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Table of Proportional Logarithms. By Captain Robert 

 Shortrede. 



The accompanying is a Table of Proportional Logarithms, which I 

 have lately constructed with a view to diminish their size, and at the 

 same time considerably extend their use. 



Proportional Logarithms are commonly arranged in vertical columns 

 of 60 each, and the construction is such, that the larger the Logarithm 

 the less is the corresponding quantity. I have never been able to 

 perceive any great benefit resulting from such a system, but often I 

 have felt much inconvenience from the want of an arrangement analo- 

 gous to that of Tables of common Logarithms. 



The present is a specimen of what I conceive to be the most conve- 

 nient form of Table. The Logarithms here given are the arithmetical 

 complements of those in common use,* so that they increase along with 

 the quantities to which they belong, and the arrangement is such, as to 

 retain the advantages of the decimal as well as the sexagesimal subdi- 

 vision. The Table was intended primarily to facilitate the finding of 

 proportional parts for minutes and seconds, in a set of Tables in which 

 the quantities were tabulated for every 10, and it was immediately 

 obvious, that it would serve equally for seconds and decimals when the 

 quantities are tabulated, as in Button's Tables, to every minute, and 

 generally for any quantities whose subdivisions are by 6, 60, or 600, &c. 



The column marked ' " contains minutes and even ten seconds from 

 1 to 9'.50. The col. marked N contains jj^th of the seconds in the 

 former; the odd integers being found in the head line of the Table 

 exactly as in Tables of Common Logarithms. The column marked 

 common difference, gives the mean value at the middle of the line 

 opposite which it stands : and beyond this are proportional parts for 

 the decimal subdivisions of the mean common diiference. 



The use of the Table is very simple. The fractional part of 10' or 1' 

 being found in the proper column (' " or N, as the case may be) take 

 out the corresponding Logarithm; to this add the log. opposite the Ta- 

 bular Difference found by column N ; the sum of these is the logarithm 

 of a number which found in column N is the proportional part required. 



* If the term Proportional Logarithms be considered as being already definitively 

 appropriated otherwise, those here given may be called Co-proportional or Arco-propor- 

 tional Logarithms, or Anti-proportional, or Proportial, or Correctional, or any other 

 term which may be preferred. 



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