156 



THE POPULAR EDUCATOR. 



CASE I. CHANGES IN THE ORDER OF THE TERMS. 

 If four quantities are proportional, the order of the means, or 

 of the extremes, or of the terms of both couplets, may be inverted 

 without destroying the proportion. 



Thus, if a -. b -. : c : d, and 12 : 8 : : 6 : 4, then, 



the 1st is to the 3rd 

 as the 2nd to the 4th. 

 the 4th is to the 2nd 



1. Inverting the means,* 



2. Inverting the extremes, 



as the 3rd to the 1st. 

 the 2nd is to the 1st 

 as the 4th to the 3rd. 



3. Inverting the terms of 



each couplet,-^ 



4. We may change the order of the two couplets. 



Cor. The order of the whole proportion may be inverted. 



N.B. If the terms of only one of the couplets are inverted, 

 the proportion becomes reciprocal or inverse. 



If a : b . . c -. d, then a is to 6, reciprocally or inversely, as d 

 to c. 



CASE II. MULTIPLYING OR DIVIDING BY THE SAME 



QUANTITY. 



If four quantities are proportional, two analogous or two 

 homologous terms maij be multiplied or divided by the same 

 quantity, without destroying the proportion. Thus, 



If a : b . : c -. d, then, if analogous terms are multiplied or 

 divided, the ratios will not be altered. 



1. ma : mb -. -. c -. d. 2. a : b : : me ; md. 



3 - - 

 'mm 



c : d. 



4. 



d 



If homologous terms be multiplied or divided, both ratios will 

 be equally increased or diminished. 



5. ma . b . . me : d. 6. a : mb -. : c : md. 



d. 



C-. d . 



m 



Cor, All the terms may be multiplied or divided by the same 



quantity. Thus, ma : mb -. : me : md, or 



d 

 m 



then a . b -. -. c : d , or a : c -. : b : d. 

 then a : b : : c : d, or a : c : : b : d. 

 thena-.b>c-.d. (Euclid V.,Def. 13.) 



CASE III. COMPARING ONE PROPORTION WITH ANOTHER. 



If two ratios are respectively equal to a third, they are equal 

 to each other. (Euclid V., Def. 11.) 



This is nothing more than an application of the axiom, that 

 things which are equal to the same are equal to one another. 



1. If a : b : : in : n 

 And c : d . : m . n 



2. If a : b : : m : n 

 And m : n : : c : d 



Cor. If a . b : : m : n 

 m . n > c : d 



For if the ratio of m : n is greater than that of c -. d, it is 

 manifest that the ratio of a -. b, which is equal to that of m : n, 

 is also greater than that of c : d. 



N.B. In these instances, the terms which are alike in the two 

 proportions are the first two and the last two, and the resulting 

 proportion is uniformly direct. But this arrangement is not 

 essential. The order of the terms may be changed in various 

 ways, without affecting the equality of the ratios. 



The proposition to which these instances of equality belong is 

 usually cited by the words, " ex aequo," or " ex wquali." (Euclid 

 V., Def. 23.) 



Any number of proportions may be compared in the same 

 manner, if the first two or the last two terms in each preceding 

 proportion are the same with the first two or the last two in the 

 following one. 



Thus, if a : b -. -. c . d 

 And C : d : : h : I 



And h : I : -.m-.n 

 And m-.n: : x : y 



That is, the first two terms of the first proportion have the 

 aame ratio as the last two terms of the last proportion. For it 

 is manifest that the ratio of all the couplets is the same. 



But if the two means or the two extremes in one proportion 



* This is called alternation. (Euclid V., Def. 16.) 

 t This is technically called inversion. 



then a : b -. -. x : y. 



be the same with the means or the extremes in another, the four 

 remaining terms will be reciprocally proportional. 



If a . m -. : n : b 



And c : m : : n : d 



then a -. c . . - . -, or a -. c . . d : b. 

 b d 



' \ therefore ab = cd, and a : c -. : d -. b. 

 = mn \ 



For ab = mn 

 And cd 



In this example, the two means in one proportion are like 

 those in the other. But the principle will be the same if the 

 extremes are alike, or if the extremes in one proportion are like 

 the means in the other. 



If m . a -. 

 And m : c . 

 Or if a -. m ; 

 And m -. c \ 



: b : n 

 :d:n 

 -.n-.b 

 :d-.n 



then a -. c : : d : b. 



then a : c -. : d : b. 



The proposition in Geometry which applies to this case is 

 usually cited by the words, " ex cequo perturbata." (Euclid 

 V., Def. 23.) 



1. 1, 3 and 5. 



2. 3, 5 and 7. 



EXERCISE 69. 



3. 14, 10, 6 and I 4. 3, 5, 7 and 9. 

 2. I 5. 1, 5, 9 and 13 



LESSONS IN MORAL SCIENCE. III. 



INTUITIVE OR SENTIMENTAL MORALITY. 



THE innate school of moralists are further accustomed to rely 

 upon the universality of the moral faculty, or the power of dis- 

 criminating between two classes of actions, one right and the 

 other wrong, as a proof that it is rot the result of any expe- 

 rience, however extended, but was inherent in our minds at their 

 first coming into being. It is to be found, they say, amongst 

 all nations and in all ages of the world's history, always the 

 same in kind, however there may be a difference in the strictness 

 with which it is thought proper actions should conform to the 

 test which it supplies. 



It has, of course, on the other hand been argued that certain 

 nations have, at certain periods, sanctioned or even commended 

 actions which all civilised persons now look upon as heinous 

 crimes. Even down to the present day, child-murder in India 

 is openly practised without shame or scruple ; and numerous 

 instances of the same kind, if space permitted, could easily be 

 brought forward, even from the history of our own country. 

 Hence, it is said, this apparent universality of the moral sense 

 is only apparent after all. 



It may be, and often is, argued that such perverted ideas only 

 exist amongst savage and uncivilised peoples, and that these 

 are not to be taken as the test by which to pronounce an 

 opinion upon mankind in general, or upon what ideas they do 

 or do not universally possess. The true test is to be found 

 only amongst civilised nations ; and in process of time, as 

 nations develop from the savage state to a state of civilisation, 

 all this perversion gradually disappears, and they tend to the 

 adoption of one common standard of morality. Against this, 

 however, it may be fairly said, as it is stated by Professor Bain, 

 in his well-known work on " Mental and Moral Science," that 

 " this argument would have great weight, in any discussion as 

 to what is good, useful, expedient, or what is in accordance 

 with the cultivated reason or intelligence of mankind ; because 

 civilisation consists in the exercise of men's intellectual faculties 

 to improve their condition. But in a controversy as to what 

 is given us by Nature, what we possess independently of 

 intelligent search and experience, the appeal to civilisation 

 does not apply. What civilised men agree upon amongst 

 themselves, as opposed to savages, is likely to be the reverse 

 of a natural instinct; in other words, something suggested 



