OF QUANTITY AND NUMBER. 9 



39. Definition. In decimal arithmetic, each figure 

 is supposed to be multiplied by that power of iO, positive 

 or negative, which is expressed by its distance from the 

 figure before the point. 



Thus 672.53 means 6 X 102 +7 X 10' +2 xlO«, or 2x1, +5x10"', 

 or ^ or ^+3x lO^S or ^, togetlier 67^. 



Scholium. On some occasions other numbers are substituted for 

 10 in calculations : particularly 12, which has many advantages, and 

 is used in operations respecting carpenter's work ; and sometimes 

 tlie number 2 facilitates computations ; and it may be employed 

 where it is inconvenient to multiply characters ; since two different 

 marks, or a mark and a vacant place, are sufficient, when continu- 

 ally 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 divided by that number. 



Scholium. Mr. Barlow has inserted an ample table of reciprocals 

 in his very useful collection of Tables. 



41. Definition. The harmonic mean of two quanti- 

 ties is the quantity of which the reciprocal is the half sum 

 of their reciprocals. 



Thus, the harmonic mean of 3 and 6 is 4 ; for ^ Q+|)zz{. And the 

 harmonic mean is equal to the product divided by the half sum. 

 Thus f =4. 



42. Definition. The common logarithm of a num- 

 ber is that power of 10 which expresses it. 



For instance, ^ 1000=3, since 10="= 1000. 1 2=.30103, for 

 lQmmozz.2. The principal use of logarithms is derived from that pro- 

 perty of indices, by which their addition and subtraction is equivalent 

 to the multiplication and division of the respective numbers. 



43. Problem. To solve a quadratic equation. 

 Reduce the equation to the form arxdiaarzzft, add the square of half 



a; then a:2:±«a:-f— =6-}-— , whence a:±-|-= ± s/ (*+^) and xzz 



±v(H^)h:^. 



