## Thursday, November 28, 2019

### Previous Work on Logarithms

Assorted retorts from yahoo boards and elsewhere : Medieval Quadrivium and Logarithms (quora) · Φιλολoγικά/Philologica : Expressing Logarithms in Points of Royal Feet · Previous Work on Logarithms · Yes, My Method Can Close In on Logarithms

I have previously been called out for believing that expressing logarithms in feet, inches, lines (and points) would make them rational.

Not so.

I do not believe in irrational numbers, but then again, I do not call π or logarithms or square root of 2 or sine, cosine, tangent, cotangent, secant, cosecant "numbers".

My previous work involved calculating approximate logarithms for 2 and 3, and perhaps some more, so as to prove I understood them, while not considering them as "irrational numbers" because I objected to calling them numbers.

I repeated some, and came to logarithm of 9 as somewhere 21/22 or 20/21.

How so? "1021/22 = (approx.) 20/21 = (approx.) 9" would have this meaning:

1021 = (approx.) 922
1020 = (approx.) 921

 921 = 109 418 989 131 512 359 209 1020 = 100 000 000 000 000 000 000 922 = 984 770 902 183 611 232 881 1021 = 1 000 000 000 000 000 000 000

Now for the ratios:

21/22 = 0.9545454545454545
20/21 = 0.9523809523809524

And the logarithm for 9:

0.9542425094393249

Obviously halving the ratio will give an approximate logarithm for 3 (halve a logarithm and you make a root extraction for antilogarithm).

21/44 = 0.4772727272727273
10/21 = 0.4761904761904762

And the logarithm for 3:

0.4771212547196624

The one major reason why I presented my earlier results (which, remember, were not meant to improve existing logarithm tables, just to prove my approach is a valid one on the subject) was so as not to be influenced by any logarithms I happened to recall (as I calculated a logarithm for 2 and very well recalled 0.301, the risk of such influence was imminent) and there were two more reasons:

• points of a foot are finer subdivisions than 1000's of a decimetre
• expressing in a measure rather than in simple numbers underlined that on my view logarithms are geometry and not arithmetic, and that therefore logarithms being irrational doesn't prove there are irrational numbers or irrational objects of arithmetic.

The practical object of getting sth which can be put on a slide rule made at home was not really met, as I could not calculate logarithms with sufficient precision or sufficiently many. Kudos to the 17th C. arstocrat who did the calculations! So, now I made it instead using normal expression, decimal expression of logarithmic values.

Hans Georg Lundahl
Nanterre UL
St. Sosthenes
28.XI.2019

Apud Corinthum natalis sancti Sosthenis, ex beati Pauli Apostoli discipulis; cujus mentionem facit idem Apostolus Corinthiis scribens. Ipse autem Sosthenes, ex principe Synagogae conversus ad Christum, fidei suae primordia, ante Gallionem Proconsulem acriter verberatus, praeclaro initio consecravit.

PS. Let's check that 1021 = (approx.) 344 and 1010 = (approx.) 321

 344 = 984 770 902 183 611 232 881 1021 = 1 000 000 000 000 000 000 000

 321 = 10 460 353 203 1010 = 10 000 000 000

Q. E. D./HGL