Logarithms and Anti-logarithms

[Ediswan]

IIRC the original purpose of Charles Babbage's Difference Engine was to produce accurate mathematical tables, as the human compiled ones of that era were notoriously full of mistakes.

That is the usual explanation, which makes sense. One !!!!! expensive machine, via the printing press, could usefully improve everybody else's results.

 

[JB_B]

I first came across log tables at our scouse comp way back in the early 1980's. Tatty, dusty, yellowing things from the back of the stockroom which might as well as been tables for aiming cannon fire. We tried them out for a few sample calculations and they did work.

I'm fairly sure we never actually used a slide rule but I remember that they were something to do with the skeletal remains of a Mr Napier.

The concept of exponential functions (and their inverses) is, of course, a different question - it's fundamental to maths and any science with a quantative component.

It wasn't until I finally did a maths degree - just a few years ago - that I finally began to appreciate why e might be important.

 

[GusB]

I came across logarithms at some point during my school career, but I'm struggling to remember whether it was in maths or physics. I remember the teacher bemoaning the fact that we all had access to "new-fangled" calculators and in his day it was all done with log tables, before producing a book of said tables. I think we were all very much in agreement in thinking "thank ****" for new-fangled calculators after seeing that!

On the subject of calculators, and without wishing to derail the thread too much, I noted upthread that someone mentioned the Sinclair Oxford. When I was a kid my dad had one of these, still in its original box. It used a PP3 battery, if I remember correctly. At that age I didn't really have any use for one, other than having the odd titter with friends after someone taught me the number 5318008...

 

[Calthrop]

Going off at a "trivia" tangent: some fifty years in the past, I found it pleasing to happen to read that at that time, students at the Massachusetts Institute of Technology had formed a singing quartet in the American "barbershop" genre; which they appropriately and rather neatly called the "Logarhythms". Should such a music group still exist there today: with the word and the procedure now unfamiliar to the majority of young people -- one wonders whether that group has adopted a different name; or whether as a "historical" thing, they keep the same one as of half a century ago?

 

[Peter Sarf]

On the subject of calculators, I still have on my desk - and regularly use - a Casio fx-570, which I believe I purchased in the early 1990s. It's still going strong. I also have no recollection of having ever needed to change the battery - which I find pretty amazing.

A couple of years ago, I bought an fx-991ES PLUS, out of curiosity to see whether a modern calculator was any easier to use. Although the newer one has a bigger screen that displays intermediate steps and is therefore useful if I'm doing a rare extended calculation with brackets and stuff (the old one just displays a single number), I actually find I prefer the old one for simple calculations - slightly fewer button presses.

A friend at university gave me her two Casio AA batteries from er calculator. I had a new clock radio cassette and neglected to buy a AA battery for the clock to function. We wanted to see if it worked and I bought her a replacement pair of batteries. The second Casio battery is still going strong with no sign of leaking. So that is two of them lasting 21 years each !. Casio quality. Of course the batteries do not go flat in a book of log tables.

 

[Springs Branch]

Like many on here, I remember learning to use logarithms at school for long multiplication/division and calculating roots of all kinds.

This thread has resurrected a few long-forgotten terms:- four-figure-tables, mean differences, mantissas and the dreaded bar notation (for when 0 < n < 1).
I do recall antilogs (verbally we used the abbreviation, "antilogarithm" was only ever the word printed in the book) and I still remember that antilog 0.3010 = 2.

Luckily hand-held calculators came on the scene just before I was due to graduate from the four-figure tables to needing five-figures. I never got anywhere near needing @DelW's seven-figure tables!

Nervously spotting widespread use of electronic calculators on the horizon, one of my physics teachers took great pains to explain why it was always better to do calculations "by hand" using log tables. Doing it the old way, allegedly, meant you would somehow be closer and more attuned to that particular task and would intuitively know roughly what the result should come out as. Just pressing numbers into a calculator, you might make a mistake (e.g. miss a decimal point), end up with an answer orders of magnitude too large or small, and just write down the result without realising it was totally wrong. I don't thing anyone took his theory seriously!

 

[najaB]

Doing it the old way, allegedly, meant you would somehow be closer and more attuned to that particular task and would intuitively know roughly what the result should come out as. Just pressing numbers into a calculator, you might make a mistake (e.g. miss a decimal point), end up with an answer orders of magnitude too large or small, and just write down the result without realising it was totally wrong. I don't thing anyone took his theory seriously!

There is a little bit of truth to it - knowing how maths works helps you get a feel for if the answer is "about right" - which helps you avoid orders of magnitude mistakes. But doing it by hand makes it more likely that you'll make small mistakes.

 

[edwin_m]

An engineering lecturer used to tell the story of someone designing something on the then-new CADCAM systems, pressing send to get it made, and getting back a component that was 10 times too big in every dimension.

 

[telstarbox]

The inverse Stonehenge error!

 

[Snow1964]

I did my O level maths in 1981 and I am sure we had log table books in the exam. I have a vague memory that one of the 2 maths papers was no calculators to ensure you could do it the old fashioned way.

I didn’t do any CSEs so don’t know if they used the log table book (GSE and CSE were still separate in 1981, later they got merged into GCSEs)

I think the books (which were brown) also had the tables for tangents etc for working out angles, as scientific calculators were still relatively new then (and about half weeks wages for many)

 

[Springs Branch]

I think the books (which were brown) also had the tables for tangents etc for working out angles, as scientific calculators were still relatively new then (and about half weeks wages for many)

Yes, the trig. function tables, sin, cos & tan, were also contained in the Log Tables book.

Then on the back pages were the hyperbolic sine, cosine and tangents (sinh, cosh & tanh) tables.
These pages always seemed in almost pristine condition, since the hyperbolic functions were hardly, if ever, used for school-level calculations.

Question for the engineers:- aren't the hyperbolic sines / cosines somehow involved in the calculation of catenaries (i.e. the suspension for railway overhead electrification)?

 

[Ploughman]

From failing Maths 0 level 4 times between School and Southport Tech in 1976. Including use of Logs and Anti logs in the 60's

I finally passed a Maths 0 level in 1991 at York Tech and went on to pass HNC including the Maths element in 1994.

Like many other comments I have also used the Casio calculators and still have a couple on my home desk still in use on original batteries.

As to wether or not I have used the Maths I was taught, I think the answer is a No, Same for slide rule use.
I cannot think of any instance in my 20 years on track renewals as a Tech, were they were ever used.
And as for Differentiation?

 

[telstarbox]

Logarithms remain on the current A-Level maths syllabus, as shown in question 5 of this OCR paper:

https://ocr.org.uk/Images/308724-unit-h240-01-pure-mathematics-sample-assessment-material.pdf

 

[DynamicSpirit]

Question for the engineers:- aren't the hyperbolic sines / cosines somehow involved in the calculation of catenaries (i.e. the suspension for railway overhead electrification)?

If you allow any chain (or rope etc., presumably also including an overhead electric cable) to hang freely, fixed at both ends, it'll naturally form the curve of, height = cosh(horizontal position). Look at https://en.wikipedia.org/wiki/Catenary and see the pictures on the right of the page. I'm answering that wearing a mathematician's hat, rather than an engineering hat, though, so I'm not sure exactly in what manner railway engineers would use that information.

 

[PeterY]

I remember learning all about logarithms etc at school in the 70's. I had a good understanding of them but since I've left school, 48+ years ago I've never used them. Today I wouldn't have a clue.

 

[david1212]

I did logarithms at school for O level maths. Calculators may have been allowed in exams for other subjects, I really can't remember 40+ years later. By A level I'm 99.5% calculators could be used but log tables provided as a backup. Similarly Sine / Cosine / Tangent.

Dad had a slide rule in his desk and I now have it but possibly inherited as either he never knew how to use it or had forgotten so could not show me.

Dad had an Anita calculator in the office at work he brought home occasionally
Anita calculators
It was similar to this, I may be wrong after ~50 years but recall it could store a number in memory

Edit - My memory was right so this or very closely related

This was my first calculator around 1973 I think ....!!!

My first proper calculator was a Commodore like this

The next was a LED Texas TI-30 with the stiff buttons and a big appetite for batteries.

After that a Dixons Prinztronic LCD own brand which took a battering with A-levels and polytechnic. If not this similar

Next a Casio FX501P when being sold off in Dixons.
By then I was working so less heavily used but it died.

The replacement early 1990's was a Casio FX100D which I still use at home.

At work I have a couple of Poundland scientific ones and a large key basic one.

 

[Ploughman]

I recall the Magic Brain mechanical one from the late 60's.

 

[swt_passenger]

I recall the Magic Brain mechanical one from the late 60's.

I remember using one, and I confirmed by finding a few videos on YouTube, (search magic brain calculator), a belief they were not that straightforward to use…

 

[nlogax]

Anita calculators
It was similar to this, I may be wrong after ~50 years but recall it could store a number in memory

Edit - My memory was right so this or very closely related

That Anita device is quite the desirable item! Lovely piece of retro kit especially with all those nixie tubes. Thanks for sharing!

 

[ABB125]

For comparison, here is a modern calculator. This is effectively the "standard" calculator for current A-level maths. It has built-in statistical distributions (eughh! ) and can do matrix calculations (up to 4x4) and vectors.

 

[Typhoon]

Yes, the trig. function tables, sin, cos & tan, were also contained in the Log Tables book.

Then on the back pages were the hyperbolic sine, cosine and tangents (sinh, cosh & tanh) tables.
These pages always seemed in almost pristine condition, since the hyperbolic functions were hardly, if ever, used for school-level calculations.

We must have used the hyperbolic functions as we were told that sinh was pronounced 'shine' and tanh 'than' (hard 'th'), whether that is true I don't know.

I didn't mind logs and antilogs until Napierean logs came along, they were a real pig.

I also remember 'log' graph paper which was interesting just because it was different. Unfortunately we rarely used it, certainly not enough to get familiar with it, because of its cost (apparently).

 

[Old Yard Dog]

I remember tables of anti-logarithms very well, one of the best sets of tables ever invented I thought at the time. Say "Bar one point seven six two one" to the kids of today and they wouldn't know what you are talking about!

When I took my GCE's in the 1960s, an easement was introduced allowing the use of slide rules in physics and chemistry exams. But I stuck with log tables as I thought I was too cack-handed to use a slide rule accurately enough.

Move on to the 1970's and the first desktop computers. I was given one at work which had to be booted up every morning by running a folded paper tape through it. This took 50 mins. The machine was next to useless for anything practical.

(For any kids of today reading this, "Bar one point seven six two one" means -0.2379 which is log10 of 0.5782 which I checked on my Casio fx-7000GA heirloom)

 

[dakta]

I've never really got on with maths, and perhaps I should given i work (predominantly) in IT, do a lot of programming. I had a fresh crack with the Open Uni but found it an exercise of frustration.

Apart from introducing me (in terms of algebra) to the concept of variables, I can't really think of anything I was force fed that I would miss. Great thing is, with todays connected world even if there was it would be of minimal consequence. (I used to refer to the study of math as math for the sake of math, where reading a page of a book about a specific subject to get a peice of code done was more 'solving a problem')

Do often wonder if the time would have been better spent teaching softer? skills - how the tax system worked, or how self employment worked and how to do it, or practical techniques to find opportunities to build a career from bottom up.

 

[telstarbox]

I could have done with less trigonometry and probably more about algorithms or group theory, which we touched on at the very end of A Level maths.

Statistics and mechanics have proved very useful at work and in general life though.

 

[Springs Branch]

We must have used the hyperbolic functions as we were told that sinh was pronounced 'shine' and tanh 'than' (hard 'th'), whether that is true I don't know.

I also remember being told about the pronunciations as shine, cosh etc. during A-Level maths, but can't remember using the hyperbolic functions in anger, or ever looking them up in the tables at the back of a log book. Possibly it was something we touched on in Pure Maths regarding complex numbers and the good old square root of -1.


Speaking of scientific calculators, I bought a Casio fx-550S in 1981 or 1982 (I'm guessing - happy to be corrected on the vintage).

As a research scientist working in a lab, this got a daily workout for at least 10 years until desktop PCs became common (along with Lotus 1-2-3) around 1990. The calculator was then consigned to the back of a desk drawer until - inspired by this thread - I dug it out today.

Like the 200-year old VW Beetle in the Woody Allen film Sleeper, I switched on the Casio, the battery still works fine and it's ready for action again (pity I'm retired now).

 

[Grumbler]

After the waters receded, Noah told the animals to go forth and multiply.
They went off two by two, except for a pair of snakes who remained behind.
Noah asked them what the problem was. They replied they could not multiply, as they were adders.
Noah said he would consult with the Lord God.
When he returned, he saw lots of snakes in a pile of wood.
"It's OK now", they said "as we've got some logs."

 

[Typhoon]

I also remember being told about the pronunciations as shine, cosh etc. during A-Level maths, but can't remember using the hyperbolic functions in anger, or ever looking them up in the tables at the back of a log book. Possibly it was something we touched on in Pure Maths regarding complex numbers and the good old square root of -1.

Thanks, I was never certain about the Pure Maths teacher we had in the VIth form, I was always worried we would see him manning a stall at East Street Market on a Sunday.

I think we may have come across them in calculus and used the tables in definite integration; I wouldn't put it past them to give us such questions to justify the school buying a quite thorough set of tables.

 

[swt_passenger]

After the waters receded, Noah told the animals to go forth and multiply.
They went off two by two, except for a pair of snakes who remained behind.
Noah asked them what the problem was. They replied they could not multiply, as they were adders.
Noah said he would consult with the Lord God.
When he returned, he saw lots of snakes in a pile of wood.
"It's OK now", they said "as we've got some logs."

Then there was the garden centre that had almost run out of rustic outdoor furniture, all they had left were some log tables…

 

[david1212]

That Anita device is quite the desirable item! Lovely piece of retro kit especially with all those nixie tubes. Thanks for sharing!

Most likely in time it lingered in the back of a cupboard for a while then ended up in a skip. Even not working it would have a value now to a collector and / or for the nixie tubes.