VOL. XXIX.] PHILOSOPHICAL TRANSACTIONS. 81 



which consequently is the first figure of the logarithm sought. Again, dividing 

 the number proposed 1 by 1.995262315, the number found in the table, the 

 quotient is 1.002374467; which being looked for in the second class of the 

 table, and finding neither its equal nor a less, I add O to the part of the loga- 

 rithm before found, and look for the said quotient 1.002374467 in the third 

 class, where the next less is 1.002305238, and against it is 1, to be added to 

 the part of the logarithm already found; and dividing the quotient 1 .002374467 

 by 1.002305238, last found in the table, the quotient is I.OOOO6907O; which 

 being sought in the fourth class gives 0, but being sought in the fifth class 

 gives 2, to be added to the part of the logarithm already found; and dividing 

 the last quotient by the number last found in the table, viz. 1.000046053, the 

 quotient is 1.000023015, which being sought in the sixth class, gives 9 to the 

 part of the logarithm already found ; and dividing the last quotient by the new 

 divisor, viz. 1.000002072, the quotient is 1.00000021 9, which being greater 

 than 1.000000115, shows that the logarithm already found, viz. 3.3010299, 

 is less than the truth by more than half a unit; therefore adding 1, we have 

 Briggs's logarithm of 2000, viz. 3.3010300. 



If any logarithm be given, suppose 3.3010300, omit the characteristic, then 

 opposite these figures 3. . .0. . 1. . .0. . 3. . 0. .0, we have in their respective 



classes I.995262315 .1.002305238 O. . , . ,1.000069080 



O. . .0 which multiplied continually into one another, the product is 

 2.000000019966, which because the characteristic is 3, becomes 2000.00 

 0019966 &c. that is, 2000, the natural number sought. 



It is obvious, that these classes of numbers, are no other than so many 

 scales of mean proportionals : in the first class, between 1 and 10; so that the 

 last number thereof, viz. 1.258925412, is the 10th root of 10, and the rest 

 in order ascending are its powers. So in the second class, the last number 

 1.023292992 is the lOOth root of 10, and the rest in the same manner are its 

 powers. So 1.002305238 in the third class, is the 10th root of the last of the 

 second, and the rest its powers, &c. Or, which is the same, each number in 

 the preceding class, is the 10th power of the corresponding number in the 

 next following class : whence it is plain, that to construct these tables, requires 

 only one extraction of the 5th or sursolid root for each class, the rest of the 

 work being done by the common rule of arithmetic; and for extracting the 

 5th root, we find more than one very compendious rule in N° 210 of these 

 Transactions. 



The process is exactly the reverse of Mr. Briggs's Doctrine, in cap. 14, of 

 his Arithmetica Logarithmica ; and had Briggs been apprized of it, it would 



VOL. VI. M 



