Henry Briggs

Commentary on the text

1.  This statement will be explained in the next paragraph, but in a modern formulation, Briggs is saying that, given numbers in the same proportion, as

2.  The "Inventor" of which he speaks is John Napier, of course.  Recall our discussion in the Introduction.  The word logarithm comes from "logos", the Greek for "ratio", together with "arithmos", or "number".  That is, logarithms are "ratio numbers".

3.  This is the first fundamental property of logarithms: in modern notation,  log(ab) = log a  +  log b.  It is this property that allows multiplication of numbers (ab) to be converted into addition of logarithms (log a  +  log b).

4.  That is,

5.  He proves the property thusly: from the proportion in note 3, the defining principle of logarithms (see note 1) implies that
log 1 - log a = log b - log(ab).  But since log 1 = 0, this becomes log(ab) = log a  +  log b.

6.  Symbolically,  log(a/b) = log a - log b.

7.  Briggs' practice is typically to express decimal numbers with a comma as a decimal point (still common especially in Europe).  He also uses commas to separate digits of a number in groups of five.  As a result, one must use the context of the discussion to determine which of the commas in a number is the decimal point!  Here, 1,00000,00000,0000 = 1.

8.  His terminology here is not the modern sense of "absolute value", as |x| is the absolute value of x, but in the sense of "true value".

9.  Here is one of the many important applications of the use of logarithms in computation: the extraction of roots.  In this example, he computes the cube root of 979.  Suppose we call this number x.  Then x³ = 979 and, taking logarithms, he deduces that  log x³ = log 979.  From the fundamental properties of logarithms it follows that 3 log x = 979, or that  log x = (1/3) log 979.  Use of the tables once again allows him to determine the value of x.

10.  Since 2 is less than 3, one can't take "a third part" of it.  Or rather, 3 goes into 2 zero times.

11.  That is, the cube root of 979 is not a whole number.

12.  99,295 is the number in the table whose logarithm is closest to the desired 49969275936.  The word chiliad is from the Greek, meaning "thousand".  Briggs' tables gave the logarithms of all the numbers from 1 to 100,000, so 99,295 in the last chiliad.

13.  If we let V be the future value of the principal P at the end of the investment period, then V/P is their ratio.  If the investment period were one year, this ratio would be 106/100, indicating the increase of 6%.  Thus  log V- log P = B, the "annual" difference of the logarithms of 106 and 100, as indicated in Briggs' computation.  For any other period of time, the corresponding scaled difference of logarithms is used.  Since P is known, so is  log P, so  log V can be computed, hence ultimately V.
 

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