§l. Convergent sequences and series
75
Since there is no practical method of extracting nth roots numerically (by
hand ... ) apart from the case where n is a power of 2 - then it suffices to
extract successive square roots -, one restricts to integers n of the form 2P •
In this case, the error given by (8) is majorised by x 2 /2 P +1.
Napier, Briggs after his death, and then Kepler, set themselves, among
other things, to calculate the logarithms of the first thousand integers exact
to 7 or 14 decimal places; starting from this, one could find those of a large
number offractional values of x, since log(P/q) = logp-log q. (Above all, they
needed the logs of the trigonometric functions, but this is another problem
again). Of course, they did not have to calculate all these logs directly; it is
enough, for a start, to calculate those of the prime numbers 2, 3, 5, 7, 11, 13,
etc. and even though in this case there are "tricks" to reduce the labour, it
remains considerable, to express it mildly.
The first candidate is x = 2. One has to extract p successive square roots,
choosing p and performing the calculations to sufficiently many places to have
a hope of obtaining the required precision. In this particular case the error
is less than 2 2 /2 p + 1 = 2- p + 1 in addition to those committed in extracting
the p successive square roots of 2. To obtain the result to 14 places, it is
thus prudent to choose p so that 2- p + 1 < 10- 15 , i.e. 2 P > 2.10 15 . Now
2 9 = 512 < 10 3 < 2 10 = 1024, whence 2 50 > 10 15 > 2 45 , which indicates the
need to choose for p a number between 45 and 50, in other words to extract
at least 45 successive square roots of 2 to ensure having 15 places exact at
the end. One can reduce the work with the aid of the following remark.
We know, and they already knew then, that
1 + u/2 - u 2 /8 < (1 + U)I/2 < 1 + u/2 for a < u < 1
(square it all), so that, for u small, the error committed in replacing the
square root of 1 + u by 1 + u/2 is less than u 2 /8 = u 2 /2 3 ; the error is thus
< 2- 50 if u < 2- 24 . Now, in calculating the X n , one has
1 + X n +l = (1 + Xn)I/2 and Xn < 2- n - 1 < 2- 24
once n > 25. One thus sees that after having extracted the 24 or 25 first
successive square roots of 2 to 15 exact places, one may assume that (1 +
u)1/2 = 1 + u/2 for p > 25. In other words, one has to calculate only the first
25 square roots to 15 places, which is still not within reach of everyone.
In the case of logs to base 10, Briggs started by extracting 54 successive
square roots of 10, which gave him the number 14
1.00000 00000 00000 12781 9149320032 35 = 1 + h
and allowed him to calculate the number M such that 10glO(1 + h) rv Mh
since
14 See E. Hairer and G. Wanner, Analysis by Its History (Springer-New York, 1996),
p. 30, which also reproduces in facsimile the page where Briggs tabulates the 54
successive square roots of 10 and their logarithms.
75
Since there is no practical method of extracting nth roots numerically (by
hand ... ) apart from the case where n is a power of 2 - then it suffices to
extract successive square roots -, one restricts to integers n of the form 2P •
In this case, the error given by (8) is majorised by x 2 /2 P +1.
Napier, Briggs after his death, and then Kepler, set themselves, among
other things, to calculate the logarithms of the first thousand integers exact
to 7 or 14 decimal places; starting from this, one could find those of a large
number offractional values of x, since log(P/q) = logp-log q. (Above all, they
needed the logs of the trigonometric functions, but this is another problem
again). Of course, they did not have to calculate all these logs directly; it is
enough, for a start, to calculate those of the prime numbers 2, 3, 5, 7, 11, 13,
etc. and even though in this case there are "tricks" to reduce the labour, it
remains considerable, to express it mildly.
The first candidate is x = 2. One has to extract p successive square roots,
choosing p and performing the calculations to sufficiently many places to have
a hope of obtaining the required precision. In this particular case the error
is less than 2 2 /2 p + 1 = 2- p + 1 in addition to those committed in extracting
the p successive square roots of 2. To obtain the result to 14 places, it is
thus prudent to choose p so that 2- p + 1 < 10- 15 , i.e. 2 P > 2.10 15 . Now
2 9 = 512 < 10 3 < 2 10 = 1024, whence 2 50 > 10 15 > 2 45 , which indicates the
need to choose for p a number between 45 and 50, in other words to extract
at least 45 successive square roots of 2 to ensure having 15 places exact at
the end. One can reduce the work with the aid of the following remark.
We know, and they already knew then, that
1 + u/2 - u 2 /8 < (1 + U)I/2 < 1 + u/2 for a < u < 1
(square it all), so that, for u small, the error committed in replacing the
square root of 1 + u by 1 + u/2 is less than u 2 /8 = u 2 /2 3 ; the error is thus
< 2- 50 if u < 2- 24 . Now, in calculating the X n , one has
1 + X n +l = (1 + Xn)I/2 and Xn < 2- n - 1 < 2- 24
once n > 25. One thus sees that after having extracted the 24 or 25 first
successive square roots of 2 to 15 exact places, one may assume that (1 +
u)1/2 = 1 + u/2 for p > 25. In other words, one has to calculate only the first
25 square roots to 15 places, which is still not within reach of everyone.
In the case of logs to base 10, Briggs started by extracting 54 successive
square roots of 10, which gave him the number 14
1.00000 00000 00000 12781 9149320032 35 = 1 + h
and allowed him to calculate the number M such that 10glO(1 + h) rv Mh
since
14 See E. Hairer and G. Wanner, Analysis by Its History (Springer-New York, 1996),
p. 30, which also reproduces in facsimile the page where Briggs tabulates the 54
successive square roots of 10 and their logarithms.