ARITHMETICAL PROGRESSION ] 



MATHEMATICS. ALGEBRA. 



477 



From what is here said, you see that two equal fractions 

 may always be converted into a proportion, and a pro- 

 portion into two equal fractions. All the properties of 

 proportional quantities may therefore be derived from 

 those of equal fractions: the following aro the niosl 

 useful : 



1. In four proportional numbers, a : 5 : : e : d, tho 

 product of the extremes is equal to the product of the 



means : for since - = ?... ad = be. 

 b d 



Conversely, if the product of one pair of factors be 

 equal to the product of another pair, the four factors aro 

 in proportion : the factors of one product being the means, 

 and those of the other the extremes : thus, if ad = bc, 

 then o : 6 : : c : d ; because from ad = be, we get, by 



dividing each side by db, - = 1 . 

 b d 



3. If four quantities are proportional, they are propor- 

 tional when taken inversely ; that is, the second is to the 

 first as the fourth to the third : thus, if a : 6 : : c : d, then 



6 : a : : d : e ; because from T = -j we get, by taking the 



. 6 d 



reciprocals, 



' o c 



4. They aro also proportional if taken alternately, 

 provided the four quantities are all of the same kind ; 

 that is, the first Ls to the third, as the second to the 

 fourth : thus, if o : 6 : : c : d then a : c : : 6 : d if the 



quantities are alike in kind ; because from r = -3 we 



get, by mutiplying by -,-=-. If a and c were quan- 

 c c d 



titles of different kinds, as for instance, pounds and 

 yards, they could of course have no ratio ; ratio being 

 always an abstract number expressing how many times 

 the antecedent is contained in the consequent. 



5. Three quantities a, b, c are in proportion when they 



a b 

 supply the equality 7 = - ! that is, when a : 5 : : b : e. 



It is plain that three such quantities must always be of 

 the same kind ; the first of them is to tho third as the 

 square of the first to the square of tho second ; that is, 



a b 



a : c : : a 2 : 6 2 ; because from , -, we get, by multi- 



6 a 6* a 2 

 plying by -'^="^ = ^2' 



6. It is further obvious that you may multiply or 

 divide the first and second terms of a proportion by any 

 number, and tho third and fourth by any number, with- 



a c 

 out disturbing the proportion ; for -, = g is the same as 



-f = f And that the first and third may be multiplied 



or divided by any number, as also tho second and fourth ; 



a c ma me a c 



for from r = -, we get -,- =r^' Moreover, since r = -. 



a m c m 

 leads to ,- = ^t whatever bo TO, it follows that the 



same powers or roots of four proportional numbers are 

 also in proportion. 



7. The terms of two proportions, when they aro num- 

 bers, may also be multiplied together ; that is if 



db cd a 



which give ,- = -T/ or - = 



y 



and 

 then 



a 

 e 



6 : : c 

 9 



d 

 h 

 dh 



for this is only multiplying the equations r-y> - f =7; 



together. 



6 ff 





8. If a : b : : c : : <*j y f ^^ {ollowg 



andb : e : :d:f) 



from multiplying together the equations 



e b 

 y ~ 



d 

 ~f 



By thus deducing different equations from the equal 



(I C 



fractions or ratios r = ^ an endless variety of sets of 



proportionals may be obtained ; but the most general of 

 these deductions is derived from the following principle, 

 namely : 



If two fractions, r-' -7 are equal, then we may replace 



the terms a, 6 of the first by any expressions involving 

 a and 6 that are homogeneous in reference to these quan- 

 tities, provided we also replace the terms c, d by ex- 

 pressions involving c and d in the same manner. For 

 instance 



2o 2 3o& + & 2 is homogeneous as respects a and 6: 

 each term being of two dimensions ; so also is 5ab + 4ot 3 

 2b~ ; we may therefore substitute these for a and 6, 

 provided we put the similar expressions 2c 3 3cd -f- d' 1 

 and 5cd + 4e" '2d 2 for c and d ; that is, we may infer 

 that because 



? = f 2a " 3<i6 + & 3 = 2e 2 3cd + d* 

 d d "5a!> + 4a 2 26 2 5ed + 4c 2 2<i 2 ' 

 The reason of this is pretty obvious ; two fractions can- 

 not be eqtial unless one is convertible into the other by 

 multiplying num. and den. of the former by some factor 

 (m) ; so that in the above, c must be = ma and d = mb. 

 Now, if in the second of the changed fractions above,. you 

 put ma, mb for their equals c and d, you will see at once 

 that that fraction will be nothing but Hie first fraction 

 with its num. and den. multiplied by m 2 . If the homo- 

 geneous expressions chosen for the terms of the first frac- 

 tion had been of three dimensions, then after the substi- 

 tution of ma, mi, for c, d in the similar terms of the 

 second fraction, the result would have differed from tho 

 first fraction only by the num. and den. being multiplied 

 by m 3 , and so on, as is obvious. The particular case of 

 this general theorem which is most frequently employed 

 is this, namely : 



a_c . a 4-m& 



or, from a : 6 : : c : d 

 .', a + mb : a -f> nb : : c -f- md : e + nJ 



where the values of m and n aro arbitrary. In most 

 applications they are chosen each equal to 1, or one equal 

 to 0, and tho other equal to 1. Wo need scarcely men- 

 tion, thut when any of the conditions of a question are 

 expressed by a proportion, the product of tho extremes 

 equated to the product of the means converts the pro- 

 portion into on equation. 



ARITHMETICAL AND GEOMETRICAL PROGRESSIONS. 



Arithmetical Progression. 



An arithmetical progression is a row, or series of quan- 

 tities, such, that each quantity, after the first, is merely 

 the preceding quantity with some invariable value added 

 to it, or subtracted from it : thus, 1, 3, 5, 7, &c., aud 1, 

 1, 3, 5, <Jrc., are arithmetical progressions ; the 

 terms after the first in tho former are obtained, each 

 from the preceding, by adding 2 ; and in the latter by 

 subtracting 2. The constant quantity thus added or sub- 

 tracted is called the common difference : it is usually 

 denoted by d ; and as d may stand for a quantity either 

 positive or negative, tho following will be the general 

 way of expressing an arithmetical progression viz.: 



d, a + 2d, a + 3d, a + 4d, .... a +(n l)d; 



where n stands for the number of the terms : the last 

 term being evidently derived from the first by adding to 

 .t as many times d as there are terms after the first ; that 

 a n 1 times d ; so that calling the last term I, we have 

 ! = a+(v l)d. You thus see that tho problem 

 3iven tho first term, tho common difference and the 

 number of terms, to find the last term, is one of very 

 easy solution. We shall therefore proceed to ,the next, 

 which is tliis : . 



