LESSONS IN AH' 



319 



. Find tho cube root of 78314C01. 



78314601 

 64 



48} 143,14 



8 

 48 

 98 



10088 



5292) i 



3760483 

 460118 



Placing tho points as indicated in tho rule, wo observe that 

 tho oubo of 4 is tho greatest cube in tho first period 78. Sub- 

 tracting I 3 , or C4, from 78, wo got a remainder 14, to tho right 

 of which wo bring down the next period 314, to form a dividend. 

 Multiplying the square of 4 by 3, wo get for a divisor 48, which 

 will go 2 times in 143 (the dividend without its two right-hand 

 fig-arcs). Wo set down 2, therefore, to tho right of 4 as tho 

 next figure in tho root, and then proceed to form tho three lines 

 according to the rule. 



1. 8 is the cube of 2. 



2. 48 is 3 x 4 x 2*. 



3. 96 is tho product of 2, the lost obtained figure in the root ; and 

 48, the divisor. 



Placing those three lines under each other, but advancing each 



ively one place towards the left, and adUm;,', wo get 



10088, which we subtract from tho dividend 14311, leaving a 



remainder 4226. To tho right of this wo bring down the next 



period 601, thus forming another dividend. 



Tho next divisor 5292 is 3 X 42 s , and is contained 7 times in 

 42266. Putting down, then, 7 as the next figure in tho root, we 

 form three lines as before : 



1. 343 is the cubo of 7, tho "last figure in the root. 

 2. 6174 in 3 x 42 x ?. 

 3. 37044 is 7 x 5202. 



Adding these up when properly placed, wo get 3766483, 

 which we subtract from tho previous dividend 4226601, leaving 

 a remainder 460118. 



There are now no more periods left. Hence 427 is the num- 

 ber whose cubo is the nearest cube number to tho given number, 

 and loss than it. If there were no remainder, the root obtained 

 would be the exact cube root of tho 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 wo may 

 wish after the original number, and thus, by continuing tho 

 process according to tho rule, get as many decimal places as 

 may bo required as an approximation to tho cubo root. 



In finding tho cubo root of a decimal, tho periods must be 

 completed by adding ciphers, if necessary. 



15. When tho cube root of a fraction is required, the cubo 

 root of tho numerator and the cubo root of tho denominator will 

 be tho numerator and denominator respectively of the fraction 

 which is the cube root of tho original fraction. If tho numc- 



rator and the denominator aro not both perfect cubes when tho 

 fraction is reduced to its lowest terms (vide 9, 06s.), tho best 

 plan generally will be to reduce the fraction to a decimal, and 

 then to find the cubo root of that decimal. In tho caso of 

 mixed numbers, they must be reduced to improper fractions, in 

 order to see whether tho resulting improper fraction has its 

 numerator and denominator both perfect cubes. Thus, 5JJ 

 reduced to an improper fraction gives ^ s , of which tho cubo 

 root is J, or If. But if, when so reduced, tho numerator and 

 denominator are cot perfect cubes, then it will bo better to 

 reduce tho fractional part of the mixed number to a decimal, 

 and placing the integral port before it, find tho cube root by 

 the above rult. 



EXAMPLE. Find the cube root of -ilj to two 

 decimals. 



} = 44-0, . . . 



44-000000(3-54 



27) 1700 



15870 



place* of 



3C75 ) 1725000 



1480864 



238136 



And BO on to as many more decimal places aa we may desire. 

 Obs. Exactly aa in tho caso of the aqnoro root, when 000 



more than half tho nun ber of figures required of the root hare 

 been found by tho role, the rest may bo found by simply divid- 

 ing, as in ordinary division, by the last divisor. 



16. Obs. It will bo observed that although 27, the fint 

 divisor, is really contained C timoa in 176, we only put down 5 

 in tho root. Tho reason is that, on examination, we find that 6 

 would bo too large, for it would make tho sum of the three 

 linos which wo odd up greater than tho dividend 17600. This 

 explains tho noto at pago 818. We must, therefore, always be 

 careful to observe whether tho figure put down in that root wfll 

 or will not mako the Hum of the three lines too large. The 

 dividing tho dividend without its two hist figures by the divisor 

 is not, therefore, an infallible guido to the next figure of the 

 root. 



EXERCISE 40. 



Find the cubo root of tho following numbers : 



1. 2107. 6. 11543-176. 11. 376. 



2. 91125. 7. 20-570824. 12. 575. 



3. 571787. 8. -241804367. 13. 



4. 2515456. 9. 37. 11. 49fr. 



5. 10218313. 10. 6. 15. 399501-352125. 



Where tho given number is not a complete cube, the root 

 may bo found to seven decimal figures in each case, attention 

 being paid to Obs. of Art. 16. 



LESSONS IN ARCHITECTURE I. 



ARCHITECTURE is tho art of planning, constructing, and adorn- 

 ing public or private buildings according to their intended use. 

 Tho word architecture is derived from tho Greek apxv (ar'-io), 1 

 command, and rtK-ruv (tcck-tone), a vxnrkman. Thia etymology 

 indicates tho operatives engaged in tho building on the one 

 hand, and tho leader or chief, tho man of science and practical 

 skill, putting in action all his resources in order to execute 

 his plan on tho other. Such a division as this was, BO 

 doubt, established from the beginning of tho art. According, 

 therefore, to tho literal moaning of tho etymology, mankind 

 must have, at tho 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 tho art, 

 which enabled some men by tkoir experience to be the leaden 

 or directors of others. In thia way, wo may suppose that the 

 art itself, or rather tho symmetry, tho harmony of proportion*, 

 and good taste in structures, at first began to bo developed. 



Before arriving at this point, mankind must hare overleapt 

 ages. One of the first wants of society was a covering or shelter 

 from tho inclemency of tho weather, whether of he.\t or of cold. 

 Simple was tho art employed in constructions of this kind. 

 Grottoes or caves hollowed square to mako them more habitable, 

 and cottages constructed of branches of trees and blocks of 

 stone such were tho primitive constructions in wood and stone 

 which formed tho rudiments of architecture. From the simpli- 

 city of early structures men passed to the study of proportion* : 



