From Katherine Arcement’s LRB Diary on reading and writing fanfic: “But by the time I told her I had stopped spending so much time online. I got bored with having to scroll through tens of misspelled summaries to find just one story that sounded appealing.” To me, “tens of misspelled summaries” sounds wrong, like something a non-native-speaker used to words like French dizaines or Russian десятки would come up with, but the author “will graduate from the College of William and Mary in May with a bachelor’s degree in English,” so I think we can rule that out as an explanation; it’s more likely that, as happens increasingly often, I am behind the curve of a changing language. And, as usual, I turn to you, the Varied Reader: does “tens” in place of the traditional “dozens” sound OK? (It’s hard to google for examples because of the prevalence of phrases like “tens of thousands.”)


  1. Jonathan Wright says

    It doesn’t sound OK to me. I’m a literary translator into English and I routinely convert tens into dozens. Maybe it is a younger-generation thing (I’m in my late 50s).

  2. So-called “tens” is just wrong. Wrong, wrong, wrong. I just noticed that eggs in Norway are sold in sixes and dozens. Presumably they found that five rational eggs are completely impractical for packing in two rows, unlike six and twelve which are both even and so work properly. So much for decimalisation, a total failure.

  3. I’d use it, though not formally, and it’s instantly comprehensible to me: some number between 20 and 99. (I wouldn’t use “dozens” formally either, for that matter. Don’t quite see the problem here.)

  4. tens of misspelled summaries
    Unlike dozens, tens doesn’t work as a collective noun. You can maybe get away with “tens of summaries” but you can’t talk about “a ten of summaries”, or “a couple of ten”, as you can “a dozen summaries” or “a couple of dozen”.

  5. (by the way, on Google you can search for ‘”tens of” -thousands -millions -billions’ to get around the common phrase problem.)

  6. D Sky Onosson says

    Perfectly comprehensible, but not something I’ve ever run across as far as I can remember. I’d be very hesitant to call it “wrong”, but I don’t think I can imagine a situation where it would be preferable to “dozens”.

  7. i recalled my favourite cartoons where they discuss what constitutes kucha – a heap, how many nuts would make it
    the best! must be there is a lot of european influence in there cartoons/ a joke
    this one is with english subtitles, there were four uploads of the series with the english subtitles
    i mean the blog discussions sometimes remind me of them 🙂

  8. Don’t quite see the problem here.
    Well, of course you don’t, since it sounds OK to you! You’ll just have to take my word for it that to some of us it sounds quite alien—though of course I wouldn’t call it “wrong,” and if people keep using it I’ll doubtless get accustomed to it.

  9. Trond Engen says

    My wife alerted me just the other day about a subtitle using the idiomatic Norwegian titalls for English dozens. She thought I’d like it, and she was right.
    Apart from that I agree with Crown. Or rather, I accept the metric system as vastly superior in a decimal world, but whoever decided to count in base 10 rather than 12 has much to answer for.

  10. J.W. Brewer says

    Perhaps via excessive consumption of fanfic the author has picked up this usage from stories about a dystopian future in which oppressive mandatory decimalization has driven “dozens of” out of the language?

  11. Using Google NGram Viewer, it appears that your instincts are correct, in general: while tens of thousands used to account for 90% or more of all instances of “tens of [any noun]”, that figure dropped mid-century to around 60%, although it’s basically been stable since 1980 at that point.

  12. It’s certainly more marked than “hundreds” or “dozens”, but I’ve heard it often enough that the quoted sentence probably wouldn’t have caught my attention as unusual. I’m actually not sure whether I’d be more likely to use “tens” or “dozens” myself, under those circumstances.

  13. Hah! Fun. I gather that it does sound really queer to some of you. Good to know. It just sounds folksy to me, in the same register as “dozen,” really.

  14. Nathaniel says

    It would sound to me if the writer wanted to deprecatingly allude to hundreds and thousands, e.g., “I have tens of subscribers to my blog!” and I have noticed the usage before in that sense at least. That doesn’t seem to be the case here.

  15. It sounds funny to me. Not as funny as, say, “nines of summaries” would sound.
    “Tens of thousands” and “tens of millions” and “hundreds of thousands” sound perfectly fine. “Fives of thousands” is bad. We don’t say “three hundreds of thousands”, but we do say “hundreds of thousands”.
    We say “a dozen oysters” and “two dozen oysters” but “dozens of oysters”. Are there dialects in which you say “a dozen of oysters”?

  16. marie-lucie says

    Trond: whoever decided to count in base 10 rather than 12 has much to answer for.
    Of course mathematically Base 12 would be better as 12 is divisible more ways than 10, but Base 10 derives from what we can count with our own bodies – most obviously our fingers. So Base 12 is the more sophisticated base (used by the ancient Mesopotamian astronomers and still used for time calculations), Base 10 (or even Base 5) the more primitive one.

  17. I’d prefer “dozens” or “scores” of summaries, but “tens of summaries” sounds only mildly odd to me. I don’t think I’d say it myself, except perhaps in the sense Nathaniel mentions above. But I didn’t realize from the title of this post that the point was going to be strangeness of “tens of summaries,” so that says something too.

  18. Was going to agree with Nathaniel (tens as a joking paraprosdokian), but it sounds like the joke had begun to be lost on some speakers. You might compare this with “Déjà vu all over again.”

  19. More than once I’ve had the problem of how to translate десятки. I feel like I’d be lying or cheating if I were to use “dozens,” and I think I’ve tried crappy alternatives like “many tens of.” So therefore if this is really a new expression I’d welcome it.

  20. Sounds fine to me in the plural, but it doesn’t work in the singular (though it could become OK).
    Since we are familiar with dozens and scores, the mere numeral seems, well, naked. What comes to mind is decads (no, not decades). What language am I in here?
    Actually I would prefer base nine, or neniad, since it’s the cube of the number of the goddess, and like the ancients, I find the multiples of three very powerful. A favourite superstition should be enjoyable; believability is irrelevant.

  21. AJP says ‘Decimalisation is a failure.’ Only because we fogeys are still alive.
    In Canada, the Federal Metricifation Committee, back in the day, made a ‘conscious decision’ (is there any other kind?) not to metrify, or decimalise, the clock and the compass. They are still base 12, like selling eggs. Can you imagine an hour with 100 minutes? The compass would have 100 grades, not degrees. The Committee was very wise. The fogeys would have marched on Ottawa, in ranks of a dozen, of course.

  22. COCA suggests that non-numerical collocates of “tens of” are pretty rare. The list includes various units of length (“meters,” “kilometers,” “nanometers,” “centimeters,” “miles,” “microns,” “light-years”) and other measurements (“seconds,” “milliseconds,” “degrees,” “kilowatts,” “tons,” “centuries”). For decimalized units of the metric system, at least, “tens” makes sense.

  23. But can we be sure she didn’t mean to write “tons of” and, you know, _misspelled_ it? 🙂

  24. I am almost certain that the use of “tens” was tongue in cheek, and the author suspected it would baffle linguistic fuddy-duddies.

  25. AJP says ‘Decimalisation is a failure.’ Only because we fogeys are still alive.
    I know it’s hard to tell sometimes, but I’m only joking. Of course decimalisation is better is most cases, and I really couldn’t care less if people say tens of instead of dozens of. It’s worth noting, though, that we have a word “twelve” and a word “dozen” but no such equivalent word with the decimal system, only “decade” (and century) for years, and the above-mentioned score.

  26. I got bored with having to scroll through tens of misspelled summaries
    “Thousands of misspelled summaries” would have sounded fine. I think she was using “tens of” as an exponential equivalent of “thousands of”. It had nothing directly to do with decimalising “dozens”.

  27. Trond Engen says

    marie-lucie: Maybe it’s wrong to blame the poor fellow who chose to count his fingers, but the one who chose to have five fingers had an unforgivable lack of foresight.
    AJP: I know you’re joking. I think there’s a tongue in every cheek until I see the foam and feel the gall.

  28. I had heard “tens of X” before, and it didn’t sound particularly odd to me. It did take me back to elementary school math class (in the mid-1980s), where we routinely talked about “tens” as a unit.

  29. tetri_tolia says

    I hear tens a lot at my university: both in scientific literature and from the 95% of students who aren’t native speakers. It sounded quite strange at first but I’ve gotten used to it; still not quite comfortable with using it.

  30. Thinking of the mathematicians I’ve known – shy, introverted, even autistic – I reckon that we count in tens because we have ten toes not ten fingers.

  31. More than once I’ve had the problem of how to translate десятки. I feel like I’d be lying or cheating if I were to use “dozens”
    No you wouldn’t: “dozens” is the English equivalent of десятки, and there’s no other way to translate it unless and until “tens” becomes a great deal more common. Don’t forget that we’re not talking about exact numbers; if we’re looking at a pile of logs that would turn out, if counted, to contain exactly 54, then “tens of logs” and “dozens of logs” are equally accurate and equally inaccurate. It’s a question of idiom, and (for most English-speakers) there is no idiom “tens of.”
    But can we be sure she didn’t mean to write “tons of” and, you know, _misspelled_ it? 🙂
    No, we can’t, and I should have thought of that possibility myself. I hope it’s the case! I can deal with typos far more easily than annoying changes to the language. (Annoying to me personally, of course, and to my fellow fuddy-duddies.)

  32. @dearieme – did they go around barefoot (wouldn’t surprise me)? I guess your observation might be correct, and what we’ve got is binary 🙂

  33. I agree with the upthread comments that it sounds like a mild joke, an understatement concerning exactly how much effort Arcement put into scrolling through summaries.

  34. marie-lucie says

    in French there are a several words for approximate numbers, derived from the words for round numbers. Unlike the round numbers, tne derived numbers tend to be restricted to specific contexts: eggs for a dozen, otherwise distances, time periods, and other things where accuracy is either not required or difficult to attain. They also constitute a fixed class, since no new approximate numbers can be formed in the same manner.
    une douzaine (usually ‘exactly 12’, for eggs, but also ‘about 12’ for other things including age in years); this is the word which was borrowed into English as dozen;
    une dizaine ‘about 10’, for years or days, distances, or objects
    une huitaine ‘about or exactly 8’, almost always for years of age or days – with the latter, refers to a week, by counting both the first and the last day
    une quinzaine ‘about or exactly 15’, in contexts similar to those for dizaine; with days, refers to a two-week period, counting both the first and last days)
    une vingtaine ‘about or exactly 20’
    une centaine ‘about or exactly 100’
    Une neuvaine (from neuf ‘9’) also exists, but with the special meaning of ‘nine-day period of prayer’ in the Catholic Church.
    For numbers from approximately 30 to 60 the words trentaine, quarantaine, cinquantaine, soixantaine are used mostly in reference to groups of things or persons and to approximate ages. (From 70 to 90 there is no single word in -aine because those numbers are compound words). In France these words refer to the start of a decade of age (as in friser la quarantaine ‘to have almost reached forty), but in Canada to the whole decade (like forties, fifties, etc.
    In addition there is un millier ‘about 1000’, next to mille ‘1000’. This word is most often used in the plural (like thousands), without an actual number.

  35. marie-lucie says

    iakon: In Canada, the Federal Metrification Committee, back in the day, made a ‘conscious decision’ (is there any other kind?) not to metrify, or decimalise, the clock and the compass.
    I did not know that they had even thought of it. It would have been unbelievably disruptive to be the only country in the world to refer to time in that way! But such a thing was considered during the French Revolution, which established the decimal system of measurement: in some museums you can see clocks made at the time, marking ten divisions instead of twelve. At the time, decimalization of most measurements was a very practical decision as units of length, weight, volume, etc were different not only in different countries but in the various provinces, creating obstacles (among many others) to commercial transactions within the country as well as outside. But time and angles had been measured in the same way internationally for millennia already, starting with the Mesopotamian astronomers, so there was no need for reform in that area.

  36. m-l, I heard the anecdote about the Metrification Committee from my supervisor at the time, who was on the provincial committee. This of course puts it in the realm of ‘unsubstantiated’.

  37. marie-lucie says

    iakon, “officially” unsubstantiated. I am sure someone proposed it without thinking of the consequences and was quickly silenced.

  38. Off-topic: Over at Universe Today today there is an article about astronomical April Fool’s Day hoaxes. One of last year is reported from the Jet Propulsion Laboratory, which monitors spacecraft Messenger, orbiting Mercury. I have a couple of grammatical queries about it.
    In this snippet ‘…a purported moon of the planet Mercury’s…’ is the redundant possessive a feature only of the American Standard Dialect?
    The sentence at the end of the paragraph states ‘The “moon” also bared a suspicious resemblance to the asteroid 243 Ida…’. This is the first time I have encountered this form of the past tense of the verb to bear. It sent a shudder down my spine all the way from cranium to coccyx. It also seemed to be a synonym for revealed. I would have used the form ‘bore’. Is anybody with me?

  39. In a comment at the end of the obove-mentioned article is a link
    to an article in italian which is a similar hoax, or rather joke. Some of you may wish to practice reading italian. Even if you click for a translation, can you spot the error (a number) in position? This is just a test to see if anyone is paying attention.

  40. I’m sure you’re right, m-l.

  41. Given how much fanfic there is, especially in popular universes (of discourse), I really do think tons of was intended.

  42. dearieme may be thinking of the old joke (which my wife and I first heard from a fairly shy mathematics student who did his PhD under my supervision):
    Q: What’s the difference between an introverted mathematician and an extroverted mathematician?
    A: The introvert looks at his shoes when he talks to you. The extrovert looks at your shoes when he talks to you.

  43. Someone should ask Arcement about this. Not about mathematicians and their shoes, but the other thing.

  44. in French there are a several words for approximate numbers, derived from the words for round numbers.
    @Marie-Lucie: This also happens in Spanish, which has near-parallel expressions to those of French:
    docena: a dozen, also applied for eggs.
    decena: ten units, applied to the composition of large figures(miles, centenas, decenas, unidades: thousands, hundreds, tens, units), approximate number of years – decenas de años. “Docena” would sound not quite idiomatic, in this context.
    década: a period of ten years, frequently designating a distinctive period, such as “la década de los 20” (the twenties.
    veintena: score, or 20 units. Mostly applied to years, or numbers of people. The same goes for treintena (thirty).
    cuarentena: a group of 40, but this exclusively has come to be equated with quarantine, no longer to designate 40 years or a group of 40 people.
    centena: a group of 100 units. With an article, may designate a group of people, years. Without an article, refers to the composition of figures, as explained above.
    cientos: literally, hundreds. Used as you would in English.
    Were it not for @languagehat’s note that the user of the “tens-of-summaries” expression would be majoring in English with a bachelor’s degree, I would have considered it a calque from Spanish. I would have also rather used dozens or scores instead.

  45. I would have used the form ‘bore’. Is anybody with me?
    I’m pretty sure just about all English-speakers are with you; I have no idea where the writer got “bared,” but since it doesn’t bear even a superficial resemblance to the present tense (if you were going to form a regular past, it would be “beared”) it’s presumably a one-off error.

  46. iakon seems to suggest that the writer was thinking also of “bare” in the sense of “reveal”. It is conceivable that this is someone who has always heard “bear a resemblance” as an instance of “bare”.

  47. Katherine Arcement says

    Bear with me, as I’m writing with one hand, but I found this through twitter, and had to comment.
    In retrospect I agree ‘tens’ sounds a bit weird, but it was not a misspelling. Looking now at ff.net, the stories actually come up as twenty to a page, which was why I went with ‘tens.’
    I also was a fluent Spanish speaker as a child, learning the metric system, which may have had something to do with it.
    That said, it now reads as a bit of a slangy joke. One which the LRB editors obviously did not catch, so thank you.

  48. Heh. Thanks very much for stopping by and clearing that up—my mind is now at ease!

  49. Oh, and I should add that I enjoyed your piece quite a lot.

  50. “Tens” strikes me as a playfully, delibrately askance variation.

  51. Ø
    Thanks for the math joke. I shall do my best to remember to share it with my husband, who likes math jokes, and with our son,who ought to find it amusing since he is extremely gaze-aversive as a result of Asperger’s Syndrome.
    By the way, how is your moniker pronounced? Is it the symbol for null or is it the Danish and Norwegian front rounded vowel? I’m guessing the former since I now know that you are a mathematician.

  52. I’ve heard that, in British university libraries, the catalog numbers of what we would now call ‘adult’ books were prefaced with the character ‘Ø.’

  53. Isidora: I pronounce it “Empty”.
    Some people will tell you that the mathematical symbol for the empty (or null) set is supposed to be different from that Scandinavian letter–to have a different shape, I mean–but that distinction is too subtle for me.
    You have to really practice that joke before you tell it. It’s very easy to go wrong. For example, you’ve already screwed it up if you say, with emphasis, “The introvert looks at his shoes …”

  54. I would read ‘tens of comments’ as (inaccurately in this case) marking someone with a science background. There, it’s specifically used to distinguish an estimate from ‘a few’ or ‘hundreds’, and meaning some number between 10 and, say, 40. I just googled ‘tens of photons per second’ and found a few millions of hits.
    Physicist-types are encouraged to be able to do ‘order of magnitude’ estimates quickly: the famous ‘how many piano tuners are there in large-city-X?’ puzzle is essentially an exercise in keeping track of powers of ten. Thus London is 10 millon people (rather than 1 or 100 million), so there might be 100,000 pianos (1 per 100 people, as opposed to 10,000 pianos or 1 million), and so on. With that mindset, ‘tens’ — meaning ?’a few times ten’ — is a natural contrast with 100s — meaning ‘a few times ten to the two’.
    But the LRB contributor information about this author suggests that this reason doesn’t really hold water. Oh well….

  55. I doubt that there is anything close to one piano for every hundred people in London.

  56. marie-lucie says

    AJP, I agree, that sounds way too high. But there must have been more pianos a century or two ago, before everybody had means of “playing” music without using human-operated musical instruments.

  57. Well, that wasn’t really my point…, but one per 1000 seemed a bit low, and ‘few per 1000’ would be straying into the High Precision regime.
    The trick with these things is to keep a tab of probably-high and probably-low estimates, and aim for them to roughly balance out. If you can get to within an order of magnitude or two of the right answer, that counts as a win (Wikipedia on your phone can bring this to an earlier end, but doing so prematurely is generally regarded as unsporting).

  58. Instead of decimalizing the clock or the compass, wouldn’t it be more amusing to radianize them?

  59. I’m sure you’re right, m-l, pianolas too.
    But Mr Gray – Norman – in the case of piano tuners doesn’t it depend on which two orders of magnitude you are within? If I were to order spare piano wire for three hundred piano tuners instead of thirty I would have extra wire that I could store in my garage and use up later; however, if I ordered piano wire for thirty-thousand piano tuners and it turned out there were in fact only three thousand, and I was stuck with twenty-seven thousand rolls of piano wire, then I’d have to rent a warehouse and try to sell off the remainder as picture-hanging wire and decorative string. Maybe you could use it for car repairs, carry a roll in your glove compartment for when your exhaust pipe falls off.

  60. @Bill W
    Hmm. “So, we’ll get together at about 3π/2.” Works for me.

  61. For what little it’s worth (native speaker of American English, no more than the required amount of science background), I’ve used “tens” both in the self-deprecating way – “I have literally tens of dollars at my disposal!” – and also sometimes sincerely to indicate a nonspecific amount where I feel, for some reason I can’t articulate, like “dozens” would be a little too big. However, I’m aware that the usage is uncommon and therefore still has a slightly playful (hopefully) affect.

  62. Trond Engen says

    Medrawt: For what little it’s worth (native speaker of American English, no more than the required amount of science background)
    For answering the general question it’s hard to imagine a better background.

  63. By ‘within an order of magnitude’ I mean ‘no more than one order of magnitude out’: wrong by no more than a factor of ten.
    So continuing the estimate: If there is indeed one piano per 1000 Londoners, then that means 10,000 pianos which need tuning (say) once a year. If it takes a day to tune a piano, and there are 100 working days in a year (near enough), then that brings us down to 100 tuners. I think I’ve made two underestimates there, so I think that the real number is probably on the high side of 100 – between 100 and 1000 (or, returning to the language issue, ‘a few times 10 to the two’). And that sounds pretty plausible; I think I’m probably right to within an order of magnitude.
    And… clickety-click… the Yellow Pages lists 86 entries for piano tuners there . Since that’s also likely to be an underestimate (it’s pretty certainly not an _over_estimate), that’s a second route to few x 10^2). Not bad; wey-hey!
    Cosmology used to be notorious for estimates within a couple of orders of magnitude – for not being able to measure things to better than a factor of hundreds or thousands or more. Having error bars on your powers of ten is either rather desperate or raffishly stylish, depending on your point of view.

  64. AJP: There are other options. You could sell it to terrorists, er, freedom fighters, to help them stave off cavalry attacks, for example.
    Empty, Isadora: Lower case o with slash is ø, upper-case o with slash is Ø, the empty-set symbol is ∅.

  65. JC: Your “empty-set symbol” is wider than it is tall. Sez who?

  66. Norman, thanks. I see what you’re getting at and I like the way you brought it back to the language question. I find physicists are inherently stylish and certainly never desperate.
    JC, a piano wire stretched tightly across the road works against bicycles too (so I’m told). I wonder if terrorists and piano tuners are of the same order of magnitude in London. Say there’s one for every ten piano tuners, or ten. That’s not a whole lot of piano wire. I’d be better off selling it to the cyclists for emergency spoke replacement.

  67. Empty: Etymologically (if that word can be applied to symbols) ∅ is a slashed circle rather than a slashed “o” or “O”. Thus it should be equally wide and tall, but YMMV. In any case, it is wider than O relative to its height in most fonts. There is a graphical variant which is a slashed zero, 0̸ (may not look right on your browser).

  68. mollymooly says

    I too would always use “dozens” rather than “tens” in contexts such as the OP quote. The order-of-magnitude specificity of “tens” is not restricted to highly mathematical language; “tens rather than hundreds” can occur in any type journalism.
    I wonder whether, for some speakers, “dozen(s)” is following “(a) couple (of)” in drifting away from its initial specific-number sense further into the realms of vagueness; in which case, “tens” might become a more attractive choice precisely because of its specificity.
    French “douzaine” applies to oysters as well as eggs.

  69. Tens is very common in the IT world. It may have gained currency in a jocular context.
    After this we’ll have – oh, tens of visitors to our website.

  70. I have literally tens of dollars at my disposal
    Jocular or not, it certainly doesn’t have the same connotations as I have literally dozens of dollars at my disposal.
    ‘Tens of dollars’, because dollars are decimal and you’d count sums of money that way (further ranging up to hundreds and thousands).
    ‘Dozens of dollars’ suggests that you’re actually counting the notes (or bills) as physical objects.

  71. I just noticed that eggs in Norway are sold in sixes and dozens.
    Really? In Austria eggs are sold in 6s and 10s. I guess they take decimalisation more seriously here.

  72. I wonder how that should sound it Russian if one uses, conversely, дюжины. “Дюжины орфографически неправильных резюме?” “Дюжины аннотаций c орфографическими ошибками?” Sounds ugly. Sounds slightly better with десятки instead of дюжины.

  73. “Tens” of anything is a particularly unnecessary neologism, since not only do we have “dozens,” we also have the word “scores” to suggest a vague round number larger than 39.

  74. Indeed.

  75. marie-lucie says

    AJP: I’m sure you’re right, m-l, pianolas too.
    Actually, I was not thinking of pianolas, but of record-players, tape-recorders, CDs, etc. In older times, if you wanted to listen to music, you had to play an instrument or sing or have access to people who did.
    mollymooly: French “douzaine” applies to oysters as well as eggs.
    Yes, that’s right, I didn’t think of it because my family didn’t eat oysters and neither do I. Eggs are the most common thing sold by the dozen (or half-dozen), but other things can be bought and sold that way, such as rolls or small pastries. A dozen of whatever has the advantage of being easily divisible into two, three or four portions, unlike ten: how would you share five oysters equally among two or three persons?

  76. In Russia, in Moscow at least, eggs are sold in 10- and 6-egg packs, much like in Austria I guess. I don’t think I’ve seen twelve-packs. The half-dozens appeared recently, probably a marketing strategy appealing to small families.

  77. In the sciences and engineering, there is a thing called a “back of the envelope calculation”, or “order of magnitude estimate”. Usually it involves getting the first digit right, and the decimal in the right place. People who make these will often talk of “tens of Xs” and “thousands of Ys”. (How many piano tuners in London? how many shoes from a cow?)
    It smacks of jargon to those who don’t make them.

  78. mollymooly says
  79. John, the way I produce my slashed circle it looks quite round to me (just as I want it to be), but now you’ve got me wondering if some people using other browsers are seeing it differently, more tall than wide.

  80. Empty, how circular does O look to you? For me, O and Ø are both taller than they are wide.

  81. But not nearly so non-circular as 0, which is only maybe half as wide as it is tall.

  82. John, in the font that I see LH comments in, the letter O is circular. Also the lower-case o. And the digit 0 is taller than it is wide. Whereas in the font that I get when typing LH comments the upper-case O is taller than it is wide and the digit 0 is even more that way (while the lower-case o is either round or maybe a shade wider than tall).
    In that font that makes O and o circular, the slashed O and o (produced by typing “ampersand Oslash semicolon” and “ampersand oslash semicolon”) are also circular, while the symbol produced by typing “ampersand emptyset semicolon” comes out wider than it is tall.
    I want my empty-set symbol round, of course, and also I prefer it to be upper-case height.

  83. Your empty-set symbol looks round to me.

  84. marie-lucie says

    It depends on the font. On my computer the comments box is in a different font from the finished text, and here as I type within the box, the capital letter O is not round but taller than wide, although not as thin as the figure 0 ‘zero’. But the text previewed and posted shows circular o and O, with a thinner 0. I used the Norwegian keyboard to get the letters ø and Ø which are just o and O slashed. I don’t know how to do “emptyset”.

  85. marie-lucie says

    I guess that what I see is the same as what Ø does, but what JC sees on the screen is what Ø and I see in the comments box.

  86. For the record, you can specify the empty set symbol ∅ as “∅”. But I usually google for “empty set” and then copy and paste the symbol from Wikipedia or wherever.
    It’s also lighter in weight than Ø, at least on my screen.

  87. David Marjanović says

    I confirm that eggs are sold in Austria, and Germany, in sixpacks and tenpacks.
    It may be interesting in this context that, while Standard German has the word Dutzend, it’s not in my active vocabulary, and my dialect lacks it entirely. Sometimes -zig (from zwanzig, vierzig, fünfzig, sechzig, siebzig, achtzig, neunzig) is used in isolation, but that’s mostly a weaker version of “gazillion”; it cannot be modified as in “a few dozen”.
    On the other hand, I wouldn’t even know what to do with 12 eggs at once…
    Is “score” in common use? I thought it was only poetic anymore?

    But can we be sure she didn’t mean to write “tons of” and, you know, _misspelled_ it? 🙂

    No, we can’t, and I should have thought of that possibility myself. I hope it’s the case!

    But how would that work? The words don’t sound similar enough for a native speaker to confuse them, and e and o are in quite different places on the keyboard.

    upper-case o with slash is Ø, the empty-set symbol is ∅

    In the font I see the comments in (Verdana?), ∅ is larger than Ø in all dimensions, and both are as circular as possible (height and width are the same in number of pixels). For this purpose, ∅ has an extra pixel on each oblique line, making the line two pixels thick in those places and prevent the symbol from being a regular octogon.
    It’s different in the comment box, where a monospace (typewriter-like) font (Courier New?) is used: ∅ is much larger than Ø and lacks the extra pixels on the oblique lines, so it is octogonal. Both are still as tall as wide (in number of pixels).

    But not nearly so non-circular as 0, which is only maybe half as wide as it is tall.

    6 pixels wide, 8 tall. O is 8 wide and also 8 tall. In the comment window, the horizontal and vertical lines have 3 px instead of 4, so it’s 7 x 7; the same holds for Ø. Lowercase o is 5 x 5 in both fonts.

  88. astrotter says

    Another vote for “tens” being common and normal-sounding in science — but in my experience (astronomy), only when used in conjunction with units of measurement. “A few tens of parsecs”, or “tens of photons/cm2/sec” is fine. “Tens of planets” sounds weird.

  89. Score is not in common use. Note that it doesn’t mean simply 20, but 20 things; “ten plus ten is a score” is unacceptable, at least to me. It has to be used with some units such as years: “fourscore and seven years ago” etc.

  90. As a child I briefly thought, based on that one example, that “score” meant “twenty years”.

  91. “In the past, I’ve referred people to local prosecutors and the attorney general for noncitizens voting,” he said. “It’s like tens or dozens of people, not hundreds. There’s no acceptable level of voter fraud and we take every one of those cases seriously.”


  92. Good find! It’s clearly in use, if not exactly common.

  93. David Marjanović says

    I’m surprised I didn’t reply to this back in 2013…

    marie-lucie: Maybe it’s wrong to blame the poor fellow who chose to count his fingers, but the one who chose to have five fingers had an unforgivable lack of foresight.

    Tulerpeton was clearly superior.

  94. What the hell kind of word is Tulerpeton? Sounds like a new drug name.

  95. David Marjanović says

    The study of reptiles and amphibians is herpetology, from (zôon) herpetón = (animal) reptile. And so, lots of four-footed animals up to and including salamanders have names that end in -(h)erpeton… even though that one didn’t creep much, living as it apparently did on a coral reef 200 km from the nearest land.

    Tula is familiar to you. 🙂

  96. Makes sense, thanks!

  97. January First-of-May says

    I wonder how that should sound it Russian if one uses, conversely, дюжины.

    I’m reminded of the Max Frei series, where the locals in Yekho commonly refer to (e.g.) dozens of thousands – a regular reminder that the culture of Yekho is at least slightly alien.

    The people of Arvarokh (an even more alien culture) apparently count in half-hundreds and half-thousands instead.

    Lower case o with slash is ø, upper-case o with slash is Ø, the empty-set symbol is ∅.

    For the record, in my current font (as I write this, in Chrome), Ø is far larger than ∅, which looks only slightly wider than ø but significantly taller.
    The two Scandinavian letters are round, or very nearly so, with (seemingly) 45-degree slashes; the slash on the empty set symbol is very visibly not 45 degrees (I’d estimate about 60 from horizontal, though it could be a little less). The independent slash (which I see in the italicizing tags) is even more slanted, but only slightly.

  98. January First-of-May says

    I’m reminded of the Max Frei series, where the locals in Yekho commonly refer to (e.g.) dozens of thousands – a regular reminder that the culture of Yekho is at least slightly alien.

    I decided to check (just in case), and found out, to my surprise, that Google, at least, couldn’t find any references in the Max Frei series to either dozens of thousands or thousands of dozens, though plenty of both dozens and thousands. (Admittedly, the search results were so polluted by references to Jack London’s The One Thousand Dozen that I might well have missed actual Max Frei quotes.)

    In any case, the inhabitants of Yekho do tend to count in dozens (дюжины), and yes, it does sound noticeably unusual. So I suppose this answers that question.

    (…I wonder if the author of the Max Frei series came up with that particular idea from her experience with English.)

  99. Rigid body motion is a topic of great practical importance, but which virtually no physicist enjoys teaching. Goldstein’s Classical Mechanics* (which has been the authoritative graduate-level text of the subject for at least two generations) includes two entire chapters on the subject in its early core—one just setting up the kinematics to describe the full three-dimensional motion of asymmetric bodies, and another for studying the dynamics of such bodies under the influences of nontrivial torques. When I took advanced classical mechanics as an undergraduate myself, Professor Michel Baranger made it abundantly clear that he disliked the topic; even his syllabus, which for the bulk of the course listed a detailed breakdown of topics that would be covered in each lecture period, simple listed the last six classes together on one line as “rigid body motion,” with no further elaboration. I myself have taught graduate classical mechanics only once, and the time I spend on rigid bodies was quite unpleasant for me (and, although I hope not, probably also for my students); my desire not to have to teach that again is by far the biggest reason that I have avoided teaching that class again.

    I did not do the full description of a freely precessing three-dimensional body in class. I covered axisymmetric tops and the linear stability of the full three-dimensional problem (the “tennis racket theorem”). However, I had one of the students come to my office hours, wanting to get more information about the description of the general motion and how it could be represented using two curves called the “polhode” and “herpolhode.” (The two curves, which roll against one another, can be seen in on the graphic for this seminar announcement.) To someone fluent in Greek, I am sure those two names are quite transparent, but even Goldstein seemed to have been confused by “herpolhode”—taking it to mean something like “snaking pole path,” rather than “creeping pole path.”

    * A few years before his death, when he was in ill health, Herbert Goldstein turned the book over to two of my colleagues at the University of South Carolina, John Safko and Charles Poole. The third (2005) edition of the book thus lists all three of them as authors, although everyone still refers to the book as “Goldstein,” the same way that the graduate textbook on Classical Electrodynamics is “Jackson.” Moreover, Safko and Poole have themselves both died in the last few years, and nobody seems to know who, if anyone, will be responsible for updating the book for a fourth edition, if that is ever deemed a necessity. (I would be somewhat sad to the see the third edition go, although for the purely prosaic reason that it is by far the lightest textbook of its size that I have ever had to carry around. The cheap paper it is printed on manages to be fairly resilient while being not at all dense.)

  100. “The polhode rolls without slipping on the herpolhode lying in the invariable plane.” Goldstein couldn’t resist putting this sentence in a footnote. It’s almost all that I remember from that book.

  101. David Marjanović says

    everyone still refers to the book as “Goldstein,”

    Eduard Strasburger: Lehrbuch der Botanik für Hochschulen. Fischer, Jena 1894

    Andreas Bresinsky, Christian Körner, Joachim W. Kadereit, Gunther Neuhaus, Uwe Sonnewald: Strasburger – Lehrbuch der Botanik. Begründet von E. Strasburger. Spektrum Akademischer Verlag, Heidelberg 2008 (36. Aufl.) ISBN 978-3827414557

    Joachim W. Kadereit, Christian Körner, Benedikt Kost, Uwe Sonnewald: Strasburger – Lehrbuch der Pflanzenwissenschaften. Springer Spektrum, 37. vollständig überarbeitete & aktualisierte Auflage, Berlin & Heidelberg 2014. ISBN 978-3-642-54434-7 (Print); ISBN 978-3-642-54435-4 (eBook)

    …in other words, the botanists have made it official, even through multiplying their “plant sciences” recently.

  102. Goldstein couldn’t resist putting this sentence in a footnote.

    The full footnote:

    *Hence the jabberwockian sounding statement: the polhode rolls without slipping on the herpolhode lying in the invariable plane.

  103. I wish for a Tenniel to rise to the task of picturing this.

  104. I am reminded of “The gostak distims the doshes” (which I must have learned about via Miles Breuer’s story “The Gostak and the Doshes”).

  105. It occurred to me that the sentence may sound better in American than in British, with far-back retroflexed r’s and strongly velarized l’s.

  106. Lars Mathiesen says

    If you want to see the Intermediate Axis Theorem in action, search youtube for “wing nuts in space“. Mel Brooks is not involved, sadly.

  107. I remember that this was the cover of the book Organic Chemistry used as the text for Organic Chemistry II—with no mention of the title on the front. However, that appears to have been strictly a quirk of some of the intermediate editions, both the earliest and the most recent editions looked much more normal. This seems all the more odd, given that McMurry also authored or coauthored a bunch of other textbooks, and I don’t think his Organic Chemistry (which is definitely an intermediate-level text) is anything like as ubiquitous as Goldstein or Jackson.

    Googling the text of Goldstein’s footnote (which is apparently fairly famous among physicists, but which I did not presonally remember) turned up a number of interesting pages. There is this 1974 poem by University of Michigan professor David Williams:

    “On the Poinsot Construction”

    When the polhode rolls on the herpolhode
    invariably in the plane,
    the forces tend to be free.

    When the polhode slips on the herpolhode,
    the forces are not mundane,
    they’re nonholonomic, you see!

    There is also this explanation of the Poinsot Construction (which quotes both Goldstein and Williams), apparently because the Poinsot Construction was mentioned in an episode of The Flash. (The explanation Alejandro Jenkins gives is pretty good, but I personally found it extremely amusing, since Alejandro is an old friend of mine. He begins his answer with, “Hm. I don’t watch TV…,” but twenty years ago, he was known for responding to almost any situation with, “That reminds of an episode of The Simpsons.”)

  108. wing nuts in space

    Highly recommended — I watched the whole thing (14+ min.)!

  109. Unfortunately, that Veritasium guy has gotten a lot of attention recently for claiming that it is impossible to measure the speed of light (or maybe the one-way speed of light). While that claim is not untrue—if you mean something very specific by “speed of light”—it is also quite misleading.

  110. with no mention of the title on the front.
    Reminds me medieval books like Isagoge / إيساغوجي ( that is Greek “Introduction”)

  111. Well, y’know, there is no point in measuring the speed of light any more, because since 2019 its value of 299792458 m/s defines the relationship between the meter and the second. I just adore this picture of the new situation, where everything except the second is defined in terms of everything else.

  112. Imagine a physicist in a Kafkaesque bureaucracy, who finds that the Speed of Light bureau has been shut down, and is sent to the Directorate of Meters; they say that meters are now defined by c and seconds, and issue a referral to the Ministry of Seconds…

  113. @John Cowan: Some people, with that definition in mind, claim that what speed of light measurements are “really” measuring is the length of the meter; and for some experiments, that is correct. However, the modern definition of the SI meter only makes sense as a definition if the speed of light propagation in vacuum is a position-independent, time-independent, direction-independent, polarization-independent, energy-independent quantity. Those invariances, although predicted by special relativity, always need to be verified empirically, which means measuring the speed of light relative to some other (non-SI) standard.

  114. January First-of-May says

    where everything except the second is defined in terms of everything else.

    …except the mol, which really should have been defined in terms of the kilogram (as it essentially was historically), but for some unfathomable reason they decided it was more convenient to just set a number. Which technically means that anything where you were converting molar amounts to masses (or vice versa) is no longer exact – despite this being pretty much the only reason to use molar amounts in the first place.

    In principle they could also have fixed that by redefining the atomic mass unit, but I doubt it would happen that easily.

  115. January First-of-May: There is actually a really good reason for not changing the definition of the atomic mass unit. (The mole, in spite of what the SI people now say, is not really a unit in the same sense as the others; it is merely a dimensionless conversion factor between two systems of mass measurements, one based on the kilogram, the other on the amu, or Dalton.) One thing one never wants is for the experimental precision of an experiment to be polluted by the limited accuracy of the underlying system of units. The way the mass of a particle is measured in Daltons involves placing the particle alongside a carbon nucleus in a Penning trap and measuring the ratios of their orbital frequencies. (For several decades, there were two competing standards for the atomic mass unit, a “chemical” scale based on comparison to C-12 and a “physical” scale based on O-16, until the chemical Dalton eventually won out.) The benchmark precision of the Penning trap measurements is currently at the tens of parts per trillion level, and historically, the Penning trap experiments have always been on the forefront of high-precision physics. So it makes sense to have the atomic mass unit defined directly in terms of the output of such an experiment.

    Fifteen years ago, there were two competing proposals for how to create a new mass standard, without reference to a platinum bob sitting in a refrigerator in Paris. The method that lost out was based on creating a graphite sphere containing an exactly specified number of carbon atoms. That would make the kilogram based on the Dalton, and it would make the conversion factor (1000/12 of Avagadro’s Number) an exactly defined quantitiy. (One of the reasons that the oxygen-based physical scale had previously lost out was that there is no comparable way of constructing a kilogram-mass of oxygen at close to room temperature, since oxygen is a fluid down to very low T.)

  116. January First-of-May says

    The mole, in spite of what the SI people now say, is not really a unit in the same sense as the others; it is merely a dimensionless conversion factor between two systems of mass measurements, one based on the kilogram, the other on the amu, or Dalton.

    Which is exactly my problem with the current definition: the mole is only a sensible unit at all as the conversion factor of Daltons per gram, so fixing it to a particular number without redefining either the Dalton (which is presumably unlikely) or the gram (only slightly more plausible now) is utterly nonsensical.

    I wouldn’t be especially surprised if in a few years the scientists working with sufficient precision would start using a historical mole (of whatever actual name) with the old definition (as adjusted for the new kilogram), because without an inconvenient fudge factor (that would have to be measured anyway) there’s really no advantage of the “new mole” over just working in yottaDaltons directly.

  117. @January First-of-May: That (or something equivalent to it) is exactly what will happen if the precision of the various experiments increases sufficiently without a corresponding redefinition of the SI units.

    From a technical standpoint, the candela is even more of a mess. However, it is less of a practical problem, given the current state of the art in precision metrology.

  118. David Marjanović says

    there’s really no advantage of the “new mole” over just working in yottaDaltons directly

    I still don’t understand what the issue is. The dalton is a unit of mass; Avogadro’s number is an actual natural number, and a mole is Avogadro’s number of molecules, atoms or whatever – counting, not weighing; a unit of amount, not mass.

  119. @David Marjanović: Except one wants to have Avogadro’s Number of carbon atoms weigh 12 g.

  120. January First-of-May says

    Except one wants to have Avogadro’s Number of carbon atoms weigh 12 g.

    Pretty much, and of course under the new definitions it no longer does.

    (Technically it never did because natural carbon contains a bit of carbon-13, but that’s beside the question.)

  121. a “chemical” scale based on comparison to C-12 and a “physical” scale based on O-16

    As I understand it, matters were more complex than that. Chemists used C = 12 with natural isotope ratios, whereas physicists used C-12 = 12. The matter was resolved by accepting the physicist’s principle (using a varying natural isotope ratio makes no sense) and the chemist’s value (the O-16 = 16 scale was extremely close to the C = 12 scale).

  122. @John Cowan: I think that the way the “chemical” and “physical” monikers are now typically applied is probably not historically accurate—probably originating from a misunderstanding of the history, since there is essentially no reason today to use the O-16 scale, which means any references to this scale are going to be in a historical context.

    The history on Wikipedia appears to cover the development pretty clearly. There were a lot of pitfalls in the earlier definitions, most of which the article discusses, although a few it glosses over. For example, the decision to make base the Dalton on the mass of an oxygen atom but Avogadro’s Number on the dioxygen molecule was a mistake—although it was an understandable one. It is easier to make macroscopic measurements using molecular, rather than atomic, oxygen.

  123. David Marjanović says

    @David Marjanović: Except one wants to have Avogadro’s Number of carbon atoms weigh 12 g.

    That should be a definition of the kilogram (or, why not, finally the gram) rather than the mole, then.

    (BTW, I’ve long been fond of the little-known proposal to rename the kilogram “einstein”.)

  124. Brett: It looks like I had the shape of the story correct up through 2018, but got the elements reversed:

    chemist’s scale: O = 16
    physicist’s scale: O-16 = 16
    unified scale: C-12 = 12

  125. one wants to have Avogadro’s Number of carbon atoms weigh 12 g

    The redefined SI (as of 2019) makes it a fundamental principle that units should be based on constants of nature (speed of light, Planck’s constant etc) rather than on physical artifacts. That’s why they got rid of the old kilogram, defined as a lump of metal in Paris. The carbon atom, likewise, is an artifact (albeit a more natural one) than a lump of metal in Paris, so having an Avogadro’s number of carbon-12 atoms weighing 12 g is a matter of convenience rather than physical necessity.

    Avogadro’s number has no fundamental significance, but assigning it a value had to be done carefully because there is a connection between its value and the Planck constant, involving the fine-structure constant, the electron mass and some other things.

    What it boils down to is that the revised SI makes Planck’s constant fixed (defining the kilogram); setting the Avogadro constant at a fixed value then means that the mass of a mole of carbon-12 atoms cannot be set to 12 g but is instead a quantity to be measured empirically. If you wanted to fix a mole of C-12 atoms at 12 g, then the Avogadro number would not be a fixed constant, but would have to be determined empirically.

    There were compromises, in other words.

  126. January First-of-May says

    If you wanted to fix a mole of C-12 atoms at 12 g, then the Avogadro number would not be a fixed constant, but would have to be determined empirically.

    Except that this (plus an artifact of history) is the only reason to have a “mole” as a separate unit. There isn’t really anything special about the number Nₐ=602,214,076,000,000,000,000,000 except that it is very close to the amount of C-12 atoms in 12 grams, and apparently recent measurements show that Nₐ C-12 atoms actually weigh 11.9999999958(36) grams, so the number is already known to be subtly wrong.

    (…TIL that Unicode does not appear to have subscript capitals. Any subscript capitals. There’s a bunch of subscript small letters, but no capitals.)

  127. It’s true that the admittedly arbitrary value of the Avogadro number was chosen in 2019 so that a mole of C-12 atoms would be close to 12 grams, as in the old system. But no matter what value they chose, that number would inevitably drift a little from 12 grams as the accuracy of mass measurements increased. A system of units has to be internally self-consistent, of course, so there’s a limited number of constants that can be set at fixed values.

    It’s not wrong, in the new system, to have a mole of C-12 atoms not be exactly 12 g. As I said, if you want to fix that as a requirement, then you would have to keep adjusting the value of Avogadro’s constant, which would have repercussions elsewhere.

    Section 3 of this review (behind a paywall, alas) goes into detail on the reasons for the choices made.

  128. @January First-of-May: I noticed that too, regarding the lack of subscript capitals—which meant I typed “Avogadro’s Number” way more times that I would have liked to. Normally, on second and subsequent reference I would have used “N” with a capital “A” subscript.

  129. Lars Mathiesen says

    It seems to me that moles, molar masses and Avogadro’s number are conveniences that allow us to talk about macroscopic quantities and/or put everything in SI units.

    Even the equation for the Rydberg constant (which I think is what Brett alluded to) that allows the determination of the Planck constant has the molar electron mass immediately divided by Avogadro’s constant — that is, it’s really the electron mass in kg that is used, just expressing it in daltons for extra measurement error. I assume that people are actually measuring mₑ in keV directly (and 1keV = 1.602176634e-16J by SI definition) and getting the molar mass by multiplying by today’s value of Avogadro’s number, so any error cancels out (if they even take that detour).

    (Since c and h are fixed numerically, I guess the equation now gets you a value for the fine-structure constant from the electron mass and Rydberg’s constant, both experimentally determined to high precision. And hence the numerical value of the vacuum permittivity).

  130. Here is what the Particle Dara Group has to say about the electron mass:

    The primary determination of an electron’s mass comes from measuring the ratio of the mass to that of a nucleus, so that the result is obtained in u (atomic mass units). The conversion factor to MeV is more uncertain than the mass of the electron in u; indeed, the recent improvements in the mass determination are not evident when the result is given in MeV.

    Until 2008, the PDG did not include these kinds of obliquely critical prefatory comments in their data tables. Such technical niceties were only printed much further down, where they might more easily be missed. I think the first time they placed a warning note like this before a data block was when they were updating their value for the charge of the photon, in response to my work:

    OKUN 2006 has argued that schemes in which all photons are charged are inconsistent. He says that if a neutral photon is also admitted to avoid this problem, then other problems emerge, such as those connected with the emission and absorption of charged photons by charged particles. He concludes that in the absence of a self-consistent phenomenological basis, interpretation of experimental data is at best difficult.

    For the record, I agree completely with Lev Okun’s analysis; we know of no viable theory that describes a charged photon. However, if one is going to quote a limit on the photon charge, I believe it should be the best limit possible.

  131. Lars Mathiesen says

    Oh, that’s interesting. So if you can measure the ratio to a C-12 nucleus you have daltons, no ifs or buts.

    But then the complaint about Avogadro’s number boils down to not knowing the amu in MeV with the precision you’d like innit. As far as I can see, the two numerical values must be inversely proportional (exactly, c being fixed), and measuring one measures the other — if using the keV value for mₑ in the Rydberg formula loses you precision, using Avogadro’s number will lose you just as much.

    (And with SI fixing Avogadro’s number, the old constant .001kg/mol/da to convert amus to molar masses becomes a measurable quantity, proportional to the amu in MeV. No practical difference).

  132. Normally, on second and subsequent reference I would have used “N” with a capital “A” subscript.

    I think that anyone technical enough to follow this is also TeXnical enough to know how N_A (or $N_A$, if you’re picky) should be read. I certainly would be, though I wouldn’t know without being told by either the author or Dr. Google that it stands for Avogadro’s number.

  133. Here is a genuine news lede where “tens” is exactly the right word; “dozens” would lose much of the effect and affect:

    Noted local criminal Mark McCloskey played host to a barbecue/political rally on Sunday afternoon, drawing tens of admirers to the sweltering parking lot of a closed outlet mall in St. Louis County to celebrate the one-year anniversary of the time he pulled a gun on a crowd of people who otherwise would never have noticed or cared he existed.

  134. That’s the same ironic usage Nathaniel referred to back in 2013:

    It would sound to me if the writer wanted to deprecatingly allude to hundreds and thousands, e.g., “I have tens of subscribers to my blog!” and I have noticed the usage before in that sense at least. That doesn’t seem to be the case here.

    And the whole reason it works is that “tens” is strongly marked so that it stands out; if it were just an alternative to “dozens” the ironic phrasing wouldn’t work.

  135. Rodger C says

    “Sex is something not expected of us [polio victims]. That’s the secret part: and I know it’s astonished more than one person. But Harriet’s had lovers, dozens of them (tens, she says, not dozens; lovers by the tens).”–John Crowley, “The Girlhood of Shakespeare’s Heroines”

  136. Huh. That’s certainly interesting, but I have no idea what effect is intended.

  137. Prax if you want to be both decimal and idiomatic you could say “I’ve had lovers by the half-score.” Admittedly the OED does not list this word, offering you definitions for half sole, half-sword, halicore ‘dugong’, albacore, and (bizarrely) halesome

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