Calculating Logarithms by Hand (2010) [pdf]
fiziko.bureau42.com
fiziko.bureau42.com
http://eulerarchive.maa.org/hedi/HEDI-2005-07.pdf
How good was Euler at that? I checked his results with a multiple precision library (mpmath) and Euler got all the correct digits for 7 out of 10, he rounded off the last decimal in the wrong direction in 2 cases. Just for log(7) he got 21 correct decimals and the last 4 were wrong.
Here's the link to Euler's book [1], where these logarithms are on page 118. His values are
l1 = 0,00000 00000 00000 00000 00000
l2 = 0,69314 71805 59945 30941 72321
l3 = 1,09861 22886 68109 69139 52452
l4 = 1,38629 43611 19890 61883 44642
l5 = 1,60943 79124 34100 37460 07593
l6 = 1,79175 94692 28055 00081 24773
l7 = 1,94591 01490 55313 30510 54639
l8 = 2.07944 15416 79835 92825 16964
l9 = 2,19722 45773 36219 38279 04905
l10 = 2,30258 50929 94045 68401 79914
l1 = 0,00000 00000 00000 00000 00000
l2 = 0,69314 71805 59945 30941 72321
l3 = 1,09861 22886 68109 69139 52452
l4 = 1,38629 43611 19890 61883 44642
l5 = 1,60943 79124 34100 37460 07593
l6 = 1,79175 94692 28055 00081 24774
l7 = 1,94591 01490 55313 30510 53527
l8 = 2.07944 15416 79835 92825 16964
l9 = 2,19722 45773 36219 38279 04905
l10 = 2,30258 50929 94045 68401 79915
This isn't the right denominator; you don't get any credit for calculating digits of ln(1).
the numerator is also "wrong"; of his 3 errors, 2 of them are wrong only "off by one" in the last digit, same direction both times, so without understanding the rounding rules in effect it's hard to know whether what happened is an actual error. ln(7) is a mess though but still at the least significant end.
Here's another fun one, although much longer.
https://www.math.ksu.edu/~cjbalm/570s14/briggs.pdf
It describes how Briggs created his table of logarithms with 14 decimal places, published in 1624, only 10 years after Napier published the first logarithms in 1614. Those times mush have felt like an incredible leap forward, probably similar to LLMs now.
Kepler published his 3rd law of planetary motion in 1619. It would not have been possible to get that without logarithms, and it's quite likely that he knew of logarithms independently of Napier (who came up with them after some discussions with Kepler's boss, Tycho Brahe).
For log10....
Scale X to be between 1 and 10
For each time you divide by 10, add 1.0 to the log
each time you have to multiply by 10, subtract 1.0 from the log
Add the decimal point to the log
Loop:
Take X' (the scaled value) to the 10th power
square it (store to memory, if you have it, this is x'^2)
square that (which is x'^4)
square that (now x'^8)
multiply that by x'^2
you now have X'^10
the number now has to be divided N times to be between 1.0 and 10, and N to the log
if you need more digits, goto Loop
Here it is, taking log10(3)
3 0.
59,049 0.4
51,537,752.07320113 0.47
13,220,708.19480807 0.477
16.31350185342626 0.4771
133.4971414230401 0.47712
17.97710116675744 0.477121
352.530441082974 0.4771212
296,460.0951963823 0.47712125
52,439.97032955288 0.477121254
15,726,220.94397862 0.4771212547I converted this into python to understand it better
def log10(x):
"""
Find 10 decimal places of log10(x) using calculator functions only
"""
assert x > 0, "can't log <= 0"
log = 0
while x >= 10:
log += 1
x /= 10
while x < 1:
log -= 1
x *= 10
# x is now [1, 10)
decimal = 0.1
for i in range(10):
x10 = x**10 # or calculate as above so as not to use power function
while x10 >= 10:
log += decimal
x10 /= 10
x = x10
decimal /= 10
print(f"log10(x) = {log:.10f}, x = {x:.10f}")
log10(3)
log10(x) = 0.4000000000, x = 5.9049000000
log10(x) = 0.4700000000, x = 5.1537752073
log10(x) = 0.4770000000, x = 1.3220708195
log10(x) = 0.4771000000, x = 1.6313501853
log10(x) = 0.4771200000, x = 1.3349714142
log10(x) = 0.4771210000, x = 1.7977101167
log10(x) = 0.4771212000, x = 3.5253044107
log10(x) = 0.4771212500, x = 2.9646009506
log10(x) = 0.4771212540, x = 5.2439970092
log10(x) = 0.4771212547, x = 1.5726220230On the other hand, (hand-waving) it takes many generated digits before the “far away” get shifted left of the decimal point, so numerical analysis probably can show you don’t need them all to reach a target number of digits.
I have in the past, this time I used the 4 function calculator mode of Windows 10's calculator. x^2, MS, x^2, x^2 * MR = on the loops, then divide by 1+n zeros, repeat.
I used NotePad++ to record the data as I went.
Doing it for a binary logarithm would be a lot easer, because then it's square and optionally divide by 2.
AFAIK, that calculator has “Infinite precision for basic arithmetic operations (addition, subtraction, multiplication, division) so that calculations never lose precision” (https://github.com/microsoft/calculator)
A few moments of Googling turned this up. Thought some other folks might find this topic interesting as well.
One fun aspect of all this, is that it forces you to consider that thinking of exponentiation as "repeated multiplication" is a bit dodgy when you allow for decimal exponents. I mean, given 3^2.5, what does it mean to say "multiply 3 by itself 2.5 times?" Actually, thinking about that was the thing that really set me off on an exploration of how the heck they worked out these log values in the beginning.
It means 3 · 3 · √3. If you write the exponent in binary, you can approximate any such power as a product of various repeated square roots.
> how the heck they worked out these log values in the beginning
If you're really curious about the very beginning, here's an English translation of Briggs's (1624) book: https://17centurymaths.com/contents/albriggs.html
And here are English translation of Napier's (1614, 1619) books: https://17centurymaths.com/contents/napiercontents.html
[1] https://en.wikipedia.org/wiki/History_of_logarithms#Napier
Going from 2^n to 2^(n+1) is ultimately about scaling the result up by a factor of x2. So, 2^(n+0.5) ought to be about scaling up halfway to x2, so that if you were to repeat that operation again, you got x2; and that's precisely sqrt(2). That reasoning gets you all the rationals. Since rationals are dense in the reals, impose continuity and you get all reals. This is the root of the algebraic definition of exponentials.
But where there are reals there is calculus, and we also get a lot of insight from a differential definition. The key insight that leads to this definition is that if we could break down the exponential to 2^(n+epsilon) for a very small epsilon, we could stack up O(1/epsilon) of them to get wherever we like on the real continuum. So it makes sense that the definition dy/dx = a.y should produce the same function.
This notion of "break down an operation into infinitesimal bits that stack up" can be taken seriously and formalised, and if you do that you end up with the theory of Lie groups and Lie algebras.
Hartzler, "A two-and-one-half-place logarithm table", The Mathematics Teacher, Vol. 53, No. 3 (Mar 1960) https://www.jstor.org/stable/27956101
Bayer, "Setting up an approximate antilog table", The Mathematics Teacher, Vol. 55, No. 3 (Mar 1962) https://www.jstor.org/stable/27956560
Doerfler, "Dead Reckoning: Calculating Without Instruments", Taylor Trade Publishing (Sep 1993) https://www.amazon.com/Dead-Reckoning-Calculating-Without-In...
> “Euler” is pronounced “oiler,” which is why virtually every sports team assembled by the University of Alberta’s math department has been named “the Edmonton Eulers.”
"Euler’s mathematical brilliance" is footnoted: "I’m not exaggerating. I’ve taken no less than 13 University level math courses, and Euler’s name showed up in at least 10 of them. The man’s contributions to mathematics are probably more significant than Einstein’s contributions to physics. Find him on Wikipedia when you have a minute or thirty."
If you’re Euler, that’s 3 conjectures, 11 equations, 9 formulas, 4 functions, 2 identities, 10+ constants, 11 theorems, and 2 laws, some of them shared with others (https://en.wikipedia.org/wiki/List_of_things_named_after_Leo...)
That includes https://en.wikipedia.org/wiki/Euclid–Euler_theorem, which must have been one of the ‘cooperations’ (Euclid died ±270BC, Euler lived 1707-1783, and his proof was published posthumously) on a theorem that took the longest time in history.
Mr Briggs had computed the first table of base 10 logarithms for the first few thousand natural numbers: https://en.wikipedia.org/wiki/Henry_Briggs_(mathematician)
https://www.feynmanlectures.caltech.edu/I_22.html#:~:text=Bu...