402 



THE POPULAR EDUCATOR. 



had the relative positions of the parties been reversed, they 

 would very much have deprecated these inferior races acting 

 towards them. In America, at the Cape of Good Hope, in 

 Australia and elsewhere, it has been so, and thus many tribes of 

 mankind are being thrust prematurely from the world. The 

 Esquimaux are passing away, and the great mass of American 

 Indians, and the islanders of the Pacific, including the New 

 Zealand Maoris and the Hottentots at the Cape of Good Hope. 

 In all likelihood most of these would ultimately have perished, 

 whatever kindness they had received from the ruling races, but 

 the injustice with which they have been too frequently treated 

 has tended to effect in a short time what else would have 

 required a longer period for its accomplishment. We trust 

 that, year by year, the treatment of the uncivilised races by 

 their more cultured brethren will be less open to objection, and 

 that special kindness in the future may partly atone for its too 

 frequent absence in the past. Still, whatever may be done, 

 many races will disappear ; and whoever is in a position more 

 definitely to ascertain their physical characteristics, their lan- 

 guage, their historic traditions, and their religious beliefs, 

 should regard himself as bound, in the interests of science, to 

 make the best use of his opportunities, so that knowledge which 

 else would be lost may be preserved to the world. 



Though mourning the human guilt which has so often been 

 shown in the treatment of the inferior by the superior races, 

 we still most deeply admire the wisdom and the goodness dis- 

 played in the natural law which makes that struggle for exist- 

 ence, so plainly discernible among the plants and animals of the 

 world, operative also in the case of our species. Its tendency is 

 slowly, and, where man does not criminally intervene, almost 

 painlessly to extinguish tribes unpleasing in person, low in 

 understanding and morality, and unlikely, within a moderate 

 period of time, to rise to the level at which they might be able 

 markedly to benefit humanity. While these are passing away, 

 their places are being rapidly taken by races of better organisa- 

 tion and higher mental and moral development. The ultimate 

 effect produced by the perpetual elimination of whatever is less 

 perfect must be to raise the general level of humanity, and 

 conduct it ultimately to heights which, but for this tinceasing 

 and dire struggle for existence, it would for ever be forbidden to 

 reach. 



LESSONS IN ALGEBRA. X.LIV. 



EXERCISE 77. 



1. Given r* + $x*= 28Jr, to find .r by quadratics. 



2. Given x:y.:y.t, x + 11 + z = 42, and "+ ij'+ z= 1092, to find 

 ', y, and z. 



3. What number is that, the double of which is as much above 40 

 as its half is below it ? 



4. A had 80, and B 30. B gave away a certain sum, and A twice 

 as much ; and then A had 3 times as much as B had. What did A give 

 away? 



5. Tea at 5s. 3d. per Ib. is mixed with tea at 4s. 3d. per lb., and 

 10 Ibs. of the mixture are sold for 44s. 6d. How much was there 

 of each ? 



6. Divide 153 between A and B, giving B 1J times A's share. 



7. Divide 77 into two parts, such that the sum of the quotients of 

 the one by 4, and the other by 11, shall be 14. 



8. A father's age is 49, and the son's is 11 ; in how many years will 

 the father's age be treble the son's ? 



9. A farm of 2,850 acres is divided between three sons (A, B, and Q), 

 so that A's share is to B's as 6 : 11 ; and C has 300 acres more than A 

 and B together. Find their shares. 



10. A garrison, consists of 2,600 men, of which there are 9 times as 

 many foot soldiers and 3 times as many artillerymen as cavalry, Find 

 the number of each. 



' 11. A bill of 7 19s. has been paid with 51 coins ; some are crowns, 

 the rest are florins. Tind the number of each. 



12. There is a number of 2 digits ; their sum is 10, and if these digits 

 be transposed, we obtain a number greater by 15 than 4 times the 

 original number. Find the original number. 



13. The sum of two numbers is 23, and 3 times their difference is 21. 

 Find the numbers. 



14. Sold a watch for 24, and by so doing lost as much per cent, as 

 the watch cost. Find the cost of the watch. 



15. The area of a triangle is 6 square feet, and the base is known to 

 be 3 times the height. Find the base and height. 



16. Compound the ratios of b" : b* x" 1 , b + x : b x, and b 3 x 3 : b 3 . 



17. Show that VH + V7 is greater than V19 + V2. 



18. Which is greater, V5 + V14 or V3 + 3 72 ? 



19. Show that the ratio compounded of a : x, x -. y, and y : b, is the 

 same as the ratio compounded of x + a : x + b, and a(x + b) : 'h(x + a). 



20. Find the number to which, if 2 and 5 be successively added, the 

 resulting numbers are in the proportion of 3:8. 



21. Find two numbers in the proportion of 3 : 4. and their sum : the 

 sum of their squares as 7 : 50. 



22. Find the 64th term of the series 4, 6J, 9. etc. 



23. Find the 7th term, and the sum of 7 terms of the series 

 -j, ^, etc. 



21. Find the sum of 5 + 4| + 4), etc., to 21 terms. 



25. How many terms of the series 19, 18, 17, etc. , amount to 124. 



26. Two hundred stones are placed at the distance of a yard from 

 each, other, in a right line with a basket, which is one yard from that 

 next to it. A person starts from the b isket, and brings them one by one 

 into it. What space does he travel over ? 



27. Insert 4 arithmetical means between 5 and 6. 



28. Given the first term of an arithmetical series = 2 ; and the sum 

 of 17 terms = 102. Find the common difference. 



29. The first term of an arithmetical series is 3 ; the 13th term is 55. 

 Find the common difference. 



30. The sum of three numbers in arithmetical progression is 21, and 

 the sum of their squares 179. Find them. 



31. Find the Stti term, and the sum of 8 terms, of the geometrical 

 series 81, 27, 9, etc. 



32. Find tho sum of 3, 6, + 12 , etc., to 6 terms. 



33. Find the limit of the sum of the series 1 + % + J, etc. 



34. Find a geometrical series whose 1st term is 2, and 7th term is -$. 



35. Insert 3 geometrical means between 2 and 10|. 



36. The perimeter of a piece of ground in the form of a right-angled 

 triangle = 96 rods, and tho radius of its inscribed circle = 41 yards. 

 Find the sides of the triangle, the area of the inscribed circle, and the 

 area of the ground. 



37. If a candl^, in the form of a cone 12 inches hi'^h, burns 12 hours, 

 and the bottom inch burns 1 hour longer than the top one, what time- 

 will the fourth inch from the top burn ; and also find the time the top 

 inch will last ? 



38. At what height must a person be to see J^ of the earth's surface, 

 supposing it to be perfectly spherical, and its diameter 7,960 miles ? 



39. Given fl_L = , to find x. 



40. Given 



X' +1 



ab 



to find x. 



41. Given x a = Jx* 1+ */x* 1, to find x by quadratics. 



42. Given 3u: + 5y = 73, to find intsgral values of ;u and y. 



43. In how many ways may 80 be paid with sovereigns and guinea? ? 



44. What number is that which, if divided by 5, 7, and 9, leaves the- 

 remainders 1, 1, and 0. 



45. Divide 150 into three parts, so that one of them being divided 

 by 9, another by 7, and the other by 2, the quotients will together 

 amount to 25. 



40. What number is that which when divided by 2, 3, 4, 5, etc., to 12 

 has for its remainder 1 less than its divisor ? 



47. How must I mix three kinds of spirits at 2s. 4d., 2s. 6d., and 

 3s. 4d. per gallon, to make 100 gallons at 3s. ? 



48. Find the side of a square, inscribed in a given semicircle, whose 

 diameter is (a). 



49. Find the side of an equilateral triangle, inscribed in a circle 

 whose radius is (a), and that of another circumscribed about the same 

 circle. 



50. Find the sides of a rectangle, the perimeter of which is equal to 

 that of a square whose side is (a), and its area equal to j the area of the 

 square. 



51. An ingot of gold was sold at a loss for 420. If it had been sold 

 for 570, then the gain would have been exactly 4 times as much as tho 

 loss is at present. What did it cost ? 



52. Find a number such that when it is added to 15, 27, and 45, there 

 arise three numbers which are in geometrical progression. 



53. A, B, and C wanted to buy a horse, but neither of them had. 

 money enough for the purpose ; A begged of B and C the half of their 

 money, in order to enable him to buy it. On the other hand, B asked A 

 and C only for the J of their money, because he then would be able to 

 buy it himself ; on which C said to A and 15, " Lend me i of your money 

 each, and then I can buy it." How much money had each, and how 

 mucn did the horse cost, supposing we know that they had no other 

 money than sovereigns ? 



54. Five friends, A, B, C, D, and E, jointly spent a certain sum at an 

 inn. This sum is to be paid by one of them, bub on counting the 

 sovereign's they had in their pockets (for none of them had smaller 

 coin) no one had enough to pay it alone. If one pay it alone the others 

 must add a part of their money ; so that A must contribute i ; B, * ; 

 C, J ; D, |; and E, i of the others' money. How much did they spend, 

 and how much had each ? 



55. Find two numbers such that their product is equal tf> their sum, 

 and their sum, added to the sum of their squares, is equal to 15|. 



56. A traveller starts from a certain place, and goes 1 mile the first 

 day, 2 the second, 3 the third, 4 the fourth, etc. Five days after another 

 traveller starts from the same pi ice, takes the same road, and goes 12 

 miles daily. On what day after the departure of the first will they be 

 together ? 



57. A bookseller sold me 2 bound books, the one which contained 48 

 sheets for 14s., the other of 78 sheets for 19s. The binding and paper 

 were the same in both. What was the price of the binding ? 



53. A person buys a piece of cloth, and pays 7 for every 5 ells ; he- 

 then sells every 11 ells for 16, and gams by the bargain 24. How 

 many ells did the piece contain ? 



59. A sum of 156 was to be divided amongst 16 poor children, in 

 proportion to their ages, in such a way that each of the elder ones 

 received exactly twice a* much as the next younger. If, therefore, the 

 youngest, according to this division, received 6, how much more did 

 each preceding one receive, and how much the eldest ? 



60. A schoolmaster gave his pupil two numbers to multiply, one of 

 which was greater than the other by 75. When the scholar had finished 

 the multiplication, he proved it ; and the product divided by the least 

 factor gave the quotient 227, and remainder 113. The schoolmaster 

 then found that it was multiplied wrong, and ordered the error to be 

 corrected. When the pupil had found out the error, he said that he 

 had calculated only 1 too little in the multiplication. " No," said the 



