ON MATHEMATICAL TABLES. i) 



in this llcport, except in very exceptional circumstances. "Where, however, 

 tlic table's are not numbered or otherwise denoted, they have been marked 

 [T. I.], [T. II.], &c., as it was necessary to have the means of referring to 

 them, invariably, therefore, where the number of the table is not included 

 in square brackets, it is to be understood that it is the author's own number. 

 Thus T. VII. in any particular work implies that the table in question is 

 numbered A'll. in that Avork, while [T. VII.] implies either that the table 

 has no number, or that the classification in the work is different from that 

 adopted in this Ileport. Whenever logarithms arc mentioned Avithout tho 

 epithet hyperbolic or Napierian, common or Eriggian logarithms (viz. to base 

 lU) are intended. In some cases, where there might bo some doubt, the 

 adjective " common " is introduced. By hyperbolic logarithms are always 

 meant logarithms to the base e (2-71828 . . . ); and these are never called 

 Napicfiaii, this word being reserved for logarithms of exactly the same kind 

 as those introduced by Napier (see § 3, art. 1 7). Such a sentence as " Five- 

 figure logarithms to lOOO," is always to be understood as meaning " logarithms 

 of numbers from unity to 1000, at intervals of unity to five decimal places ;" 

 viz., wlien the lower limit of a table is not expressed, it is always to be taken 

 as unity ; and when the intervals are not mentioned, they are always unity'. 

 The term "places" is used throughout for " decimal places " or " decimals," 

 a number " to 3 places " meaning a number given to 3 2''^(^ces of decimals 

 (not 'djir/ares). The only exception made to this rule is in the description of 

 tables of common logarithms ; the words " seven-figure logarithms, six-figure 

 logarithms," <fec., have become by usage so completely recognized as meaning 

 logarithms to seven places, to six places, &c., that it did not seem worth while 

 disturbing the established mode of expression, as it could lead to no error. 



The contents of old works have been described in the language and nota- 

 tion of the present day, and not in the manner adopted by their authors ; 

 any peculiarities of notation &c. in a table, however, are pointed out. It was 

 long universal, and is still very common, to describe trigonometrical tables as 

 being computed to a certain radius ; these are translated into the language 

 of decimals ; thus a table " to radius 10,000,000 " is described as a table 

 " to seven places," and so on. As a rule the characteristics of the logarithms 

 have been ignored in describing a table ; i. e. it has not been stated whether 

 the characteristic was given or no, or, if given, what was the understanding on 

 which it was added. In many tables, contained in works intended for a special 

 purpose (as in collections of nautical tables, &c.), arbitrary numbers are added 

 to or subtracted from the characteristics to facilitate their use in working 

 some particular formula ; to have included details of this kind would have 

 taken much room, and been really superfluous, as in most eases all that is 

 required to be known in the description of a table of logarithms, is the range 

 of the table, and the number of places to which the mantissa} are given. 



We m!iy here mention that an ambiguity occurs in the description of propor- 

 tional-part tables ; thus a *' table of proportional parts to tenths " may mean 

 cither that the proportional parts are given for one, two, three, &c. tenths of 

 the difference, or else that the numbers that form the proportional-part table 

 are given to one place of decimals. The former is tlie meaning generally in- 

 tended ; and it would be better if in this ease the words " to tenths " were 

 replaced by " for every tenth." 



A good many tables had been' described before the ambiguity was noticed; 

 but it is believed the context will generally show the true meaning ; when 

 the words to tenilis, to hundredths, &c. arc italicized, the latter interpreta- 

 tion (viz. results given to onO; two, &c. decimal places) is to be assigned. 



