- 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
Precisely likewise, what is useful for the mathematician is not the irrational ratio itself of the logarithm, except in the geometry of a slide rule, but in tables the useful things are approximations, which are rational ratios.
|If I wanted to reinvent logarithms (not to correct Napier, just to know how he did, or replace if his work is lost or so), I might use base two:|
|6||a, 1||56||c, 3|
|10||b, 1||60||a, b, 2|
|12||a, 2||62||j, 1|
|15||a, b||65||b, e|
|16||4||66||a, d, 1|
|18||2a, 1||68||f, 2|
|20||b, 2||70||b, c, 1|
|22||d, 1||72||2a, 3|
|24||a, 3||74||k, 1|
|26||e, 1||76||g, 2|
|28||c, 2||78||a, e, 1|
|30||a, b, 1||80||b, 4|
|34||f, 1||84||a, c, 2|
|35||b, c||85||b, f|
|36||2a, 2||86||m, 1|
|38||g, 1||88||d, 3|
|40||b, 3||90||2a, b, 1|
|42||a, c, 1||92||h, 2|
|44||d, 2||94||n, 1|
|45||2a, b||95||b, g|
|46||h, 1||96||a, 5|
|48||a, 4||98||2c, 1|
|50||2b, 1||100||2b, 2|
|35 = circa = 28|
|37 = circa = 211|
|a = 8/5 / 11/7?|
|a = 56/35 / 55/35?|
|a = 224/140 ... 220/140|
|3140 = 6,26...*1066|
|2224 = 2,69...*1067|
|2223 = 1,34...*1067|
|2222 = 6,74...*1066|
|= circa = 3140 = 6,26...*1066|
|a = 222/140 = 111/70|
|And so on for all other logarithmic components (all being incommensurable if totally exact, which they never get anyway), then add them up, which will involve making a common denominator for two such components (like .../35 was for 8/5 and 11/7).|
|Now, whichever base you calculate your logarithms in, you will get equal distances between those for 2, 4, 8, 16, 32, 64, 128 and so on, and also between 10, 100, 1000 - "and so on" if you go on. And the logarithm for two added to the logarithm for ten will give you the logarithm for twenty. True for base ten logarithms, in which twenty gets 1+logarithm mantiss for 2, true for base two logarithms, in which 20 gets 1+logarithm mantiss for 10.|
|This means that once you have enough logarithms - like for all the whole number antilogarithms between 1 and 1000 - you can set out to convert any of these into any unit you chose. If you start out with base 2 logarithms, in order to get an order where 10=logarithmic one, all you need to do is give the "one" of above table a value like 30.1 millimetres so that the value of ten becomes somthing close to a decimetre. If you chose to make two the unit and each unit an inch you will get a very similar though slightly off and incompatible scale.|
|And the fact that logarithms are the same relation whichever base you chose, illustrates that as they are irrational, they are also no numbers. Like lengths the "unit" is arbitrary rather than a real unit. Meaning, their place is not in arithmetic but in geometry. It is only their application (tables or slide rules) which is useful in arithmetic - or for calculating without overusing your knowledge of arithmetic. Which, as I have said elsewhere, is not the same as understanding arithmetic correctly in a philosophical way./HGL|