1600s binary multiplier (Napier's Location Arithmetic) Review / HowTo

A board for Napier's location arithmetic, as described in Napier's "Rabdology" (rabdologia), published 1617. It multiplies binary numbers!
This is episode 80 of my video series about old calculating devices.
End song inspired by "Hotter than a Molotov" by The Coup. • The Coup: Hotter Than ...
Chris Staecker webpage with location board free PDF download: faculty.fairfi...
Chris Staecker shop with location board poster for purchase: chris-staecker...

[@FiveTrackTape]

"Hold like the seagull thing close to you with the baby devil on the left side" Love it. The gift of teaching is strong in this one!

[@ChrisStaecker]

You can tell I have deep respect for the horoscope.

[@MrEMeat-kk9tc]

@@ChrisStaeckerMath and astronomy do hand in glove after all

[@Sedgewise47]

@@ChrisStaecker 🤔[??]….Umm-By any chance-might this-or “something(s?)” like this?-have _anything_ to do with how the UK’s “Chancellor of the Exchequer” got its name? ( 🤷‍♂️ Just asking!…)

[@evanbarnes9984]

I wish we could bring Napier forward in time for a bit to show him just how important his ideas turned out to be. There are so many people in history I would love to do that with!

[@retrogiftsuk4812]

I think it would be easier to do the diagonal moves (and the carrying) if the squares were coloured like a chess board. As you slide along a diagonal the piece would stay on its colour (it would be easy to see where it's going). When you carry, the piece changes colour. Obviously this would make it less authentic, but hey I'm assuming those buttons are made of animal bone or ivory... Great video as always. I'm always excited when I see you've posted a new one. Can't wait for the sequel...

[@redoktopus3047]

I was thinking the same thing! or simply writing the product in each box. then if you use rings instead of buttons so you can see the product through the middle it becomes easy as pie.

[@ChrisStaecker]

@@redoktopus3047 yes I agree the colors would be a good idea. He specifically likens it to a chessboard in the book, so I’m sure he thought of this. I would guess he really wanted to write the letters, which would get messed up by coloring the squares. In the book he says that we should imagine that EVERY square has a letter in it, with diagonal squares having the same letters. (So all squares diagonally equivalent to ‘x’ have a ‘x’ in them.) According to his location numeral system (no spoilers!) this would be equivalent to putting the products in every square.

[@ClausB252]

6:50 Wait, what? 20 binary digits is over 1 million decimal, not 52,000.

[@ChrisStaecker]

Yes you're right- I'm not sure what I was thinking. Maybe I accidentally did 2^19 which is around 520k, and THEN accidentally read it as 52k?

[@ClausB252]

Otherwise excellent video on an interesting topic.

[@jjrubes1880]

Shouldn't it also be 19 binary digits too? 20 double counts the corner

[@nmmm2000]

when I saw this board for the first time 2 years ago, it was like magic :) I think you should explain square roots as well. the "rectangle" squares used there are also magical.

[@geckoman1011]

As a historian with a passing interest in math, this really hit the spot. Thanks for the great video. Also, kind of had This old Tony vibes to the video, and I liked it

[@michaeldamolsen]

Thank you Chris, this is a wonderful video. I am really looking forward to the follow-up.

[@rlamacraft]

Now I'm just thinking what would a wooden one look like, with marbles for counters. Could such a thing be designed so that tipping the device would cause the balls to roll across, and as the balls pile up they begin a cascade of carrying?

[@ChrisStaecker]

I actually had a similar thought- a rack of some kind where you load it up by hand, and then pick it up or shake it out, and the answer appears. I'm not sure exactly how it would work, but it's probably possible. Maybe arranged like a Turing Tumble?

[@RobertMilesAI]

Looking at the latin, the squares on the board are called 'areolae'. Turns out 'areola' just means 'a small area/space'! The anatomical term is short for 'areola papillaris', so basically 'the small area around the nipple', and now we just call it 'the small area'

[@ChrisStaecker]

I noticed this too, thought for a while about how I could work this weird vocabulary fact into the video, and gave up after I couldn't do it justice.

[@crowndotm3866]

BTW our family has been placing and lighting birthday candles in binary for YEARS. Think of the money we have saved on birthday candles!

[@BROTRRer]

Damn, my boy Napier thought of everything!

[@wmeisel]

Great video, Chris. I thought I knew a lot about Napier but this is the first I’ve heard of this.

[@austinerickson8901]

The follow up video will be epic, indeed!

[@pafnutiytheartist]

0:35 Arguably, still does, floating point numbers implicitly use logarithms base 2 if you squint a little.

[@Keldor314]

Floating point numbers are really just scientific notation in base 2. I guess you could say they're still related to logarithms.

[@malpollyon]

You can also easily extend the board to negative numbers with base "-2" and even to complex numbers with base "i-1" as both only need the digits 0 and 1. The only difference is that each base has a different rule for carries.

[@ChrisStaecker]

More trivial and probably more useful, label half of the rows/columns with negative exponents (still base 2). Then you'll be able to handle non-integers.

[@yogimew]

I converted my magnetic reversi/othello board for location arithmetic once before. It was rather a low effort way of demonstrating the key concepts of the thing.

[@Pallethands]

They are buttons made of bones. Use them with this board and you can have your bones and math 'em too!

[@tajmahalfred]

Wooooo another craft project for the weekend

[@redoktopus3047]

If you wanted to measure the mass of the observable universe using up quarks as your unit, you'd need 275 binary digits. To measure the volume of the observable universe using cubic planck lengths you'd need 614 binary digits. So if Napier just had a playground and some chalk (and maybe some kids to order around this big grid) he could have as big a board as he'd ever need.

[@Laundry_Hamper]

You could use some tin snips to cut a square out of a baking tray, glue that to the back of the canvas and then it'd be compatible with magnet technology - which would unlock the wall-mount feature without compromising the rustic aesthetic :P

[@maxmn5821]

Suggestion to viewers without printer and buttons: try it with a chess board and rice grains. The chess board makes diagonal moves easier and rice grains pay the tribute to an ancient legend about a geometric progression with common ratio equal to the binary system base.

[@ompizz]

People see buttons. I see convolution.

[@rodneylives]

Nicely explained. I'm still surprised by the fact that computers, when they do math on binary values, essentially do it the way we did back in school, but just with binary digits. Yet the process feels different, in a way. Such arithmetic turns out to be simpler in binary.

[@MattMcIrvin]

I think what makes it feel different is that in base 10, we think of counting in one digit as the "normal" case and carrying as an exception that happens sometimes, but in binary, the digit counting is trivial ("0,1") and carrying is most of what you do. The emphasis is all different.

[@grantolson]

Honestly I feel Napier had some mental breakthroughs that were like E=MC² breakthroughs.

[@redoktopus3047]

I hope one day you can get an in at a museum that has an Antikythera replica! or maybe the Abakion or reckoning tables! There is a really cool book that I think is right up your alley called "Computation and Its Limits".

[@MichaelBernhardArpSrensen]

I love your videos. You inspire me. Again.

[@biquinary]

Yessssssss I'm so thrilled!!

[@jurjenbos228]

Any modern microprocessor contains a microscopic electronic version of this board to do its multiplications.

[@algorithminc.8850]

Thanks. I look forward to scoping your channel. Subscribed. Great stuff. Cheers

[@tinacordon7866]

A modern reprint of Napier's Rabdology forms volume 15 of the Charles Babbage Institute reprint series for the History of Computing

[@ChrisStaecker]

Not so fun fact: I was holding that book when a pivotal life event happened! A time I shudder to remember, but I won’t forget that it happened when I was reading Napier.

[@timetraveller6643]

Topic Suggestion: The weird MONIAC Phillips Machine that modeled the economy with plumbing.

[@Nashvillain10SE]

Ahh, well worth the wait!! 😀👍

[@esaa7256]

Hohoo, he’s got a follow-up video for us!

[@Vallee152]

Can't wait for the locational number system video!

[@BillRicker]

sweet. Napier had Four of the top two multiplication systems !

[@davegraham7550]

great work...thanks

[@timetraveller6643]

Lovely ! Thank you. I too want to see the sqrt function. I tried to work it out from the booklet with no success. Has there ever been a clever square root finder that didn't involve logarithms and nomograms?

[@AngrocSound]

Awesome videos as always. Looking forward to the follow-up!

[@GeorgMierau]

Thank you very much! Besides a very cool video (as usual) I found a matchbox from my childhood at 7:25 ;)

[@itwasrightthere]

Very cool. I believe that I’ll play around with this.

[@Gelatinocyte2]

Now, if only there's a simplified logic circuit that does exactly this process...

[@joehopfield]

This feels like antikithythera-level time travel magic - it could have been mechanized with 1600s clock-work tech (division is ugly in binary cpus as well)

[@StephenRansom47]

Bravo 👏 … lol … that baby devil thing. They are actually the first four Glyphs in the Zodiac and the primal expressions of ♈️Fire, ♉️Earth, ♊️Air & ♋️Water 🤔 … does he use them for orientation? I am very curious.

[@michaeldamolsen]

My guess is he just ran out of Greek letters and randomly opted to use the zodiacal signs. If he had given it careful thought, I think he would have used Aries, Cancer, Libra & Capricorn instead, as those are the traditional symbols for the equinoxes and solstices and are thus 90 degrees apart. Back in those days, most people with a keen interest in mathematics would also be familiar with astronomy, and the equinox/solstice orientation would have made immediate sense to them, much more so than the 4 sequential signs Napier used.

[@StephenRansom47]

@@michaeldamolsen that was my thinking, but the chart could be a mnemonic of a flame ♈️ with sums accumulating to the left (grounding) ♉️ … numbers rise into higher Air ♊️ and flow diagonally in streams ♋️ 🤔 PS - thx for the comment, These things fascinate me. The old mnemonics of a Mind Palace (or calculator)

[@ChrisStaecker]

You can see the full text of the original book with some commentary here: www.sliderulemuseum.com/Papers/Napier_John.Rabdologiae.1617.Edinburgh.pdf Not exactly a translation, but it's usually not too hard to make out what he's saying. He uses the signs often in the text, but usually just to specify what he's talking about on the diagram- like "the side ♈️ ♉ of the board" kind of like how we'd say "side AB in triangle ABC" in high school geometry. I didn't see any explanation of why he was using these symbols, but I kinda skimmed those parts.

[@StephenRansom47]

@@ChrisStaecker 🙏 … as always, brilliant work.

[@michaeldamolsen]

@@ChrisStaecker Thanks for the pdf link! "I didn't see any explanation of why he was using these symbols, but I kinda skimmed those parts." - Yes me too, I spent a lot longer than I thought hacking my way through the latin, and arrived at the same conclusion as you. He doesn't attribute the symbols to anything in particular, but he might still have thought about it in the way you describe.

[@ewthmatth]

7:12 ok but in general isn't binary the easiest base with which to do square roots by hand?

[@tracefleemangarcia8812]

Hey Chris! I'm currently preparing a manuscript for publication on Ramón Llull and Leibniz, especially their indebtedness to Arab astronomical (really astrological) calculating devices. Do you plan to do a video on Llull's ars magna or Leibniz's ars combinatoria? Both have been called the grandfather of modern computers.

[@ChrisStaecker]

Thanks so much! I usually don't do straight books, but maybe... I'm not familiar with Llull at all- I'll check it out.

[@tracefleemangarcia8812]

@@ChrisStaecker Llull and Leibniz definitely published their work as books, but in the case of Llull at least it usually was full of little paper wheels that spun around to produce combinations of terms. It spawned a whole centuries-long fad for books with moving discs in them. It IS incredibly opaque though...as one should expect from a system allegedly that can prove all the facts of reality and was given to Llull by none other than a mystical vision of Jesus. I don't fully understand it myself as I've only been studying it for a year. Another possible ancient precedent is the I Ching (Yijing) that inspired Leibniz to publish his work on binary arithmetic. Definitely easier to learn to use than Llull's art.

[@ChrisStaecker]

@tracefleemangarcia8812 spinning wheels you say? Now I’m really interested…

[@tracefleemangarcia8812]

@@ChrisStaecker Yes, spinning wheels -- sometimes in very complex configurations! This is the reason Llull is sometimes understood as an early computer scientist: he was making machines (very simple paper machines) to "think" and it was this that inspired Leibniz to, you know, start building calculators. Arguably the idea of combining terms algorithmically is how Leibniz developed binary. Lots of wackiness going on if you're mainly interested in the math though! There's a reason Llull became seen as a magician / sorcerer in later centuries.

[@ChrisStaecker]

@tracefleemangarcia8812 is there a modern edition in English you can recommend?

[@dp29931]

Please, please, please make another retorts video!

[@ChrisStaecker]

I will make retorts for any video that goes to a million!

[@olegtarasovrodionov]

Division and square root is just backwards multiplication. How they could be less successful?

[@johnsrabe]

0:29 Ha

[@dolata000]

3:38 32 + 16 + 4 + 1 is not 55

[@ChrisStaecker]

You are correct. I don't know what I was thinking. I suppose I was thinking that 32 + 16 + 4 + 1 = 55.

[@dolata000]

@@ChrisStaecker To err is human! Good video overall with one minor ooopsie.

[@PraxZimmerman]

Fuck yeah, math shit 🥰

[@tajmahalfred]

Don't tell me you have a Soviet matchbox collection you're hiding

[@ChrisStaecker]

Well I’m hardly hiding it!

[@tajmahalfred]

Maybe "teasing"

[@tajmahalfred]

Anyway glad to see you're making the most of the summer break

[@YEWCHENGYINMoe]

14h ago

[@YZoxK52m]

These guys like buttons, right?! 😊

[@twixerclawford]

The real magic is realizing that binary is simply the best number system because doing this exact procedure on paper is so much faster than doing the same multiplication by hand in base 10

[@lafcursiax]

I really appreciate the long ſ!

[@NonTwinBrothers]

ae ct st ligatures my beloved