PRO 



PRO 



word London, the answer may be, " no, 

 I shall go to Richmond only." 



Quintilian advises his pupils to study 

 the principles of pronunciation under a 

 comedian. There are three things which 

 come under the pronunciation, viz. the 

 memory, voice, and gesture. 



PROOF, in arithmetic, an operation 

 whereby the truth and justness of a calcu- 

 lation is examined and ascertained. The 

 proper proof is always by the contrary 

 rule : thus subtraction is the proof of ad- 

 dition, and multiplication of division ; and 

 vice versa. 



PROOF, in military affairs, is a trial 

 whether the piece will stand the quantity 

 of powder allotted for that purpose. 



PROPAGO, in botany, properly a slip, 

 layer, or cutting of a vine or other tree. 



PROPORTION. When two quantities 

 are compared one with another, in respect 

 of their greatness or smallness, the com- 

 parison is called ratio, reason, rate, or 

 proportion; but when more than two 

 quantities are compared, then the com- 

 parison is more usually called the propor- 

 tion that they have to one another. The 

 words ratio and proportion are frequent- 

 ly used promiscuously. When two quan- 

 tities only are compared, the former term 

 is called the antecedent, and the latter 

 the consequent. The relation of two ho- 

 mogeneous quantities one to another, 

 may be considered either, 1. By how 

 much the one exceeds the other, which 

 is called their difference. Thus 5 exceeds 

 3 by the difference 2. Or, 2. What part 

 or parts one is of another, which is called 

 ratio. Thus the ratio of 6 to 3 is ^=, 

 or double ; and the ratio of 3 to 6 is =, 



er subduple. 



When two differences are equal, the 

 terms that compose them are said to be 

 arithmetically proportional. Thus, sup- 

 pose the term to be a and b, their differ- 

 ence d. If a be the last term, then a-f d 

 And it <i be the greatest, then a 



is found by dividing the antecedent by the 

 consequent, and the quotient is the expo- 

 nent or denominator of the ratio. 



If, when tour quantities are considered, 

 you find that the first hath as much great- 

 ness or smallness in respect to the second, 

 as the third hath in respect to the fourth : 

 those four quantities are called propor- 

 tionals, and are thus expressed : 



But when two ratios are equal, the terms 

 that compose them are said to be geome- 

 trically proportional. For suppose a and 

 b to be the terms of any ratio ; if a be the 



least term, put r=-,thenar=byequal 

 multiplication : but if b be the least term, 

 put r=j, then b r==a by equal multipli- 

 cation, and-=6 by equal division. 



Thus the ratio of two quantities, or of 

 two numbers, in geometrical proportion, 



tains B=2 four times, so C=16 contains 

 D.=4, four times and, therefore, A has 

 the same ratio to B as C has to D; and, 

 consequently, these four quantities having 

 equal ratios, are proportionals. 



Proportion consists of three terms at 

 least, whereof the second supplies the 

 place of two. 



When three magnitudes, A, B, C, are 

 proportional, the first, A, has a duplicate 

 ratio to the third, C, of that it hath to the 

 second, B: but when four magnitudes, 

 A, B, G, D, are proportional, the first, A, 

 has a triplicate ratio to the fourth, D, ot* 

 what it has to the second, B ; and so al- 

 ways in order one more, as the proportion 

 shall be extended. 



, A A 



Duplicate ratio is thus expressed, Q= g 



twice ; that is, the ratio of A to C is du- 

 plicate of the ratio of A to B. For let 

 A 2, B=4, C=8 : then the ratio of 2 to 

 8 is duplicate of the ratio of 2= A to 

 B=4, or as the square of 2 to the square 

 of 4. 



Triplicate ratio is thus expressed, =g 



thrice ; that is, the ratio of A again == 2, 

 to D = 16, is triplicate of the ratio of A 

 = 2 to B = 4, or as 8 the cube of 2, to 

 64 the cube of 4. Wherefore duplicate 

 ratio is the proportion of squares, and tri- 

 plicate that of cubes. 



And the ratio of 2 to 8 is compounded 

 of the ratio of that of 2 to 4, and of 4 to 

 8. From what has been said of the na- 

 ture of ratio and proportion, the six ways 

 of arguing, which are often used by ma- 

 thematicians, will evidently follow. 



1. Alternate proportion is the compar- 

 ing of antecedent to antecedent, and con- 

 sequent to consequent. As if 



therefore alternate- 

 ly, or by permutation, as 

 A : C :: B : D 

 2 : 8 :: 4: 16 



2. Inverse ratio, is when the consequent 

 is taken as the antecedent, and so compar- 



