228 Notices respecting New Books. 



bing briefly our author's treatment of the question of multiplication, 

 to which his third chapter is devoted. In the first place he explains, 

 by means of two examples, Oughtred's method of contracted mul- 

 tiplication ; but instead of leaving the last figure uncertain as is 

 usually done, he notices that the process gives an approximation 

 in defect, and points out that if the sum of the digits in the mul- 

 tiplier, which give partial products, be increased by the first un- 

 used digit of the multiplier, and by unity if there be a second 

 unused digit in the multiplier, and then this sum be added to the 

 product, we now have an approximation in excess ; and by compa- 

 ring the two we obtain a result in which a certain number of digits 

 are known to be exact. Moreover the rule, as usually stated, 

 directs that if, for instance, the result were required to be true for 

 two places of decimals, the unit digit of the multiplier should be 

 placed under the second decimal digit of the multiplicand ; our 

 author notices that it should usually be placed under the third 

 decimal digit, and in certain circumstances under the fourth or 

 fifth, and so on in other cases. He next enters on the question, 

 Given that the factors are incommensurable numbers, to what 

 degree of approximation must they be known that the first n digits 

 of the product may be exact ? He first shows that when the factors 

 are approximate by defect, the relative error of their product is less 

 than the sum of the relative errors of the factors ; and then 

 reasons as follows : — Suppose that there are^> factors (p being less 

 than 10), calculate each factor to n+1 digits; then the relative 



error of each factor is less than — — - , and consequently the sum 



of their relative errors will be less than —— , and their product will 



have the first n digits exact. He also observes that if the first 

 digit of the required product is known before hand, it is, under 

 certain circumstances (which he specifies), enough to calculate 

 some of the factors to n digits. 



It will be evident from this that the author is quite justified in 

 thinking that "he has given completeness, in the present work, to 

 methods laid down by other writers." He states that the work 

 was originally published as an appendix to that on curved lines 

 noticed above, that it has been carefully revised, and contains 

 several important additions ; amongst others is a complete solution 

 of the question, How many digits of a number must be known in 

 order that its with, root may have its first n digits exact ? 



