Bertelsen's Number

The number of primes less than x is usually denoted by pi(x). In 
Hardy & Wright's book "An Introduction to the Theory of Numbers", 
5th edition, page 9, it says the vale of pi(10^9) is 50,847,478.  
However, on page 179 of Paulo Ribenboim's book "The Book of Prime 
Number Records" it gives the value 50,847,534, which exceeds H&W's 
count by 56.  Since number theory is a somewhat empirical science, 
should we consider the possibility that the number of primes has 
increased since 1938 (when H&W was first written)?

Looking for a tie-breaker, I remembered seeing a table of pi(x) in 
"The Mathematical Experience" by Reuben and Hersh.  Actually, the 
book mentions pi(10^9) in two places.  On page 175 it says pi(10^9) 
is known to equal 50,847,478, in agreement with Hardy and Wright, 
whereas the table on page 213 lists pi(10^9) as 50,847,534, in agreement
with Ribenboim!  Curious.  At least two of these three books has an 
error in it.  Perhaps Reuben and Hersh got their first value from 
Hardy and Wright, and then constructed their table from some other 
(more recent) source? 

Out of curiosity I checked a few more books and found the same 
erroneous value of pi(10^9) in each of the following:

An Introduction To the Theory of Numbers      Hardy/Wright  1938/1988
The Mathematical Experience                   Davis/Hersh   1981/1981
Number Theory and Its History                 Ore           1948/1988
Elementär talteori                            Nagell        1950/1950
Numbers and Infinity                          Sondheimer    1981/1981
The Nature and Growth of Modern Mathematics   Kramer        1970/1982
Introduction to Algorithms                    Cormen, et al 1993/1993     

The error evidently goes back further then Hardy and Wright.  In 
Oystein Ore's book "Number Theory and Its History" (1st Dover edition,
1988) the table on page 77 gives the old value pi(10^9) = 50,847,478.
Furthermore, it says the values of pi(x) for x up to 10^7 were 
obtained by actual count from tables, whereas the values for x=10^8
and 10^9 "are the values of pi(x) calculated by Meissel and Bertelsen
which we have mentioned previously."

On page 69 Ore discusses a formula for computing pi(N) in terms of 
all the primes less than sqrt(N).  He goes on to say that Meissel 
(1870) improved the method and, through various shortcuts, succeeded 
in finding pi(10^8) = 5,761,455.

  "These computations were continued by the Danish mathematician
   Bertelsen [Niels Peder Bertelsen (1869-1938)], who applied them 
   for the determination of errors in tables.  He announced the 
   following result (1893):

                     pi(10^9) = 50,847,478

   which represents our most extended knowledge of the number
   of primes."

From this account one would infer that there was an error in Bertelsen's
calculation back in 1893 (rather than just a typo), and that it has 
been propogated onwards, appearing in books printed nearly 100 years
later. This strikes me as interesting, especially since Bertelsen applied 
his results "for the determination of errors in tables".

S. Sorrento then called my attention to the 1968 book "En bok om primtal" 
by Riesel, which gives the correct value, and remarks on the earlier
incorrect value appearing in various texts. Then I heard about a paper
entitled "Calculating pi(x): The Meisell-Lehmer Method" by Lagarias, 
Miller, and Odlyzko, Math Comp, 1985, which reports that the correct 
value of pi(10^9) is 50,847,534, but contrary to Ore's account it says
the erroneous value 50,847,478 originated with Meisell in 1885, and
was merely repeated by Bertelsen in 1893.  Remarkably, the erroneous 
value (which I had dubbed "Bertelsen's number") was still being propagated
in text books a century later, with at least seven citations in books 
published between 1938 and 1993.

Apparently there has been a long hazardous history of computing the number
of primes.  Often the greatest "known" value of pi(x) has later turned out
to be wrong. As another illustration of this, the value of pi(10^10) listed
in the "modern" table in Reuben and Hersch is 455052512, which is the 
uppermost value they quote, exceeds by 1 the value in Ribenboim's Book of 
Prime Number Records (1988).

Anyway, for the record, based on Ribenboim's book, the number of primes of
the first several orders of magnitude are listed (correctly I hope!) below:

                         number of primes
              x          less than x
           -------       ----------------
             10                        4
             10^2                     25
             10^3                    168
             10^4                   1229
             10^5                   9592
             10^6                  78498
             10^7                 664579
             10^8                5761455
             10^9               50847534
             10^10             455052511
             10^11            4118054813
             10^12           37607912018
             10^13          346065536839
             10^14         3204941750802
             10^15        29844570422669
             10^16       279238341033925
           2*10^16       547863431950008
           4*10^16      1075292778753150