## Sunday, September 8, 2013

### ... on Classic Mathematics Logarithms - Revisited

1) ... on Classic Mathematics Logarithms - Revisited, 2) ... on Napier - with Admiration

Video commented on:
numberphile : Log Tables - Numberphile
Hans-Georg Lundahl
(essay type multicomment)
Before getting to Napier, I would like﻿ to state I have:

a) reformulated definition of logarithms (esp. fractional exponents)

b) used that definition to work out a table of base ten formulated in feet, inches and lines and points (12 points = 1 line, obviously, French subdivision)

c) translated that very scarce table to decimals and found it agrees with usual table fairly well

thereby proving I was right in my reformulation.

a decimal series is a fraction is a ratio

a﻿ ratio as exponent is a ratio between exponents

100 to exp 3/2 = 1000

=

100 to exp 3 = 1000 to exp 2

So, look here:

10 to 1 = 2 to 3 (roughly)

10 to 2 = 2 to 7 (roughly)

Is the log for 2 6/21 or 7/21? Between, rather.

If we multiply both sides by 2 we get 12-13-14/42 [a less binary choice]

4398046511104 (4 rounds﻿ down) = 2 to 42

1000000000000 = 10 to 12

9223372036854775808 (9 rounds up) = 2 to 63

10000000000000000000 = 10 to ...

Not 18, but 19. So 19/63 is an approximation to base ten log of 2. And so on.

0,3010299956639 ... base ten log for two according to calculator

0,3015873015873 ... 19/63

My goal was not to make useful logarithmic tables, just to check if my understanding of what logarithm means could help me make a sufficiently accurate one to check I was on the right track.

As you saw, it could.

Meaning﻿ I can also dispose of logarithms as an argument for irrational numbers. A log is not the number of times that ten is multiplied by itself, it is a ratio.

And in my book, so to speak, irrational ratios are quite ok, it is only irrational numbers that are out of the possible.

Pi﻿ and Sqrt of 2 are not there in the arithmetic of Boethius, but they are sure there in his Geometry.

And a logarithm is really a geometric size to size ratio, even if the most famous ones are so for size ratios there are whole numbers for.

So are of course sine, cosine, tangent (remember soh-cah-toa).

Look at this quote:

Objection﻿ 1. It seems that there can be something actually infinite in magnitude. For in mathematics there is no error, since "there is no lie in things abstract," as the Philosopher says (Phys. ii). But mathematics uses the infinite in magnitude; thus, the geometrician in his demonstrations says, "Let this line be infinite." Therefore it is not impossible for a thing to be infinite in magnitude. ...

Reply to Objection 1. A geometrician does not need to assume a line actually infinite, but takes some actually finite line, from which he subtracts whatever he finds necessary; which line he calls infinite.

Summa Theologica﻿ I, Q7, A3