OF QVANTITY AND NUMBER. 



36. Definition. A series of numbers 

 formed by the continual addition of the same 

 number to any given number, is called an 

 arithmetical progression. 



2, 5, 8, 11, 14, 17, 20, by adding 3. 

 20, 17, 14, 11,' 8, 5, 2, by adding— 3. 



o, a+b,a+il;a+3t, a+ (n— l). I, in 



genetal. 



Scholium. It may be observed that the sura of each 

 pair of the numbers of these equal progressions is 

 22=2+20=a+a+ (n— l).t=2a + (m— l).i; the whole 

 sum 22X7= (20 + (n— l). V). n, and the sum of each, 



na + ^^. b, a being the first term, i the difference, 



2 



and n the number of terms. 



37. Definition. A series of numbers 

 formed by continual multiplication by a 

 given number, is called a geometrical pro- 

 gression. 



As 2, 0, 18, 54 ; multiplying 2 continually by 3. 

 a, ah, alb, ablb ; multiplying a by b. 



38. Definition. If one of the terms of 

 a geometrical progression is unity, the other 

 terms are called powers of the common mul- 

 tiplier. 



As 1^. ■^. h h v 1' 2' ■•' 8, 16, 32. Each term is de- 

 noted by placing obliquely over the common multiplier a 

 number expressive of its distance from unity, as 8r:2': ne- 

 gative numbers, implying a contrary situation to positive 

 ones, denote that the term precedes instead of follovfing the 

 unit, ■jl% i='2"'. 



By reversing the series it is obvious that |^(i)', and 

 2=(i)-'. 



It appears that the addition of the indices denoting the 

 places of any terms will point out a term which is their 

 product, as 2' X a'xia', or 8 X 4::=32 ; and that the subtrac- 

 tion of the index is equivalent to division by the term. 

 Hence if a'=:i'— i", a'«must be equal to I'i in order that 

 i 2 -f i 2 may make J'rra*. So that simple fractional num- 

 bers serve as indices of the number of times that the quan- 

 tity must be multiplied together, in order that the product 

 may be the common multiplier of the series, or the simple 

 number b. 



Scholium. Fractional powers are sometimes denoted 

 by the mark \/, meaning root; thus ^ono2,'v'a^a' 

 The second power of a number a, being called its square, 

 and the third its cube, the fractional powers arc called 

 square and cube roots. 



The sums of geometrical progressions may be thus com- 

 puted, if a-t-at-^-ai' . . . +ai"~':=x, ab-xab^+ab'. . . + 

 ab'^bx, and subtracting the former equation from the lat- 



ab" — a 



ter at" — a'^.bx — x, therefore .r:i: -; . Which, when 



b — 1 



i< 1 and nzzos , or infinite, becomes — — . 



i—b 



The binomial theorem, for involution, is (a4-i)"=a"-|- 



a^-^b'-i- . . 

 In simple cases, its truth may be shown by induction. 



, , n — 1 „., n — 1 n — 2 



n.a"~' b+n. . a''~^b-+n. . 



2 2 3 



POWERS OF NUMBERS. 



39. Definition. In decimal arithmetic, 

 each figure is supposed to be multiplied by 

 that power of lOi positive or negative, which 

 is expressed by its distance from the figure 

 before the point. 



Thus 672.53 means 6Xl0H7Xl0'-f-2Xl0'', or2Xl, 

 -1-5X10-', or^or-j''<^-|-3Xlo-«, or ife, together 672,Vo- 



Scholium. On some occasions other numbers are sub- 

 stituted for 10 in calculations: particularly 12, which has 

 many advantages, and is used in operations respecting car- 

 penter's work ; and sometimes the number 2 facilitates 

 computations; and it may be employed where it is incon- 

 venient to multiply characters; since two different marks, 

 or a mark and a vacant place, are sufficient, when conti- 

 nually repeated, to express all numbers. The powers of 

 60 arc also used in the subdivisions of time, and of angles. 



40. Definition. The reciprocal of a 

 number is the quotient of a given unit tU- 

 vided by that number. 



