Oldalképek
PDF
ePub

below the line b b, and finished with a loop like the elementary stroke in Copy-slip No. 52. This stroke and the various letters into whose composition it enters, one of which is the letter j, in Copy-slip No. 53, will form the subject of our next lesson.

LESSONS IN LATIN.-VIII.

THE THIRD DECLENSION.

WE pass on to the third declension. In the third declension we find, in the nominative case, so great a variety of terminations, that we must endeavour to arrange the nouns in certain classes. The genitive singular, however, is the characteristic case, and it ends in is.

Before classifying these nouns, I must give you some explanations. Parisyllabic is a word I have to use. It consists of three words, which I will mark thus

1 2 3
pari syl lab(ic);

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small]

The genders of the nouns of the third declension may be stated thus, though the rules are not without exceptions:First, nouns ending in o, or, os, er, and imparisyllabics in es, are masculine; second, nouns ending in as, is, aus, us (gen. utis or udis) and a, and those which end in s blended with the preceding consonant, as well as parisyllabics in es, are feminine; third, nouns ending in a, e, c, l, en, ar, ur, ut, and us (gen. ōris, ĕris, ŭris), are neuter. By practice you will in time become familiar with these somewhat complex facts.

I proceed to set down specimens in classes.

CLASS I.

NOUNS WITH CONSONANTAL STEMS; IMPARISYLLABIC. 1.-Without the termination s.

(i.) The stem and the nominative are the same; stems end in r and l.

Cases.

of these the two latter are of Greek origin. The former is
Latin. As the word is thus made up of terms from two lan-
guages, it is a sort of hybrid. No. 2 signifies with; No. 3
signifies to take; the ic is merely the termination. If you put
2 and 3 together you have syllab, which with the termination
ble makes syllable. A syllable, then, is so much of sound as
may be taken or uttered at once. No. 1 means equal (pari
found in the English par and pair); parisyllabic, then, signifies
that which is equal in its syllables; and nouns are called
parisyllabic which have the same number of syllables in all the
cases of the singular number. I say of the singular number,
because the plural of all nouns is not parisyllabic, inasmuch as
the genitive plural, as in the cases of arum and orum, has a
syllable more than the other cases. Now nouns which have in
the genitive singular a syllable more than they have in the
nominative singular are called imparisyllabic. In this word, as
here given, you find an additional syllable, namely, im from in-
then becoming m, before the p-which signifies not. Impari- Cases.
syllabic, then, is not-parisyllabic; and the words denote those N.
nouns which in the genitive singular have not the same number G.
of syllables as they have in the nominative. Piscis, a fish, is
parisyllabic; for in the genitive it is piscis, having two syllables Ac.
as in the nominative. But cantor, a singer, is imparisyllabic,
for in the genitive it is cantoris, having three syllables, whereas

the nominative has but two. Here then we have one distinc-
tion-namely, nouns of the third declension are either pari-
syllabic or imparisyllabic.

N.

G.

[merged small][merged small][ocr errors]

D.

Ac.

dolorem, grief.

V.

Ab.

dolor, O grief!
dolore, by grief.

dolores, griefs.
dolorum, of griefs.
doloribus, to griefs.
dolores, griefs.
dolores, O griefs!

D.

V.

Ab.

Ab.

MASCULINES.

Singular.

anser, a goose. ansĕris, of a goose. anseri, to a goose. anserem, a goose. anser, O goose! ansere, by a goose. Plural. anseres, geese. anserum, of geese. anseribus, to geese. anseres, geese. anseres, O geese!

doloribus, by griefs, anseribus, by geese.

Cases.

N.

G.

D.

Ac.

V.

guttur, a threat. gutturis, of a throat, gutturi, to a threat. guttur, a throat. guttur, O throat! gutture, by a throat,

guttura, throats.
gutturum, of throats,
gutturibus, to throats,
guttura, throats.
guttura, O throats!

Cases.

N.

G.

D.

Ac.

V.

NEUTERS.

Singular.
calcar, a spur.
calcaris, of a spur.
calcari, to a spur.
calcar, a spur.
calcar, O spur!
calcari, by a spur.
Plural.

calcaria, spurs.
calcarium, of spurs.
calcaribus, to spurs.
calcaria, spurs.
calcaria, O spurs !

vomer, a ploughshare. voměris, of a ploughshare. vomeri, to a ploughshare. vomerem, a ploughshare. vomer, O ploughshare! vomere, by a ploughshare,

vomeres, ploughshares. vomerum, of ploughshares. vomeribus, to ploughshares. vomeres, ploughshares. vomeres, O ploughshares! vomeribus, by ploughshares

animal, an animal. animalis, of an animal, animali, to an animal, animal, an animal. animal, O animal! animali, by an animal,

animalia, animals, animalium, of animals. animalibus, to animals. animalia, animals. animalia, O animals! animalibus, by animals.

Now, inquiry has shown that parisyllabic nouns have a vowel stem, and imparisyllabic nouns a consonant stem; that is, that the stem of the former ends in a vowel, and the stem of the latter ends in a consonant. Of the stem of a noun and a verb I have already said something. It is better to repeat than not to be understood. Take nubes, a cloud, and form the genitive; the genitive is nubis. You get the stem by cutting off the sign of the genitive, which in this case is s (as in the English cloud, cloud's). You thus obtain nubi. Nubi has two syllables, the same as the nominative nubes. It is therefore parisyllabic, and ends in a vowel. Take also dolor, grief; genitive, doloris. Cut Ab. off is, the sign of the genitive, and you obtain dolor. Dolor ends, you see, in a consonant, and is a consonantal stem. The word is also imparisyllabic, because it increases in the genitive singular. Imparisyllabic nouns, then, have consonantal stems. In this case the stem and the nominative are the same, both being dolor. But in nomen, a name, genitive nominis, stem nomin, the nominative and the stem are unlike. Of consonantal stems, then, there are two classes: first, those of which the stem is identical with the nominative; second, those in which it is different. The consonants in which the stem terminates Agger, aggeris, m., a Fulgur, fulguris, n., Passer, passèris, m., a

[blocks in formation]

gutturibus, by throats. calcaribus, by spurs. Here observe, that as in the neuter nouns of the second declension, the neuter nouns of the third declension have in both the singular and the plural three cases alike, namely, the nominative, the accusative, and the vocative. In animal, the nominative plural is ia, instead of a. This is owing to its being originally from a vowel stem-as, nominative, animal; genitive, animalis; stem, animali.

[blocks in formation]

like. Error,

error.

mother. erroris, m., Mihi, to me. Nobis, to us.

Frater, fratris, m.,

a brother.

Odor, odoris, m., odour,

smell.

[blocks in formation]

OBS.-Est mihi, I have, used with the noun as nom, to est; thus, guttur est mihi, I have a throat; so in the plural, guttura 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 est illi (a throat is to him), he has a throat; guttura sunt vobis (throats are to you), you have throats; guttura sunt illis (throats are to them), they have throats.

EXERCISE 25.-LATIN-ENGLISH.

1. Magnus dolor est mihi. 2. Nonne tibi est magnus dolor? 3. Sunt magni dolores matribus. 4. Color pulvinaris pulcher est. 5. Estne pulcher pulvinaris color? 6. Funestus error est illi. 7. Cur funesti errores sunt patri? 8. Frater est mihi. 9. Fratribus sunt magni dolores. 10. Fulgura terrent animalia. 11. Nonne matres terrent fulgura? 12. Fulgura terrent passeres.

EXERCISE 26.-ENGLISH-LATIN.

[blocks in formation]

Cases.

N. leo, a lion.

G. leonis, of a lion.
D. leoni, to a lion.

Ac. leonem, a lion.
V. leo, O lion!
Ab. leone, by a lion.
Cases.

MASCULINES AND FEMININES.

N. leones, lions.
G. leonum, of lions.
D. leonibus, to lions.
Ac. leones, lions.
V. leones, 0 lions!
Ab. leonibus, by lions.

Canes.

N. corpus, a body.
G. corporis, of a body.
D. corpori, to a body.
Ac. corpus, a body.
V. corpus, O body!
Ab. corpore, by a body.
Cases.

N. corpora, bodies.

G. corporum, of bodies.
D. corporibus, to bodies.
Ac. corpora, bodies.
V. corpora, O bodies!

Singular. homo, a man. hominis, of a man. homini, to a man. hominem, a man. homo, O man! homine, by a man. Plural.

homines, men. hominum, of men." hominibus, to men. homines, men. homines, O men! hominibus, by men. NEUTERS. Singular. nomen, a name. nominis, of a name. nomini, to a name. nomen, a name. nomen, O name! nomine, by a name. Plural. nomina, names. nominum, of names. nominibus, to names, nomine, names.

pater, a father. patris, of a father. patri, to a father. patrem, a father. pater, O father! patre, by a father.

patres, fathers. patrum, of fathers. patribus, to fathers. patres, fathers. patres, O fathers! patribus, by fathers.

genus, a race. generis, of a race. generi, to a race. genus, a race. genus, O race! genere, by a race.

genera, races.

generum, of races. generibus, to races. genera, races. nomina, O names! genera, O races! Ab. corporibus, by bodies. nominibus, by names. generibus, by races. In two of the words declined above, corpus, corporis, corpor, and genus, generis, gener, the stems-namely, corpor and gener -seem to end in r. The r, however, is only the representative of s, for between two vowels, as in corporis, the s by the laws of pronunciation passes into r. Thus, instead of corpus, corpusis, we have corporis, the s being changed into r and the u into o. Similar changes take place in tellus (tellūsis) tellūris, the earth; pulvis, pulveris, dust; mas, maris, a male; æs, æris, brass; flos, floris, a flower.

charcoal. Cardo, cardinis, m., a hinge. Carmen, carminis, n., a poem.

VOCABULARY.

Pectus, pectoris, n., a

breast.

Pignus, pignoris, n., a pledge.

Pulvis, pulveris, m., dust.

Carbo, carbonis, m., | Littus, litteris, a shore. Lumen, luminis, n., light. Occasio, occasionis, f., an opportunity. Opus, opěris, n., work. Cinis, cineris, .m., Ordo, ordinis, m., order, Regio, regionis, f., a ashes. series. region or district. Decus, decoris, n., Pavo, pavonis, m., a Vulnus, vulueris, n., becomingness. peacock.

a wound.

EXERCISE 27.-LATIN-ENGLISH.

1. Carbonem timeo. 2. Pavones ferit puer. 3. Pulchræ sunt regiones. 4. Occasio est tibi. 5. Movemus cineres. 6. Cardo movetur. 7. Ordinis decus delectat matres. 8. Magnus est pulvis cineris. 9. In littore sunt pavones. 10. Carmina non sunt nobis. 11. Vulnus est in pectore. 12. Regionis magnum est lumen. 13. Illi est nomen magnum. 14. Pignora non laudantur.

EXERCISE 28.-ENGLISH-LATIN.

1. Dost thon fear charcoal? 2. Why does the mother strike the boy? 8. They have no becomingness. 4. Thou hast a wound. 5.

Thy fathers have wounds. 6. Wounds frighten mothers. 7. Poems flourish in the region. 8. Thou hast a great name. 9. I have not a pledge. 10. They have an opportunity. 11. The man's opportunity is great.

KEY TO EXERCISES IN LESSONS IN LATIN.-VII.
EXERCISE 21.-LATIN-ENGLISH.

1. Good men love good boys. 2. Good boys are loved by good men. 3. A good boy loves school. 4. The good masters of good boys are loved. 5. Hast thou a good master? 6. The war is deadly. 7. I have a good female friend. 8. The boys are in school. 9. Are not the boys in school? 10. Many foreigners sail into Britain. 11. The boar of my friend is great. 12. There is play on the river's bank. 13. Scholars love (like) letters. 14. There are frogs on the banks. 15. The goat is great. 16. There are deadly wars in the island.

EXERCISE 22.-ENGLISH-LATIN.

8. In 10. O viri,

1. Bonos discipulos amo. 2. Boni discipuli a bonis viris amantur. 3. Amasne amicum ? 4. Aper est mihi. 5. Tibi est caper. 6. Capri sunt in ripâ. 7. Est in insulâ magnum et funestum bellum. Britannia sunt agri multi. 9. Funesti sæpe sunt apri? amatisne pueros? 11. Amici mei peregrinos non amant. 12. Ludum amant pueri. 13. Amantne pueri ludum ? 14. Estne tibi amica? 15. Magnus aper non est mihi. 16. Amica epistola est in horto.

EXERCISE 23.-LATIN-ENGLISH.

1. The horse neighs. 2. The horse's mane is beautiful. 3. The flies are troublesome. 4. Are the flies troublesome? 5. Good scholars are not troublesome. 6. Long wars are troublesome. 7. Horses rum quickly. 8. A man guides the horse. 9. A horse is guided by a mau. 10. I am delighted by a beautiful horse. 11. The fields are fruitful.

12. The herbs of the fields are various. 13. The husbandman commits to the fields grains of corn. 14. The husbandman tills the fields. 15. How beautifully the fields flourish. 16. Various herbs flourish in the fields.

EXERCISE 24.-ENGLISH-LATIN.

1. Fecundus est ager. 2. Suntne agri fecundi? 3. Bella fecunda non sunt. 4. Agri coluntur. 5. Deos colis. 6. Dii coluntur a Tullio. 7. Equus et equa a viro reguntur. 8. Celeriter currunt apri. 9. Curruntne capri celeriter ? 10. In pulchro horto sunt muscæ. 11. Equum agro committis. 12. Boni discipuli coluntur. 13. O mi fili! diis et deabus committuntur templa. 14. O Antoni! dii deæque in templis coluntur. 15. O bone Deus! in fecundis agris coleris. 16. Boni viri a filiis et filiabus coluntur.

LESSONS IN DRAWING.-VIII.

HAVING gone thus far in our instructions for drawing an outline, we think it necessary to detain the pupil a little longer upon this early and most important part of our subject, for reasons that will be apparent as we proceed. So essential is good drawing, that without a correct outline the most laboured performance in other respects will be a failure; it may be very neat in its execution, carefully shaded, or perhaps cleverly coloured; but if it fail in the outline by not giving a truthful representation of the form of the object, it is then for all practical purposes useless. We know what a great temptation it is to the young to begin to paint, but they do not consider that to be able to paint well they must be able to draw well. Painting, in its practice—that is, the execution is nothing more than placing colours, as we have said of lines, in their right places, and the power of handling the brush successfully depends upon the pupil's ability for handling the pencil. Of course we make no allusion to the arrangement of colours themselves, their harmony and tones; we mean simply the power of using the brush where it is necessary to perfect the form of the object being painted, without having to lay down the brush to resume the pencil. We wish also to warn the pupil against that slovenly, dangerous, and unsatisfactory manner of drawing which is generally termed sketching, that is, producing a hurried, careless outline, its correctness being the last thing considered. Sketching, with an imperfect power of drawing, in the majority of cases amounts to nothing more than scribbling; there may be thousands of individuals who can sketch, but amongst these there are comparatively few who can draw. The dogs, horses, and ships with which the schoolboy adorns the pages of his dictionary, or the margins of his exercises, may, on the whole, bear a strong resemblance to the general character of the class of animals or objects intended; but this is not drawing: it is quite another thing to give the individuality of these objects: in this is the test of 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 learner to begin the art by sketching is altogether a mistake. We once heard an eminent landscape painter say that "sketching is the ruin of hundreds of young artists; it is beginning at the wrong end; let them draw well first and secure the power, then afterwards they may sketch." Sketches are clever, and valuable only when they are done by men who can really draw well; the unfortunate result of the habit of sketching by an inexperienced hand may be compared to that of the very objectionable system which compels schoolboys to write out pages of Latin or English for punishment. There are many who acknowledge in after years that their handwriting was

c

under our notice, to draw which we shall be materially assisted by principles borrowed from geometry. But though we cannot employ compasses to draw the forms of flowers and leaves, yet by the practice of geometry we easily associate lines, angles, and centres with curves, although they are not visible upon the object. Instruments are usually depended upon for drawing architectural curves, mouldings, and the like, because they must be constructed according to received proportions. We propose now to place before our readers some examples of architectural curves, with the rules for constructing them; our reason for doing so being simply to show the pupil a way of making his eye familiarwhit the construction of curves on geometrical prinb

α

[merged small][ocr errors]
[ocr errors][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small]

roded by these "tasks" or "impositions," and who were Lever able afterwards, with all their efforts, to write well. Let the papal therefore give up all idea of sketching, and seek 6 fem well, if he at all hopes to make the art useful for practical purposes, or to secure in its practice a pleasurable rere in bisure hours.

There is much to be said upon the advantage to be gained by a knowledge of geometrical drawing, a branch which depends for its accuracy upon the use of compasses, scales, and rulers. We have already explained a method of drawing curves by hand, that is, by previously placing points in the course of the intended curve, and then drawing the line through these points. There are innumerable instances of curves which may be better drawn without the aid of instruments than with them. Leaves and flowers, for instance, afford an inexhaustible supply of car to copy which we usually depend entirely on the d; while there are curves which frequently come

ciples. From long experience we have found it to be the case that they always make the best and quickest draughtsmen, and do their work with the least labour, who have dipped deeply into geometrical drawing and lineal perspective. In their practice they have acquired a habit of precision, and have learnt the means to arrive at it readily, and have become fully impressed with its importance; they know the reasons why in such and such directions lines must be drawn; the mind and the eye have acquired a keener perception of the principles of proportion; a feeling for arrangement has grown from the use of instruments in geometrical exercises, and then in the end the hand readily takes up the practice.

The curve called the Scotia (Fig. 58).-Let a b and c d be the two lines between which the curve is to be formed. Draw b d perpendicular to cd, and divide it into three equal parts; through e draw the line gf parallel to a b; from e, with the radius e b, draw the arc b g, and at the same time mark the

[ocr errors][ocr errors]

point f; from f, 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 more easily effaced when done with.

The curve called the Cyma Recta (Fig. 59).-Let the curve be formed between the lines a b and c d; draw the line b d, and divide it into five equal parts; mark the second division from b, viz., e; upon be describe the equilateral triangle bef, and upon ed describe the equilateral triangle de g; from f, with the radins fb, draw the arc b e, and from g, with the radius ge, draw the arc e d; bed will be the curve required.

The pupil can draw an equilateral triangle upon a given line by the following method. Let a (Fig. 60) be the line upon which the triangle is to be described; from a and b as centres, with the radius ba, describe two arcs intersecting each other in the point c; join e a and c b; the triangle a b c is an equilateral triangle. (See Lessons in Geometry, VII., page 209.) The curve called the Ogee (Fig. 61).-Let it be drawn between the lines ab and c d; draw d e perpendicular to c d, and divide it into four equal parts; through the first from e-namely, hdraw the line hig parallel to a b, make h i equal to he; draw the line ki parallel to e d, and from i, with the radius i k,

To draw the pear (Fig. 63), we will first draw a line to represent the length or axis, and from this line "offsets" on each side as shown by dotted lines. The pupil may please himself as to the number of these "offsets" and their whereabouts; he will not be long before he finds that such lines are best arranged opposite, and to meet, angles, and the greatest distance of curvature from the axis. He will then proceed to draw the outline through the extremities of these offsets, especially observing the kind of line requisite between each point: in some parts the outline is more outwardly curved than in others, in some it is nearly straight, in others the curve is inward. If the pupil will exercise his observation in this way when looking at solids and natural objects, which he can do at all times, whether he has a pencil in his hand or not, even when out for a walk, he will be not a little surprised, should he make this his general practice, to find how rapidly he will gain confidence and power, and be able to produce truthful

[graphic][subsumed][subsumed][subsumed]

draw the semicircle k gl; join l d, and upon it draw the equilateral triangle 1 m d; from m as centre, with the distance md or ml as radius, draw the arc d n l; the line d n 1 gk will be the curve required. By recommending the practice of geometrical drawing, we only wish to direct the pupil where to find further assistance in free-hand drawing; we will now show, by a few examples, how these principles may be applied. An oval or egg-shaped figure (Fig. 62) would be very difficult to draw, if the boundary line only were to be attempted without some assistance from geometry; there would be a great deal of rubbing out and alteration before it was finished. Let the pupil try the figure in the following manner: first by the help of compasses, then by hand only. Draw the straight line a b, and divide it into two equal parts in the point d. Through d draw ede at right angles to a b, and make d c equal to a d or d b. Construct upon a b the equilateral triangle a e b, and take the point g at one-third of the distance rom e to b, and determine the point ƒ in the same way. Then from the points f, g, draw the lines fi, g h, perpendicular to a e and eb respectively, and make each of them equal to one-half of e f or eg. After this arrangement has been made, draw the semicircle a c b and the arcs be and a e through h and i. It will be necessary to repeat it a few times, when the pupil will begin to see the advantage

and useful drawings. We will give him another example (Fig. 64), for which he must arrange the scaffolding himself, with one exception, because it includes a principle which we will merely allude to now, as we shall have better and more frequent opportunities by-and-by to enlarge upon it. The exceptional assistance we offer in this case, is that of the dotted line which runs through the centre of the handle of the trowel, and passes in a direct course to the point of the blade. We may here observe that an implement of this kind, to be really useful, ought to be so constructed; and if we look at it with an artistic eye, the composition of lines which make up this very simple subject, must strike any one as being more symmetrical than if the handle and the blade had been united at any other angle. This remark upon so insignificant an object as a garden trowel may appear trivial, but it is the principle we contend for, and which is, in reality, of the greatest importance. It is true we might have selected a more noble object, but it would not have better illustrated our meaning, or have made it more evident, and at the same time have provided the pupil with an example for his practice more suited to the experience he has at present attained as a draughtsman. Nature teaches us thi lesson, and it is evident everywhere that harmony of line proportion always accompany the greatest utility and stre

[blocks in formation]

3679 is the quotient arising from dividing the dividend by the divisor as if they were whole numbers, and the denominator 100 shows that there must be two decimal places in the quotient. These two decimal places arise, as will be seen by the fraction from the fact of there being two decimal places more in the dividend than in the divisor.

CASE 2.-If the number of decimal places in the divisor and dividend were the same, the result would be exactly the same as if the divisor and dividend were whole numbers. Thus, 1203 033327 = 1203033321 = 1203033 X 1008 = 3679.

1000

327

CASE 3. Suppose that there are more decimal places in the divisor than in the dividend.

Take, for example, 120303-3327.

1203033 ÷ 327 = 1203033 ÷ 127 = 1203033 × 1000 = 367900. The true quotient in this example is an integer, but it will not be so in all cases.

It will be better in practice, before commencing the operation, to annex ciphers to the dividend sufficient to make the number 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 been in whole numbers.

ADDITIONAL EXAMPLE OF CASE 2.-Divide 411.95 by 1.25.

[blocks in formation]

231

If more

The part of the true quotient already obtained is an integer, the division being in fact the same as that of 19. sphere be annexed to the dividend, we shall get decimal places the quotient, and the more we obtain the nearer to the true yatient shail we arrive.

11. These examples will sufficiently illustrate and explain the flowing

Sie for the Division of Decimals.

Dhexe as if the divisor and dividend were whole numbers. If the number of decimal places in the dividend exceed the amber the divisor, cut off from the quotient as many

as the expression true quotient to indicate the total 7 the division of one number by another, thus distinthe quotient defined in Lesson V., Art. 1 (page 69), ntegral part arising from a division.

decimal places as are equal in number to this excess, prefixing ciphers if necessary.

If the number of decimal places in the dividend and divisor be equal, the division will be the same as in whole numbers.

If the number of decimal places in the dividend be less than the number in the divisor, annex as many ciphers to the dividend as will make the number equal to the number in the divisor, and then proceed as in whole numbers.

12. We subjoin other examples of division of decimals. EXAMPLE.-Divide 1 by 10-473, carrying the quotient to 5 places of decimals.

We are at liberty to write 1 thus-1-00000, putting as many ciphers after the decimal point as may be required. Since there are to be 5 decimal places in the quotient, and since there are 3 in the divisor, we must add 8 ciphers.

10.473) 1 00000000 (9548 94257

57430

52365

50650

41892

87580

83784

3796

Hence the required answer is 09548, prefixing a cipher in order to get 5 decimal places in the quotient. 13. EXAMPLE.-Divide 8 by '00002. Annexing 4 ciphers to 8, since there are 5 decimal places in the divisor, we have

00002) 80000 (40000 80000

the division by the rule being, in fact, the same as that of 80000 by 2.

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 remainderi.e., that the quotient can exactly be found in the form of a decimal. But if by continually dividing we cannot arrive at a stage where there is no remainder, then we can only get what is termed an approximation to the result. The more figures of the quotient we take, the nearer we shall be to the value of the true quotient.

Thus, in the division above performed in Art. 12, if we stopped at four decimal places in the quotient, the result would that 8 is the next figure of the quotient, and therefore this 8 be 0954. Carrying on the operation one step further, we see meaning we are nearer to the true quotient by UD

Where we are required to find a quotient to a given number of places, it is customary to carry on the division to one place more than is actually required, in order to see whether the next figure is greater or less than 5. If it is greater than 5, then we shall be nearer to the true result if we increase the last figure of the required number of places by unity.

Thus, in the case above given, finding that the fifth decimal 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. Now 09550 is more than '09548; whereas 09540 is

[blocks in formation]
« ElőzőTovább »