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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 offic

by pacaro 6y ago

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.

pacaro 6y ago
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
- nine_k 6y ago
  Maybe they are not considered obsolete?
jacobolus 6y ago
> 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.
- willyt 6y ago
  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.
  - pacaro 6y ago
    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
- acqq 6y ago
  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.
  - segfaultbuserr 6y ago
    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.
xioxox 6y ago
It's also commonly used in astronomy to measure the distance on the sky between two objects: https://en.wikipedia.org/wiki/Haversine_formula 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).
segfaultbuserr 6y ago
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 https://en.wikipedia.org/wiki/Prosthaphaeresis
- jacobolus 6y ago
  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 https://en.wikipedia.org/wiki/Complex_logarithm
  - HPsquared 6y ago
    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 https://mathworld.wolfram.com/ProsthaphaeresisFormulas.html
II2II 6y ago
> 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.)
quietbritishjim 6y ago
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...
HeyLaughingBoy 6y ago
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.

by pacaro 6y ago

pacaro 6y ago

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

nine_k 6y ago
Maybe they are not considered obsolete?

jacobolus 6y ago

> 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.

willyt 6y ago
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.
- pacaro 6y ago
  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
acqq 6y ago
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.
- segfaultbuserr 6y ago
  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.

xioxox 6y ago

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

segfaultbuserr 6y ago

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 https://en.wikipedia.org/wiki/Prosthaphaeresis

jacobolus 6y ago
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 https://en.wikipedia.org/wiki/Complex_logarithm
- HPsquared 6y ago
  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 https://mathworld.wolfram.com/ProsthaphaeresisFormulas.html

II2II 6y ago

> 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.)

quietbritishjim 6y ago

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...

HeyLaughingBoy 6y ago

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.