aujourd'hui ou demain ? 14. On nous dit qu'il doit avoir lien are brought to me (R. 2] every day, but I have no time to read cette après-midi. 15. Il aura lieu à cinq heures et demie. 16. them. 7. What should one do (doit on faire) when one is sick ? Avez-vous envie de venir au lieu de votre frère ? 17. Mon frère 8. One should send for a physician. 9. Do you send for my doit venir au lieu de notre cousin. 18. Avez-vous l'intention brother? 10. I am to send for him this morning. 11. Do you de lui dire ce qu'il doit faire ? 19. Il sait ce qu'il doit faire. | hear from your son every day? 12. I hear from him every time 20. Savez-vous ce qu'on dit de nouveau ? 21. On ne dit rien de that your brother comes. 13. Does the sale take place to-day ? nouveau. 22. Trouve-t-on beaucoup d'or en Californie ? 23. 14. It takes place this afternoon. 15. At what time does it On y en trouve beaucoup. 24. Y trouve-t-on aussi des diamants ? take place ? 16. It takes place at half after three. 17. I have 25. On n'y en trouve point, on n'y trouve que de l'or. 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 EXERCISE 64. on), is very sick. 20. Is he to leave for Africa ? 21. He is to 1. What do people say of me? 2. People say that you are leave for Algiers. 22. Do you come instead of your father ? not very attentive to your lessons. 3. Is it said that much gold 23. I am to write instead of him. 24. Does the concert take is found in Africa ? 4. It is said that much gold is found in place this morning ? 25. It is to take place this afternoon. 26. California. 5. Do they bring you books every day? 6. Books Do you know at what hour? 27. At a quarter before five. COPY-SLIP NO. 73.—THE LETTER f. COPY-SLIP NO. 74.—THE WORD frog. COPY-SLIP NO. 75.-ELEMENTARY STROKES FORMING THE LETTER k. LESSONS IN PENMANSHIP.-XX. made in the form of a loop, the pressure of the pen being relaxed, and the down-stroke narrowed gradually until it is The simplest method of writing the letter f, and that which is turned at the bottom in a hair-stroke, which is carried upwards most generally used in writing large-hand copies, is shown in and across the down-stroke about the line cc, or centre of Copy-slip No. 73. In this form, which is repeated in Copy-slip the letter, in a small loop. Sometimes the loop at the upper No. 74, where f is given in conjunction with other letters, it is part of the letter is omitted, the down-stroke being commenced commenced with a fine hair-stroke a little above the line cc, at the line ee (see Copy-slip No. 10, p. 60, for the height of which is carried upwards until it reaches the line kk, where it this line above a a), and thickened very gradually until it reaches is turned towards the left and brought downwards across the its thickest part about the line bb, when the pressure on the fine up-stroke, the pressure on the pen being gradually increased pen is immediately lessened to narrow the stroke into the fine until a thick down-stroke is formed, which terminates at the line that forms the loop below the line bb. Examples of the line gg. The letter is finished with a hair-stroke carried out methods of making the letter f that have just been described from the back of the letter, about the line cc, to the left, and will be found in future copy-slips. In Copy-slip No. 75 then brought to the right in a curve across the down.stroke. the learner will find the elementary strokes that form the In small-hand writing, the lower part of the letter f is generally letter k. HR POPULAR EDUCATOR. LESSONS IN ARITHMWITO, XX, nywven, wo get the most by dividing as in ordina the ima divisor, 2898123. 11. We might oxtract the square root of a pe 2. Th e fact wat wy taking the plaing it into its prime factors, but unless the Bure pou w luw # SHARE, and the woro Inree this would be a tedions method. that of the en te fue # Aminuut willow eneo IXAMPLIFind the square root of 441. frem we w e n t the multicuttum trations in Following the method given in Lesson VIII i will the lov i numarulaw, and the Senminste minuter When wither the numerator 3)16 e ile ruuhu mimointer and denominatap She sy in the EXAMPLE. Find the cube root of 78314601. EXAMPLE.—Find the cube root of 443 to two places of decimals. 7831460i ( 427 443 = 44:6. 44:600000 (3:54 48) 143,14 27) 17600 root. 48 125 225 135 10088 15875 3675) 1725000 64 1680 14700 3766483 1486864 460118 238136 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. A And so on to as many more decimal places as we may desire. tracting 43, or 64, from 78, we get a remainder 14, to the right Obs.—Exactly as in the case of the square root, when one of which we bring down the next period 314, to form a dividend. more than half the number of figures required of the root have Multiplying the square of 4 by 3, we get for a divisor 48, which been found by the rule, the rest may be found by simply dividwill go 2 times in 143 (the dividend without its two right-hand ing, as in ordinary division, by the last divisor. figures). We set down 2, therefore, to the right of 4 as the 16. Obs.-It will be observed that although 27, the first Dext figure in the root, and then proceed to form the three lines divisor, is really contained 6 times in 176, we only put down 5 according to the rule. 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 1. 8 is the cube of 2. lines which we add up greater than the dividend 17600. This 2. 48 is 3 X 4 X 29. explains the note at page 318. We must, therefore, always be 3. 96 is the product of 2, the last obtained figure in the root; and careful to observe whether the figure put down in that root will 4, the divisor. or will not make the sum of the three lines too large. The Placing these three lines under each other, but advancing each dividing the dividend without its two last figures by the divisor successively one place towards the left, and adding, we get is not, therefore, an infallible guide to the next figure of the 10088, which we subtract from the dividend 14314, leaving a remainder 4226. To the right of this we bring down the next EXERCISE 40. period 601, thus forming another dividend. Find the cube root of the following numbers :The next divisor 5292 is 3 X 422, and is contained 7 times in 1. 2197. 6. 11543.176. I 11, 376. 42266. Putting down, then, 7 as the next figure in the root, we 2. 91125. 7. 20:570824. 12. 575. form three lines as before : 3. 571787. 8. •241804367. 13. 114. 4. 2513456, 9, 37 14. 4975. 1. 343 is the cube of 7, the last figure in the root. 5. 10218313. 15. 399501.352125. 2 6174 is 3 X 42 X 79. 3. 37044 is 7 X 5202. Where the given number is not a complete cube, the root may be found to seven decimal places in each case, attention Adding these up when properly placed, we get 3766483, being paid to Obs. of Art. 16. which we subtract from the previous dividend 4226601, leaving remainder 460118. There are now no more periods left. Hence 427 is the num. LESSONS IN ARCHITECTURE.—I. er whose cube is the nearest cube number to the given number, nd less than it. If there were no remainder, the root obtained ARCHITECTURE is the art of planning, constructing, and adorn"ould be the exact cube root of the given number. ing public or private buildings according to their intended use. 14. In such an example as that worked out above, we could The word architecture is derived from the Greek apxw (ar'-ko), I lace a decimal point and as many periods of ciphers as wo may command, and TEKT WV (teck-tone), a workman. This etymology ish after the original number, and thus, by continuing the indicates the operatives engaged in the building on the one rocess according to the rule, get as many decimal places as hand, and the leader or chief, the man of science and practical ay be required as an approximation to the cube root. skill, putting in action all his resources in order to execute In finding the cube root of a decimal, the periods must be his plan on the other. Such a division as this was, no mpleted by adding ciphers, if necessary. doubt, established from the beginning of the art. According, 15. When the cube root of a fraction is required, the cube therefore, to the literal meaning of the etymology, mankind st of the numerator and the cube root of the denominator will must have, at the origin of architecture, possessed a degree of • the numerator und denominator respectively of the fraction civilisation sufficient for the organisation of different kinds of Tich is the cube root of the original fraction. If the nume- industrial operations, and acquired a degree of skill in the art, *tor and the denominator are not both perfect cubes when the which enabled some men by their experience to be the leaders "action is reduced to its lowest terms (vide 9, Obs.), the best or directors of others. In this way, we may suppose that the in generally will be to reduce the fraction to a decimal, and art itself, or rather the symmetry, the harmony of proportions, en to find the cube root of that decimal. In the case of and good taste in structures, at first began to be developed. xed numbers, they must be reduced to improper fractions, in Before arriving at this point, mankind must have overleapt Her to see whether the resulting improper fraction has its ages. One of the first wants of society was a covering or shelter merator and denominator both perfect cubes. Thus, 533 from the inclemency of the weather, whether of heat or of cold. duced to an improper fraction gives 343, of which the cube Simple was the art employed in constructions of this kind. *is, or 19. But if, when so reduced, the numerator and Grottoes or caves hollowed square to make them more habitable, nominator are not perfect cubes, thẽn it will be better to and cottages constructed of branches of trees and blocks of luce the fractional part of the mixed number to a decimal, stone—such were the primitive constructions in wood and stone placing the integral part before it, find the cube root by which formed the rudiments of architecture. From the simpliabove rule. city of early structures men passed to the study of proportions ; periods. LESSONS IN ARITHMETIC.XX. figures, we get the rest by dividing as in ordinary division by the last divisor, 2828423. SQUARE AND CUBE ROOT (continued). 11. We might extract the square root of a perfect square by 9, THE square root of a fraction is obtained by taking the splitting it into its prime factors, but unless the number is not square root of the numerator for a numerator, and the square large this would be a tedious method. root of the denominator for a denominator. This follows at once EXAMPLE.—Find the square root of 441. from the consideration that the multiplication of fractions is Following the method given in Lesson VIII., Art. 5— effected by multiplying the numerators for a numerator, and the 3) 441 denominators for a denominator. When either the numerator or the denominator is not a complete square, in which case the 3) 147 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 Therefore 441 = 32 x 7o; of which the square root is 3 x 7, first to reduce the fraction to a decimal, and then to take the or 21. Obs.—Unless a number is made up of prime factors, each of square root. EXAMPLE.—To find the square root of 3. which is repeated an even number of times, it is not a perfect square. Reducing , to a decimal, we find it to be .285714 (seo Lesson EXERCISE 39. 1. Find the square root of the following numbers :- 11. •81796 to 4 places. of .28571428571428 ... to as many decimal places as we please, 2. 5329. 12, 1169 64. by continually taking in more and more figures of the recurring 3. 784. 13. 3-172181 to 4 places. 4. 4761. 14. 10342656. Similarly, in finding the square root of %, we should proceed 5. 7056. 15. H, Hart 6. 9801. 16. to 4 places. thus : =:4, and then find the square root of .400000, etc., to 7. 27889. 17. 177 to 4 places. as many places as we please. 8. 961. 18. 964:5192360241. Obs.-It does not follow that because the numerator and 9. 97 to 4 places of decimals. 19. 00000625. denominator of a fraction are not complete squares, that the 10. 190 to 5 places. fraction has no square root; for the division of numerator and 2. Find the square root of the following numbers by the denominator by some common measure may reduce them to abbreviated method :perfect squares. Thus, as, when numerator and denominator are i 1. 365 to 11 figures in the root. 1 3. 3 to 17 figures. divided by 7, gives y, the square root of which is. A fraction 2. 2 to 12 figures. must be reduced to its lowest terms to determine whether it be 3. Extract the square root of 2116, 21316, and 7056, by a complete square or not. 10. Abbreviated Process of Extraction of Square Root. splitting them into their prime factors. When the square root of a number is required to a consider-) 12. Extraction of the Cube Root. able number of decimal places, we may shorten the process by To extract the cube root of a given number is the same thing the following as resolving it into three equal factors. Rule for the Contraction of the Square Root Process. . As in the case of the square root, we must content ourselves Find by the ordinary method one more than half the number with giving, without explanation of the reason of its truth, the of figures required, and then, using the last obtained divisor as Rule for the Extraction of the Cube Root of a given number. a divisor, continue the operation as in ordinary long division. Mark off the given number into periods of three figures each, EXAMPLE.—Find the square root of 2 to 12 figures. by placing a point over the figure in the unit's place, and then 2.0000, etc. (1•414213 | 56237 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 24.) 100 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 281 ) 400 of the root already obtained by 3 to form a divisor, and then, 281 having determined how many times this divisor is contained in 2824) 11900 the dividend without its two right-hand figures, annex this 11296 quotient to the part of the root already obtained.* Then deter mine three lines of figures by the following processes :28282) 60400 1. Cube the last figure in the root. 56564 2. Multiply all the figures of the root except the last by 3, and the result by the square of the last. 282841 ) 383600 3. Multiply the divisor by the last figure in the root. 282841 Set down these lines in order, under each other, advancing 2828423) 10075900 each successively one place to the left. Add them up, and 8485269 subtract their sum from the dividend. Bring down the next period to the right of the remainder, to form a new dividend, 15906310 and then proceed to form a divisor, and to find another figure of 14142115 the root by exactly the same process, continuing the operation until all the periods are exhausted. 17641950 13. In decimals, the number of decimal places in the cubo 16970538 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. 6714120 5656846 018.-If, finally, there be a remainder, then the given nunber has no exact cube root, but, as in the case of the square root, an 10572740 approximation can be carried to any degree of nearness by 8485269 adding ciphers, and finding any number of decimal places. The rule will be best understood by following the steps of an 20874710 example. 19798961 • It will be found necessary sometimes, as will be seen by the 1075749 example given in Art. 15, to set down as the next figure in the root, Here, having obtained by the ordinary process the first seven one less than this quotient. 125 225 135 10088 15875 3675) 1725000 64 1680 14700 3766483 1486864 460118 238136 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. An period 78" Soh. And so on to as many more decimal places as we may desire. tracting 43, or 64, from 78, we get a remainder 14, to the right Obs.-Exactly as in the case of the square root, when one of which we bring down the next period 314. to form a dividend. | more than half the number of figures required of the root have Multiplying the square of 4 by 3, we get for a divisor 48, which be 248 which been found by the rule, the rest may be found by simply dividwill go 2 times in 143 (the dividend without its two right-hand d ing, as in ordinary division, by the last divisor. figures). We set down 2, therefore, to the right of 4 as the 16. Obs.--It will be observed that although 27, the first Dext figure in the root, and then proceed to form the three lines divisor, is really contained 6 times in 176, we only put down 5 according to the rule. 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 1. 8 is the cube of 2. lines which we add up greater than the dividend 17600. This 2. 48 is 3 X 4 X 29. explains the note at page 318. We must, therefore, always be 3. 96 is the product of 2, the last obtained figure in the root; and careful to observe whether the figure put down in that root will the divisor. or will not make the sum of the three lines too large. The Placing these three lines under each other, but advancing each dividing the dividend without its two last figures by the divisor successively one place towards the left, and adding, we get is not, therefore, an infallible guide to the next figure of the 10088, which we subtract from the dividend 14314, leaving a root. remainder 4226. To the right of this we bring down the next EXERCISE 40. period 601, thus forming another dividend. Find the cube root of the following numbers :The next divisor 5292 is 3 X 42, and is contained 7 times in 1. 2197. 6. 11543.176. 11. 376. 42266. Putting down, then, 7 as the next figure in the root, we 2. 91125. 7. 20:570824. 12. 575. form three lines as before : 3. 571787. 8. *241804367. 13. 1311. 4. 2513456. 9. 37. 14, 49,7. 1. 313 is the cube of 7, the last figure in the root. 5. 10218313. 15. 399501-352125. 2 6174 is 3 x 42 x 7o. 3. 37044 is 7 x 5292. Where the given number is not a complete cube, the root may be found to seven decimal places in each case, attention Adding these up when properly placed, we get 3766483, | being paid to Obs. of Art. 16. which we subtract from the previous dividend 4226601, leaving a remainder 460118. There are now no more periods left. Hence 427 is the num LESSONS IN ARCHITECTURE.-I. ber whose cube is the nearest cube number to the given number, and less than it. If there were no remainder, the root obtained ARCHITECTURE is the art of planning, constructing, and adornwould be the exact cube root of the given number. ing public or private buildings according to their intended use. 14. In such an example as that worked out above, we could The word architecture is derived from the Greek apxw (ar'-ko), I place a decimal point and as many periods of ciphers as we may command, and TEKT WV (teck-tone), a workman. This etymology wish after the original number, and thus, by continuing the indicates the operatives engaged in the building on the one process according to the rule, get as many decimal places as hand, and the leader or chief, the man of science and practical may be required as an approximation to the cube root. skill, putting in action all his resources in order to execute In finding the cube root of a decimal, the periods must be his plan on the other. Such a division as this was, no completed by adding ciphers, if necessary. doubt, established from the beginning of the art. According, 15. When the cube root of a fraction is required, the cube therefore, to the literal meaning of the etymology, mankind root of the numerator and the cube root of the denominator will must have, at the origin of architecture, possessed a degree of be the numerator and denominator respectively of the fraction civilisation sufficient for the organisation of different kinds of which is the cube root of the original fraction. If the nume- industrial operations, and acquired a degree of skill in the art, rator and the denominator are not both perfect cubes when the which enabled some men by their experience to be the leaders fraction is reduced to its lowest terms (vide 9, Obs.), the best or directors of others. In this way, we may suppose that the plan generally will be to reduce the fraction to a decimal, and art itself, or rather the symmetry, the harmony of proportions, tien to find the cube root of that decimal. In the case of and good taste in structures, at first began to be developed. mixed numbers, they must be reduced to improper fractions, in Before arriving at this point, mankind must have overleapt order to see whether the resulting improper fraction has its ages. One of the first wants of society was a covering or shelter numerator and denominator both perfect cubes. Thus, 523 from the inclemency of the weather, whether of heat or of cold. reduced to an improper fraction gives 343, of which the cube Simple was the art employed in constructions of this kind. root is j, or 19. But if, when so reduced, the numerator and Grottoes or caves hollowed square to make them more habitable, dezoininator are not perfect cubes, then it will be better to and cottages constructed of branches of trees and blocks of reduce the fractional part of the mixed number to a decimal, stone—such were the primitive constructions in wood and stone and placing the integral part before it, find the cube root by which formed the rudiments of architecture. From the simpli. the above rule. city of early structures men passed to the study of proportions ; 10. 6. |