THE POP UTAH EDUCATOR 



6. We add, for the sake of illustration, a table of the powers 

 of the number 3 : 

 3 x 1 = 3 l = 3, the 1st power. | 3 6 x 3 = 3' = 2187, the 7th power. 



729- = 3 = 6th root of 729. 

 243* = 3 = 5th 243. 



81* = 3 = 4th 81. 



27* = 3 = 3rd 27. 

 9- - 3 = 2nd 9. 



7. The following is a table of the fractional indices by which 

 the relation of the root 3 to its powers is indicated : 



531441" = 3 = 12th root of 531441. 



177147^ = 3 = llth 177147. 



59349T" = 3 = 10th 59049. 



19383* = 3 = 9th 19683. 



6561' = 3 = 8th 6561. 



2187^ = 3 = 7th 2187. 



8. We have pointed out that G 5 indicates that five sixes have 

 been multiplied together to form the quantities which it repre- 

 sents ; and similarly with G 4 . Hence it is obvious that to 

 multiply G 5 by 6 4 we should have to multiply the product of 

 five sixes by the product of four sixes obtaining, obviously, 

 the product of nine sixes, or 6 9 . Hence a simple rule to multiply 

 two powers of the same number : Add their indices. 



9. The same rule applies for fractional indices that is, for 

 roots; thus 



729* = 3 ; 729* = 9. 



* = 



729 x 729 = 9 x 3 ; or 729 



* 



= 729 = 27. 



10. In a similar way, the division of quantities expressed in 

 the form of powers of the same number is accomplished by the 

 subtraction of the less from the greater index. Thus G 5 indi- 

 cates five sixes multiplied together ; 6 4 the same for four sixes. 

 Hence, if 6 4 be written as a denominator, it is evident that 

 the four sixes of which it is composed will cut out four of those 

 of which 6 3 is composed, and leave in the numerator only 1 (or 

 5 - 4) ; thus 



6 6x6x6x6x6 



6x6x6x6 



_ 



Hence the above rule To divide one power of a number ly 

 another, subtract the lesser from the greater index. 



11. The same rule holds for fractional indices that is, for 

 roots ; thus 



729* +.729* = 9 -;- 3; or 729* ~ * = 729* = 3. 



12. We have seen that the multiplication or division of powers 

 of a number is effected by the addition or subtraction of their 

 indices. We naturally ask, what is the effect if indices be 

 multiplied together ? We shall answer this question most easily 

 by remembering that multiplication is only an abbreviated form 

 of addition. Thus, if we multiply 2 by 4, we do in reality only 

 add together four twos : 



4x2 = 2 + 2 + 2 + 2. 



And so, if we have 6 2 and 6 4 , and multiply together their indices, 

 we have in reality done the same as if we had added the indices 

 of G 2 , 6 2 , (J 2 , 6-, or of 6 4 , G 4 that is, we have done the same as 

 raise 6 2 to its fourth power, or 6' to its second power. Hence 

 it is obvious that when the index of any power is multiplied by 

 any quantity, that power is itself raised to the power of that 

 quantity ; thus 



6" x * = (6 s )* = 36*; and 6* X2 = (G 1 ) 2 = 12962. 



13. We are now in a position to determine the meanings of 

 fractional indices whose numerators are not unity ; thus 



6*= (6*J* = (6,* s 

 that is, the fourth power of the cube root of 6, or the cube root 

 of the fourth power of 6. As an example, take 



27^ = (27 s )* = (729)* = 9 ; or 27* = (27*' 2 = (3) a = 9 



14. It will be observed that we have made no reference to 

 the index 0. Remembering that any number divided by itself 

 gives unity as a quotient, we have 



6 6 + 6 = 6' = 6 = 1. 



Hence we arrive at the apparent paradox that any number raised 

 to the zero power is equal to unity an arithmetical curiosity, 

 which the reader must be content to receive without further 

 explanation. 



LESSONS IN MORAL SCIENCE. I. 



RISE OF SCIENCE OF ETHICS PLATO AKISTOTLE LEADING 



PRINCIPLES OF MORAL SCIENCE MODERN WRITERS ON 

 ETHICS. 



FROM the very earliest times there have been recognised in 

 every country, both by society at large and by the individual, 

 certain rules of right and wrong. No doubt these have not 

 been the same everywhere ; the actions which in one age and in 

 one country have been looked upon with little, if any, disfavour, 

 have been in another age or in another country viewed in a very 

 different light. But yet it is true that, universally, there has 

 existed a standard of some sort by which actions have been 

 judged, and that, according to their conformity or non-conformity 

 to this, they are visited with praise or blame. Now, what is the 

 true nature of this rule ? whence are men led to seek after it and 

 to judge actions by it ? what is its origin, and within what 

 limits is it confined ? These, and questions like them, are those 

 which the science of Ethics investigates and seeks to determine ; 

 and it is in the answers given that the different ethical systems 

 differ from each other. 



Although of ancient origin, Ethics did not engage the atten- 

 tion of mankind at as early a period in the history of the world 

 as many other sciences of less real importance. The phenomena 

 of external nature, and the sciences which aim at turning them 

 to practical account, naturally engaged the attention of man 

 before he commenced to turn his mind in upon itself and 

 examine the laws by which he thinks, or in conformity to whick 

 he acts. Still, Ethical Science was cultivated very largely in 

 ancient times. Many of the Greek philosophers not only wrote 

 much, but thought deeply, upon the subject ; and the germs of 

 many, if not of all, the most elaborate modern systems of Ethics 

 may be found in Plato, the earliest moral philosopher whose 

 writings have come down to us ; though he quotes many prede- 

 cessors of whom nothing has survived except their names. 



Aristotle, who came next, seems, indeed, to have differed with 

 Plato more in words than in reality ; and, after him, the progress 

 of ethical philosophy was considerably advanced in the disputes 

 between the rival schools of the Stoics and Epicureans. 



The Eomans borrowed their notions of moral philosophy, as 

 they did so much else, from the Greeks ; but contributed little, 

 if anything, themselves to the most important principles of the 

 science. 



For nearly two thousand years after the close of ancient 

 philosophy, the history of Ethics presents a blank, until the 

 time of the schoolmen, who, directly or indirectly, first revived 

 in modern times many of the most important qiiestions in 

 morals ; although they hardly treated at all, or left entirely out 

 of view, many of its most essential features. It is, however, 

 from the writings of Hobbes, in the early part of the seventeenth 

 century, that the commencement of modern Ethics must be 

 dated ; and from that period downwards there has been a con- 

 tinuous succession of ethical writers, comprising Cudworth, 

 Malebranche, Butler, Hume, Smith, Hartley, Bentham, Mackin- 

 tosh, and a multitude of others, some of whom we may subse- 

 quently have occasion to refer to more particularly. 



In order to obtain a general conception of the leading prin- 

 ciples of Ethical or Moral Science, we may consider it aa 

 involving two great questions: Firstly, "What is virtue?" 

 Secondly, " What is it in our minds which recommends virtus to 

 us for our adoption ?" 



If we thoroughly understand the meaning of these two ques- 

 tions, and the different answers which have been from time to 

 time given to them by different systems, we shall have become 

 acquainted with the chief features of Ethical Science. Before, 

 however, we proceed to consider these questions in detail, it is 

 well that we should see how far the various principles of our 

 nature can be classified. 



The Epicurean theory, which was, perhaps, the oldest of at, 

 admitted only one principle of human nature, into which every 

 other was capable of being ultimately resolved. In the system 

 of Epicurus our sole principles of action were the appetites, 

 which had their origin in the body, and the means of gratifying 

 which were furnished by the senses ; and the sole object of 

 man's existence was to gratify these sensual appetites. Such a 

 theory as this is contradicted by the simple but obvious fact 

 that men do not gratify such appetites only : for instance, 

 they are often influenced by ambition and compassion. True, 



