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I'm not that old, but I learned the haversine formula in a marine navigation class I took through (boy) scouts. Our instructor was a retired merchant navy officer who was somewhat upset that the curriculum didn't include it, and deemed that we should know it.

My A-level (≈ AP) maths teacher was mildly amused that I knew it and digressed about log tables. We were (I believe) the last cohort to be issued log-tables, but given the choice in examinations between log-tables, slide-rule, or calculator, we all made the same choice.

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I recently learned about the existence of the Prosthaphaeresis algorithm [0], an even more ancient and obsolete trigonometric calculation technique. It allows you to use a trig table to do calculations when you don't have a log table. To calculate a x b, find arccos(a) and arccos(b), then use the product-to-sum formula: cos(a) x cos(b) = 1/2 x [cos(a + b) + cos(a - b)]. For example, to calculate 4294967296 x 2147483648. First, scale both numbers down by a factor of 1e10, a ≈ 0.4295, b ≈ 0.2147. Next, do inverse trig table lookup: x = arccos(a) ≈ 1.1269, y = arccos(b) ≈1.3544. Now, sum and difference: y + x = 2.4813, y - x = 0.2275. Finally, do trig table lookup and scale up, (cos(2.4813) + cos(0.2275)) x 1/2 x 1e20 ≈ 9221026434613711000. The true answer is 9223372036854775808, error is less than 0.03%. It was used by astronomers in the 16th century before logarithm was invented.

I learned it while reading the Wikipedia article Slide Rule, it mentions that two math educators David B. Sher and Dean C. Nataro invented a Prosthaphaeresis slide rule in 2004 - it's what a slide rule may look like in an alternative universe. Unfortunately, I still haven't seen it - I couldn't find the original paper, it doesn't seem to have a DOI, can't use Sci-Hub.

[0] https://en.wikipedia.org/wiki/Prosthaphaeresis

Prosthaphaeresis works because an angle measure inherently is a type of logarithm. It’s the logarithm of a rotation, instead of the logarithm of a scale. Like other kinds of logarithms, this converts multiplication (composition of rotations) to addition (addition of angle measures).

If you want to multiply two numbers, you can treat them as rotations, take the logarithm of each (i.e. find the angle measure), add the two, then take the exponential.

cf. https://en.wikipedia.org/wiki/Complex_logarithm

Is there a more general definition of logarithms than being the inverse of the exponential function?

Mathworld says Prosthapharesis formulas "convert a product of functions into a sum or difference" [0], which resembles a property of logarithms (i.e. log(ab) == log(a) + log(b)). However I don't quite get how angle measurement itself is a type of logarithm.

Maybe something to do with Euler's formula?

[0] https://mathworld.wolfram.com/ProsthaphaeresisFormulas.html

Starting for Euler's formula you can write cos(x) = (exp(ix) + exp(-ix))/2 so the cosine is in some sense a "type of exponential". If the cosine is a type of exponential then its inverse (the function that takes you to the angle measurement) is inherently a "type of logarithm".

This is a very handwavey statement. I would say a more precise statement is that the proper exponential and log maps are the "correct" thing to use to map multiplication to adidtion and vice-versa. They are exactly the maps that do this properly. The fact that cosine is a sum of exponentials means that you can write the inverse of cosine as a (slightly ugly) formula in terms of logarithms (arcos(z) = -i log(sqrt(1-z^2) + z). This is the sense in which the map to angle measures is a "type of log".

The angle measure i acos(x) is the logarithm of the rotation x + i√(1 − x²).

If you want to compute the product of two rotations, you can take logarithms and rewrite it as a sum of angle measures:

[x + i√(1 − x²)][y + i√(1 − y²)] = exp(i acos(x) + i acos(y))

[x + i√(1 − x²)][y − i√(1 − y²)] = exp(i acos(x) − i acos(y))

Now if you just look at the real parts, you can write:

xy − √(1 − x²)√(1 − y²) = cos(acos(x) + acos(y))

xy + √(1 − x²)√(1 − y²) = cos(acos(x) − acos(y))

Taking the sum on each side:

2xy = cos(acos(x) + acos(y)) + cos(acos(x) − acos(y))

This method is rather more cumbersome (due to rescaling and then converting the factors to rotations as an intermediate step) but fundamentally based on the same concept as:

xy = exp(log(x) + log(y))

There is an axiomatic approach that defines a logarithm as any function f(x) such that f(x•y) = f(x) + f(y). But it's easy to prove that this reduces to the usual definition for logarithms on the real numbers.

This is funny because when I was in school being taught navigation, we were told that celestial navigation would basically tell you where you were half an hour ago, because doing all the math with trig tables and a slide rule or log tables (forced to learn how to do it without electronic calculators), it would take about 30 minutes to reduce a star sight to a position on Earth surface.

So now we use that algorithm and you've just calculated where the ship was a few hours ago :-)

Is that really true? Maybe it takes 30 min to calculate from first principles, but I'd expect you could have a book of tables that tells you with just one or two lookups.

From first principles would take a lot longer. 30 minutes is roughly what I remember to reduce about 4 stars. I could be off a bit since it's been a really long time and I forget all the steps, but I remember a lot of table lookups and interpolations to get to a final fix.

It's also commonly used in astronomy to measure the distance on the sky between two objects: https://en.wikipedia.org/wiki/Haversine_formula

I could imagine these could be useful to be defined by computer libraries, as (see article) it would help reduce large reductions in accuracy for some calculations (e.g. by subtracting similar numbers, or wanting to calculate sin^2).

We were apparently allowed to ask for log tables in our A-level physics exams, even if we already had a calculator with us (which of course we all did). Our teacher took the step of proactively putting them on our desks before the start of the exam because that seemed to be consistent with the rules and didn't hurt.

Oh and by the way the log tables supplied by the exam board also had some useful formulae printed in them we'd otherwise have to memorise. Not that that had anything to do with it...

I'm also mildly amused that the author failed to mention the hyperbolic functions, of which there is one for each of the regular circular trig functions

Maybe they are not considered obsolete?

> given the choice in examinations between log-tables, slide-rule, or calculator, we all made the same choice

Students would benefit greatly if given a slide rule instead of an electronic calculator for their exams. The former is an effective teacher which viscerally reveals crucial insights, while the latter is pedagogically almost useless; using an electronic calculator to solve problems consists of nothing beyond punching keys and looking at the result, given as a disembodied string of digits entirely out of any context. The slide rule probably even ends up slightly more efficient for the type of problems students typically see in high school, after a bit of practice. Precision is more than sufficient.

Any assigned problem which needs a calculator would be more effective if the calculator were removed and the numbers involved were simplified to the point students could manage them mentally or with a trivial amount of scratch work.

Where electronic computers really shine is in their programmability, and for that students should be working with a full programming language and a real keyboard.

What I liked is the awareness of the inexactness which is inherent in everything but which slide rules made more obvious. The joke from these times was "ask an engineer what is 2 times 3 and he'll take out his slide rule and say between 5.9 and 6.1" Whoever thought that up maybe wanted to show how the engineers just use tools and don't think, but I've seen the opposite: it was some with just a theoretical mathematical training who blindly believed in every digit of the result they saw or produced, even when it was really meaningless or where it was obvious to me that the process resulted in an error.

The Art of Electronics warns the reader in the first chapter:

> There is a tendency among beginners to want to compute resistor values and other circuit component values to many significant places, and the availability of inexpensive calculators has only made matters worse. There are two reasons you should try to avoid falling into this habit: (a) the components themselves are of finite precision (typical resistors are 5%; the parameters that characterize transistors, say, frequently are known only to a factor of two); (b) one mark of a good circuit design is insensitivity of the finished circuit to precise values of the components (there are exceptions, of course). You'll also learn circuit intuition more quickly if you get into the habit of doing approximate calculations in your head, rather than watching meaningless numbers pop up on a calculator display.

There's a small book called "Errors of Observation and Their Treatment", ISBN 978-94-011-6928-8, which seemed dull in class but it emphasized being able to express and defend a numerical result as x +/- y . I did that thereafter and it made a lot of difference in my understanding of the result. It isn't done by many engineers any more, and I always look with suspicion on results claiming absurd precision. There isn't so much understanding there and people who engage in it are much more likely to submit work with multiple order-of-magnitude errors. They do not -know- the answer, they accept the line on the screen un-critically.

Those interested in this thread should also consider the posts by the user "anonytrary" while commenting this article in this context.

When I did my A-levels, we were taught to give answers as surds the exams usually asked for this when we were doing trig and it was generally accepted to give answers as fractions so I think I ended up using my expensive graphic calculator once or twice in 6 hours of maths exams.

This matches my recollection too. I had a calculator, I used it almost exclusively for "wasting" time programming in class, I don't believe I ever used it in an exam.

While the graphing calculators were available we weren't told we needed them so, being on a tight family budget, I didn't have one. My sister, a few years later was told that she had to have a graphing calculator, which she claims she never used

This seems to imagine that the only thing being taught/worth teaching on the mathematics curriculum is number sense.

Firstly, that's not the case. If you want students to answer a problem with any practical application (e.g. money/finance, statistics), or more complex problems where the point is to follow the logic, being able to use the calculator in your pocket is great.

A perfectly reasonable assigned problem might be: add the 7.1% sales tax to this ticket price, do you have enough cash?

And secondly, number sense is easier to teach without depending on particular algorithms.

This is the same complaint as people grumbling about "common core" math, or saying we should go back to the basics of just rote learning times tables and long division.

Students are learning about number sense by different routes now - routes that don't end up with most students blindly trying to follow a particular algorithm with no sense of what is actually happening.

I once taught in a physics curriculum that went a step beyond calculators. Students never had to do numerical calculations in the exams. For example, a question asking you to find a force, could have an answer like "F=ma where m=3 kg and a=2m/s". There were higher grades for showing insight into the physics that you could get by looking at the form of the symbolic equations you'd derived but would have trouble noticing if you'd substituted numbers early on.

I wish I had learned times tables properly as a kid though. I struggle with things like 6*7 on a near daily basis for my work and general life. Not really for number sense but for not having to shift my attention to a basic math problem while I'm working on something else.

I think you are misunderstanding me / underestimating students. The slide rule teaches in a very direct physical way how logarithms work and how they can be used. This is something that most people never learn, but is extremely valuable.

> This seems to imagine that the only thing being taught/worth teaching on the mathematics curriculum is number sense

First of all, number sense is extremely important. Probably the most important thing taught in primary/secondary math courses. But it is certainly not the only important thing.

Students can learn to use an electronic calculator in very little time. A person of average intelligence who understands the relevant math should be able to learn to use their calculator pretty much independently, and become fluent at it with a tiny bit of practice over a short time. There are very quickly diminishing returns to teaching “calculator skills”, because those are extremely shallow.

> And secondly, number sense is easier to teach without depending on particular algorithms.

I really have no idea which ‘particular algorithms’ you are talking about. Have you ever used a slide rule? It is a very flexible general-purpose tool.

> This is the same complaint as people grumbling about "common core" math, or saying we should go back to the basics of just rote learning times tables and long division.

No, it is precisely the opposite recommendation to those.

> A perfectly reasonable assigned problem might be: add the 7.1% sales tax to this ticket price, do you have enough cash?

This is a reasonable problem to teach students about for like 1 week at age ~10. If they learn how to do it using pen and paper, or a soroban, or mental arithmetic, or a slide rule, or a pile of loose pebbles, they’ll have no trouble accomplishing the same with a calculator. It is not a reasonable problem to spend 5 more years on. We are talking about exams for 15-year-olds.

> This is a reasonable problem to teach students about for like 1 week at age ~10.

I'm afraid you have a completely unrealistic expectation of the median student (which is reflected in your other comments as well). I have taught hundreds of students and I would think a handful of them could answer this at age 10.

This type of question first appears on the Khan Academy in Grade 7, i.e. targeted at 12-13 year-olds.

There is a significant body of prerequisite work in the earlier grades, some of which is limited by what is developmentally appropriate.

Students will need dedicated practice to recall this and most won't remember it after seeing it or mastering it on one occasion. It needs to be supported in the rest of the curriculum. You might be able to intensively teach it earlier, but it is hardly worth it, because they will completely forget when you intensively teach the next topic.

Many 15-year-olds will continue to struggle with this and, if they are well-supported, might be taught it as a step-by-step procedure in order to best score marks in the exam they need to. These students are unlikely to finish without a good concept of what they were doing - and might be more likely to pick it up as adults out of necessity.

«Add 7.1% sales tax to the $19.95 ticket» (or whatever) is a 1-step (or generously 2-step) arithmetic problem, of the kind most students in many parts in the world learn to understand conceptually at age 6–7 with no problem. It additionally involves multi-digit multiplication, of the type people typically learn by about age 9–10 (?).

If students have not practiced solving problems enough to handle combining these things until age 13, and many are still struggling with it at age 15 (with a calculator!), that’s a serious indictment of the entire education system.

I checked some well-respected national curriculums.

Singapore was the only one I found where percentage increase is introduced earlier as a national standard.

It's mentioned in the curriculum for Singapore Primary 6, i.e. 11–12 year olds, for students in the higher of two streams for mathematics.

As I mentioned in a comment upthread, I was trained in navigation as a Merchant Marine officer and we were not allowed to use electronic calculators during exams. The reason given was "what if you're in the middle of the ocean and your batteries die?" or something stupid like that. But the real reason was that our instructor wanted us to think about what we were doing. Celestial navigation involves a lot of steps and errors compound. Forcing you to think clearly about the intermediate results means that you're more likely to realize you pulled the wrong number out of a table, or your star sight was just plain wrong and you're not off the coast of Madagascar, but somewhere near the Azores instead.

Same here: haversines seemed normal to me since I was trained as a merchant marine officer and we used them in Great Circle calculations. I still have my old copy of American Practical Navigator ("Bowditch") with pages and pages of trig tables.

> log-tables, slide-rule

Is there a difference?

I went to high school well after the rise of calculators. Indeed, only one of my maths teachers had even used a slide rule, so I had to figure the thing out myself. One of the first things was that the multiplication scale on a slide rule is a low precision log-table. The other is why most math and physics problems only asked for solutions to two or three significant figures. (While physics problems do deal with precision, the consistency of the precision is definitely suspect.)

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