constructed by the Assyrians, if they did erect any, and those LESSONS IN LATIN-No. XX. By Joux R. BEARD, D.D. while the Grecian arts were in their cradle; and it is difficult OX PARSING. to aunit that the Greeks had not learned their first lessons in 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 włat are seen in the frieze and entablature, belong exclusively to you have done with the strictest regard to the forms and rules. 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 sentence but the polished freeze, and consequently, the continued subject, are in this order very rare. Still this order does not exclude Tullia patrem amat. all 'decoration; and in buildings of a common character it The first thing I have to do is to construe it, or put it into loses its heaviness, and becomes very elegant; the mouldings corresponding English words. On looking at it, I see that then becomo finer, and some are decorated with various orna- Tullia is in the nominative case. Consequently, Tullia is the ments. See fig. 12. subject, and with it I must begin. But, patrem comes next : According to Vitruvius, it was in the temple of Juno, at am I to take patrem in the second place. This I cannot do; Argor, where the Doric order of architecture first rose to a for patrem is in the accusative case, and consequently must marked ominence, and became the model for the magnificent be dependent on some verb. The verb is there. Amat than odificos afterwards erected throughout Greece. It was next comes after Tullia, Putting the two together I have, Tullia employed in the templo of Jupiter Nemeus, at Nemea, be- amat. Tullia loves. What does Tullia love? Patrem, her twoen Argos and Corinth ; of Jupiter Olympius, at Olympia, in father. The whole then is, Tullia loves her father. Here you Elis, in a splendid triple portico in the city of Elis; and in three see a departure in the English from the Latin idiom. With temples of the same city, - viz., those of Juno, Minerva, and such deviations you should familiarise your mind by constaut Dindymene, or Cybele; at Eleusis, in the great temple to Ceres : and careful observation. The departure here is this, that to in thó templo of Minerva at Sunium; and in the temple of the make good or idiomatic English, I am obliged to add the prosamo goddons at Athens, called the Parthenon (see fig. 13); noun her, “her father," there being in the Latin no word corin the entrance to the Acropolis, and in other public buildings responding to her. Do not hence suppose that it would be of great magnitudo and splendour at Athens. In many of bad Latin to say Tullia amat patrem suum, her father ; but tho Islands of Greece and Magna Græcia, there were also it is not customary to employ the pronoun in such cases, temples of the Doric style of architecture; as that of Apollo, except it is wanted for the sake of emphasis. in Dolom; of Juno, in Samos; of Jupiter Panellenius, of Having translated the sentence, I must now parse it. I Agina, and of Silenus, in Sicily; and many others in places shall take each word in its grammatical order. of inferior noto. Many of thoso temples were of great magni- Tullia, Tulliae, a noun feminine of the first declension, tude ; they were univorsally of an oblong form ; in some the nominative case, the subject to the verb amat. porticoes were only at the end ; in others they were extended The stem is Tullia (thus Tullia, G. Tulliae, the e of the quito round the interior of tho building, some in single, genitive being removed, Tullia remains as the stem). and others in doublo rungos; some were covered with After giving the parts and relations of a noun as above, you roolu, others were left partly uncovered, and some were di- should * go through” or decline the noun. So with all vided by ranges of pillars along the middlo of the interior. The nouns, pronouns, verbs, and adjectives. superstructure will placed upon a platform composed of three Amat, from amo, is a verb transitive of the first conjugation, *L*pe, which surrounded the whole building, and upon which indicative mood, present tense, third person singular, agreeing the columns were all placed without bases. Tho number of with its subject Tullia, according to the rule " a subject must endurna were either wix along the endy, and thirtcen along the agree with its verb in number and person." The four chief Rixtes, oreight along the end., and seventeen along the sides. parts of amo are-amo, amari, amatum, amare. The stem of When built upon wo largo a scale, with the ranges of columns amo is am, the stemr of the present tense is ama, the personwo diutimatly imunud), the oncontinl parts of the Doric order pro- endings are o, as, at, amus, atis, ant. Amao is contracted into duwun fel wit surpassed for simplicity and majesty ; and amo. Then go through the tense uniting the stem with the even when important roinain which have oscaped the ravages of person-endings. You would act wisely if, in addition, you uma and tretmanity appear to have far exceeded the expecta- made amat the subject of inquiry; thus, what would amai be Iwnik till mineur, In the earlier examples of this order in the subjunctive mood? In the passive voice? In the sub1. Ainuters at the Dunia duon were very considerable in junctive passive? By what change is amat made plural ? win mo their handgents for instance, the column of the What is the corresponding second person singular plural ? Wie ist danum, in sily, was only five diameters in height; What does amat become in the future tense? In the pluWein aumwa ulim, the relative dimensions were changed, perfect indicative? Go through the imperfect of amo. Give # * preten we lapted to the production of delicate the perfect subjunctive first person singular; third person All win well, The Doric style of architecture was, plural, write what he waxbellions, the only one employed in Greece or These things may seem minute and troublesome to you: Wa Wami culmies in Sicily, and Italy, and in Asia Minor, they would, however, be required by any good teacher; and wrond wine the period of the Macedoninn conquest. In Asia attention to them is, I assure you, requisite to make a sound Wwwe, watende presticularly in Lonia, there speedily arose, subse- scholar ; it is also requisite for that mental discipline which - ' W. gened, an order of architecture more elegant than the study of language may give, and which, in its perfect form, nimeid. Hul our notice of this order must form the subject is of very high value. Another word remains-patrem ; patrem from pater, patris an imparisyllablic noun, of the masculine gender, the third option); pecco 1, 1 sin ; rădămo, I love again; rogo 1, I ask; sano 1, declension, consonantal stem patr, in the singular number I heal (E. R. sanatory); supero 1, I surpass, overcome; tracto 1, I accusative case, being the object of the transitive verb amat, handle, treat ; evěnit, it happens; conscientia, ae, f. consciencc; uva, by which it is governed, according to the rule, “transitive ae, f. a grape; adbileo 2, I apply; adhibere curam, to take care; mediverbs require their object to be in the accusative case." cus, i, a physician (E. R. medical); honestas, átis, f. honesty, honour Observe, that in thus setting before you a specimen of aegrotus, i, a sick man; immortalitas, átis, f. deathlessness; utilitas átis, f. utility; statio, onis, f. a station, a post; maturus, a, um, ripe parsing, I have given you two rules in Syntax; thus mature ; immaturus, a, um, unripe; religiose, conscientiously; acer I. A subject must agree with its verb in number and person. rime, very bravely; vide, see thou; vide ne, see thou do not. II. Transitive verbs require their object to be in the accusa RULE : The conjunctions ut, that, in order that, so that, so as tive case. Of these rules you will forth with have need to and ne not to, so that not, to prevent, require after them the make constant application. Commit them to memory, and subjunctive mood. repeat them memoriter whenever applied. A verbal and exact LATIN-ENGLISHI. repetition of them, and of all rules, is desirable at first; after Saepe evenit ut utilitas cum honestate certet; vide ne pecces wards, I wish that you should give the substance rather than the words of a rule, for if you express its substance you show diligenter tractent ; ita vivere debemus ut in omni re rectam con contra virtutis praecepta ; omnes parentes optant ut filii litteras that you understand its import, scientiam servemus; me redicus omnem curam adhibet ut aegrotum VOCABULARY. sanet; amo te ut me redămes; dux imperavit ut milites stationes Compăro 1, I get together, acquire; emigro 1, I go out, quit (E. R. exercitus noster acerrime pugnabat ut urbem ab interitu servaret. suas servarent; heri ambulabam ut tristem animum exbilarem; emigration); flo 1, I blow; intro 1, I go into, enter (E. R. entrance); judico 1, I judge ; la ro 1, I bark ; libero 1, I set free (E. R. liberation); ENGLISH-Latin. numéro 1, I number ; observo l, I keep under my eye, observe; occŭpo See that your son does not sin ; dost thou sin against the pre1, I fall upon, take possession of (E. R. Occupation); vigilo 1, I watch, keep awake, guard (E. R. vigilant); ventus i, m. wind; terror, oris, m? cepts of virtue ? a wise father takes care that his children do not terror ; timor, oris, m. fear (E. R. timid); narratio, onis, f.'a narra- sin; the generals take care that the soldiers keep their posts ; you tive; interitus, ùs, m. ruin ; 'placidus, a, um, placid, tranquil; ingens: do not (ne) sin); good mothers take care that their children obey take care to prevent your children from sinning (literally, that they ingentis, very great; veněmens, vehementis, vehement, very strong; their commands; thou oughtest to live so as to (so that you may) jam, adv. already ; nuper, adv. lately. preserve honour. LATIN-ENGLISH. Rule: After non dubito, the conjunction quin is used reEgo te laudabam, tu me vituperabas, frater judicabat; ego te quiring the subjunctive mood; thus non dubito quin, 1 doubt laudabo, tu me vituperabis, frater judicabit; ego ambulavi, tu vigilavisti, rentus flavit; ego ambulaveram, tu vigilaveras, ventus doubts that; quis dubitat quin ? who doubts that? dubium non not but, or that. In the same way, nemo dubitat quin, no one Aaverat; ego te laudavero, tu me vituperaveris, frater judicaverit. Quum milites urbem intrabant, omnes cives timoris pleni erant; est quin, there is no doubt that. quam in silvâ ambulabamus, vehemens ventus per altas quercus LATIN-ENGLISH. fiabat, dum nos placidus somnus recreabat. Vos vigilabatis; quamdiu eris felix, multos numerabis amicos ; bonos semper bam quin milites nostri hostes superavissent; quin milites nostri Non dubito quin milites nostri superaverint hostes; non dubitalaudabo, improbos semper vituperabo ; si acriter pugnabitis, o milites, patriam interitu liberabitis; si virtutem amabis, omnes superavissent hostes non dubitabam; non dubito quin milites nostri boni te amabunt. hostes superaturi sint; non dubitabam quin milites nostri hostes superaturi essent; non dubitabam quin vos patrium servitute libeRemark that sometimes an abbreviation takes place in the raiuri essetis ; dubium non erat quin exercitus noster omnes perfect tense, and the tenses formed from the perfect tense. labóres et aerumnas facile toleraturus esset: quis dubitat quin Thus, instead of saying in full, vigilavisti, as above, the Latins Hannibal contra Romanos fortissime pugnavěrit? non dubitabitis shortened the word into vigilasti, leaving out the vi. This quin ego vos semper amaverim; quis dubitat quin bonos semper ion est dubium process is called syncopation, and verbs thus contracted (drawn laudaverimus, malos semper vituperaverimus ? together) are said to be syncopated. Other syncopated forms quin semper fidem servaritis (syocop. for servaveritis) nemo dubiensue; as laudasti for laudavisti; amasti for amavisti ; amasse tabat quin hostes urbem expugnavissent; nulli civium dubium for amavisse; also in other conjugations, as complesti for urbem oppugnabant, non erat dubium quin ingens terror omnium erat quin pro patriâ libertate acerrin e pugnavissetis ; quum hostes complevisti ; audieram for audiveram ; audierunt for audive civium animos occupavisset (sync. occupasset). ENGLISH-Latin. No one doubts that you will fight bravely for the liberty of your magnam vobis laudem comparastis; cur per totam noctem vigi- I country, no one doubts that he fought bravely; no one will doubt lasti ? Præceptores meos semper amavi; nonne amasti tuos ? that he will fight bravely; no one doubted that he had fought bravely; Acri:er contra hostes pugnastis ; quum milites urbem intraverant, who doubts that the soldiers will capture the city? There is no ingens terror omnium civium animos occupabat; narratio quam | doubt that my father will come; who duubis that I shall con doubt that you endeavour (studeo) to preserve honour; I do not mihi nuper narraveras, vehementer exercitus hostilis urbem oppugnaverat, nos jam emigraverámus ; scientiously preserve the city? si animum virtutibus ornaveris (urnaris) semper beatus eris; quum Though non dubito quin, &c., requires the sulijunctive mood hostes urbis nostrae agroe devastaverint, urbem ipsam oppug. in Latin, the verb must be Englished by an indicative mood; Dabant. as may be seen in the English exanıples just given. In order ENGLISH-LATIN. to make this quire plain, I will give another instance : We praised thee, thou didst blame me, father was judging; thou Non dubito quin bonus sit avunculus tuus, wilt praise me, he will praise thee, father will judge us; thou hast I doubt not that thy uncle is good. walked (syncopated form!, I have watched, the winds blew ; I will Here, then, you see the verb which in Latin must be in walk abroad; thou art watching; the wind was blowing; the the subjunctive mood, must stand in the indicative mood in Boldiers will enter the city; the soldiers were entering the city; English. Such is a by no means unusual fact. the soldiers are entering the city, the soldiers have entered the city; the soldiers had entered the city a very strong wind the negative non, as ne crede, do not believe. Rule: With the imperative the negative ne is used, and not blows through the house; dost thou number many soldiers ? I have numbered many friends; he has liberated (set free) his LATIN-ENGLISH, country from ruin; hast thou watched all night? Love thy preceptors ; let them love their parents; O boys, love virtue.' The O pueri! Ne dubitato de animarum immortalitate; semper serva Diligenter cura, mi amice, valetudinem tuam ; anate litteras narrative delighted my brother ; the narrative delights the girls; ini fili, conscientiam rectam ; discipulus amato praeceptores the narrative will delight father and mother; thou hast acquired laudatote probos, vituperatore imprulos; ne lauda malus pueros fame by the narrative of the ruin. mnes homincs amanio Deum, VOCABULARY. My sons, take care of your mother; Charles, do not doubt of the Advento 1, I come to, arrive at; exhilăro 1, 1 exhilarate, rejoice; tolero deathlessness of the good; preserve, children, an upright con1, I endere, bear; expugno 1, I capture ; gusto 1, I taste; obtempero love Gou; do not blame thy sister, Charles. scienoc; let children love (iheir) preceptors; do you, my friends, 1, I obey (with the object in the dative); opto 1, I wish (E. R. Rule: The first supine, that is, the supine in um, is used ܝܢ me . after verbs denoting motion ; the second supine, that in u, is factor, and so on till you have multiplied by all the factors, The used after certain adjectives. last product will be the answer required. Remark: The supines are supposed to have been nouns of The factors into which a number may be resolved, must not die the fourth declension, that in um, a noun in the accusative confounded with the parts into which it may be separated. In case; that in u, a noun in the dative case (u for ui). former have reference to multiplication, the latter to additica; that is, facto LATIN-ENGLISH. must be multipted together, but parts must be added together to produce the given number. Thus, 56 marts Parentes mei in urbem migraverunt habitatum ; exercitus resolved into two factors, 8 and 7; it may be separated into in hostilis adventarit agros nostros devastatum; uva immatura est parts, 5 tens or 50, and 6. Now, 8X7=56, and 50+636. acerba gustatu; aerumnae sunt durae toleratu; sitis difficillima est toleratu ; pira sunt dulcia gustatu; luscinia cantans animos EXERCISES. nostros delectat; hostes adventant expugnaturi urbem ; non est 1. What will 24 horses cost, at 74 crowns apiece. dubitandum de immortalitate animarum; obtemperandum est 2. What cost 45 hogsheads of tobacco, at 128 crowns 2 baş praeceptis virtutis. head? ENGLISH-LATIN. 3. What cost 54 acres of land, at 150 crowns per acre? The soldiers approach to deliver (supine in um) the king ; the hos'fle army approaches to capture the city; ripe grapes are board 49 weeks? 4. At 118 shillings per week, how much will it cost a fast » 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 5. If a man travels at the rate of 372 miles a day, how far ri by singing (gerund) delight the mind; we must fight bravely for he travel in 64 days? our native land (patria), the soldiers have approached to fight for 6. At 163 crowns per ton, how much will 72 tons of leados their homes and their wives. 7. What cost 81 pieces of broadcloth, at 245 shillings apieski 8. What cost 84 carriages, at 38 1 crowns apiece? CASE II.—When the multiplier is 1 with ciphers arnered to it LESSONS IN ARITHMETIC.-No. XV. It is a fundamental principle of notation, that each removal o'ı CONTRACTIONS IN MULTIPLICATION. figure one place towards the left, increases its value ten tima; di Although the general rules previously given be adequate to the sequently, annesing a cipher to a number will increase its value is solution of all examples that occur in multiplication, yet in many value a hundred times, or multiply it by 100; annexing to times, or multiply it by 10; annexing two ciphers will increzen instances, by the exercise of judgment in applying the preceding ciphers will increase it a thousand times, or multiply it by pri principles, the operation may be very much abridged. Definition.—Any number which may be produced by multi- &c. Thus, 15 with a cipher andexed, becomes 150, and ist plying two or more numbers together, is called a Composite Num. same as 15x10; 15 with two ciphers annexed, becomes 1500 , e ber: Thus, 4, 15, 21, are composite numbers; for 4=2X2 ; 15= is the same as 15x 100; 15 with three ciphers annexed, bernama 5X3; 21=7X3. 15000, and is the same as 15 x 1000, &c. The factors which, being multiplied together, produce a com. To multiply by 10, 100, 1000, &c. Rule : posite number, are sometimes called the component parts of the in the multiplier, and the number thus formed will be the prota Annex as many ciphers to the multiplicand as there are curar number. The process of finding the factors of which a given required. number is composed, is called resolving the number into factors. To annex means to place after, or at the right hand. EXERCISES. EXERCISES. 1. Resolve 9, 10, 14, 22, into their factors. 1. What will 10 boxes of lemons cost, at 63 shillings per der 2. What are the factors of 35, 54, 56, 63 ? Ans. 630 shillings. 3. What are the factors of 45, 72, 64, 81, 96 ? Some numbers may be resolved into more than two factors ; and at 100 bushels per acre ? 2. How many bushels of corn will 465 acres of land produr also into different sets of factors. Thus, 12=2 X 2X3; also 12= 3. Allowing 365 days for a year, how many days are then i 4X3=6x2. 1000 years 4. What are the different factors and sets of factors of 8, 16, 4. Multiply 153486 by 10000. 18, 20, 24? 5. Multiply 3120467 by 100000. 6. What are the different factors and sets of factors of 27, 32, 6. Multiply 52690078 by 1000000. 36, 40, 48? 7. Multiply 689063457 by 10000000. We have seen that the product of any two numbers is the same, 8. Multiply 4946030506 by 100000000. whichever factor is taken for the multiplier, and that the product 9. Multiply 87831206507 by 1000000000. of any three or more factors will be the same, in whatever order 10. Multiply 67856005109 by 10000000000. they are multiplied. For, the product of two factors may be considered as one number, and this may be taken either for the multi Case III.— When the multiplier has ciphers on the right Lab plicand, or the multiplier. Again, the product of three factors Examplk.—What will 30 wagons cost at 45 crowns apie? may be considered as one number, and be taken for the multipli- Any number of ciphers on its right hand, is obviously & cand, or the multiplier, &c. Thus , 24–3x2x2x2=6x2x2= site number ; the significant figure or figures being one factor, el 12 X2=6X4=4X2X3=8X3. 1, with the given ciphers annexed to it, the other factor. Thuis. CASE 1.-When the multiplier is a composite number. may be resolved into the factors 3 and 10. We may Example 1:-- What will 27 bureaus cost, at 31 crowns apiece? first multiply by 3 and then by 10, by annexing a cipher Solution. - Here 45 x 3 = 133, and 135 x 10 = 1.60 9X3, it is manifest that 27 bureaus will cost three times as much crowns. Ans. as 9 bureaus. EXAMPLE.-How many acres of land are there in 3000 faras ? Operation. Erplanation. each farm contains 475 acres ? Crowns 31 cost of 1 bureau, Having resolved 27 into the *Analysis. Operation. 9 factors 9 and 3, we find the cost 475 X 3 = 1425 ; and adding three Multiplier First, 3000 = 3 X 1000. Now Multiplicand 475 Crowns 279 cost of 9 bureaus. of 9 bureaus, then multiplying ciphers to this product, multiplies it Product 3 that by 3, we have the cost of 27 by 1000. Crowns 837 cost of 27 bureaus. To multiply when there are ciphers on the right hand of the EXAMPLE 2:- What will 36 oxen cost, at 43 crowns per head ? multiplier, Rule : Solution.-36=9X4; and 43X9X4=1548 crowns. Ans. Multiply the miltiplicand by the significant figures of the early Or, 3633X3X4; and 43X3X3X4=1548 crowns. Ans. To multiply by a composite number. Rule : on the right of the multiplier. tiplier, and to this product annex as many ciphers, as are for Resolve the multiplier into two or more factors ; multiply the It will be perceived that this case combines the principles of the multiplican i by one of these factors, and this product by another and one of its factors is 1 with ciphers annexed to it. 1 there 3000 1425000 crowns. EXERCISES. To multiply by 21, 31, 41, &c., or by 1 with either of the other 1. How much will 50 hogs weigh, at 375 pounds apiece? significant figures prefixed to it. Rule: 2. If i barrel of flour weighs 192 pounds, how much will 500 Multiply by the tens' figure of the multiplier, and write the first barrels weigh? figure oj ihe partial product in the tens' place ; finally add this 3. Multiply 14376 by 25000. partial product to the multiplicand, and the result will be the 4. Multiply 350634 by 410000. answer required. 5. Multiply 4630425 by 6200000. The reason of this method of contraction is similar to that of the CASE IV.- When the multiplicand has ciphers on the right preceding. hand. 1. Multiply 4275 by 31. 3. Multiply 7504 by 41. Example 1.---What will 37 ships cost, at 29000 crowns apiece ? 2. Multiply 38256 by 61. 4. Multiply 70267 by 81. Analysis. Operation. First, 29000=29 X 1000. But Crowns 29000 multiplicand CASE VII.- When the multiplier is a decimal numberless unity. the product of two or more factors 37 multiplier EXAMPLE.-How much will 99 carriages cost, at 235 crowns is the same in whatever order they 203 apiece? are multiplied. We therefore mul. Analysis 87 Operation. tiply 29 by 37, and this product by Crns. 1073000 product. Since one carriage costs 235 33500 cost of 100 carriages 1000 by adding three ciphers to it. crowns, 100 carriages will cost 100 235 cost of l carriage To multiply when there are ciphers on the right band of the times as much, which is 23500 23265 cost of 99 carriages. multiplicand. Rule : But we wished to find the Multiply the significant figures of the multiplicand by the mul- cost of 99 carriages only. Now 99 is one less than 100; theretiplier, and to the product annex as many ciphers as are found on fore, if we subtract the price of 1 carriage from the price of 100, the right hand of the multiplicand. it will give the price of 99 carriages. Hence, EXERCISES. To multiply by 9, 99, 999, or a number composed of any num1. Multiply 2370000 by 52. ber of 9's. Rule : 2. Multiply 48120000 by 48. Annex as many ciphers to the multiplicand as there are 9's 3. Multiply 356300000 by 74. in the multiplier; from the result subtract the given multiplicand, 4. Multiply 1623000000 by 89. and the remainder will be the answer required. EXAMPLE 2.-Multiply 540000 by 700. The reason of this method is obvious from the fact that annexing Analysis. Operation. as many ciphers to the multiplicand as there are 9's in the multiplier, 540000=54 x 10000, and 700= 540000 multiplicand multiplies it by a number one more than is required; consequently, 7x 100; we therefore multiply the 700 multiplier subtracting the multiplicand from the number thus produced, must significant figures, or the factors 378000000 product. give the true answer. 54 and 7 together, and to this product 1. Multiply 4791 by 99. 3. Multiply 6034 by 999. annex six ciphers. 2. Multiply 7301 by 999. 4. Multiply 463 by 9999. When both the multiplier and multiplicand have ciphers on the CASE VIII.- When the multiplier is composed of a digit and right, multiply the significant figures together as if there were no ciphers, and to their product annex as many ciphers as are found its multiples. on the right of both factors. EXAMPLE.—What is the product of 867 multiplied by 84? EXERCISES. Analysis. Operation. 1. Multiply 1563800 by 20000. We first multiply by 4 in the usual way. 867 multiplic ind 2. Multiply 31230000 by 120000. Now, since 8=4X2, it is plain, if the par. 84 multiplier 3. Multiply 5310200 by 3400000 tial product of 4 is multiplied by 2, it will 3168X2 4. Multiply 82065000 by 8100000. give the partial product of 8. But as 8 de 6936 5, Multiply 210909000 by 5100000. notes tens, the first figure of its product There are other methods of contracting the operations in multi- will also be tens. The sum of the two par 72828 product. plication, which, in certain cases, may be resorted to with advan- tial products will be the answer required. iage. Some of the most useful are the following : For the sake of convenience in multiplying, the factor 2 is placed Case V.-When the multiplier is any digit preceded by unity. at the right of the partial product of 4, with the sign X between Example.—How many gallons of water will a hydrant discharge them. in 13 hours, if it discharges 2325 gallons per hour ? What is the product of 987 by 186 ? Operation. Multiplicand 987 Since 48–6X8, we multiply the Multiplier 486 partial product of 6 by 8, and set the Product 30225 gallons. the right of the multiplicand. Now, 5922 X 8 first product figure in tens' place as since multiplying by 1 is taking the mul 47376 before. tiplicand once, we add together the multiplicand and the partial pro. duct already obtained, and the result 30225 gallons is the answer. Product 479682 To multiply by 13, 14, 15, &c., or by 1, with either of the other To multiply when part of the multiplier is a composite number digits annered to it. Rule: of which the other figure is a factor. Rule : Multiply by the units' figure of the mulliplier, and write each First multiply by the figure that is a factor; then multiply this figure of the partial product one place to the right of that from partial product by the oilier factor, or factors, taking care to write which it arises; finally, add the partial product to the multiplicand, the first figure oj" each partial product in its proper order, and and the result will be the answer required. their sum will be the answer required. This method is the same, in effect, as if we actually multiplied by When the figure in thousands, ten thousands, or any other the one ten, and placed the first figure of the partial product under column, is a factor of the other part, or parts of the multiplier, care the figure by which we multiply. must be taken to place the first figure of its product under the 1. Multiply 3251 by 14. 3. Multiply 4028 by 17. factor itself, and the first figure of each of the other partial pro. 2. Multiply 25039 by 16. 4. Multiply 50389 by 18. ducts in its own order. Case VI.-When the multiplier is any digit followed by unity. EXAMPLE 1. EXAMPLE 2. EXAMPLE, If 21 men can do a job of work in 365 days, how 2378 multiplicand 256841 multiplicand long will it take 1 man to do it? 936 multiplier 85632 multiplier Operation. Explanation. 21402 X 4 2054728 X7X4 14383096 2225808 product. 8218912 Product 7665 days. adding this partial product to the multiplicand, we bave 7665 days for the answer. 21993808512 product, 2 Par. Products 2 EXERCISES. 13. 42304 x 999. 27. 5234 X 2435, 1. Multiply 665 by 82. 4. Multiply 783 by 93. 14. 50421 x 9999. 28. 48743000 x 637. 15. 67243 X 99999. 5. Multiply 69412 by 95436. 29, 31890420 X 85672. 2. Multiply 876 by 396. 3. 324325 X 54426. 6. 256721 X 85632. 16. 78563 X 93. 30, 80460000 X 2763. EXAMPLE.- What is the product of 63 multiplied by 45? 17. 34054 X 639. 31. 2364793 X 8485672. 18. 52156 x 756. 32. 1256702 X 999999. By multiplying the figures which produce the same order, and 19. 41907 x 54486. 33, 6840005 X 91 x 61. adding the results mentally, we may obtain the answer without 20, 26397 X 24648. 34. 45067034 x 17 x 51, setting down the partial products, Operation, Explanation. 21. 12900 X 14000. 35. 788031245 X 81x16. 22. 64172X42432. First, multiply the units into units, we set down 36. 61800000 X 23000. 63 37. 12563000 X 4800000. 45 23. 26815678X81. the result and carry as usual. Now, since the 6 24, 85 X 85. tens into 5 units, and 3 units into 4 tens will pro 38. 91300203 X 1000000, 2835 Ans. duce the same order, viz. teps, we multiply them 25. 256 X 256. 39. 680040000 x 1000000. 26. 322 X 325. 40. 4000000000 x 1000000. 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 LESSONS IN FRENCH.—No. XXI. partial products. Rule : First multiply the units together ; then multiply the figures By Professor Louis FASQUELLE, LL.D. which produce lens, and adding the products mentally, set down the result and carry as usual. Next multiply the figures which Section XLIV, produce hundreds, and add the products, déc., as before. In like (Sce Section 35.) wanner, perform the multiplications which produce thousands, ten thousands, dc., adding the products of cach order as you proceed, and thus continue the operation until all the figures are multiplied. auxiliary ($ 46]: 1. The reflective or pronominal verb always takes être as its ExamPLE.-What is the product of 23456789 into 54321 ? Votre cousin s'est promené. Your cousin has taken a walk. Nos amis se sont flattés. Our friends have flattered themselves. 5 2. Although the past participle of a reflective verb, be con1 jugated with être, it agrees with its direct regimen, when that 2x 13X14X 15x16x17X18X119X1 regimen precedes it, and is invariable when the regimen fol2X 2 3X2 4X25X26X27X28X29X2 lows it. The student should be careful to see, if the reflective 2X33X34X35X36X37X38X39X3 pronoun be a direct or an indirect regimen :135] :2X43X4 4X45X46X47X48X49X4 Vous vous êtres flattées, Mesde. You have flattered yourselves, young 2 x 53 x 5 4 X55X56X5|7x58X59X5 moiselles. ladies. 12 7 4 1 9 6 2 3 5 6 9 Elles se sont donné la main. They have given (to) cach other the Explanation.—Having multiplied by the first two figures of the hand. multiplier, as in the last example, we perceive that there are three It will be easily perceived that se in the first sentence is a multiplications which will produce hundreds, viz., 7X1,8 X 2, direct regimen, and that the same word in the second repreand 9 X 3; we therefore perform these multiplications, add their sents an indirect object. products mentally, and proceed to the next order. Again, there 3. Verbs essentially unipersonal, i. e., verbs which cannot be are four multiplications which will produce thousands, viz., 6 x 1, conjugated otherwise, take avoir as an auxiliary :7X2, 8 X 3, and 9 X 4. We perform these multiplications as before, and proceed in a similar manner through all the remaining Il a plu, il a neigé, il a gelé. It rained, it snowed, it froze. orders. Ans. 1274196235269. 4. Verbs occasionally unipersonal, take être as an auxiliary :In the solution above, the multiplications of the different figures Il lui est arrivé un malheur. A misfortune has happened to him. are arranged in separate columns, that the various combinations which produce the same order, may be seen at a glance. In prac. take the auxiliary avoir : 5. Faire [4 ir.] used unipersonally, and Y avoir, to be there, tice it is unnecessary to denote these multiplications. The principle being understood, the process of multiplying and adding may A-t-il fait beau temps le mois passé ? Was it fine weather last month.1 easily be carried on in the mind, wbile the final product only is set Y a-t-il eu beaucoup de monde ? Were there many people there! down. When the factors contain but two or three figures each, this able [§ 135 (6)]: 6. The participle past of a unipersonal verb is always in varimethod is very simple and expeditious. A little practice will enable the student to apply it with facility when the factors con: Les pluies qu'il y a eu cet été. The rains which we have had this 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 Résumé of EXAMPLES. to extend it so far is not profitable. Les Italiennes se sont elles pro- Did the Italian laclics acalk:1 menées ? Oui, Monsieur, elles se sont pro- Yes, Sir, they have taken a walk. 1. Multiply 25 x 25. 5. Multiply 54 X 54. menées. 2. Multiply 81X64. 6. Multiply 45 X 92. Nous nous sommes aperçus de We perceived that, or we took notice 3. Multiply 194 X 144. 7. Multiply 1234X 125. cela. of that. 4. Multiply 4825X2352. 8. Multiply 6521x5312. Votre mère s'est elle bien portée ? Has your mother been well:: By suitable attention, the critical student will discover various Vos souirs se sont elles assises ? Did your sisters sit dolor! other methods of abbreviating the processes of multiplication, Cette marchandise s'est elle bien Did that merchandise sell well ? vendue ? Exercises. Vos enfants se sont ils appliqués Did your children apply to study : Solve the following examples, contracting the operations when à l'étude ? practicable, and prove them by the general rule, or by other Il s'y sont appliqués. They applied to it. wethods. Nous nous sommes donné de la }} ( jute (to) ou. At-tes Pich toiba 1. 12634 x 63. 7. 4300450 X 19. peine (5 135 11)] 2.0, 15 56. 8. 9803401 X41. Quei temps a-t-il fut ce matin ? Il'ht. wcuuther wus it this norning? 3. 7216X1000. 9. 6710015 x 71. Nr til pas fait beau temps ? as it lot fine weather! 4. 42000 x 40300. 10. 3456710 x 18. Quel malheur vous est il arrivé ? 11% at waisfurtune hus kappened to 5, 80000 x 25000. 11. 7000541x91. 6. 2567345 X 17. Vous est il arrivé quelque chose? 12, 4102034 X 99. Has anything kappened to you? Il ne n'est rien arrivé. Nothing has happened to me. Sunimer. you? |