74 REPORT 1873. 



There seems a tendency to keep the name logistic logarithms for those tables 

 in which a = 3600" = 1° (so that the table gives log 3600 — log .r, x being 

 expressed in minutes and seconds), and to use the ndirae proportional logarithms 

 when a has any other value. We have not met with any modern table of 

 this kind forming a separate work ; and such tables are usually of no great 

 extent. They are to be found, however, in many collections of tables ; and the 

 logistic logarithms from Callet were published separately at Nuremberg in 

 a tract of 9 pp. in 1843 (see title in § 5). 



It may be remarked that tables of log - often extend to values of x 



X 



greater than a ; and then, in the portion of the table for which this is the 

 case, the mantissae are rendered positive (by the supposed addition of the 

 characteristic — 1, which is omitted) before tabulation, 



Kepler, 1624. We cannot do better than foUow De Morgan's example, 

 and give a specimen of the table, which contains five columns : — 



53- 36-36 



5-48 



80500-00 



19- 19-12 



21691-30 

 124-15 



48-18 



The sinus or numerus dhsolutus is 805, which (to a radius 1000) is the 

 gine of 53° 36' 36", and the Napierian logarithm is 2169130. The third and 

 fifth columns are explained as follows : — if 1000 represent 24'', then 805 

 represents 19*" 19"" 12' ; and if 1000 represents 60°, then 805 represents 

 48° 18' ; there are intorscript differences for the first and fourth columns. 

 Thus, as De Morgan remarks, Kepler originated logistic logarithms. Kepler's 

 tract is reprinted by Maseres in vol. i. of his ' Scriptores Logarithmici ' 

 (1791); and there is also reprinted there " Joannis Keplcri .... supple - 

 mentum chiliadis logarithmorum . . . .Marpurgi, 1625," the original of which 

 we have not seen, but it contains no table. The copy of the 1624 work Ave 

 have described is iu the Cambridge University Library, For an account of 

 Kepler's ' Tabulae Rudolphinae,' see De Morgan. 



Proportional logarithms for every second, a being 3°, are given almost 

 invariably in collections of nautical tables, usually to four places, but some- 

 times to five. T. 74 of Raper, so frequently referred to in § 4, is a four- 

 place table of this kind, and was, as we have seen stated in several places, first 

 computed by Maskelyne. The reference was made to Raper rather than 

 to any other of the numerous places where it occurs, as his work on 

 Navigation is one of the best-known, and has been through numerous 

 editions. Prof. Everett (Phil. Mag, Nov. 1866) says, quoting Raper, that 

 proportional logarithms as at present used are a source of perpetual mis- 

 takes even to expert computers ; but this must be intended to apply 

 rather to practical men, as for the mathematical calculator they are very 

 convenient. 



The following is a list of tables on the subject of this article, which are 

 described more fiiUy in § 4. 



Logistic logarithms for every second to 1°, viz. log 3600 — log x. — (To 4 

 places) Gakmnek, 1742 and (Avignon) 1770, T. III. (to 4800") ; Dodson, 

 1747, T. XXXVI. (to 4800") ; Schtjlze, 1778 [T. IV.] (to 3600") ; Vega, 

 1797, Vol. II. T. IV, (to 3600") ; Gordon, 1849, T. XXI. (to 3600") ; 

 Callet, 1853 [T. XI.] (to 5280") ; Htttton, 1858, T. VII. (to 5280") ; 

 Inman, 1871 [T. I.] (to 3600", intervals of 2"). 



Proportional logarithms for every second to 3°, viz. log 10,800 — log x. — 

 (To 5 places) Rios, 1809, T. XIV. ; Lax, 1821, T. XIV. ; -Galbeaith, 



