436 On Multiplications, i^c. by a Table of Single Entry. 



of this is in Mr. J. T. Graves's valuable mathematical library at 

 Cheltenham. 



By logarithms the author intends the same quantities as I 

 term respondents, certainly a less objectionable and safer term 

 to employ. There appears to be an error in the title in affirming 

 that any two numbers, not separately exceeding 20,000, may be 

 multiplied by aid of these tables, as the sum of the two factors 

 ought not to exceed 20,000. Mr. Peter Gray, so favourably 

 known to an important section of the public as the author of 

 many useful tables,has informed me that Major Shortredd, now in 

 India, has computed a table of quadratic respondents extending to 

 the argument 200,000, which he is taking measures to have pub- 

 lished. Such tables would be very useful to computers, as they 

 would serve for the multiplication of any two numbers whatever 

 not containing more than five figures each. I should like to 

 see a table of cubic respondents up to 30,000 appended to this 

 work*. 



26 Lincoln's Inn Fields, 

 May 12, 1854. 



* The best practical mode of using and arranging such a table I find, after 

 much thought and consideration, would be as follows. It is easy to add two 

 quantities and subtract their sum from a third by a single operation. If, 

 then, a, b, c are the three numbers whose product it is required to find, 

 they should be written under one another ; and against (o) should be set 

 the value oia—b—c; against (i), that of 6— a— c ; and against(c), that of 

 c — a — b ; under these three last results should be written the value of 

 a + b + c; of the three former, two at least must be, all may be negative; 

 their values arithmetically expressed will be of the form K(10,000) + N, 

 where K is 0, 1 or 2. In order that the final process of combining the 4 

 cubes may be made piu*elv additive, the tables should show the values of 

 (10,000)3 less the respoiident to K(10,000)— N, when K is 1 or 2 for all 

 values of N from 1 to 9999. These complements to the respondents of the 

 simple or augmented complements of N may be termed respectively the 

 simply and doubly affected respondents of N, but in using the tables no 

 distinction need be tbawn between the respondents and the afiected re- 

 spondents. The arrangement of the tables will be as follows. In each 

 page there will be a column for the arguments, which will extend from 1 to 

 9999, and five other columns containing respondents and bearing respect- 

 ively for their headings the numbers 2, 1, 0, 1, 2. The four quantities 

 formed by addition, or by addition and subtraction, from a, b, c, will all be 

 of the form K vi v^ v^ v^ (fj v.2 v-^ v^ denoting respectively some one or other 

 of the digits from to 9), and K being one of the five symbols 2, 1, 0, 1, 2, 

 the value corresponding to v^ v^ v^ v^ will then be sought for in its proper 

 column (according to the value of the guiding figure K), and the sum of 

 the four values so found will be taken (the last figure to the left, which 

 will be 2 or 3, being rejected). This result, aff'ected, if uecessaiy, with the 

 proper correction of +^, will express the value of a X 6 x c. 



