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aujourd'hui ou demain? 14. On nous dit qu'il doit avoir lieu cette après-midi. 15. Il aura lieu à cinq heures et demie. 16. Avez-vous envie de venir au lieu de votre frère ? 17. Mon frère doit venir au lieu de notre cousin. 18. Avez-vous l'intention de lui dire ce qu'il doit faire? 19. Il sait ce qu'il doit faire. 20. Savez-vous ce qu'on dit de nouveau ? 21. On ne dit rien de 22. Trouve-t-on beaucoup d'or en Californie? 23. On y en trouve beaucoup. 24. Y trouve-t-on aussi des diamants? 25. On n'y en trouve point, on n'y trouve que de l'or.

nouveau.

EXERCISE 64.

1. What do people say of me? 2. People say that you are not very attentive to your lessons. 3. Is it said that much gold is found in Africa? 4. It is said that much gold is found in California. 5. Do they bring you books every day? 6. Books

are brought to me [R. 2] every day, but I have no time to read them. 7. What should one do (doit on faire) when one is sick? 8. One should send for a physician. 9. Do you send for my brother? 10. I am to send for him this morning. 11. Do you hear from your son every day? 12. I hear from him every time that your brother comes. 13. Does the sale take place to-day? 14. It takes place this afternoon. 15. At what time does it take place? 16. It takes place at half after three. 17. I have a wish to go there, but my brother is sick. 18. What am I to do? 19. You are to write to your brother, who, it is said (dit on), is very sick. 20. Is he to leave for Africa ? 21. He is to leave for Algiers. 22. Do you come instead of your father? 23. I am to write instead of him. 24. Does the concert take place this morning? 25. It is to take place this afternoon. 26. Do you know at what hour? 27. At a quarter before five.

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LESSONS IN PENMANSHIP.-XX.

THE simplest method of writing the letter f, and that which is most generally used in writing large-hand copies, is shown in Copy-slip No. 73. In this form, which is repeated in Copy-slip No. 74, where f is given in conjunction with other letters, it is commenced with a fine hair-stroke a little above the line cc, which is carried upwards until it reaches the line kk, where it is turned towards the left and brought downwards across the fine up-stroke, the pressure on the pen being gradually increased until a thick down-stroke is formed, which terminates at the line gg. The letter is finished with a hair-stroke carried out from the back of the letter, about the line cc, to the left, and then brought to the right in a curve across the down-stroke. In small-hand writing, the lower part of the letter f is generally

made in the form of a loop, the pressure of the pen being relaxed, and the down-stroke narrowed gradually until it is turned at the bottom in a hair-stroke, which is carried upwards and across the down-stroke about the line cc, or centre of the letter, in a small loop. Sometimes the loop at the upper part of the letter is omitted, the down-stroke being commenced at the line ee (see Copy-slip No. 10, p. 60, for the height of this line above a a), and thickened very gradually until it reaches its thickest part about the line bb, when the pressure on the pen is immediately lessened to narrow the stroke into the fine line that forms the loop below the line bb. Examples of the methods of making the letter f that have just been described will be found in future copy-slips. In Copy-slip No. 75 the learner will find the elementary strokes that form the letter k.

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Placing the points as indicated in the rule, we observe that the cube of 4 is the greatest cube in the first period 78. Sub. tracting 43, or 64, from 78, we get a remainder 14, to the right of which we bring down the next period 314, to form a dividend. Multiplying the square of 4 by 3, we get for a divisor 48, which will go 2 times in 143 (the dividend without its two right-hand figures). We set down 2, therefore, to the right of 4 as the next figure in the root, and then proceed to form the three lines according to the rule.

1. 8 is the cube of 2. 2. 48 is 3 x 4 x 22.

3. 96 is the product of 2, the last obtained figure in the root; and 48, the divisor.

Placing these three lines under each other, but advancing each Buccessively one place towards the left, and adding, we get 10088, which we subtract from the dividend 14314, leaving a remainder 4226. To the right of this we bring down the next period 601, thus forming another dividend.

The next divisor 5292 is 3 × 422, and is contained 7 times in 42266. Putting down, then, 7 as the next figure in the root, we form three lines as before :

1. 343 is the cube of 7, the last figure in the root.

2 6174 is 3 x 42 x 79.

3. 37044 is 7 x 5202.

Adding these up when properly placed, we get 3766483, which we subtract from the previous dividend 4226601, leaving remainder 460118.

There are now no more periods left. Hence 427 is the numer whose cube is the nearest cube number to the given number, ad less than it. If there were no remainder, the root obtained ould be the exact cube root of the given number.

14. In such an example as that worked out above, we could ace a decimal point and as many periods of ciphers as we may ish after the original number, and thus, by continuing the rocess according to the rule, get as many decimal places as ay be required as an approximation to the cube root. In finding the cube root of a decimal, the periods must be mpleted by adding ciphers, if necessary. 15. When the cube root of a fraction is required, the cube st of the numerator and the cube root of the denominator will the numerator and denominator respectively of the fraction ich is the cube root of the original fraction. If the numeor and the denominator are not both perfect cubes when the etion is reduced to its lowest terms (vide 9, Obs.), the best in generally will be to reduce the fraction to a decimal, and en to find the cube root of that decimal. In the case of xed numbers, they must be reduced to improper fractions, in ler to see whether the resulting improper fraction has its merator and denominator both perfect cubes. Thus, 5 duced to an improper fraction gives 343, of which the cube tis, or 1. But if, when so reduced, the numerator and nominator are not perfect cubes, then it will be better to uce the fractional part of the mixed number to a decimal, placing the integral part before it, find the cube root by Jabove rule.

And so on to as many more decimal places as we may desire. Obs. Exactly as in the case of the square root, when one more than half the number of figures required of the root have been found by the rule, the rest may be found by simply dividing, as in ordinary division, by the last divisor.

in the root.

16. Obs. It will be observed that although 27, the first divisor, is really contained 6 times in 176, we only put down 5 The reason is that, on examination, we find that 6 would be too large, for it would make the sum of the three lines which we add up greater than the dividend 17600. This explains the note at page 318. We must, therefore, always be careful to observe whether the figure put down in that root will or will not make the sum of the three lines too large. The dividing the dividend without its two last figures by the divisor is not, therefore, an infallible guide to the next figure of the

root.

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LESSONS IN ARCHITECTURE.-I. ARCHITECTURE is the art of planning, constructing, and adorning public or private buildings according to their intended use. The word architecture is derived from the Greek apxw (ar'-ko), I command, and TEKTWV (teck'-tone), a workman. This etymology indicates the operatives engaged in the building on the one hand, and the leader or chief, the man of science and practical skill, putting in action all his resources in order to execute his plan on the other. Such a division as this was, no doubt, established from the beginning of the art. According, therefore, to the literal meaning of the etymology, mankind must have, at the origin of architecture, possessed a degree of civilisation sufficient for the organisation of different kinds of industrial operations, and acquired a degree of skill in the art, which enabled some men by their experience to be the leaders or directors of others. In this way, we may suppose that the art itself, or rather the symmetry, the harmony of proportions, and good taste in structures, at first began to be developed.

Before arriving at this point, mankind must have overleapt ages. One of the first wants of society was a covering or shelter from the inclemency of the weather, whether of heat or of cold. Simple was the art employed in constructions of this kind. Grottoes or caves hollowed square to make them more habitable, and cottages constructed of branches of trees and blocks of stone-such were the primitive constructions in wood and stone which formed the rudiments of architecture. From the simplicity of early structures men passed to the study of proportions ;

318

LESSONS IN ARITHMETIC.-XX.

SQUARE AND CUBE ROOT (continued).

9. THE square root of a fraction is obtained by taking the square root of the numerator for a numerator, and the square root of the denominator for a denominator. This follows at once from the consideration that the multiplication of fractions is effected by multiplying the numerators for a numerator, and the denominators for a denominator. When either the numerator or the denominator is not a complete square, in which case the fraction itself evidently has no exact square root, instead of finding an approximate root of both numerator and denominator in decimals, and then dividing one by the other, it will be better first to reduce the fraction to a decimal, and then to take the square root.

EXAMPLE.-To find the square root of 2.

Reducing to a decimal, we find it to be 285714 (see Lesson XVI., Art. 21).

Hence we should find by the previous method the square root of 28571428571428... to as many decimal places as we please, by continually taking in more and more figures of the recurring periods.

Similarly, in finding the square root of, we should proceed thus: 4, and then find the square root of 400000, etc., to as many places as we please.

=

Obs. It does not follow that because the numerator and denominator of a fraction are not complete squares, that the fraction has no square root; for the division of numerator and denominator by some common measure may reduce them to perfect squares. Thus, , when numerator and denominator are divided by 7, gives, the square root of which is . A fraction must be reduced to its lowest terms to determine whether it be a complete square or not.

10. Abbreviated Process of Extraction of Square Root. When the square root of a number is required to a considerable number of decimal places, we may shorten the process by the following

Rule for the Contraction of the Square Root Process.

Find by the ordinary method one more than half the number of figures required, and then, using the last obtained divisor as a divisor, continue the operation as in ordinary long division. EXAMPLE.-Find the square root of 2 to 12 figures.

2-0000, etc. (1-414213 | 56237

1

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or 21.
Therefore 441 = 32 x 72; of which the square root is 3 x 7,

Obs. Unless a number is made up of prime factors, each of
which is repeated an even number of times, it is not a perfect
Equare.
EXERCISE 39.

1. Find the square root of the following numbers :-
1. 529.

2. 5329.

3. 784.

4. 4761.

5. 7056.

6. 9801.

7. 27889. 8. 961.

9. 97 to 4 places of decimals. 10. 190 to 5 places.

11. 81796 to 4 places.

12. 1169 64.

13. 3-172181 to 4 places.

14. 10342656.

15. 28, 11, 198, POST.

16. to 4 places.

17. 17 to 4 places.

18. 964 5192360241.
19. 00000625.

abbreviated method:-
2. Find the square root of the following numbers by the

1. 365 to 11 figures in the root.
2. 2 to 12 figures.

3. 3 to 17 figures.

3. Extract the square root of 2116, 21316, and 7056, by splitting them into their prime factors.

12. Extraction of the Cube Root.

To extract the cube root of a given number is the same thing as resolving it into three equal factors.

with giving, without explanation of the reason of its truth, the As in the case of the square root, we must content ourselves Rule for the Extraction of the Cube Root of a given number. Mark off the given number into periods of three figures each, by placing a point over the figure in the unit's place, and then over every third figure to the left (and to the right also, if there be any decimals). Put down for the first figure of the root the figure whose cube is the greatest cube in the first period, and subtract its cube from the first period, bringing down the next period to the right of the remainder, and thus forming a number which we shall call a dividend. Multiply the square of the part of the root already obtained by 3 to form a divisor, and then, having determined how many times this divisor is contained in the dividend without its two right-hand figures, annex this quotient to the part of the root already obtained.* mine three lines of figures by the following processes :-Then deter

1. Cube the last figure in the root.

2. Multiply all the figures of the root except the last by 3, and the result by the square of the last.

3. Multiply the divisor by the last figure in the root.

Set down these lines in order, under each other, advancing each successively one place to the left. Add them up, and subtract their sum from the dividend. Bring down the next period to the right of the remainder, to form a new dividend, and then proceed to form a divisor, and to find another figure of the root by exactly the same process, continuing the operation until all the periods are exhausted.

13. In decimals, the number of decimal places in the cube root will be the same as the number of points placed over the decimal part, i.e., as the number of periods in the decimal part.

Obs.-If, finally, there be a remainder, then the given number has no exact cube root, but, as in the case of the square root, an approximation can be carried to any degree of nearness by adding ciphers, and finding any number of decimal places.

The rule will be best understood by following the steps of an example.

It will be found necessary sometimes, as will be seen by the example given in Art. 15, to set down as the next figure in the root, one less than this quotient.

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Placing the points as indicated in the rule, we observe that the cube of 4 is the greatest cube in the first period 78. Subtracting 43, or 64, from 78, we get a remainder 14, to the right of which we bring down the next period 314, to form a dividend. Multiplying the square of 4 by 3, we get for a divisor 48, which will go 2 times in 143 (the dividend without its two right-hand figures). We set down 2, therefore, to the right of 4 as the next figure in the root, and then proceed to form the three lines according to the rule.

1. 8 is the cube of 2. 2. 48 is 3 x 4 × 2a.

3. 96 is the product of 2, the last obtained figure in the root; and 48, the divisor.

Placing these three lines under each other, but advancing each successively one place towards the left, and adding, we get 10088, which we subtract from the dividend 14314, leaving a remainder 4226. To the right of this we bring down the next period 601, thus forming another dividend.

The next divisor 5292 is 3 X 42%, and is contained 7 times in 42266. Putting down, then, 7 as the next figure in the root, we form three lines as before:

1. 343 is the cube of 7, the last figure in the root.

2 6174 is 3 x 42 x 72.

3. 37044 is 7 x 5292,

Adding these up when properly placed, we get 3766483, which we subtract from the previous dividend 4226601, leaving a remainder 460118.

There are now no more periods left. Hence 427 is the number whose cube is the nearest cube number to the given number, and less than it. If there were no remainder, the root obtained would be the exact cube root of the given number.

14. In such an example as that worked out above, we could place a decimal point and as many periods of ciphers as we may wish after the original number, and thus, by continuing the process according to the rule, get as many decimal places as may be required as an approximation to the cube root.

In finding the cube root of a decimal, the periods must be completed by adding ciphers, if necessary.

15. When the cube root of a fraction is required, the cube root of the numerator and the cube root of the denominator will be the numerator and denominator respectively of the fraction which is the cube root of the original fraction. If the numerator and the denominator are not both perfect cubes when the fraction is reduced to its lowest terms (vide 9, Obs.), the best plan generally will be to reduce the fraction to a decimal, and then to find the cube root of that decimal. In the case of mixed numbers, they must be reduced to improper fractions, in order to see whether the resulting improper fraction has its numerator and denominator both perfect cubes. Thus, 5 reduced to an improper fraction gives 33, of which the cube root is, or 1. But if, when so reduced, the numerator and denominator are not perfect cubes, then it will be better to reduce the fractional part of the mixed number to a decimal, and placing the integral part before it, find the cube root by the above rule.

238136

And so on to as many more decimal places as we may desire. Obs. Exactly as in the case of the square root, when one more than half the number of figures required of the root have been found by the rule, the rest may be found by simply dividing, as in ordinary division, by the last divisor.

16. Obs. It will be observed that although 27, the first divisor, is really contained 6 times in 176, we only put down 5 in the root. The reason is that, on examination, we find that 6 would be too large, for it would make the sum of the three lines which we add up greater than the dividend 17600. This explains the note at page 318. We must, therefore, always be careful to observe whether the figure put down in that root will or will not make the sum of the three lines too large. The dividing the dividend without its two last figures by the divisor is not, therefore, an infallible guide to the next figure of the

root.

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LESSONS IN ARCHITECTURE.-I.

ARCHITECTURE is the art of planning, constructing, and adorning public or private buildings according to their intended use. The word architecture is derived from the Greek apxw (ar'-ko), I command, and TekTwv (teck'-tone), a workman. This etymology indicates the operatives engaged in the building on the one hand, and the leader or chief, the man of science and practical skill, putting in action all his resources in order to execute his plan on the other. Such a division as this was, no doubt, established from the beginning of the art. According, therefore, to the literal meaning of the etymology, mankind must have, at the origin of architecture, possessed a degree of civilisation sufficient for the organisation of different kinds of industrial operations, and acquired a degree of skill in the art, which enabled some men by their experience to be the leaders or directors of others. In this way, we may suppose that the art itself, or rather the symmetry, the harmony of proportions, and good taste in structures, at first began to be developed.

Before arriving at this point, mankind must have overleapt ages. One of the first wants of society was a covering or shelter from the inclemency of the weather, whether of heat or of cold. Simple was the art employed in constructions of this kind. Grottoes or caves hollowed square to make them more habitable, and cottages constructed of branches of trees and blocks of stone-such were the primitive constructions in wood and stone which formed the rudiments of architecture. From the simplicity of early structures men passed to the study of proportions;

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