170 



THE POPULAR EDUCATOR. 



nately, ended the life of tho greatest navigator of modern times. 

 Captain Clerke, who was second in command, took charge of 

 the expedition, and sailed to the north-east in search of tho 

 passage to the Atlantic ; but the same obstacles compelled him 

 to abandon the enterprise, and he died on the voyage home. 



To attempt to describe all the benefits which the discoveries 

 if Captain Cook have conferred on the sciences of geography and 

 hydrography, is more than can be done in this historical sketch 

 of these memorable expeditions. The accuracy with which 

 this illustrious navigator determined the geographical positions 

 of the places which he discovered or visited, rectified numerous 

 errors in tho maps and charts of the century in which he 

 flourished, and accelerated the progress of the science to which 

 these remarks form our introduction, in a degree hitherto un- 

 known. Mathematical geography has, since this time, taken 

 her place among the exact sciences. 



In concluding this lesson, we may remark that Cook lifted 

 the veil of darkness which hung over the extremities of the 

 Pacific Ocean, and the junction of the continents of Asia and 

 America. His last voyage, by disclosing the vast breadth of 

 America at the latitude of Behring Strait, made the hopes of 

 discovering the north-western passage darker than ever. That 

 continent had, previous to the time of the English navigator, 

 been considered as terminating to the north in a point or cape, 

 after passing which, the navigator would find himself at once 

 in the South Seas, and in full sail to China or Japan. But the 

 discovery of Cook showed that there was found intervening a 

 space of land of nearly three thousand miles in breadth, a very 

 large portion of the circumference of the globe. Hence, geo- 

 graphers viewing the coast running northward from Behring 

 Strait, Hudson Bay, and Baffin Bay, all enclosed by land, 

 received tho impression, and constructed their maps accordingly, 

 that an unbroken mass of land reached onwards to the pole, 

 and that all these boundaries were for ever barred against the 

 enterprising navigator. 



LESSONS IN ARITHMETIC. XI. 



FRACTIONS (continued). 



19. To reduce fractions to equivalent fractions having tlie same 

 denominator. 



RULE. Find the least common multiple of all the denomina- 

 tors. Multiply the numerator and denominator of each fraction 

 by the quotient obtained from dividing the least common mul- 

 tiple by that denominator. 



EXAMPLE. Reduce |, f, ^, J 2 , to a common denominator. 

 1260 is the least common multiple of 9, 7, 10, 12 (see page 134), 

 and the quotients of 1260 by these respectively are 140, 180, 

 126, 105. Multiplying each numerator and each denominator 

 by these numbers respectively, we get ^jj{j, -ggj, ^jgj, j^ which 

 are fractions equivalent to the given ones, and" all of which have 

 the same denominator. 



It may be observed that the common denominator found in 

 this case is the least. Any common multiple of the denominator 

 of the original fractions would have given fractions with the 

 same common denominator ; but the least common multiple gives, 

 of course, the least common denominator. 



11.' Fractions may also often conveniently be made to have 

 the same denominator by the following method : Multiply each 

 numerator into all the denominators except its own for a new 

 numerator, and all the denominators together for a common 

 denominator. The reason of this will be clearly seen from an 



EXAMPLE. Reduce |, -jj, 5, |, to fractions having the same 1 

 common denominator. 



Following the rule, we get for the first fraction 



2x6x5 x 9 

 3x6x5x9 



where we have multiplied the numerator 2, and denominator 3, 

 by 6 X 5 X 9, the product of tho denominators of the other 

 fractions. The fractions will therefore be 



2 x 6 x 5 x 



5x3x5x9 



3x6x5x6 3 x 6~x 0x9 



Or, 540, 675, 



810 810 



Here, evidently, the common denominator is not the 



3 x 3 x 6 x 9 ? 

 3 x 6x5 x 9 



486 ( 630. 

 8To' 8lo' 



7x3x6x5 



3x6x5x9 



much as 3 X 6 X 5 X 9, the common multiple of 3, 6, 5, and 

 9, which we have taken, is not the least common multiple. 



12. Wo are enabled by this means to find which of two 

 fractions is the greater'. For instance, if we wished to know 

 which of the four fractions given in Art. 11 is the greatest, 

 having reduced them to a common denominator, 810, we are 

 able to say that the second fraction, |, is the greatest, because 

 it contains the greatest number of the 810 parta into which the 

 unit is divided, viz., 675 ; and in the same way we- see that the 

 order of magnitude of the four fractions is f, J, , |. 



EXEECISE 24. 

 Place in order of magnitude the following sets of fractions : 



1. 



2. ?-, |, I, J-. 



3. 



5- A, , I. 

 6. ft,*f,2!,i 



t. tt,#V,tt. 



13. Addition of Fractions. 



Required to add and \ together. Reducing the fractions to 

 a common denominator, = -J'j , and {! = ^ : 



Therefore } + * = ^9 + ^ = 1 ; 



or, as it could be written, 1^ (Art. 3). We have here effected 

 the addition, i.e., found a single fraction which is equal to the 

 sum of tho two given on9s, by reducing the fractions to a com- 

 mon denominator, 15. 



The same method will apply to any other two or more frac- 

 tions. Ho'nce we are able to enunciate the following 



Rule for the Addition of Fractions. 



Reduce the fractions to a common denominator, add the new 

 numerators so formed for a numerator, and take the common 

 denominator for a denominator. The single fraction so formed 

 will bo the sum of the given fractions. 



Obs. It will generally be most convenient to reduce each 

 fraction, before commencing the operation, to its lowest terms, 

 if it is not already in them, and then to take the least common 

 denominator. 



14. Subtraction of fractions. 



The operation of subtracting one fraction from another will 

 evidently be effected in the same manner. Thus, to subtract | 

 from |, we have, aa above, 



and 9 fifteenths subtracted from 10 fifteenths is 1 fifteenth. 



Hence f -J = ^V- 

 Hence the following 1 



Rule for tlie Subtraction of Fractions. 



Reduce the two fractions to a common denominator, subtract 

 the less numerator from the greater for a numerator, and take 

 the common denominator for a denominator. The fraction so 

 formed will be the difference of the given fractions. 



The same observation with respect to the least common deno- 

 minator, which was made with reference to the rule for Addi- 

 tion, evidently applies equally to that for Subtraction. 



N.B. In all cases a whole number must be treated as a 

 fraction having a denominator unity. For instance, to subtract . 

 | from 2. 2 = f = s. 



Therefore 2 - f = - = J = 1J. 



EXERCISE 26. x 



1. Find the difference between 



1. ? and i. 5. *J and T "r- 9. 83i, and y 5 l . 



2. f and f. 6. -"- and |f . 10. 230/ T and 16Q? S . 



3. f and -J. 7. f and |f. 11. 1 and **-. 



4. 5; and *. 8. 6-> s and |. 12. 5 and f . 



2. Simplify the following expressions : 



1. 3 - * + l - f 



2. 9-J - 7-J - J-. 



3. 12711- - "?} + 57863-'- + 1428|-J. 



4. 41} - J + 105* - I + 300J + 41J + 472J- - 230J. 



