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below the line 5 b, and finished with a loop like the elementary
THIRD DECLENSION. stroke in Copy-slip No. 52. This stroke and the various letters
Sign is. into whose composition it enters, one of which is the letter j, in
m. and F.
å or is LESSONS IN LATIN.-VIII.
um or ium um or iam THE THIRD DECLENSION,
X or ix We pass on to the third declension. In the third declension V.
(like the nom.)
å or is we find, in the nominative case, so great a variety of termina Ab.
8 or i.
Ibús. tions, that we must endeavour to arrange the nouns in certain
The genders of the nouns of the third declension may be classes. The genitive singular, however, is the characteristic stated thus, though the rules are not without exceptions :case, and it ends in is.
First, nouns ending in o, or, os, er, and imparisyllabics in es, Before classifying these nouns, I must give you some expla.
are masculine ; second, nouns ending in as, is, aus, us (gen. nations. Parisyllabic is a word I have to use. It consists of
utus or udis) and x, and those which end in s blended with the three words, which I will mark thus
preceding consonant, as well as parisyllabics in es, are feminine; 1 2 3
third, nouns ending in a, e, c, l, en, ar, ur, ut, and us (gen. pari syl lab(ic);
āris, ēris, iris), are neuter. By practice you will in time become of these the two latter are of Greek origin. The former is familiar with these somewhat complex facts. Latin. As the word is thus made up of terms from two lan I proceed to set down specimens in classes. guages, it is a sort of hybrid. No. 2 signifies with; No. 3
CLASS I. signifies to take; the ic is merely the termination. If you put NOUNS WITH CONSONANTAL STEMS ; IMPARISYLLABIC. 2 and 3 together you have syllab, which with the termination
1.-Without the termination s. ble makes syllable. A syllable, then, is so much of sound as
(i.) The stem and the nominative are the same ; stems end may be taken or uttered at once. No. 1 means equal (pari in 7 and I. found in the English par and pair); parisyllabic, then, signifies
Thus : nom. dolor; gen. doloris ; stem, dolor. that which is equal in its syllables; and nouns are called parisyllabic which have the same number of syllables in all the
. EXAMPLES, cases of the singular number. I say of the singular number,
MASCULINES, because the plural of all nouns is not parisyllabic, inasmuch as
Singular, the genitive plural, as in the cases of arum and orum, has a
N. dolor, grief. anser, a goose. vomer, a ploughshare.
G. doloris, of grief. ansēris, of a goose. syllable more than the other cases. Now nouns which have in
voměris, of a ploughshare,
D. dolori, to grief. anseri, to a goose. romeri, to a ploughshare. the genitive singular & syllable more than they have in the
Ac. dolorem, grief. anserem, a goose. vomerem, a ploughshare. nominative singular are called imparisyllabic. In this word, as v.
dolor, O grief! anser, O goose ! vomer, O ploughshare! here given, you find an additional syllable, namely, im from in-| Ab. dolore, by grief. ansere, by a goose. vomere, by a ploughshare, the n becoming m, before the p-which signifies not. Imperi
Plural. syllabic, then, is not-parisyllabic; and the words denote those n. dolores, griefs. anseres, geese. vomeres, ploughshares. nouns which in the genitive singular have not the same number G. dolorum, of griefs. anserum, of geeso. vomerum, of ploughshare. of syllables as they have in the nominative. Piscis, a fish, is D. doloribus, to griefs. anseribus, to geese. vomeribus, to ploughshers. parisyllabic; for in the genitive it is piscis, having two syllables
Ac. dolores, griefs. anseres, geese, romeras, ploughshares.
dolores, as in the nominative. But cantor, a singer, is imparisyllabic,
anseres, O geese! vomeres, o ploughshares ! for in the genitive it is cantoris, having three syllables, whereas
Ab. doloribus, by griefs, anseribus, by geese. vomeribus, byploughshares the nominative has but two. Here then we have one distinc
NEUTERS. tion--namely, nouns of the third declension are either pari.
Singular. syllabic or imparisyllabic.
N. guttur, a threat. calcar, a spur. animal, an animal. Now, inquiry has shown that parisyllabic nouns have a vowell
guttüris, of a throat. calcarts, of a spur. animühs, of an animal,
D. stem, and imparisyllabic nouns a consonant stem; that is, that
gutturi, to a throat, calcari, to a spur. animali, to an animal.
guttur, a throat. the stem of the former ends in a vowel, and the stem of the
calcar, a spur. animal, an animal.
V. guttur, O throat! calcar, O spur! animal, o animal! latter ends in a consonant. Of the stem of a noun and a verb
gutture, by a throat. calcari, by a spur. animali, by an animal, I have already said something. It is better to repeat than not
Plural. to be understood. Take nubes, a cloud, and form the genitive;
N. guttura, throats. calcaria, spurs. animalia, animals. the genitive is nubis. You get the stem by cutting off the sign
gutturum, of throats. calcarium, of spurs. animalium, of animals, of the genitive, which in this case is 8 (as in the English cloud, D. gutturibus, to throats, calcaribus, to spurs. animalibus, to enunads, cloud's). You thus obtain nubi. Nubi has two syllables, the
Ac. guttura, throats. calcaria, spurs, animalia, animals. same as the nominative nubes. It is therefore parisyllabic, and V. guttura, O throats! calcaria, O spurs ! animalia, o animals! ends in a vowel. Take also dolor, grief; genitive, doloris. Cut Ab. gutturibus, bythroats. calcaribus, by spurs, animalibus, by animals. off is, the sign of the genitive, and you obtain dolor. Dolor Here observe, that as in the neuter nouns of the second de ends, you see, in a consonant, and is a consonantal stem. The clension, the neuter nouns of the third declension have in both word is also imparisyllabic, because it increases in the genitive the singular and the plural three cases alike,-namely, the singular. Imparisyllabio nouns, then, have consonantal stems. nominative, the accusative, and the vocative. In animal, the In this case the stem and the nominative are the same, both nominative plural is ia, instead of a. This is owing to its being being dolor. But in nomen, a name, genitive nominis, stem originally from a vowel stem-as, nominative, animal; genitive, nomin, the nominative and the stem are unlike. Of consonantal animalis; stem, animali. stems, then, there are two classes: first, those of which the
Agger, aggěris, m., a Fulgur, fulguris, 1., | Passer, passéris, m., a
sparrow. are :
Color, colöris, m., | Illi, to him.
Pulvinar, pulvipäris, c. 6. t. d. p. b. mutes,
n., couch. I. m. . r.
Diligo, 3, I love or Mater, matris, f., a Rumor, rumoris, m., the sibilant.
report, From these stems the nominative is formed with or without Error, erroris, m.,
.Mihi, to me.
Tibi, to thee. the addition of 8. An instance of the formation of the nomina
Vectigal, vectigalis, 11., tive with the addition of s, is found in nom. rex, a king, gen.
Frater, fratris, m., Odor, odoris, m., odour, a ter,
Vobis, to you regis, stem reg; add & and you have regs, which is pronounced rex. An instance of the formation of the nominative without OBS.-Est mihi, I have, used with the noun as non, to est; the addition of s you find in nom. leo, a lion, gen. leonis, stem, thus, guttur est mihi, I have a throat ; so in the plural, guttura shortened into leo.
sunt nobis (throats are to us), we have throats. In the same way,
guttur est tibi (a throat is to thee), thou hast a throat; guttur Thy fathers have wounds. 6. Wounds frighten mothers. 7. Poems est illi (a throat is to him), he has a throat; guttura sunt vobis flourish in the region. 8. Thou hast a great name. 9. I have not a (throats are to you), you have throats; guttura sunt illis (throats pledge. 10. They have an opportunity. 11. The man's opportunity
is great. are to them), they have throats. EXERCISE 25,- LATIN-ENGLISH.
KEY TO EXERCISES IN LESSONS IN LATIN.–VII. 1. Magnus dolor est mihi. 2. Nonne tibi est magnus dolor ? 3. Sunt
EXERCISE 21.–LATIN-ENGLISH. magni dolores matribus. 4. Color pulvinaris pulcher est. 5. Estne
1 1. Good men love good boys. 2. Good boys are loved by good men. pulcher pulvinaris color ? 6. Funestus error est illi. 7. Cur funesti
3. A good boy loves school. 4. The good masters of good boys are errores sunt patri? 8. Frater est mihi. 9. Fratribus sunt magni dolores. 10. Fulgura terrent animalia. 11. Nonno matres terrent
6. The war is deadly. 7. I loved. 5. Hast thou a good master?
have a good female friend. 8. The boys are in school. 9. Are fulgura? 12. Fulgura terrent passeres.
not tho boys in school? 10. Many foreigners sail into Britain. EXERCISE 26.- ENGLISH-LATIN.
11. The boar of my friend is great. 12. There is play on the river's 1. I have a spur. 2. Hast thou a goose? 3. They have geese. 4.
bank. 13. Scholars love (like) letters. 14. There are frogs on the Have you a mound ? 5. The odour of the lightning is on the cushion. / banks. 15. The goat is great. 16. There are deadly wars in the
island. 6. I do not like taxes. 7. Rumours are troublesome. 8. Have they a couch? 9. They have not a goose. 10. You have a father, a
EXERCISE 22.-ENGLISH-LATIN. brother, and a mother. 11. They have griefs. 12. Thou hast a great 1. Bonos discipulos amo. 2. Boni discipuli a bonis viris amantur, cushion,
| 3. Amasne amicum ? 4. Aper est mihi. 5. Tibi est caper. 6. Capri (i.) The stem and the nominative are different; stem in nand r. sunt in ripa. 7. Est in insulâ magnum et funestum bellum. 8. In
Britanniâ sunt agri multi. 9. Funesti sæpe sunt apri? 10. O viri, MASCULINES AND FEMININES.
amatisne pueros ? 11. Amici mei peregrinos non amant. 12. Ludum Cases. Singular.
amant pueri. 13. Amantne pueri ludum ? 14. Estne tibi amica ? N. leo, a lion.
homo, a man. pater, a father. 15. Magnus aper non est mihi. 16. Amicæ epistola est in horto. G. leónis, of a lion. hominis, of a man. patris, of a father. D. leoni, to a lion. homini, to a man. patri, to a father.
EXERCISE 23.–LATIN-ENGLISH. Ac. leonem, a lion. hominem, a man. patrem, a father.
1. The horse neighs. 2. The horse's mane is beautiful, 3. The V. leo, O lion !
homo, O man! pater, O father! flies are troublesome. 4. Are the flies troublesome? 5. Good scholars Ab. leone, by a lion. homine, by a man. patre, by a father. are not troublesome. 6. Long wars are troublesome. 7. Horses Cases, Plural.
quickly. 8. A man guides the horse. 9. A horse is guided by a man. N. leones, lions. homines, men, patres, fathers.
10. I am delighted by a beautiful horse. 11. The fields are fruitful. G. leopum, of lions. hominum, of men patrum, of fathers.
12. The herbs of the fields are various. 13. The husbandman commits D. leonibus, to lions. hominibus, to mon. patribus, to fathers. to the fields grains of corn. 14. The husbandman tills the fields. Ac. leones, lions. homines, men. patres, fathers.
15. How beautifully the fields flourish. 16. Various herbs flourish in V. leones, O lions! homines, O men! patres, O fathers ! the fields. Ab. leonibus, by lions. hominibus, by men. patribus, by fathers.
1. Fecundus est ager. 2. Suntne agri fecundi ? 3. Bella fecunda
non sunt. 4. Agri coluntur. 5. Deos colis. 6. Dii coluntur a Singular.
Tullio. 7. Equus et equa a viro reguntur. 8. Celeriter currunt apri. N. corpus, a body. nomen, a name, genus, a race.
9. Curruntne capri celeriter ? 10. In pulchro horto sunt muscæ. G. corporis, of a body. nominis, of a name. genèris, of a race.
11. Equum agro committis. 12. Boni discipuli coluntur. 13. O mi D. corpori, to a body. nomini, to a name. generi, to a race.
fili! diis et deabus committuntur templa. 14. O Antoni! dii deæque Ac. corpus, a body. nomen, a name. genus, a race.
in templis coluntur. 15. Obone Deus! in fecundis agris coleris. V. corpus, O body! nomen, 0 name! genus, O race!
16. Boni viri a filiis et filiabus coluntur.
LESSONS IN DRAWING.–VIII.
we think it necessary to detain the pupil a little longer upon this V. corpora, o bodies! nomina, 0 names! genera, O races!
early and most important part of our subject, for reasons that Ab. corporibus, by bodies. nominibus, by names, generibus, by races.
will be apparent as we proceed. So essential is good drawing, In two of the words declined above, corpus, corporis, corpor, that without a correct outline the most laboured performance in and genns, generis, gener, the stems-namely, corpor and gener other respects will be a failure ; it may be very neat in its execu
-seem to end in r. The r, however, is only the representation, carefully shaded, or perhaps cleverly coloured; but if it fail tive of s, for between two vowels, as in corporis, the s by the in the outline by not giving a truthful representation of the form laws of pronunciation passes into r. Thus, instead of corpus, of the object, it is then for all practical purposes useless. We corpüsis, we have corporis, the s being changed into r and the u know what a great temptation it is to the young to begin to into o. Similar changes take place in tellus (tellūsis) telluris, paint, but they do not consider that to be able to paint well the earth ; pulvis, pulvéris, dust; mas, maris, a male; æs, æris, they must be able to draw well. Painting, in its practice that is, brass ; flos, floris, a flower.
the execution-is nothing more than placing colours, as we have VOCABULARY.
said of lines, in their right places, and the power of handling
the brush successfully depends upon the pupil's ability for Carbo, carbônia, m., | Littus, littoris, a shore. Pectus, pectoris, n., a charcoal. Lumen, luminis, n., breast.
handling the pencil. Of course we make no allusion to the Cardo, cardinis, m., a light.
arrangement of colours themselves, their harmony and tones; we Pignus, pignoris, n., a Occasio, occasionis, f., pledge.
moan simply the power of using the brush where it is necessary Carmen, carminis, 1., | an opportunity. Pulvis, pulveris, m., to perfect the form of the object being painted, without having a poem, Opus, opěris, n., work. dust.
to lay down the brush to resume the pencil. We wish also to Cinis, cineris, m., | Ordo, ordinis, m., order, Regio, regionis, f., a warn the pupil against that slovenly, dangerous, and unsatisfacashes. series.
region or district. tory manner of drawing which is generally termed sketching, Decas, decoris, n., Pavo, pavonis, m., a Vulnus, vulneris, n., that is, producing a hurried, careless outline, its correctness becomingnesa. I peacock.
I a wound.
being the last thing considered. Sketching, with an imperfect EXERCISE 27.-LATIN-ENGLISH.
power of drawing, in the majority of cases amounts to nothing 1. Carbodem timeo. 2. Pavones ferit puer. 3. Pulchræ sunt more than scribbling; there may be thousands of individuals regiones. 4. Occasio est tibi. 5. Movemus cineres. 6. Cardo move who can sketch, but amongst these there are comparatively tur, 7. Ordinis decus delectat matres. 8. Magnus est pulvis cineris. few who can draw. The dogs, horses, and ships with which the 9. In littore sunt pavones. 10. Carining non sunt nobis. 11. Vulnus schoolboy adorns the pages of his dictionary, or the margins of est in pectore. 12. Regionis magnum est lumen. 13. Illi est nomen toagnur, 14. Pignora non laudantur.
his exercises, may, on the whole, bear a strong resemblance to
the general character of the class of animals or objects inEXERCISE 28.-ENGLISH-LATIN.
tended; but this is not drawing: it is quite another thing 1. Dost thon fear charcoal ? 2. Why does the mother strike the to give the individuality of these objects : in this is the test of boy? 8. They have no becomingness. 4. Thou hast a wound. 5. ability. It is true that the hand of a master may by a few
lines express an idea with great force and power, but for a under our notice, to draw which we shall be materially assisted learner to begin the art by sketching is altogether a mistake. by principles borrowed from geometry. But though we cannot We once heard an eminent landscape painter say that "sketch. employ compasses to draw the forms of flowers and leaves, ing is the ruin of hundreds of young artists ; it is beginning yet by the practice of geometry we easily associate lines, angles, at the wrong end; let them draw well first and secure the ' and centres with curves, although they are not visible upon the power, then afterwards they may sketch." Sketches are clever object. Instruments are usually depended upon for drawing and valuable only when they are done by men who can really i architectural curves, mouldings, and the like, because they must draw well; the unfortunate result of the habit of sketching by be constructed according to received proportions. We propose an inexperienced hand may be compared to that of the very now to place before our readers some examples of architectural objectionable system which compels schoolboys to write out curves, with the rules for constructing them; our reason for pages of Latin or English for punishment. There are many doing so being simply to show the pupil a way of making his who acknowledge in after years that their handwriting was eye familiarwhit the construction of curves on geometrical prin.
Filed by these “tasks” or “impositions," and who were ciples. From long experience we have found it to be the case nerer able afterwards, with all their efforts, to write well. Let that they always make the best and quickest draughtsmen, te prapl therefore give up all idea of sketching, and seek and do their work with the least labour, who have dipped to draw rell, if he at all hopes to make the art useful for deeply into geometrical drawing and lineal perspective. In their practical purposes, or to secure in its practice a pleasurable practice they have acquired a habit of precision, and have Pesource in leisure hours.
learnt the means to arrive at it readily, and have become fully There is much to be said upon the advantage to be gained by impressed with its importance; they know the reasons why in
nowledge of geometrical drawing, a branch which depends such and such directions lines must be drawn; the mind and for its socuracy upon the use of compasses, scales, and rulers. the eye have acquired a keener perception of the principles of We have already explained a method of drawing curves by proportion; a feeling for arrangement has grown from the use hand, that is, by previously placing points in the course of the of instruments in geometrical exercises, and then in the end the intended curve, and then drawing the line through these points. hand readily takes up the practice. There are innumerable instances of curves which may be better. The curve called the Scotia (Fig. 58).-Let ab and c d be the drawn without the aid of instruments than with them. Leaves two lines between which the curve is to be formed. Draw bd and flowers, for instance, afford an inexhaustible supply of perpendicular to cd, and divide it into three equal parts; curved lines, to copy which we usually depend entirely on the through e draw the line g f parallel to a b; from e, with the egepas hand; while there are curves which frequently come ! radius eb, draw the arc b g, and at the same time mark the point f; from s, with the radius fg, draw the arc gi; bgi of a little help from geometry; we advise him also to draw all will be the curve required.
these lines of arrangement with a light hand, that they may be The curve called the Cyma Recta (Fig. 59).—Let the curve be more easily effaced when done with. formed between the lines a b and cd; draw the line b d, and To draw the pear (Fig. 63), we will first draw a line to divide it into five equal parts; mark the second division from b, represent the length or axis, and from this line “ offsets” on viz., e; upon b e describe the equilateral triangle b e f, and upon each side as shown by dotted lines. The pupil may please hime d describe the equilateral triangle deg; from f, with the self as to the number of these “offsets ” and their whereabouts; radias f b, draw the arc be, and from g, with the radius g e, he will not be long before he finds that such lines are best draw the arc e d; bed will be the curve required.
arranged opposite, and to meet, angles, and the greatest disThe pupil can draw an equilateral triangle upon a given line tance of curvature from the axis. He will then proceed to by the following method. Let a b (Fig. 60) be the line upon draw the outline through the extremities of these offsets, which the triangle is to be described; from a and b as centres, especially observing the kind of line requisite between each with the radius ba, describe two arcs intersecting each other point: in some parts the outline is more outwardly curved than in the point c; join c a and cb; the triangle a b c is an equi. in others, in some it is nearly straight, in others the curve lateral triangle. (See Lessons in Geometry, VII., page 209.) is inward. If the pupil will exercise his observation in this
The curve called the Ogee (Fig. 61).—Let it be drawn between way when looking at solids and natural objects, which he can the lines ab and c d; draw d e perpendicular to c d, and divide do at all times, whether he has a pencil in his hand or not, it into four equal parts; through the first from e-namely, h, even when out for a walk, he will be not a little surprised, draw the line h ig parallel to a b, make h i equal to he; draw should he make this his general practice, to find how rapidly he the line k i l parallel to e d, and from i, with the radius i k, will gain confidence and power, and be able to produce truthful
draw the semicircle k gl; join 1 d, and upon it draw the equi. and useful drawings. We will give him another example lateral triangle Im d; from m as centre, with the distance (Fig. 64), for which he must arrange the scaffolding himself, md or m l as radius, draw the arc d n l; the line d nlgk with one exception, because it includes a principle which we will be the curve required. By recommending the practice of will merely allude to now, as we shall have better and more geometrical drawing, we only wish to direct the pupil where to frequent opportunities by-and-by to enlarge upon it. The find further assistance in free-hand drawing; we will now show, exceptional assistance we offer in this case, is that of the dotted by a few examples, how these principles may be applied. An line which runs through the centre of the handle of the trowel, oval or egg-shaped figure (Fig. 62) would be very difficult to and passes in a direct course to the point of the blade. We draw, if the boundary line only were to be attempted without may here observe that an implement of this kind, to be really some assistance from geometry; there would be a great deal of useful, ought to be so constructed ; and if we look at it with an rubbing out and alteration before it was finished. Let the pupil artistic eye, the composition of lines which make up this very try the figure in the following manner : first by the help of com- simple subject, must strike any one as being more symmetrical passes, then by hand only. Draw the straight line a b, and than if the handle and the blade had been united at any other divide it into two equal parts in the point d. Through d draw angle. This remark upon so insignificant an object as a garden cde at right angles to a b, and make d c equal to a d or d b. trowel may appear trivial, but it is the principle we contend for, Construct upon a b the equilateral triangle a e b, and take the and which is, in reality, of the greatest importance. It is true point g at one-third of the distance from e to b, and determine we might have selected a more noble object, but it would not the point f in the same way. Then from the points f, g, draw have better illustrated our meaning, or have made it more the lines fi, gh, perpendicular to a e and eb respectively, and evident, and at the same time have provided the pupil with an make each of them equal to one-half of e f or e g. After this example for his practice more suited to the experience he has at arrangement has been made, draw the semicircle a c b and the present attained as a draughtsman. Nature teaches us this arcs be and a e through h and i. It will be necessary to repeat lesson, and it is evident everywhere that harmony of line 2 it a few times, when the pupil will begin to see the advantage proportion always accompany the greatest utility and stren
LESSONS IN ARITHMETIC.—XV. decimal places as are equal in number to this excess, prefixing
ciphers if necessary. DECIMALS (continued).
If the number of decimal places in the dividend and divisor 10. Division of Decimals.
be equal, the division will be the same as in whole numbers, CASE 1.—Divide 120-3033 by 3.27.
If the number of decimal places in the dividend be less than
the number in the divisor, annex as many ciphers to the dividend 120-3033 • 3.27 = 4203033 = 323 = 12930123 X TUBO = 300.
as will make the number equal to the number in the divisor, 3679 is the quotient arising from dividing the dividend by the and then proceed as in whole numbers. divisor as if they were whole numbers, and the denominator 100 12. We subjoin other examples of division of decimals. shows that there must be two decimal places in the quotient. EXAMPLE.-Divide 1 by 10:473, carrying the quotient to 5 These two decimal places arise, as will be seen by the fraction places of decimals.
1000, from the fact of there being two decimal places more in We are at liberty to write 1 thus-1.00000, putting as many the dividend than in the divisor.
ciphers after the decimal point as may be required. Since there CASE 2.-If the number of decimal places in the divisor and are to be 5 decimal places in the quotient, and since there are dividend were the same, the result would be exactly the same 3 in the divisor, we must add 8 ciphers. as if the divisor and dividend were whole numbers. Thus,
10.473) 1.00000000 ( 9548 1203.033 = 327 = 1908033 = 334 = 1.293033 x 1808 = 3679.
94257 CASE 3.—Suppose that there are more decimal places in the
57430 divisor than in the dividend.
52365 Take, for example, 120303:3 = .327.
50650 120303-3 = 327 = 1208083 = = 1293932 x 2900 = 367900.
41892 The true* quotient in this example is an integer, but it will not be so in all cases.
87580 It will be better in practice, before commencing the operation,
83784 to annex ciphers to the dividend sufficient to make the number
3796 of decimal places equal to the number in the divisor, in which case the result will be exactly the same as if the division had Hence the required answer is .09548, prefixing a cipher in been in whole numbers.
order to get 5 decimal places in the quotient. ADDITIONAL EXAMPLE OF CASE 2.- Divide 411.95 by 1.25. 13. EXAMPLE.—Divide 8 by .00002.
Annexing 4 ciphers to .8, since there are 5 decimal places in 1.25 ) 411.95,00 ( 329-56
the divisor, we have
*00002) .80000 ( 40000
the division by the rule being, in fact, the same as that of 1195
80000 by 2. 1125
14. It will be observed that we are not required in some cases to find more than a certain number of figures of the quotient when it is a decimal. Sometimes, by continuing the division far enough, we shall find that there is no remainder
i.e., that the quotient can exactly be found in the form of a 750
decimal. But if by continually dividing we cannot arrive at & stage where there is no remainder, then we can only get what is
termed an approximation to the result. The more figures of the Dividing as in whole numbers, we get a quotient 329, and a
quotient we take, the nearer we shall be to the value of the remainder 70. Now annex ciphers to the dividend, which will
true quotient. not alter its value, and continue the division. We now find
Thus, in the division above performed in Art. 12, if we that the true quotient is 329-56.
stopped at four decimal places in the quotient, the result would ADDITIONAL EXAMPLE OF CASE 3.—To divide 356.7 by 2:31.
be .0954. Carrying on the operation one step further, we see Annering a cipher to 356-7 before commencing the operation,
that 8 is the next figure of the quotient, and therefore—this 8 we have
meaning —we are nearer to the true quotient by www. 2-31 ) 356-70 ( 154
Where we are required to find a quotient to a given number of 231
places, it is customary to carry on the division to one place
more than is actually required, in order to see whether the next 1257
figure is greater or less than 5. If it is greater than 5, then we 1155
shall be nearer to the true result if we increase the last figure
of the required number of places by unity. 1020
Thus, in the case above given, finding that the fifth decimal 924
place is 8, the quotient to four decimal places will be more accurately written .0955 than .0954, because .0955-or, what is
the same thing, .09550—is nearer to .09548 than ·09540 is. The part of the true quotient already obtained is an integer, Now .09550 is no more than .09548 ; whereas .09540 is toda the Tio being in fact the same as that of 1979. If more less than .09548. aber be annexed to the dividend, we shall get decimal places. The same method is applied whenever a limited number of is the gootient, and the more we obtain the nearer to the true decimals is employed. We shall return to this subject hereafter. quoties sal we arrive.
EXERCISE 33. 11. These examples will sufficiently illustrate and explain the
1. Find the quotients of the following examples in division of Bule for the Division of Decimals.
decimals : DES the divisor and dividend were whole numbers. 1. 5.64 + 4. 10. 4.32067 < .001. I 19. 684234-6 + 2682. the saber o decimal places in the dividend exceed the
2. 5.64 4.
11. 167342 + 002. 20. 7.231068 – 12. er is the divisor, cat off from
12. 000045 – 9.
3. 5.64 + the quotient as many
21. 26-3845 • 125. 4, 46.84 – 7.9. 13. 4• *00001.
22. 6 • 0000001.
5. 1.658 + 25. 14, 018769 = '0000137. 23. 8 -0000002. 1 ss the expresion truss quotient to indicate the total 6. 4.00334 • 6:31. 15. 67284 – 85.
24. 6541234567 — 21. by the dinion of one number by another, thus distin.
7. 00033 + 011. 16. 73-8243 + .061. 25. 7461-30765 + 112. the quotient defined in Lesson V., Art. 1 (page 69),
236-011 175. 17. 300 002 - 12.1. 26. 325-67543 - 20-02. integral part arising from a division.
9. 60.0001 + 1.01. 18. .00006 * *003. 27. 2186-054 1993.