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Sat, 21 Nov 2020
Testing for divisibility by 19
[ Previously, Testing for divisibility by 7. ] A couple of nights ago I was keeping Katara company while she revised an essay on The Scarlet Letter (ugh) and to pass the time one of the things I did was tinker with the tests for divisibility rules by 9 and 11. In the course of this I discovered the following method for divisibility by 19:
The result will be a smaller number which is a multiple of 19 if and only if the original number was. For example, let's consider, oh, I don't know, 2337. We calculate:
76 is a multiple of 19, so 2337 was also. But if you're not sure about 76 you can compute 2·6+7 = 19 and if you're not sure about that you need more help than I can provide. I don't claim this is especially practical, but it is fun, not completely unworkable, and I hadn't seen anything like it before. You can save a lot of trouble by reducing the intermediate values mod 19 when needed. In the example above above, after the first step you get to 17, which you can reduce mod 19 to -2, and then the next step is -2·2+3 = -1, and the final step is -1·2+2 = 0. Last time I wrote about this Eric Roode sent me a whole compendium of divisibility tests, including one for divisibility by 19. It's a little like mine, but in reverse: group the digits in pairs, left to right; multiply each pair by 5 and then add the next pair. Here's 2337 again:
Again you can save a lot trouble by reducing mod 19 before the multiplication. So instead of the first step being 23·5 + 37 you can reduce the 23·5 to 4·5 = 20 and then add the 37 to get 57 right away. [ Addendum: Of course this was discovered long ago, and in fact Wikipedia mentions it. ] [ Addendum 20201123: An earlier version of this article claimed that the double-and-add step I described preserves the mod-19 residue. It does not, of course; the doubling step doubles it. It is, however, true that it is zero afterward if and only if it was zero before. ] [Other articles in category /math] permanent link [Other articles in category /misc] permanent link Mon, 02 Nov 2020
A better way to do remote presentations
A few months ago I wrote an article about a strategy I had tried when giving a talk via videochat. Typically:
I thought I had done it a better way:
This, I thought, had several advantages:
When I brought this up with my co-workers, some of them had a very good objection:
Fair enough! I have done this. If you package your slides with page-turner, one instance becomes the “leader” and the rest are “followers”. Whenever the leader moves from one slide to the next, a very simple backend server is notified. The followers periodically contact the server to find out what slide they are supposed to be showing, and update themselves accordingly. The person watching the show can sit back and let it do all the work. But! If an audience member wants to skip ahead, or go back, that works too. They can use the arrow keys on their keyboard. Their follower instance will stop synchronizing with the leader's slide. Instead, it will display a box in the corner of the page, next to the current slide's page number, that says what slide the leader is looking at. The number in this box updates dynamically, so the audience person always knows how far ahead or behind they are.
At left, the leader is displaying slide 3, and the follower is there also. When the leader moves on to slide 4, the follower instance will switch automatically. At right, the follower is still looking at slide 3, but is detached from the leader, who has moved on to slide 007, as you can see in the gray box. When the audience member has finished their excursion, they can click the gray box and their own instance will immediately resynchronize with the leader and follow along until the next time they want to depart. I used this to give a talk to the Charlotte Perl Mongers last week and it worked. Responses were generally positive even though the UI is a little bit rough-looking. Technical detailsThe back end is a tiny server, written in Python 3 with Flask. The server is really tiny, only about 60 lines of code. It has only two endpoints: for getting the leader's current page, and for setting it. Setting requires a password.
The front end runs in the browser. The user downloads the front-end
script,
The On page switching, a tiny amount of information is stored in the
browser window's If you want page-turner to display the leader's slide number when the
follower is desynchronized, as in the example above, you include an
element with class
The password for the
to
or whatever. Many improvements are certainly possible. It could definitely look a lot better, but I leave that to someone who has interest and aptitude in design. I know of one serious bug: at present the server doesn't handle SSL,
so must be run at an Source code downloadPatches welcome. LicenseThe software is licensed under the Creative Commons Attribution 4.0 license (CC BY 4.0). You are free to share (copy and redistribute the software in any medium or format) and adapt (remix, transform, and build upon the material) for any purpose, even commercially, so long as you give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. Share and enjoy. [Other articles in category /talk] permanent link Sun, 18 Oct 2020
Newton's Method and its instability
While messing around with Newton's method for last week's article, I built this Desmos thingy: The red point represents the initial guess; grab it and drag it around, and watch how the later iterations change. Or, better, visit the Desmos site and play with the slider yourself. (The curve here is !!y = (x-2.2)(x-3.3)(x-5.5)!!; it has a local maximum at around !!2.7!!, and a local minimum at around !!4.64!!.) Watching the attractor point jump around I realized I was arriving at a much better understanding of the instability of the convergence. Clearly, if your initial guess happens to be near an extremum of !!f!!, the next guess could be arbitrarily far away, rather than a small refinement of the original guess. But even if the original guess is pretty good, the refinement might be near an extremum, and then the following guess will be somewhere random. For example, although !!f!! is quite well-behaved in the interval !![4.3, 4.35]!!, as the initial guess !!g!! increases across this interval, the refined guess !!\hat g!! decreases from !!2.74!! to !!2.52!!, and in between these there is a local maximum that kicks the ball into the weeds. The result is that at !!g=4.3!! the method converges to the largest of the three roots, and at !!g=4.35!!, it converges to the smallest. This is where the Newton basins come from: Here we are considering the function !!f:z\mapsto z^3 -1!! in the complex plane. Zero is at the center, and the obvious root, !!z=1!! is to its right, deep in the large red region. The other two roots are at the corresponding positions in the green and blue regions. Starting at any red point converges to the !!z=1!! root. Usually, if you start near this root, you will converge to it, which is why all the points near it are red. But some nearish starting points are near an extremum, so that the next guess goes wild, and then the iteration ends up at the green or the blue root instead; these areas are the rows of green and blue leaves along the boundary of the large red region. And some starting points on the boundaries of those leaves kick the ball into one of the other leaves… Here's the corresponding basin diagram for the polynomial !!y = (x-2.2)(x-3.3)(x-5.5)!! from earlier: The real axis is the horizontal hairline along the middle of the diagram. The three large regions are the main basins of attraction to the three roots (!!x=2.2, 3.3!!, and !!5.5!!) that lie within them. But along the boundaries of each region are smaller intrusive bubbles where the iteration converges to a surprising value. A point moving from left to right along the real axis passes through the large pink !!2.2!! region, and then through a very small yellow bubble, corresponding to the values right around the local maximum near !!x=2.7!! where the process unexpectedly converges to the !!5.5!! root. Then things settle down for a while in the blue region, converging to the !!3.3!! root as one would expect, until the value gets close to the local minimum at !!4.64!! where there is a pink bubble because the iteration converges to the !!2.2!! root instead. Then as !!x!! increases from !!4.64!! to !!5.5!!, it leaves the pink bubble and enters the main basin of attraction to !!5.5!! and stays there. If the picture were higher resolution, you would be able to see that the pink bubbles all have tiny yellow bubbles growing out of them (one is 4.39), and the tiny yellow bubbles have even tinier pink bubbles, and so on forever. (This was generated by the Online Fractal Generator at usefuljs.net; the labels were added later by me. The labels’ positions are only approximate.) [ Addendum: Regarding complex points and !!f : z\mapsto z^3-1!! I said “some nearish starting points are near an extremum”. But this isn't right; !!f!! has no extrema. It has an inflection point at !!z=0!! but this doesn't explain the instability along the lines !!\theta = \frac{2k+1}{3}\pi!!. So there's something going on here with the complex derivative that I don't understand yet. ] [Other articles in category /math] permanent link
Fixed points and attractors, part 3
Last week I wrote about a super-lightweight variation on Newton's method, in which one takes this function: $$f_n : \frac ab \mapsto \frac{a+nb}{a+b}$$ or equivalently $$f_n : x \mapsto \frac{x+n}{x+1}$$ Iterating !!f_n!! for a suitable initial value (say, !!1!!) converges to !!\sqrt n!!: $$ \begin{array}{rr} x & f_3(x) \\ \hline 1.0 & 2.0 \\ 2.0 & 1.667 \\ 1.667 & 1.75 \\ 1.75 & 1.727 \\ 1.727 & 1.733 \\ 1.733 & 1.732 \\ 1.732 & 1.732 \end{array} $$ Later I remembered that a few months back I wrote a couple of articles about a more general method that includes this as a special case: The general idea was:
We can see that !!\sqrt n!! is a fixed point of !!f_n!!: $$ \begin{align} f_n(\sqrt n) & = \frac{\sqrt n + n}{\sqrt n + 1} \\ & = \frac{\sqrt n(1 + \sqrt n)}{1 + \sqrt n} \\ & = \sqrt n \end{align} $$ And in fact, it is an attracting fixed point, because if !!x = \sqrt n + \epsilon!! then $$\begin{align} f_n(\sqrt n + \epsilon) & = \frac{\sqrt n + \epsilon + n}{\sqrt n + \epsilon + 1} \\ & = \frac{(\sqrt n + \sqrt n\epsilon + n) - (\sqrt n -1)\epsilon}{\sqrt n + \epsilon + 1} \\ & = \sqrt n - \frac{(\sqrt n -1)\epsilon}{\sqrt n + \epsilon + 1} \end{align}$$ Disregarding the !!\epsilon!! in the denominator we obtain $$f_n(\sqrt n + \epsilon) \approx \sqrt n - \frac{\sqrt n - 1}{\sqrt n + 1} \epsilon $$ The error term !!-\frac{\sqrt n - 1}{\sqrt n + 1} \epsilon!! is strictly smaller than the original error !!\epsilon!!, because !!0 < \frac{x-1}{x+1} < 1!! whenever !!x>1!!. This shows that the fixed point !!\sqrt n!! is attractive. In the previous articles I considered several different simple functions that had fixed points at !!\sqrt n!!, but I didn't think to consider this unusally simple one. I said at the time:
but I never did get around to the Möbius transformations, and I have long since forgotten what I planned to say. !!f_n!! is an example of a Möbius transformation, and I wonder if my idea was to systematically find all the Möbius transformations that have !!\sqrt n!! as a fixed point, and see what they look like. It is probably possible to automate the analysis of whether the fixed point is attractive, and if not to apply one of the transformations from the previous article to make it attractive. [Other articles in category /math] permanent link Tue, 13 Oct 2020
Newton's Method but without calculus — or multiplication
Newton's method goes like this: We have a function !!f!! and we want to solve the equation !!f(x) = 0.!! We guess an approximate solution, !!g!!, and it doesn't have to be a very good guess. Then we calculate the line !!T!! tangent to !!f!! through the point !!\langle g, f(g)\rangle!!. This line intersects the !!x!!-axis at some new point !!\langle \hat g, 0\rangle!!, and this new value, !!\hat g!!, is a better approximation to the value we're seeking. Analytically, we have: $$\hat g = g - \frac{f(g)}{f'(g)}$$ where !!f'(g)!! is the derivative of !!f!! at !!g!!. We can repeat the process if we like, getting better and better approximations to the solution. (See detail at left; click to enlarge. Again, the blue line is the tangent, this time at !!\langle \hat g, f(\hat g)\rangle!!. As you can see, it intersects the axis very close to the actual solution.) In general, this requires calculus or something like it, but in any particular case you can avoid the calculus. Suppose we would like to find the square root of 2. This amounts to solving the equation $$x^2-2 = 0.$$ The function !!f!! here is !!x^2-2!!, and !!f'!! is !!2x!!. Once we know (or guess) !!f'!!, no further calculus is needed. The method then becomes: Guess !!g!!, then calculate $$\hat g = g - \frac{g^2-2}{2g}.$$ For example, if our initial guess is !!g = 1.5!!, then the formula above tells us that a better guess is !!\hat g = 1.5 - \frac{2.25 - 2}{3} = 1.4166\ldots!!, and repeating the process with !!\hat g!! produces !!1.41421\mathbf{5686}!!, which is very close to the correct result !!1.41421\mathbf{3562}!!. If we want the square root of a different number !!n!! we just substitute it for the !!2!! in the numerator. This method for extracting square roots works well and requires no calculus. It's called the Babylonian method and while there's no evidence that it was actually known to the Babylonians, it is quite ancient; it was first recorded by Hero of Alexandria about 2000 years ago. How might this have been discovered if you didn't have calculus? It's actually quite easy. Here's a picture of the number line. Zero is at one end, !!n!! is at the other, and somewhere in between is !!\sqrt n!!, which we want to find. Also somewhere in between is our guess !!g!!. Say we guessed too low, so !!0 \lt g < \sqrt n!!. Now consider !!\frac ng!!. Since !!g!! is too small to be !!\sqrt n!! exactly, !!\frac ng!! must be too large. (If !!g!! and !!\frac ng!! were both smaller than !!\sqrt n!!, then their product would be smaller than !!n!!, and it isn't.) Similarly, if the guess !!g!! is too large, so that !!\sqrt n < g!!, then !!\frac ng!! must be less than !!\sqrt n!!. The important point is that !!\sqrt n!! is between !!g!! and !!\frac ng!!. We have narrowed down the interval in which !!\sqrt n!! lies, just by guessing. Since !!\sqrt n!! lies in the interval between !!g!! and !!\frac ng!! our next guess should be somewhere in this smaller interval. The most obvious thing we can do is to pick the point halfway in the middle of !!g!! and !!\frac ng!!, So if we guess the average, $$\frac12\left(g + \frac ng\right),$$ this will probably be much closer to !!\sqrt n!! than !!g!! was: This average is exactly what Newton's method would have calculated, because $$\frac12\left(g + \frac ng\right) = g - \frac{g^2-n}{2g}.$$ But we were able to arrive at the same computation with no calculus at all — which is why this method could have been, and was, discovered 1700 years before Newton's method itself. If we're dealing with rational numbers then we might write !!g=\frac ab!!, and then instead of replacing our guess !!g!! with a better guess !!\frac12\left(g + \frac ng\right)!!, we could think of it as replacing our guess !!\frac ab!! with a better guess !!\frac12\left(\frac ab + \frac n{\frac ab}\right)!!. This simplifies to $$\frac ab \Rightarrow \frac{a^2 + nb^2}{2ab}$$ so that for example, if we are calculating !!\sqrt 2!!, and we start with the guess !!g=\frac32!!, the next guess is $$\frac{3^2 + 2\cdot2^2}{2\cdot3\cdot 2} = \frac{17}{12} = 1.4166\ldots$$ as we saw before. The approximation after that is !!\frac{289+288}{2\cdot17\cdot12} = \frac{577}{408} = 1.41421568\ldots!!. Used this way, the method requires only integer calculations, and converges very quickly. But the numerators and denominators increase rapidly, which is good in one sense (it means you get to the accurate approximations quickly) but can also be troublesome because the numbers get big and also because you have to multiply, and multiplication is hard. But remember how we figured out to do this calculation in the first place: all we're really trying to do is find a number in between !!g!! and !!\frac ng!!. We did that the first way that came to mind, by averaging. But perhaps there's a simpler operation that we could use instead, something even easier to compute? Indeed there is! We can calculate the mediant. The mediant of !!\frac ab!! and !!\frac cd!! is simply $$\frac{a+c}{b+d}$$ and it is very easy to show that it lies between !!\frac ab!! and !!\frac cd!!, as we want. So instead of the relatively complicated $$\frac ab \Rightarrow \frac{a^2 + nb^2}{2ab}$$ operation, we can try the very simple and quick $$\frac ab \Rightarrow \operatorname{mediant}\left(\frac ab, \frac{nb}{a}\right) = \frac{a+nb}{b+a}$$ operation. Taking !!n=2!! as before, and starting with !!\frac 32!!, this produces: $$ \frac 32 \Rightarrow\frac{ 7 }{ 5 } \Rightarrow\frac{ 17 }{ 12 } \Rightarrow\frac{ 41 }{ 29 } \Rightarrow\frac{ 99 }{ 70 } \Rightarrow\frac{ 239 }{ 169 } \Rightarrow\frac{ 577 }{ 408 } \Rightarrow\cdots$$ which you may recognize as the convergents of !!\sqrt2!!. These are actually the rational approximations of !!\sqrt 2!! that are optimally accurate relative to the sizes of their denominators. Notice that !!\frac{17}{12}!! and !!\frac{577}{408}!! are in there as they were before, although it takes longer to get to them. I think it's cool that you can view it as a highly-simplified version of Newton's method. [ Addendum: An earlier version of the last paragraph claimed:
Simon Tatham pointed out that this was mistaken. It's true when !!n=2!!, but not in general. The sequence of fractions that you get does indeed converge to !!\sqrt n!!, but it's not usually the convergents, or even in lowest terms. When !!n=3!!, for example, the numerators and denominators are all even. ] [ Addendum: Newton's method as I described it, with the crucial !!g → g - \frac{f(g)}{f'(g)}!! transformation, was actually invented in 1740 by Thomas Simpson. Both Isaac Newton and Thomas Raphson had earlier described only special cases, as had several Asian mathematicians, including Seki Kōwa. ] [ Previous discussion of convergents: Archimedes and the square root of 3; 60-degree angles on a lattice. A different variation on the Babylonian method. ] [ Note to self: Take a look at what the AM-GM inequality has to say about the behavior of !!\hat g!!. ] [ Addendum 20201018: A while back I discussed the general method of picking a function !!f!! that has !!\sqrt 2!! as a fixed point, and iterating !!f!!. This is yet another example of such a function. ] [Other articles in category /math] permanent link Wed, 23 Sep 2020
The mystery of the malformed command-line flags
Today a user came to tell me that their command
failed, saying:
This is surprising. The command looks correct. The branch name is
required. The The
But it still didn't work:
I dug in to the script and discovered the problem, which was not actually a programming error. The crappy shell script was behaving correctly! I had written up release notes for the In an earlier draft of the release notes, when they were displayed as an HTML page, there would be bad line breaks:
or:
No problem, I can fix it! I just changed the pair of hyphens ( But then this hapless user clipboard-copied the option string out of
the release notes, including its U+2011 characters. The parser in the
script was (correctly) looking for U+002D characters, and didn't
recognize One lesson learned: people will copy-paste stuff out of documentation, and I should be prepared for that. There are several places to address this. I made the error message more transparent; formerly it would complain only about the first argument, which was confusing because it was the one argument that wasn't superfluous. Now it will say something like
which is more descriptive of what it actually doesn't like. I could change the nonbreaking hyphens in the release notes back to regular hyphens and just accept the bad line breaks. But I don't want to. Typography is important. One idea I'm toying with is to have the shell script silently replace all nonbreaking hyphens with regular ones before any further processing. It's a hack, but it seems like it might be a harmless one. So many weird things can go wrong. This computer stuff is really complicated. I don't know how anyone get anything done. [ Addendum: A reader suggests that I could have fixed the line breaks with CSS. But the release notes were being presented as a Slack “Post”, which is essentially a WYSIWYG editor for creating shared documents. It presents the document in a canned HTML style, and as far as I know there's no way to change the CSS it uses. Similarly, there's no way to insert raw HTML elements, so no way to change the style per-element. ] [Other articles in category /prog/bug] permanent link Sun, 13 Sep 2020The front page of NPR.org today has this headline: It contains this annoying phrase:
Someone has really committed to hedging. I would have said that the race would certainly help determine control of the Senate, or that it could determine control of the Senate. The statement as written makes an extremely weak claim. The article itself doesn't include this phrase. This is why reporters hate headline-writers. [Other articles in category /lang] permanent link Fri, 11 Sep 2020
Historical diffusion of words for “eggplant”
In reply to my recent article about the history of words for “eggplant”, a reader, Lydia, sent me this incredible map they had made that depicts the history and the diffusion of the terms: Lydia kindly gave me permission to share their map with you. You can see the early Dravidian term vaḻutanaṅṅa in India, and then the arrows show it travelling westward across Persia and, Arabia, from there to East Africa and Europe, and from there to the rest of the world, eventually making its way back to India as brinjal before setting out again on yet more voyages. Thank you very much, Lydia! And Happy Diada Nacional de Catalunya, everyone! [Other articles in category /lang/etym] permanent link
A maxim for conference speakers
The only thing worse than re-writing your talk the night before is writing your talk the night before. [Other articles in category /talk] permanent link Fri, 28 Aug 2020This morning Katara asked me why we call these vegetables “zucchini” and “eggplant” but the British call them “courgette” and “aubergine”. I have only partial answers, and the more I look, the more complicated they get. ZucchiniThe zucchini is a kind of squash, which means that in Europe it is a post-Columbian import from the Americas. “Squash” itself is from Narragansett, and is not related to the verb “to squash”. So I speculate that what happened here was:
The Big Dictionary has citations for “zucchini” only back to 1929, and “courgette” to 1931. What was this vegetable called before that? Why did the Americans start calling it “zucchini” instead of whatever they called it before, and why “zucchini” and not “courgette”? If it was brought in by Italian immigrants, one might expect to the word to have appeared earlier; the mass immigration of Italians into the U.S. was over by 1920. Following up on this thought, I found a mention of it in Cuniberti, J. Lovejoy., Herndon, J. B. (1918). Practical Italian recipes for American kitchens, p. 18: “Zucchini are a kind of small squash for sale in groceries and markets of the Italian neighborhoods of our large cities.” Note that Cuniberti explains what a zucchini is, rather than saying something like “the zucchini is sometimes known as a green summer squash” or whatever, which suggests that she thinks it will not already be familiar to the readers. It looks as though the story is: Colonial Europeans in North America stopped eating the zucchini at some point, and forgot about it, until it was re-introduced in the early 20th century by Italian immigrants. When did the French start calling it courgette? When did the Italians start calling it zucchini? Is the Italian term a calque of the French, or vice versa? Or neither? And since courge (and gourd) are evidently descended from Latin cucurbita, where did the Italians get zucca? So many mysteries. EggplantHere I was able to get better answers. Unlike squash, the eggplant is native to Eurasia and has been cultivated in western Asia for thousands of years. The puzzling name “eggplant” is because the fruit, in some varieties, is round, white, and egg-sized. The term “eggplant” was then adopted for other varieties of the same plant where the fruit is entirely un-egglike. “Eggplant” in English goes back only to 1767. What was it called before that? Here the OED was more help. It gives this quotation, from 1785:
I inferred that the preceding text described it under a better-known name, so, thanks to the Wonders of the Internet, I looked up the original source:
(Jean-Jacques Rosseau, Letters on the Elements of Botany, tr. Thos. Martyn 1785. Page 202. (Wikipedia)) The most common term I've found that was used before “egg-plant” itself is “mad apple”. The OED has cites from the late 1500s that also refer to it as a “rage apple”, which is a calque of French pomme de rage. I don't know how long it was called that in French. I also found “Malum Insanam” in the 1736 Lexicon technicum of John Harris, entry “Bacciferous Plants”. Melongena was used as a scientific genus name around 1700 and later adopted by Linnaeus in 1753. I can't find any sign that it was used in English colloquial, non-scientific writing. Its etymology is a whirlwind trip across the globe. Here's what the OED says about it:
Wowzers. Okay, now how do we get to “aubergine”? The list above includes Arabic bāḏinjān, and this, like many Arabic words was borrowed into Spanish, as berengena or alberingena. (The “al-” prefix is Arabic for “the” and is attached to many such borrowings, for example “alcohol” and “alcove”.) From alberingena it's a short step to French aubergine. The OED entry for aubergine doesn't mention this. It claims that aubergine is from “Spanish alberchigo, alverchiga, ‘an apricocke’”. I think it's clear that the OED blew it here, and I think this must be the first time I've ever been confident enough to say that. Even the OED itself supports me on this: the note at the entry for brinjal says: “cognate with the Spanish alberengena is the French aubergine”. Okay then. (Brinjal, of course, is a contraction of berengena, via Portuguese bringella.) Sanskrit vātiṅgaṇa is also the ultimate source of modern Hindi baingan, as in baingan bharta. (Wasn't there a classical Latin word for eggplant? If so, what was it? Didn't the Romans eat eggplant? How do you conquer the world without any eggplants?) [ Addendum: My search for antedatings of “zucchini” turned up some surprises. For example, I found what seemed to be many mentions in an 1896 history of Sicily. These turned out not to be about zucchini at all, but rather the computer's pathetic attempts at recognizing the word Σικελίαν. ] [ Addendum 20200831: Another surprise: Google Books and Hathi Trust report that “zucchini” appears in the 1905 Collier Modern Eclectic Dictionary of the English Langauge, but it's an incredible OCR failure for the word “acclamation”. ] [ Addendum 20200911: A reader, Lydia, sent me a beautiful map showing the evolution of the many words for ‘eggplant’. Check it out. ] [Other articles in category /lang/etym] permanent link Mon, 24 Aug 2020Ripta Pasay brought to my attention the English cookbook Liber Cure Cocorum, published sometime between 1420 and 1440. The recipes are conveyed as poems:
(Original plus translation by Cindy Renfrow) “Conyngus” is a rabbit; English has the cognate “coney”. If you have read my article on how to read Middle English you won't have much trouble with this. There are a few obsolete words: sere means “separately”; myed bread is bread crumbs, and amydone is starch. I translate it (very freely) as follows:
Thanks, Ripta! [Other articles in category /food] permanent link |