294 



THE POPULAB EDUCATOE. 



chased by Charles I., and have ever since remained in England. 

 Though they lack, of course, the finer touches of Raffael's 

 finished works, they yet exhibit very well the main broad fea- 

 tures of his style ; though it is to be feared that a misappre- 

 hension of their true object and nature has often led hasty 

 observers to misunderstand the character of Raffael's paintings. 

 Those who have the opportunity should not fail to correct their 

 impressions derived from these large and free designs by a study 

 of the few other works by Raffael which may be seen in the 

 National Gallery. 



Nor must we omit to mention that Raffael, like most of the 

 other great men of the Italian Renaissance, did not confine his 

 attention to painting alone. As an architect, he had for a time 

 direction of the building of St. Peter's, at Rome, that vast em- 

 bodiment of the Renaissance ideas round which all the artistic 

 work of the period centres. When we consider that he died 

 just as he had completed his thirty- seventh year, an age at 

 which men are now considered almost as still beginners, we can 

 estimate once more the greatness of this great artistic epoch, 

 which could number as contemporaries three such men as 

 Raffael, Michel Angelo, and Lionardo. 



The position which Raffael has obtained in the history of art 

 is twofold. In the first place, his is one of the great epoch- 

 making names, at least as great in this respect as Phidias in 

 Greek or as Giotto in early Italian art. He forms, as it were, 

 the boundary figure between mediaevalism and the modern 

 European school. All before him we still describe as pre- 

 Raffaelite, all after him seems to us purely of our own type in 

 art. It is true, the world might in some respects with greater 

 justice have pitched upon Lionardo as forming the first among 

 the strictly modern painters, but the fame of the younger artist 

 has not unnaturally eclipsed that of the elder, partly because we 

 have more of Raffael's pictures still existing than of Lionardo' s, 

 and partly because Raffael made a real and great advance even 

 upon the manner of his vigorous predecessor. Mankind gener- 

 ally love to seize upon a single personage as the central figure 

 to symbolise and sum up a movement or an epoch, and their 

 instinct has rightly caused them to single out the personality of 

 Raffael as such a central figure for the whole grand artistic 

 movement of the Italian Renaissance. But besides all this, in 

 the second place, even apart from his historical importance in 

 the development of art, Raffael stands out also as the greatest 

 name in all the annals of art on the strength of his actual per- 

 formances taken by themselves. He ranks first by the same 

 sort of general consent as that by which Shakespeare ranks first 

 among poets, or Newton among scientific thinkers. There may 

 be many, whose opinions are entitled to the highest respect, who 

 would place Raffael second to some other favourite painter, but 

 almost all' would place him second to one alone. And as in 

 the old Greek story, Themistocles was adjudged by all alike the 

 second place, while many others were adjudged the first, we 

 may take this general consent of opinion as justly entitling 

 Raffael to rank at the head of all the artistic roll. In his union 

 of every high quality of technical execution and of vivid con- 

 ception, in his admirably harmonious combination of all those 

 qualities in mind, hand, and eye, which go to make up the 

 perfect artist, he has never been equalled. Art, of course, has 

 learned some new lessons since his time, especially in technique, 

 but no subsequent painter has possessed such a grandly- 

 tempered union of all the chief artistic gifts as that which dis- 

 tinguished Raffael. It is his breadth of grasp even more than 

 uis subtle sweetness that has given him his unique power of 

 extorting our admiration and our applause throughout so many 

 centuries. 



LESSONS IN ARITHMETIC. XXXII. 



RULE OP THREE SINGLE AND DOUBLE. 

 1. THIS is a name given to the application of the principles of 

 Simple Proportion to concrete quantities. We have shown 

 (Art. 5, Lesson XX., Vol. I., page 343) that if any three num- 

 bers be given, a fourth can always be found such that the four 

 numbers shall be proportionals. Hence, if three concrete 

 quantities be given, two of which are of the same kind, and the 

 third of another kind, a fourth quantity of the same kind as 

 the third can be found such that it shall bear the same ratio 

 to the third quantity as the first two bear to each other; 



or, what is the same thing, so that the four quantities shall be 

 proportionals. 



It is evident, since a concrete quantity can only be compared 

 with another of the same kind (06s. 11, Lesson XXVII.. Vol. II., 

 page 102), that the fourth quantity determined must be of the 

 same kind as the third quantity. In order that the ratios of 

 the two pairs of quantities may be equal, either two must be 

 of one kind and two of another, or all four must be of the same 

 kind. 



2. Suppose we have the following question proposed : 

 EXAMPLE. If the rent of 40 acres of land be .95, what will 



be the rent of 37 acres ? 



It is evident that the sum required must bear the same ratio 

 to 95 that 37 acres do to 40 acres. 



Hence we have, writing the ratios in the form of fractions, 



Sum required. 37 acres 37 



- = ^ = the abstract number 



95 40 acres 40. 



Therefore the sum required = |-J x 95, which can be reduced 

 to pounds, shillings, and pence. 



3. The last question might also have been solved thus : 



Since 40 acres cost 95, 

 1 acre costs |H ; 



And therefore 37 acres cost 



37 x 95 

 40 



pounds. 



4. In solving such a question by the Rule of Three, the state- 

 ment of the proportion is generally written thus : 



acres, acres. 

 40 : 37 : : 95 : sum required. 



Then, by equating the product of the extremes and means, we 

 get the result. We have put the first example, however, in 

 the fractional form, in order to indicate clearly the fact that 

 the ratio of the two quantities of the same kind (acres in this 

 case) is an abstract number, by which the other quantity, the 

 .95, is multiplied. When we state the question in the second 

 way, and talk about multiplying the means and extremes 

 together, some confusion might arise from the idea of multi- 

 plying 37 acres by .95. The fact to be borne in mind is that 

 the rule is merely the expression of the fact that the ratios of 

 two pairs of quantities are equal. 



5. The example we have given is what is called a case of 

 direct Proportion that is to say, if one quantity were increased, 

 the corresponding quantity of the other kind would be increased. 

 Thus, if the number of acres were increased, the number of 

 pounds they cost would be increased. 



If, however, the case be such that, as one of these corre- 

 sponding quantities be increased, the other is proportionally 

 diminished, the case is one of what is called Inverse Proportion.. 

 For instance : 



EXAMPLE. If 35 men eat a certain quantity of bread in 20- 

 days, how long will it take 50 men to eat it ? 



Here, evidently, the more men there are, the less time will 

 they take to eat the bread ; hence, as the number of men 

 increases, the corresponding quantity of the other kind viz., 

 the number of days decreases. 



Hence, since 50 men are more than 35 men, the required 

 number of days will be fewer than the 20 days which corre- 

 spond to the 35 men. 



In stating the proportion, therefore, in order to make the 

 ratios equal, if we place the larger of the two terms of one ratio 

 in the first place, we must place the larger of the two terms of 

 the other ratio in the third place. 



Thus, placing 50 men in the first place, we must put 20 

 days (which, we can see, will be larger than the required 

 answer) in the third place, and then the statement would be 

 correctly made thus : 



50 : 35 : : 20 days : required number of days. 



Therefore the required number of days = 20 x f J- days = 14 days. 



N.B. We might reduce the example to a case of Direct 

 Proportion thus, which will, perhaps, explain the above method 

 more clearly : 



35 men eat 

 50 



of the bread in one day. 



required number of days 

 Hence, since the quantity eaten in one day will increase with 

 the number of men, we have 



As 35 : 50 : : 5 \? : 3 ' 



required number of days ; 



Therefore required time = 20 x -fj = 14 days, as before. 



