94 



THE POPULAR EDUCATOR, 



EXERCISE 67. 



1. Which is the greater, the ratio of 11 : 9, or that of 44 : 35 ? 



2. Which is the greater, the ratio of a + 3 : Ja, or that of 2o, + 7 : jjd ? 



3. If the antecedent of a couplet be 65, and the ratio 13, what is the 

 consequent ? 



4. If the consequent of a couplet be 7, and the ratio 18, what is the 

 antecedent ? 



5. What is the ratio compounded of the ratios of 3 : 7, and 2a : 5b, 

 and 7x + 1 : 3y - 2 ? 



6. What is the ratio compounded of ai + y : 6, and x y : a + b, and 

 o+b: h? 



7. If the ratios of 5o: + 7 : 2# 3, and se + 2 : ^.t + 3 be compounded, 

 will they produce a ratio of greater inequality, or of less inequality ? 



8. What is the ratio compounded of x + y : a, and x y -. b, and 

 a; 4 y 1 



a 



9. What is the ratio compounded of 7 : 5, and the duplicate ratio of 

 4 : 9, and the triplicate ratio of 3 : 2 ? 



10. What is the ratio compounded of 3 : 7, and the triplicate ratio 

 of x : y, and the subduplicate ratio of 49 : 9 ? 



PROPORTION. 



When four quantities are related to one another in such a 

 manner that the first divided by the second is equal to the third 

 divided by the fourth in other words, when the ratio of the 

 first to the second is equal to the ratio of the third to the fourth, 

 the four are said to be in direct proportion. From this definition 

 it will be seen that proportion is simply the equality of ratios. 

 Though we have only spoken of two equal ratios, there may be 

 any number, and in all cases the terms of these ratios are said 

 to be in direct proportion. 



Care must be taken not to confound proportion with ratio. 

 This caution is the more necessary, as in common discourse the 

 two terms are used indiscriminately, or rather, proportion is 

 used for both. The expenses of one man are said to bear a 

 greater proportion to his income than those of another. But 

 according to the definition which has just been given, one pro- 

 portion is neither greater nor less than another. For equality 

 does not admit of degrees. One ratio may be greater or less 

 than another. The ratio of 12 : 2 is greater than that of 6:2, 

 and less than that of 20 : 2. But these differences are not ap- 

 plicable to proportion, when the term is used in its technical 

 sense. The loose signification which is so frequently attached 

 to this word, may be proper enough in familiar language ; for 

 it is sanctioned by general usage. But for scientific purposes, 

 the distinction between proportion and ratio should be clearly 

 drawn and cautiously observed. 



Proportion may be expressed, either by the common sign of 

 equality, or by four points between the two couplets. 



' 8 " 6 = 4 " 2, or 8 " 6 : : 4 " 2 | are arithmetical 

 d, or a " b : : c d ) proportions. 

 : 4, or 12 : 6 : : 8 : 4 \ are geometrical 

 : h, or a :b . : d :h) proportions. 



The latter is read, " the ratio of a to b equals the ratio of 

 d to h ;" or more concisely, " a is to b as d to h." 



The first and last terms are called the extremes, and the 

 other two the means. Homologous terms are either the two 

 antecedents or the two consequents. Analogous terms are the 

 antecedent and consequent of the same couplet. 



As the ratios are equal, it is manifestly immaterial which of 

 the two couplets is placed first. 



Thus 



. , 



a " b = c " 



( 12 : 6 = 8 



(_ a : b = d 



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



T-I .. a c ,-, c a 



For if - = -, then - = . 



b d d b 



The number of terms in a proportion must be at least four. 

 For the equality is between the ratios of two couplets ; and each 

 couplet must have an antecedent and a consequent. There 

 may be a proportion, however, between three quantities; for 

 one of the quantities may bo repeated, so as to form two terms. 

 In this case the quantity repeated is called the middle term, or 

 a mean proportional between the two other quantities, especially 

 if the proportion is geometrical. 



Thus the numbers 8, 4, 2, are proportional. That is, 8 : 4 : : 

 4 : 2. Here 4 is both the consequent in the first couplet, and 

 the antecedent in the last. It is therefore a mean proportional 

 between 8 and 2. 



The last term is called a third proportional to the two other 

 quantities. Thus 2 is a third proportional to 8 and 4. 



Inverse or reciprocal proportion is an equality between a 

 direct ratio and a recijn-ocal ratio. 



Thus 4 : 2 : : | : i ; that is, 4 is to 2 reciprocally, as 3 to 6.. 

 Sometimes, also, the order of the terms in one of the couplets is 

 inverted, without writing them in the form of a fraction. 



Thus 4 : 2 : .- 3 : 6 inversely. In this case, the first term is 

 TO the second, as the fourth to the third; that is, the first 

 divided by the second, is equal to the fourth divided by the 

 third. 



When there is a series of quantities, such that the ratios of 

 the first to the second, of the second to the third, of the third 

 to the fourth, etc., are all equal, the quantities are said to ba 

 in continued proportion. The consequent of each preceding 

 ratio is then the antecedent of the following one. 



N.B. Continued proportion is also called progression. 



In the preceding articles of this section, the general pro 

 perties of ratio and proportion have been defined and illustrated. 

 It now remains to consider the principles which are peculiar to 

 each kind of proportion, and attend to their practical appli- 

 cation in the solution of problems. 



ACOUSTICS. I. 



DIFFERENCE BETWEEN SOUND AND NOISE SOUND PRODUCED 

 BY VIBRATIONS HOW CONVEYED TO THE EAR CONDUC- 

 TION OF SOUND TELEPHONIC CONCERT CAUSES AFFECT- 

 ING INTENSITY. 



THE science of Acoustics, on the study of which we are now 

 about to enter, is concerned with inquiries as to the nature and 

 properties of sound, and the vibrations of elastic bodies. In 

 the human body a special nerve, called the auditory nerve, is 

 given off from the brain, and spreads out into a number of 

 minute filaments, which are distributed over the surface of one 

 of the cavities of the ear. When any elastic substance is put 

 in rapid vibration, it causes certain tremors or vibrations of the 

 air around ; these are conveyed to the ear, and acting upon this 

 nerve give rise to the sensation of sound. 



These sounds are very varied in character, and the science of 

 Music treats of the different effects they produce upon the 

 emotions. Acoustics is merely occupied with ascertaining the 

 nature and causes of these differences. 



Whenever the air or any elastic body is set in vibration, a 

 sound will be produced, provided that the vibrations be suf- 

 ficiently rapid : if they be too slow, the ear will be unable to 

 distinguish the sound ; different ears have, however, different 

 powers in this way, as will be explained shortly. The simplest 

 illustration of this fact is seen by fixing one end of a string to 

 a hook (Fig. 1), and suspending a weight from its lower end, at 

 the same time limiting the motion of the cord by means of a 

 ring, A. If now the cord be plucked near the middle by the 

 finger and thumb it will be set in vibration, forming the curves 

 shown by the dotted lines ; at the same time a distinct musical 

 note or tone will be produced. These vibrations will gradually 

 diminish in amplitude, and the sound will become weaker and 

 weaker till it ceases. The pitch of the note depends upon 

 various causes which will hereafter be explained. 



Frequently, however, a sound is produced which is net ^ 



