507 by 1000000000.
10. Multiply 67856005109 by 10000000000.
CASE III.-When the multiplier has ciphers on the right inh
EXAMPLE.—What will 30 wagons cost at 45 crowns apier?
Any number of ciphers on its right hand, is obviously a
site number; the significant figure or figures being one facit s
1, with the given ciphers annexed to it, the other factor. Thus 3
may be resolved into the factors 3 and 10. We may thera
first multiply by 3 and then by 10, by annexing a cipher
Solution. Here 45 × 3 = 135, and 135 X 10 &
EXAMPLE.-How many acres of land are there in 3000 farms,

crowns. Ans.

each farm contains 475 acres?
Analysis.

475 X 3=1425; and adding three
First, 3000 3 x 1000. Now
ciphers to this product, multiplies it
by 1000.

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To multiply when there are ciphers on the right hand of the

multiplier. Rule:

tiplier, and to this product annex as many ciphers, as are on the right of the multiplier.

Multiply the multiplicand by the significant figures of the

It will be perceived that this case combines the principles of the

multiplican by one of these factors, and this product by another

Resolve the multiplier into two or more factors; multiply the two preceding cases; for, the multiplier is a composite numer

and one of its factors is 1 with ciphers annexed to it.

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1. Multiply 1563800 by 20000.
2. Multiply 31230000 by 120000.
3. Multiply 5310200 by 3400000

4. Multiply 82065000 by 8100000.

5, Multiply 210909000 by 5100000.

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Since one carriage costs 235 crowns, 100 carriages will cost 100 times as much, which is 23500

crowns. But we wished to find the

Operation. 33500 cost of 100 carriages 235 cost of 1 carriage 23265 cost of 99 carriages.

cost of 99 carriages only. Now 99 is one less than 100; therefore, if we subtract the price of 1 carriage from the price of 100, it will give the price of 99 carriages. Hence,

To multiply by 9, 99, 999, or a number composed of any number of 9's. Rule:

Annex as many ciphers to the multiplicand as there are 9's in the multiplier; from the result subtract the given multiplicand, and the remainder will be the answer required.

The reason of this method is obvious from the fact that annexing as many ciphers to the multiplicand as there are 9's in the multiplier, multiplies it by a number one more than is required; consequently, subtracting the multiplicand from the number thus produced, must give the true answer.

1. Multiply 4791 by 99. 2. Multiply 7301 by 999.

3. Multiply 6034 by 999. 4. Multiply 463 by 9999.

CASE VIII.- When the multiplier is composed of a digit and its multiples.

EXAMPLE.-What is the product of 867 multiplied by 84?

Analysis.

We first multiply by 4 in the usual way. Now, since 8=4×2, it is plain, if the partial product of 4 is multiplied by 2, it will give the partial product of 8. But as 8 denotes tens, the first figure of its product tial products will be the answer required.

There are other methods of contracting the operations in multi-will also be tens. The sum of the two parplication, which, in certain cases, may be resorted to with advantage. Some of the most useful are the following:

CASE V.-When the multiplier is any digit preceded by unity. EXAMPLE.-How many gallons of water will a hydrant discharge in 13 hours, if it discharges 2325 gallons per hour?

Operation. 2325×13 6975

Product 30225 gallons.

Explanation.

Multiplying by the 3 units, we set the first figure of the product one place to the right of the multiplicand. Now, since multiplying by 1 is taking the multiplicand once, we add together the multiplicand and the partial pro. duct already obtained, and the result 30225 gallons is the answer. To multiply by 13, 14, 15, &c., or by 1, with either of the other digits annexed to it. Rule:

Multiply by the units' figure of the multiplier, and write each figure of the partial product one place to the right of that from which it arises; finally, add the partial product to the multiplicand, and the result will be the answer required.

This method is the same, in effect, as if we actually multiplied by the one ten, and placed the first figure of the partial product under the figure by which we multiply.

1. Multiply 3251 by 14.

2. Multiply 25039 by 16.

3. Multiply 4028 by 17. 4. Multiply 50389 by 18.

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Operation. 867 multiplicand 84 multiplier 3468×2 6936 72828 product.

For the sake of convenience in multiplying, the factor 2 is placed at the right of the partial product of 4, with the sign X between them.

What is the product of 987 by 486?
Operation.
Multiplicand 987
Multiplier 486

5922×8

47376

Product 479682

Since 48 6X8, we multiply the partial product of 6 by 8, and set the first product figure in tens' place as before.

To multiply when part of the multiplier is a composite number of which the other figure is a fuctor. Rule:

First multiply by the figure that is a factor; then multiply this partial product by the other factor, or factors, taking care to write the first figure of each partial product in its proper order, and their sum will be the answer required.

When the figure in thousands, ten thousands, or any other column, is a factor of the other part, or parts of the multiplier, care must be taken to place the first figure of its product under the factor itself, and the first figure of each of the other partial products in its own order.

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Operation. 63

45

2835 Ans.

Explanation.

First, multiply the units into units, we set down the result and carry as usual. Now, since the 6 tens into 5 units, and 3 units into 4 tens will produce the same order, viz. tens, we multiply them and add their products mentally. Thus, 6x5= 30, and 3X4-12; now 30+12=42, and 1 (to carry) makes 43. Finally, 6X4=24, and 4 (to carry) make 28.

To multiply any two numbers together without setting down the partial products. Rule:

First multiply the units together; then multiply the figures hich produce lens, and adding the products mentally, set down the result and carry as usual. Next multiply the figures which produce hundreds, and add the products, &c., as before. In like manner, perform the multiplications which produce thousands, ten thousands, &c., adding the products of each order as you proceed, and thus continue the operation until all the figures are multiplied. EXAMPLE.-What is the product of 23456789 into 54321 F Analytic Operation.

Par. Products

Multiplicand Multiplier

2 3 4 5 6 7 8 9 5 4 3 2 1

2 6 9

2X13X14X15X16X17X18X19X1 2X23×24X25X26×27X28X29×2 2X33X34X35X36X37x38x39X3 2X43X44X45X46X47X48X49X4 2X53X54X55X56X57x58x59x5 12 7 4 1 9 6 2 3 5 Explanation.-Having multiplied by the first two figures of the multiplier, as in the last example, we perceive that there are three multiplications which will produce hundreds, viz., 7 x 1, 8 x 2, and 9 X3; we therefore perform these multiplications, add their products mentally, and proceed to the next order. Again, there are four multiplications which will produce thousands, viz., 6 X 1, 7 X 2, 8 x 3, and 9 X 4. We perform these multiplications as before, and proceed in a similar manner through all the remaining orders. Ans. 1274196235269.

In the solution above, the multiplications of the different figures are arranged in separate columns, that the various combinations which produce the same order, may be seen at a glance. In practice it is unnecessary to denote these multiplications. The principle being understood, the process of multiplying and adding may easily be carried on in the mind, while the final product only is set down.

When the factors contain but two or three figures each, this method is very simple and expeditious. A little practice will enable the student to apply it with facility when the factors con⚫ tain six or eight figures each, and its application will afford an excellent discipline to the mind. It has sometimes been used when the factors contain twenty-four figures each; but the attempt to extend it so far is not profitable.

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17. 34054X639.

18. 52156×756. 19. 41907X54486. 20. 26397×24648. 21. 12900X14000. 22. 64172X42432. 23. 26815678×81. 24. 85x85. 25. 256X256. 26. 322×325.

27. 5234 X 2435. 28. 48743000 × 637. 29. 31890420 × 85672. 30, 80460000×2763. 31. 2364793×8485672. 32. 1256702×999999. 33. 684000591×61. 34. 45067034×17×51. 35. 788031245 × 81 × 16. 36. 61800000X23000. 37. 12563000X4800000. 38. 91300203X1000000, 39. 680040000×1000000. 40. 4000000000×1000000.

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It will be easily perceived that se in the first sentence is a direct regimen, and that the same word in the second represents an indirect object.

3. Verbs essentially unipersonal, i. e., verbs which cannot be conjugated otherwise, take avoir as an auxiliary :— Il a plu, il a neigé, il a gelé. It rained, it snowed, it froze.

4. Verbs occasionally unipersonal, take être as an auxiliary :— Il lui est arrivé un malheur. A misfortune has happened to him. 5. Faire [4 ir.] used unipersonally, and Y avoir, to be there, take the auxiliary avoir :

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Les Italiennes se sont elles promenées ?

Oui, Monsieur, elles se sont promenées.

Nous nous sommes aperçus de cela.

Votre mère s'est elle bien portée ? Vos sours se sont elles assises? Cette marchandise s'est elle bien vendue ?

Did the Italian ladies walk!
Yes, Sir, they have taken a walk.
We perceived that, or we took notice
of that.

Has your mother been well?
Did your sisters sit down?
Did that merchandise sell well?

Vos enfants se sont ils appliqués Did your children apply to study ↑ à l'étude ?

Il s'y sont appliqués.

Nous nous sommes donné de la peine [§ 135 (1)]

Quei temps a-t-il tait ce matin ? N'. til pas fait beau temps? Quel malheur vous est il arrivé?

Vous est il arrivé quelque chose? Il ne m'est rien arrivé.

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