## Times more / less than

In a message about the "excruciatingly slow internet speed in China" that I privately circulated to some friends, students, and colleagues, I made the statement that "in many cases that I have personally experienced, the internet speed in China is actually hundreds of times slower than it is in the United States and elsewhere in the world."  Geoff Wade wrote back to me:  "Grammatical question: can something be hundreds of times slower than anything else?"

Thus began an intense series of exchanges that lasted over two days.  I won't repeat all of our arguments and counterarguments, but will briefly summarize their gist and give some examples of how I defended myself against Geoff's spirited opposition to the "times slower" construction.

VHM:  If a tortoise "runs" ten feet in one minute and a rabbit runs a thousand feet in one minute, the rabbit has run hundreds of times faster / farther than the tortoise; the tortoise has "run" hundreds of times slower than the rabbit.

GW:  I obviously agree with the "times faster", but am not convinced  of the  "times slower" use.  Would not one-hundreth times as fast be more accurate?  A multiple (times) and a divisor (one-hundreth) seem to have different functions.

VHM:  We'll have to get a mathematician to adjudicate this one.

GW:  In brief, I don't see how lack of speed can be represented as a multiple (times), vis-a-vis something going faster.

VHM:  Well, Geoff, before we bring in the mathematician, let me put my case another way:

go slow
go slower / more slowly
go 5 times slower / more slowly  (GW:  go one-fifth as fast, not five times slower)
go 10 times slower / more slowly (GW:  go one-tenth as fast, not 10 times slower)
go a 100 times slower / more slowly (GW:  go one-hundreth as fast, not 100 times slower)
go hundreds of times slower / more slowly

Similarly, we could talk about the rate of a chemical reaction or the rate of economic expansion slowing down at greater and greater rates

GW:  Yes, the rate of chemical reaction could be increasingly slow — but it would not be "x times slower" than before. Rather it would "1/x as fast" as before.

Maybe it is just British and American English. I don't think the Brits say, for example, "five times as slow" or "five times slower".

VHM:  Fast and slow are mirror opposites of each other.

GW:  I thought I might be isolated in my concerns, but there are others out there……  [here Geoff assembled a number of quotations from the web of people complaining about the "times smaller / less / slower..." construction]

VHM:  Train A is going 200 miles an hour, Train B is going 100 miles an hour, Train C is going 10 miles an hour.

Train A is going 20 times faster than Train C (GW:  I completely agree); Train C is going 20 times slower / more slowly than Train A. (GW:  Nope. You cannot be "times slower". Train C is going 1/20th the speed of Train A.)

I'll see if I can get my buddies at Language Log to adjudicate.

GW:  I am sure that your Language Logger friends will be on my side!

At this point, I consulted with my colleagues at Language Log headquarters, and Ben Zimmer directed me to this post by Arnold Zwicky: "Recency".

In his post, Arnold gives a masterful account of the "Times-er construction", including showing how — both in its "more" and "less" manifestations — it has been around in English (and Finnish and Latin…) for centuries.

GW:  It seems that the construction has been in use for a long time, but  there are a few of us stick-in-the-muds who still find it strange. We all grow up with certain beliefs about our language practices…..

I have submitted, but my mind may take some time to adjust…..

kòutóu 叩头 ("kowtow")

VHM:  I sympathize with you, Geoff, and can appreciate why this usage is blowing your mind away, but conceptually it's actually very much like the jiāyǐ 加以 ("increasingly") construction in Chinese, where you can — seemingly paradoxically — have expressions like this:  jiāyǐ jiǎnshǎo 加以减少 ("increasingly fewer").  Those types of expressions always used to blow my mind away too, but finally I got used to them, because they've been occurring in Chinese texts for the last two millennia and are still very common today.

Tansen Sen had been listening in on the entire debate between Geoff and me.  He brought it to a close with these brilliant words:  "Ok, as a referee, I call an end to this discussion, which has been ten times faster than anything I have seen before through email exchanges, but twenty times slower than on a tweeter forum."

GW:  We are old generation. Six times too old for tweets….

Meanwhile, around the water cooler at Language Log Central, Paul Kay observed:

My seat-of-the-pants take on this:  there are two questions here (at least). The one your friend seems most concerned with is "two times faster" versus "twice as fast".  The other is (a) given as acceptable either or both "twice as fast" and "two times as fast", (b) can one say "twice/two times as slow"?  My impression — based on no data — is that in American English all three forms are common but "two times as slow/slower" is heavily stigmatized and "two times as fast/faster" perhaps lightly stigmatized.  In contrast, in academic and literary French, "deux fois plus lent/petit/court" is just fine. The first G-hit for "deux fois plus court" (of a probably meaninglessly estimated 3.5 million) just now was "Le texte d'Eph étant deux fois plus court que celui d'He, cette différence de traitement garde exactement la proportion voulue. Le nombre des mots est identique " from Google Books.  Clearly an academic source. I remember when growing up being taught that I should stop saying things like "twice as short"; that such talk is illogical.  And being a dutiful child, that's what I did.  Imagine my surprise decades later to hear "deux fois plus court" from the mouths of linguistically fastidious French academics.

Whether either or both "two time shorter" and "deux fois plus court" is (are?!) illogical, I leave for your friend to decide.

And Barbara Partee remarked:

Arnold in his last paragraph writes, "I was taught that two times more than X really means 'three times as many as X'." (i.e. X + 2X, as it was put earlier in Arnold's post)  My own father insisted on that, and convinced me it made sense because we all (I think) see the difference between "Box A contains 50% more Cheerios than Box B" and "Box C contains 50% as many Cheerios as Box B." I'm sure that Box A does contain 150% as many Cheerios as Box B in that case. (Interesting that we say 'half again as many' vs 'half as many' for the same distinction; we can't say 'half more', I don't think.) But when it gets to bigger numbers, I don't "feel" any distinction, and I will (if I'm thinking about it) avoid "twice more than" because for me it's ambiguous between the meaning my father convinced me it should have and the meaning I can feel just as easily, maybe more easily, where it's synonymous with "twice as many as".

I have no opinion on "three times as slow" — I never thought about that before. I can see both sides — it doesn't sound bad to me and Victor's symmetry arguments sound reasonable, and evidently lots of people do it. On the other hand there is one big difference: it's the "positive" adjectives that we also use as the "neutral" forms for adding measurement modifiers: 6 feet tall, or 4 feet tall, but never 4 feet short. ('90 years young' is used as a joke.) But I don't think there's a restriction on modifying comparatives. With measure modifiers, "A is 3 inches taller than B" is a neutral description; "B is 3 inches shorter than A" may suggest (I'm not sure it has to) that they're both short, but the two are equivalent. But the OKness of those doesn't "prove" the OKness of "B is twice as short as A" — it might be odd because of implying that you should be able to tell me how short both are in some measure terms, like *B is 3 feet short and A is 6 feet short, and that's clearly impossible. Interesting – but we'll be traveling today, so I don't have time to try to think any farther than that.

One thing is certain:  all of this is much too heady for a beautiful fall day, when I should be out running in the woods, but I'll be less than content if I don't get this off my chest before I go out and enjoy the foliage.

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1. ### G said,

October 21, 2012 @ 12:24 pm

I'm sure the argument over whether it's proper to use multipliers along with a diminutive comparative to express the inverse relationship must be one of the most common and widespread usage debates.

I would argue that even if you object to "twice as small/short/low/thin/light/young, etc.," there's a commonsense interpretation of "twice as slow" that should make it acceptable. Speed is progress (distance traveled, or in this case kilobytes downloaded) over time, so to say something is twice as slow is to say it will take twice as much time to complete.

Similarly, I find no problem with something being "twice as rare" (over "half as frequent"). It means you'll have to go through twice as much chaff in order to find it.

2. ### Terry Collmann said,

October 21, 2012 @ 12:38 pm

I (southern English) don't have a problem with "hundreds of times slower", and even "three times slower", but I do start to want to say "half as fast" rather than "twice as slow", and for, eg, "twice as thin/fat/", I would need some other comparative to have been made or implied already – that is, I couldn't simply say "Bill is twice as thin as Joe," only "Joe is thin, but Bill is twice as thin."

3. ### efahl said,

October 21, 2012 @ 12:40 pm

Even impaired with an undergrad in mathematics, "times slower" make perfect sense to me; I'm sure I use "times slower/faster" quite frequently. A hundred times slower simply means "X * 0.01".

4. ### James said,

October 21, 2012 @ 12:40 pm

I certainly won't argue against the well-supported (by Zwicky) point that "n times slower than" is established and understood. But I side with GW's view that it is illogical.

For something to be ten times slower than another, there would have to be such a thing as slowness that something could have in degrees, and the first have ten times as much of it. I'm six feet tall. Is someone five feet tall 117% as short as me?

5. ### The Ridger said,

October 21, 2012 @ 12:42 pm

This is utterly standard in Russian, where expressions such as "he's twice as short" or "five times as cheap" consistently confound my translation students, most of whom feel – like Prof Pullum – that these expressions don't work.

6. ### Jeff Carney said,

October 21, 2012 @ 12:46 pm

I think Arnold Zwicky's post really nails this thing on the head. If you want to be understood in most pedestrian contexts, the construction makes perfect sense, because we are accustomed to understanding it as it is intended to be understood (anyway, I am so accustomed, and seems most of us are as well). In a more technical context, the problem of illogicality is probably moot; statisticians, chemists, linguists, etc. do not consider using such constructions when writing professional discourse. The language of numbers is replete with more exact ways of phrasing such things.

7. ### Lugubert said,

October 21, 2012 @ 1:01 pm

I could never be persuaded to write "twice as thin as" when I mean "half as thin as", or "twice as rare" for "half as frequent". But I think that I understand what is meant.

A more serious problem is when I encounter, say, "vacuum less than 1.33 mbar". Apart from "vacuum" being an absolute to me (the practically unattainable 0 mbar), I feel insecure. Does it mean that the pressure is even lower than 1.33 mbar, or that it’s less than 1.33 mbar below atmospheric pressure?

8. ### EasyArray said,

October 21, 2012 @ 1:06 pm

G makes the excellent point that "twice as X" makes perfect sense when X is a ratio, such as miles per hour. You simply switch the numerator and denominator, yielding hours per mile in this case. Twice as many hours per mile is the same as half as many miles per hour.

This is tougher for non-ratios, as in "twice as short," but all of these expressions make sense if you think of words forming pairs of multiplicative inverse scales (as James rejects above). So, "tall" measures your height in feet, while "short" measures your height in ft^-1 (inverse feet?). 4-foot tall is 1/4-ft^-1 short, which is two times the shortness of 8-foot tall (1/8-ft^-1 short), and hence a 4-ft person is twice as short as an 8-ft person. Perfectly logical.

The real question is: why can't we say "I'm 1/6 inverse feet short!"

9. ### David L said,

October 21, 2012 @ 1:24 pm

As an erstwhile theoretical physicist, I am completely mystified by the claim that there's something illogical about '100 times slower than.' We're talking about the rate of some process. A rate can be 100 times bigger than some other rate, or 100 times smaller. Smaller rate = slower. Where's the illogic?

I once worked at a magazine where there was an injunction, handed down on an ancient style sheet, against constructions such as "A is three times bigger than B." The argument, which I found perverse, was that this might be taken to mean A = B+3B rather than A = 3B. Instead we had to write "A is three times as big as B." To my way of thinking, if you're going to find the first construction ambiguous, you can find the second one ambiguous in the same way. I protested at this silliness, but the style sheet was held sacred.

10. ### Bril said,

October 21, 2012 @ 1:43 pm

Since when does a language have to be logical all over?

11. ### Jason Cullen said,

October 21, 2012 @ 1:53 pm

Finally, an interesting read on Language Log!!! Seriously guys, you've been in a rut.

12. ### mollymooly said,

October 21, 2012 @ 2:48 pm

Ngrams:

"times as much" exceeds "times more" from 1900 to 1960, then falls back

Since "n times less" has only one plausible meaning, I don't have a problem processing it. But I have to remind myself that a given instance of "n times more" may mean "n times as much".

The ambiguity usually doesn't matter in any case. If n is large the difference is negligible. And a responsible writer will have provided the underlying numbers, so readers can check the calculated ratio if they wish.

The likeliest problem value is n = 3. "three times as much" is passed by "three times more" in 1980

13. ### Steve Kass said,

October 21, 2012 @ 2:55 pm

There are several issues in this discussion, not all of which have been separated.

One: The potential difference between the expressions "X times bigger" and "X times as big" (in their meaning and in their "correctness").

Two: Expressions with "times as small" and "times smaller" instead of "times as big" and "times bigger."

Three: (No one has brought this up yet, but I think it leads to the next issue, which is important to the discussion.) Whether percents can be exchanged with numbers in these expressions. "500%" seems like an alternative way to say "5 times", so are there differences between "500% bigger," "500% as big," "5 times bigger," and "5 times as big"?

Four: The possibility that what these expressions can be safely used to mean depends on the realm of biggerness they're used in.

In the "hugely different" realm, one can say "100 times bigger," "100 times as big," "100 times smaller," or "100 times as small" without danger of miscommunication. Some folks avoid the "small/smaller" expressions, but few of them would argue that they're grossly misunderstood. One could argue that "100 times bigger" is yet a bit more bigger (by one more time) than "100 times as big," or that "times smaller" and "times as small" aren't precisely defined, but who cares? In this realm, statements like this aren't likely to be intended to communicate great accuracy. If they are, use mathematical notation, not natural language.

But then there's the "slightly different" realm. If A is 5 and B is 4, A is "one quarter bigger than B," "25% bigger than B," or "125% as big as B." A is not well served by being termed "one quarter times bigger than B." In the same situation, the "smaller" phrases might be unclear. B isn't "25%/a quarter smaller than A," though it's easy to slip up and assume that "B is X times smaller than A" is the same as "A is X times bigger than B," even though it only is (for practical purposes) when X is large.

The fact that percents and numbers are often interchangeable, and that "smaller" expressions are often used unambiguously to express the same relationships as "bigger" expressions leads to what I think is the real horror in these kinds of expressions, and what makes me wary of using any of the "small/smaller" versions: "A is 300% smaller than B," and "Profits fell 750%."

October 21, 2012 @ 4:09 pm

Geoff Wade is wrong. There you are; I've said in 4 words what's taken the rest of you about 4,000.

15. ### Layra said,

October 21, 2012 @ 4:27 pm

My first thought is that it isn't quite about scaling, but about addition. If you stack two things one of which is X tall and one of which is Y tall, you get something X+Y tall. Shortness is much harder to add. Similarly, if you're on a train going at speed X and you toss a ball in the direction of the train with speed Y, and it's before 1905, then the ball moves with total speed X+Y.
So perhaps the issue (and the issue of X bigger than) is that our intuition of multiplication is built on our intuition of addition.

The first comparison that springs to mind for me is electrical resistance and conductance, which are mirror opposites of one another and are measured in units that are inverses. Given a pair of resistors, one can combine them in series or in parallel; in series, the resistances add, while in parallel the conductances add. Resistance is the more common, perhaps more natural notion, but conductance is more akin to the positive qualities than resistance is. But then, I don't think I've ever heard anyone say either "more resistant" or "more conductant", so this provides no data points.

16. ### Q. Pheevr said,

October 21, 2012 @ 4:30 pm

I think I would avoid things like "five times slower" and, especially, "five times shorter" in careful writing, especially if the exact figures are important. But in ordinary colloquial usage, I happily accept (and probably use) them as equivalent to "slower/shorter by a factor of five."

Just out of curiosity—does anyone who rejects "five times slower" also reject "slower by a factor of five"?

17. ### Ethan said,

October 21, 2012 @ 4:36 pm

Of all the examples given, the only one that leaves me perplexed is Lugubert's "half as thin as". Lugubert wants this to mean the same thing as "twice as thin as", but to me it is at best ambiguous and more easily interpreted as being opposite in sense.

"Compared to Santa Clause, Bill is thin. But Dan is half that thin." If forced to turn this into quantitative measure, my sense is this translates to "SC is N units wide; B is N-delta units wide; D is N-delta/2 units wide". So D is actually fatter than B.

I would have the same or worse problem with a comparison using "close". "A is far away, B is close, but C is half as close". To quote an earlier blog topic: does that even mean anything? Surely that cannot be replaced by "C is twice as far". I think this pair is different from Barbara Partee's "tall/short", as neither "near" nor "far" is the default neutral measure word.

18. ### CBK said,

October 21, 2012 @ 5:24 pm

Here's a more parallel version of (and maybe slight generalization of) things that have already been said:

We measure rates such as speed of an object in distance per time, e.g., miles per hour. (We call it "speed" and not slowness.) If one speed is a positive integer multiple of another, we talk about that in terms of "times faster." It seems implicit that there's an associated rate that's measured in units of the form distance per time, e.g., 10 miles per hour is twice as fast as 5 miles per hour. (Less common: 5 mph is half as fast as 10 mph.) The analogue for internet speed is digital information per time, e.g., megabits per second.

But we could measure these rates in hours per mile or seconds per megabit. These units make "slower" a comparison that can be made in terms of integer multiples. For example, 4 hours per mile is twice as slow as 2 hours per mile. (Much less common–or maybe not used: 2 hpm is half as slow as 4 hpm.)

Distance (or information) per time and time per distance (or information) are both forms for units of rates that give the same information about how fast or slow something is going. One form associates "faster" with positive integer multiple comparisons (e.g., "twice as fast") and the other associates "slower" with those comparisons.

N.B. The terms rate and ratio get used with different meanings by different people. And "X percent increase" is sometimes ambiguous. The ban on "A times bigger than B" that David L mentions might have been due to overgeneralization of that problem. There's discussion of all of these things in the context of mathematics standards for grades 6 and 7 here.

19. ### Mark F. said,

October 21, 2012 @ 6:01 pm

In my intuition, "5 times slower" is way better than "5 times as slow". "Twice as short" sounds quite odd.

In describing the processing speed of computers, I was taught that "a 5x slowdown" is correct, rather than "a speedup of 0.2". From a pragmatic point of view, this makes sense to me, since the latter phrase seems almost intentionally confusing. But that's different from saying "a fifth as fast".

David L – People who don't like "100 times slower" also wouldn't like "100 times smaller".

20. ### Observation said,

October 21, 2012 @ 7:27 pm

I think it is true that 'two times more than X' really means 'three times as many as X', at least in Chinese.

高鐵比普通鐵路快X倍。(=high-speed trains are X times faster than normal trains)
高鐵的速度為普通鐵路的X倍。(=high-speed trains are X times as fast as normal trains)

Also, I was reminded by 加以減少 of the following sentence (spoken by 梁惠王) from Mencius:

鄰國之民不加少，寡人之民不加多，何也？

Did Eastern Zhou kings ponder over the addition of negative numbers?

21. ### nvb said,

October 21, 2012 @ 8:10 pm

Many many years ago I read about a leaflet campaign to encourage people to use energy in winter sensibly (Making sure windows are closed, blocking drafty gaps under doors etc).

Originally it had been worded in a scientific way: Do x, y and z in order to keep the heat in. But for whatever reason, they decided to reword it as keeping the cold out. Then the campaign was more successful.

I think my point is that Slowness and Coldness aren't scientific entities, but to most people make perfect sense, and perhaps are preferable. I'm British, and I'd happily say something such as " Oh no I'm not taking the bus, it's twice as slow as driving".

Same applies to money: 10 times cheaper etc.

22. ### Matthew Stuckwisch said,

October 21, 2012 @ 9:28 pm

Here's where the issue is for me.

Consider the expressions "it's twice as hot" or "it's twice as cold". Where does the doubling occur? I'd submit that, for metric folk, the doubling occurs with 0. So if something is 20º, twice as hot is 40º. If something is -5º, then twice as cold is -10º.

Switch to ºF and it gets weird. Are we still going to work from 0º? In which case twice as cold/hot in ºF is different than ºC. Or do we accept that freezing should be our break off point? I'd guess that whenever we say X is twice as cold, we often are creating an arbitrary standard point (say, 60º when it's 50º and we're commenting about a 40º place, "it's twice as cold").

When considering speeds or time, since the concepts of negative time and distance aren't really possible in the discussions (a trip can't take negative minutes, and someone can't be negative feet tall, although I'm sure physics says it's theoretically possible), when we talk about anything being twice as short, we're setting up an arbitrary point of comparison first. Consider:

John is twice as short as Jamie.

I don't interpret this as the same as he's half as tall. I imagine John and Jamie standing next to Jack. Jack is 6'2, Jamie is 5'11, and so John is twice as short, and is 5'8. Am I the only one that's seeing it this way?

23. ### John said,

October 21, 2012 @ 10:11 pm

For those who think that "500% more than" means 5 times as big, and "200% more than means twice as big, I'd ask if "100% more than" means exactly the same size.

But I'd think the real problem here is expecting mathematical logic in a natural language. Yeah, the prescriptivists like to pick one meaning from a list, and insist that the others are all illogical. But that has little if any impact on how the rest of us use the language.

If you want logic, you have to restrict your search to technical dialects. I've had some fun with the peevers over the good old less/fewer thing, by pointing out that mathematicians always use "less than" (and "greater than") for all numeric comparisons, and never use "fewer than" (or "more than") in their technical writing. This leads to much consternation, since the peevers never understand the difference between common and technical speech, and thus don't understand that I'm just trolling them for the lols (as the Web's technical language would have it ;-).

The common language isn't about to be restricted to logical forms in such situations. The fun example that math educators have come up with to illustrate the problems is that when you ask a random person in the street the value of 2+2*2, the overwhelming majority say "eight". If people can't even get that right, why would you expect them to handle any other sort of arithmetic correctly?

(It's more disturbing that you get the same wrong answer when you ask in many technical settings.)

24. ### WalterW said,

October 21, 2012 @ 10:21 pm

For expressions like 'twice as thin', I usually just see it as describing something as having twice the quality of thin… double-thin, if you will. (Not just thin, but double thin!) To me, a word like thin is just a quality which can be intensified by some fluid degree (very, unbelievably, etc) or a more concrete amount (twice, double, etc). I see the same for words like slow or short.

Which reminds suddenly of expressions like "doubly sure". I think we can say "doubly slow", can't we?

25. ### Victor Mair said,

October 21, 2012 @ 10:59 pm

@Observation

I was thinking of precisely that passage from Mencius when I mentioned jiāyǐ jiǎnshǎo 加以减少 ("increasingly fewer") as having two millennia (actually more) of usage. For those who don't read Chinese, what King Hui of Liang said is roughly / literally this: "Why is it that the people of the neighboring kingdoms are not increasingly fewer and my people are not increasingly more [numerous]?"

26. ### Ruben Polo-Sherk said,

October 21, 2012 @ 11:09 pm

It is possible to have conceptualizations of things that aren't informed or restricted by their mathematical description.

A distinction that would be useful is between 1) subjective judgments, where short, fast, etc. is oriented around some arbitrary point, and 2) comparisons, where it is oriented around the object of comparison, and additionally, 3) descriptions in "physical" dimensions. For example:

John is 7' tall. Bob is 6'10" tall.

So, while Bob is in no contexts short (1), he is still 2" shorter (2) than John. This has no connection with a notion of "the degree of Bob's shortness (3) is 2" greater than that of John's". "Bob is shorter (2) than John" does not mean "Bob's shortness (3) is greater than John's shortness (3)".*

We don't say "He is 3' short" (aside from jokes), because, unlike with height or length, there is no concept of measuring something's "shortth". The fact that there is no concept in this sense of "shortth" has nothing to do with having a concept of "the degree to which something is short", where below a certain subjective (and usually vague) point, things can be shorterer than each other.

*This does not mean that it is not possible to have mixtures of 1) and 2), as in "John is even taller than Bob".

@John (not the one who is 7' tall), maybe I don't understand what you mean, but mathematicians have a very good reason for using "less than" rather than "fewer than", and they never will say "Set A has less elements than Set B".

27. ### Oskar said,

October 21, 2012 @ 11:13 pm

I've reconciled myself to the fact that people seem to think that constructions like "one hundred times slower" is perfectly acceptable, and that the arguments for it make perfect sense, but I just can't get myself to agree with them. Those constructions just sound plain wrong to me. Who would ever say "Alice is twice as thin as Bob"? My linguistic intuition just rebels against that kind of thing.

However, if I've learned one thing from years of reading Language Log, it's that the world is full of annoying peevologists, who are invariably wrong about everything. I will not add myself to their ranks, so I say: you guys are probably right, and I'm not going to try to not be bothered by it. Maybe one day I'll even find myself saying something like that, though I seriously doubt it.

28. ### Ruben Polo-Sherk said,

October 21, 2012 @ 11:34 pm

Forgot to say:

VHM was using 2) (a comparison of speed) and GW was seeing a mixture of 2) and 3) (a comparison of the measure of the dimension of slowness), and interpreting the statement as a comment on what would be analogous to what I called "shortth" above. His objection was made having taken that "twice/four times, etc." necessitates 3). I think that not many people would object to mixtures of 1) and 2) (though often the reference point for 1) is not well-defined); I don't think GW would object to mixtures of 1) and 2), like "The internet is slow in the U.S. But it is twice as slow in China".

29. ### Kaleberg said,

October 21, 2012 @ 11:38 pm

Lugubert – Yes, there are degrees of vacuum. The vacuum in a vacuum tube or vacuum bottle is far from devoid of matter, but it is empty enough to allow electrons to flow or heat not to flow. The quality of the vacuum in a vacuum packed food item is even worse, since the goal is simply to limit oxidation.

In other words, there are different levels of vacuum, so some process might require a vacuum less than 1.3 mbar which is measured on an absolute scale from a theoretical minimum, just like temperature which also has an absolute minimum. When you get down to very low pressures, you start running into quantum effects, and at the quantum level nature does abhor a vacuum, so neither absolute zero pressure or temperature are obtainable.

P.S. I have no problem with twice as slow or ten times as slow. When you say times, you are specifying a geometric scale. Having been doing a little SAT coaching, I've found no one confuses "twice as much as" with 3x, though a lot of students slip up on 200% greater than which is 3x greater.

30. ### Michael B said,

October 22, 2012 @ 2:24 am

@Matthew Stuckwisch: I've never heard the expressions "twice as hot" or "twice as cold." IMHO, for good reason: they make no sense whether you're using Centigrade or Fahrenheit degrees. As you acknowledge, 0° C and 32° F are arbitrary reference points. The only reference point that would make any sense might be 0° K [ −273.15° C or −459.67° F], so that if yesterday's temperature was 0° C today's would have to be 273.15° C to be "twice as hot" as yesterday's.

31. ### Jukka Kohonen said,

October 22, 2012 @ 3:07 am

Steve Kass asks whether percents and "times" are interchangeable. Apparently not, even in the positive degree: while "5 times as long" is standard English, "500 % as long" is quite unusual (perhaps ungrammatical), though one might conjure up such an expression and ask what it "would" mean if it existed. If percents and times are not interchangeable in the positive degree, perhaps it is not a big surprise that they are not interchangeable in the comparative degree either.

But certainly the expectation of interchangeability is a big source of the confused arguments over the "logical" meaning of "2 times longer". I just wonder if the same people confuse "Jim has 5 apples, Joan has 2 more" with "Joan has 200 % more" since "200 % means two"? Probably not; I think people are in fact quite capable of differentiating percents from other forms of comparison.

An interesting side topic is whether percent comparisons are as unambiguous as one might think. In fact, they are notoriously often ambiguous with respect to what is the base value. For example, in a series of percent increases, one must find out whether the further increases are "% of the initial value" or "% of the current value". In tax calculations one must know whether the tax is % of the "gross" or "net" price (in fact the Finnish VAT used to be one way and now is the other way).

For another example: When comparing hard disk speeds, sometimes you report operations per second, sometimes you report how long a particular task takes (basically the inverse of the former). This is not a hypothetical example; such comparisons are common in computer magazines. If disk A is then reported as 50 % slower than disk B, what do you get? Does it take 1,5 times as long, or does it perform 0,5 times as many operations? Now, that's ambiguous. Interestingly, "2 times slower" or "1,5 times slower" avoids this ambiguity. With the "times" comparison it does not matter whether we are measuring operations per time or its inverse. It works the same both ways.

32. ### Jason said,

October 22, 2012 @ 3:30 am

I suggest this analysis:

Let speed one be A, let speed 2 be B, where A = 5B.

The fastness of a rate of change variable Q with respect to a rate of change variable P is defined as Q/P.

The fastness of A is equal to A/B = 5B/B = 5.
Informally we say A is 5 times faster than B.

The fastness of B wrt A is equal to B/A = B/5B = 1/5.
B is 1/5 times as fast as A.

Slowness is presumably the reciprocal of fastness. Therefore, the slowness of B wrt A is equal to five, or informally, B is five times as slow as A.

This dispute is precisely why mathematicians use their own specialized language, in which everything is largely (though not entirely) precisely defined and ambiguity seldom occurs. But I would suggest that the usage is long attested and perfectly clear, and can be given clear definition, and this is dispositive of the question for most contemporary mathematicians.

33. ### Chris Hunt said,

October 22, 2012 @ 3:48 am

"Maybe it is just British and American English. I don't think the Brits say, for example, 'five times as slow' or 'five times slower'."

Speaking as a Brit, I'd certainly use "five time slower" in preference to the cumbersome "one fifth as fast".

34. ### Jukka Kohonen said,

October 22, 2012 @ 3:59 am

Jeff Carney suggests that "statisticians, chemists, linguists, etc. do not consider using such constructions when writing professional discourse." An interesting hypothesis, to be sure, but let's look at actual professional discourse.

For just a quick example, try the search box on the "Science" magazine's home page. You'll find, among others,
http://www.sciencemag.org/search?site_area=sci&fulltext=%22times%20less%22&submit=yes
745 articles containing "times less"; 141 articles containing "times slower"; and 748 articles containing "times faster". Note that the "times faster" sometimes refers to a rate ("things per second"), sometimes to a time ("seconds until the rats start to eat"), in full accordance with the theory that speed can be compared both ways, as noted by some commentators above.

And if you look elsewhere in professional discourse, you'll find that "statisticians etc." use the "times" comparison all the time.

I can't see how to reconcile the hypothesis with the observations, unless perhaps by invoking the "no true Scotsman" argument. Oh, you found that scientist N.N. writes "times smaller"? I always thought he's no true scientist. (But what do then make of Newton and Cardano?)

35. ### Mary Apodaca said,

October 22, 2012 @ 4:52 am

Time for a joke: Perhaps twice as slow is used to avoid "half fast."

36. ### Daublin said,

October 22, 2012 @ 9:01 am

Steve Kass is right that percents change the interpretation. "Fred runs two times faster Bob" is the same as "Fred runs 100% faster than Bob". It is not the same as "Fred runs 200% faster than Bob".

Second, a few of the commenters are right that fractional ratios are hard on the listener. If you say "Fred is X times faster than Bob", then–if you have a degree of human empathy–you should plan for your listener to think that Fred really is faster than Bob. "X times" is expected to be a modifier, not something that reverses the sense of the sentence!

Perhaps this is why the "slower than" constructions are so common. The nearest alternative expression is to use "faster than" with a fraction, but that's hard on the listener.

37. ### Brian T said,

October 22, 2012 @ 9:05 am

The problem is that "5 times slower" warps the concept of multiplication.

If X is 100 mph and "2 times faster than 100 mph is "200 mph," then what's twice as fast as that? Multiply by "2 times" and you see that "4 times faster than 100 mph is 400 mph," which is still correct.

Using "times slower" requires a completely separate (and usually unrecognized) format for seeing what's happening: "2 times slower than 100 mph is 50 mph," apparently, and what is twice as slow as that? Multiply by "2 times" and you see that "4 times slower than 100 mph is 100 mph." Whoops.

Well, let indulge a figure of speech and grant that, for just this one usage, "times" indicates that you're supposed to divide, not multiply. So "2 times slower than 100 mph is 50 mph" and "4 times slower than 100 mph is 25 mph." Good! We got the intended result even though our method makes no sense.

The uselessness of this idiom becomes apparent when you note that a regular, constant amount separates each of the milestones as you go from "2 times faster" to "3 times faster" to "4 times faster" and so on. But from "2 times slower" to "3 times slower" is a difference of 17 mph; from "3 times slower" to "4 times slower" is a difference of 8 mph; from "4 times slower" to "5 times slower" is a difference of 5 mph. Not useful, and easily misleading.

38. ### Roger Lustig said,

October 22, 2012 @ 9:05 am

As a musician/music historian, I was initially surprised by the whole discussion. (As a statistician not so much.) "Twice as slow" means "taking twice as long," as others have noted. "Slowness" in the sense(s) we're discussing is effectively the same as "duration." Rate x time = distance, so "half/twice as slow/fast" is easy to understand in any combination.

I think the musician in me found the controversy surprising because musical notation uses two durational systems simultaneously. The first is time:space, i.e., the amount of horizontal staff space used. For reasons of economy, scribes back when didn't respect that relationship much, especially when writing out one part at a time.

But in modern score-writing (you can buy paper already divided 4 measures/line) and in individual parts that play more than one note at once (keyboard or fretboard), time-space is pretty evident and often used fairly strictly.

The other relationship is time:ink, which nowadays is reciprocal, i.e., more ink~less time. Add a stem to a whole note and get a half; fill in the note head and halve the note again. Then add flags to halve again and again… In England, older terms for the note values are still in use and involve adding to the syllable count too (hemidemisemiquaver for 64th note).

But even back in the day (say, October 20, 1412) more ink could also indicate longer duration–especially if it ate up more of the time-space. What we call a whole note in the US is a semibreve in England and in Olde; and was a lozenge before it was a circle. As the name indicates, it was half of a breve, which name implied that there was something out there called a long. Which there was. Those of us used to more modern notation, especially the last few centuries, may never have seen a long, and only rarely a breve. A breve is a rectangle with an area ca. twice that of the semibreve-lozenge; to make it into a long, add a stem. And if that's not long enough, use a maxima: a loooong long, i.e., a rectangle that takes up more of the horizontal.

So it seems to me that the time:ink and time:space relationships were originally similar and not reciprocal as they (mostly) are now. Increasing complexity of durational values on the one hand, and the cost of paper on the other, led to all sorts of notational kludges, some of which we still haven't cleaned away.

39. ### Jukka Kohonen said,

October 22, 2012 @ 9:57 am

Thanks Roger for the interesting musical aspect. On the same note, I'll chime in that Guido of Arezzo's Micrologus, dating approximately to A.D. 1026, may contain one of the earliest known instances of the equivalent Latin expression:

"et aliae uoces ab aliis morulam duplo longiorem uel duplo breuiorem."
http://www.calumcille.com/griogair/9A9.html

Note the comparative forms; literally, "twice longer or twice shorter".

40. ### Nathan said,

October 22, 2012 @ 10:41 am

In an algebraic notation:

"X is N times as much as Y": X = NY
"X is N times more than Y": X = (N+1)Y
"X is N times less than Y": I don't like this at all, but it seems you all are saying X = Y/N.

So it leaves sort of a gap in the system, I guess. "X is N times less than Y" is reciprocal to "X is N times as much as Y" instead of "X is N times more than Y", which has no idiomatic inverse.

October 22, 2012 @ 11:13 am

http://www.amazon.com/Twice-as-Less-Eleanor-Orr/dp/0393317412

42. ### Jukka Kohonen said,

October 22, 2012 @ 12:09 pm

Nathan, I don't know how you have managed not to notice that the actual meaning is (in algebraic notation if you wish)

"X is N times more than Y": X = NY

Since this is the meaning that the expression has in actual usage, there is no such asymmetry as you thought. Problem solved, nice isn't it?

43. ### Nathan said,

October 22, 2012 @ 12:21 pm

Jukka Kohonen, I don't like what you're saying, but I don't want to be prescriptive. If that's the way people besides me usually use it, then so be it.

44. ### Jukka Kohonen said,

October 22, 2012 @ 12:28 pm

Nathan, I'm not sure what you don't like. That I tell you how people use the expression, or that people use it that way?

Anyway, if asymmetry was something you did not like, that problem was nicely solved by observing there is no asymmetry. I'm sure this makes you happier.

45. ### David Walker said,

October 22, 2012 @ 1:35 pm

My favorite mis-use of this is from a display ad that was PAID FOR by a mainframe software company in the 80's. They quoted a computer programmer saying that their production jobs run in "140 percent less time" now that they are using the wonderful software that was being advertised. That's quite a feat!

It is CLEAR that "hundreds of times slower" or "four times slower" is not intuitive, and the use of this phrase should be discouraged.

46. ### Ted said,

October 22, 2012 @ 1:52 pm

I have a bit of a different take on it, which doesn't make a difference in the "twice as fast" context but I think may apply to other adjectives such as slow, hot, cold, etc.

"How fast was he going?" translates to "What was his speed?" Speed is a straightforward ratio that can be doubled, halved, etc. So if I go 40 miles an hour and you go twice as fast, you're going 80 mph. If you went "half as fast," it's pretty clear that you'd be going 20 mph, although it would be more natural to say "half the speed" instead. But in each case, "fast" just refers to speed — it doesn't imply a conclusion that either of us was necessarily "fast" in the sense of "not slow." In other words, one snail could go twice as fast as another, even though neither of them is going very fast.

"Slow" is not reciprocal because "how slow was he going?" isn't just a question about his speed. It only works if there's an understanding that he is slow. So if I'm going 80 mph and you're going 40, I wouldn't normally say you were going twice as slow as I am, even though I'm going twice as fast as you are, because 80 mph isn't slow in most contexts. (You could say, I suppose, that I have zero slowness, and since 2 x zero is still zero a multiplication factor is unsuitable as a basis for comparison.)

The key is that there has to be a reference point from which a deviation (in the direction suggested by the adjective) is measured. This works, for example, in the temperature context discussed above. If room A is twice as warm as room B, this implies first that room A is above normal room temperature by a given amount and second that room B is above room temperature by twice that amount. It doesn't matter what units you use or where the customary zero point for those units are; the relevant reference value is that below which one would begin to describe a room as cold.

So in the VHM/GW debate, I generally concur with GW's observations, but disagree with his blanket objection to "hundreds of times slower." If VHM believed, for example, that a particular web page should download in 3 seconds, and in fact the page downloaded in 5 seconds in the US and five minutes in China, I think it would be correct to describe the Chinese internet as hundreds of times slower, because it would have 397 seconds of slowness compared to only 2 seconds of slowness in the US.

47. ### Ted said,

October 22, 2012 @ 1:54 pm

Sorry, in the next-to-last paragraph above I should have said that the reference value is that above which you would describe a room as warm, not that below which you would describe the room as cold. (They are not necessarily the same, which adds another layer of complexity.)

48. ### Jukka Kohonen said,

October 22, 2012 @ 2:37 pm

I don't believe using the word "slower" in a comparison implies that either speed is especially slow. "These SPs travel slightly slower than light in free space" (Science, August 2002) seems perfectly normal language to me even though neither light or the surface plasmons are exactly snails.

49. ### Birdseed said,

October 22, 2012 @ 3:07 pm

Intrestingly, while the equivalent of "twice as short" (dubbelt så kort) sounds utterly wrong in Swedish, the opposite, still internally-illogical, equivalent of "half as short" (hälften så kort) is entirely the correct expression. "half as tall" (hälften så lång) sounds clunky in comparison.

50. ### chris said,

October 22, 2012 @ 3:14 pm

As if this wasn't confusing enough already, if something is twice as thick, it's 100% thicker, but if it's twice as thin, it's only 50% thinner. Percentages are funny that way.

51. ### LDavidH said,

October 22, 2012 @ 4:03 pm

All this proves is that language doesn't operate on logic alone. I'm sure most of us would understand these different statements intuitively, rather than mathematically, and the "strangeness" only comes out when you start analyzing. As I often say: listen to what I mean, not what I say! (Particularly important for us ESL speakers…)

52. ### Daniel Barkalow said,

October 22, 2012 @ 6:16 pm

There's a confounding factor with "twice as slow" in particular, which is that you may well mean latency when you say something is slow, rather than lack of throughput. If so, it would be "twice as slow" (takes twice as long for a response) that's obvious, and "twice as fast" would be what GW would reject ("The US connection responds twice as soon!?"). Since we work with both speed and duration all the time, we tend to have a geometric relationship that makes "twice as fast" and "twice as slow" both fine. On the other hand, I would be confused to hear that a Chinese internet connection is as fast (or as slow) as a US one and a South Korean one put together, because I'm not clear what's going on in an additive relationship.

This is different from "twice as fat"/"twice as thin", because we don't deal with inverse distances and we do additive operations with distances; and "twice as few" is very strange compared to "twice as many".

53. ### Chris C. said,

October 22, 2012 @ 7:24 pm

As a math major, I dislike the "times slower" construction exceedingly, but I've come to accept that it's how journalists indicate fractions.

54. ### Sevly said,

October 22, 2012 @ 10:30 pm

Brian T said,
The problem is that "5 times slower" warps the concept of multiplication.

Well, let indulge a figure of speech and grant that, for just this one usage, "times" indicates that you're supposed to divide, not multiply. So "2 times slower than 100 mph is 50 mph" and "4 times slower than 100 mph is 25 mph." Good! We got the intended result even though our method makes no sense.

No.

It's not that we use "times" any differently, it's that slow is the inverse of fast—that is, slowness measures inverse speed (hpm) rather than speed (mph). So "2 times slower than 100 mph" is 2 × (1/(100 mph)) = 1/50 hpm = 50 mph. And 2 times slower than that is 2 × (1/50 hpm) = 1/25 hpm = 25 mph, which is 4 times slower than 100 mph.

The idea that "times" means you divide may make no sense, but the idea that slow is the inverse of fast is quite intuitive.

55. ### Jukka Kohonen said,

October 23, 2012 @ 1:09 am

Isn't it funny how many journalists write in Science?

Seriously, it seems "journalists" are the utter out-group in terms of linguistic contempt. If any construction is perceived as bad language by someone, then journalists are likely to be the main perpetrators.

It would be interesting (if difficult) to study the mental processes leading to this perception.

56. ### Roger Lustig said,

October 23, 2012 @ 6:52 am

@Jukka (in Guidonian shorthand): I think the "journalist" rap (trope? meme?) is partly revenge, esp. when used around here. LL is a noted source of criticism of journalistic precision in science writing. "Noted" used seriously here–I've been reading Cordelia Fine's deliciously snarky Delusions of Gender, parts of which are highly Mark-L-inspired.

Also, the worst of "grammar" peevery was perpetrated by journalists when I was a lad: Edwin Newman, John Simon, Ted "Thistlebottom" Bernstein, etc. (Ever wonder why Safire got so much slack? He was a latecomer to journalism *and* better at the language punditry than most.)

57. ### joanne salton said,

October 23, 2012 @ 7:23 am

I'm sorry if someone made this point, but doesn't it really depend what we are aiming for? There is nothing wrong, it seems to me, with saying "5 times slower" etc if we are aiming for slowness. We seldom are though, and certainly not on the internet, thus Victor's original comment sounds vaguely odd.

58. ### Eric said,

October 23, 2012 @ 7:55 am

LDavidH: "listen to what I mean, not what I say!"

Isn't that precisely the problem, though? When you say something is "three times as short," what exactly do you mean?

As the purpose of language is to convey ideas from one mind to another, if one particular linguistic construction–no matter how common–garbles the message while another does not, why then would one cleave to the former?

59. ### Victor Mair said,

October 23, 2012 @ 9:49 am

@joanne salton

Sometimes the Chinese government intentionally aims to slow down or even bring to a halt the functioning of all or a part of the internet. For example, after the July, 2009 riots in Ürümqi, the entire internet was essentially shut down throughout the region; I think that it was closed for a whole year. One of my supreme frustrations (there were many) while living in China was the deliberate choking or blocking of Google. At times, the authorities would throttle it down to about 15-20 times less than (i.e., 1/15th-1/20th) its normal speed and sometimes they just wouldn't let it work at all. But the general slowness of the Chinese internet *for most users* is due to all the censorship and other types of restrictions and sheer clumsiness of the system overall. I should note, however, that Party, police, and military tend to have faster internet connections, because they are exempt from many of the operations of the Great Firewall and other measures to restrict the free flow of data and information that prove so frustrating to the common citizen.

60. ### Roger Lustig said,

October 23, 2012 @ 10:06 am

@Eric: "Jim is three times as short as Bob" means nothing I can imagine, unless we're saying that Jim has only 70% of the cost of a ticket whereas Bob has 90%.

"A sixteenth-note triplet is three times as short as an eighth note" gives me no problems whatever.

"What I mean" is conveyed by verbal and situational context as well as by a particular turn of phrase (the "what I say" component).

(Statistician's note: that's the first time a '7' has appeared in this discussion.)

61. ### JS said,

October 23, 2012 @ 11:36 pm

@Ted, etc.

Interesting… you begin with the point I had hoped to make (and more lucidly than I could have managed, I should say), but seem ultimately to convince yourself of an Obviously Wrong. "Shorter," "slower," etc., certainly needn't involve reference to "short" and "slow" comparanda respectively, for we may refer to someone as "two inches shorter than Yao Ming" (both people being tall), "a bit slower than her classmates" (not all class members being slow), etc.

(1) "He is fast." = He displays great speed (as you note)
(2) "How fast was he running?" = two possibilities, (I) At what speed was he running? (so no implication of objective "fastness" — just as you point out, "slow" does not offer the same possibility here); (II) At what great speed was he running?
(3) "He is faster than me." = two possibilities parallel to (2.I,II): (I) He displays greater speed than I do (with no necessary implication regarding my fastness/slowness); (II) I display great speed and he displays still greater speed. (This second may be indicated unambiguously by insertion of "still," "even," etc.)

(1a) "He is slow." = He displays low speed.
(2a) "How slow was he running?" = ONLY At what low speed was he running?, as you note.
(3a) "He is slower than me." = two possibilities precisely parallel to "fast," which to me is where your argument goes awry: (I) He displays lower speed than I do (with no necessary implication regarding my fastness/slowness); (II) I display low speed and he displays still lower speed.

So comparative/superlative adjectives like "slower," "shortest," etc. often simply mean "displaying less speed/less height" and need not imply that the focus comparandum, or indeed either comparandum, is objectively "slow" or "short" — but we already knew that, for again it is possible to be "two inches shorter than Yao Ming" and not only "not as tall as Yao Ming by two inches" or however that might be phrased. So we ought already to have known that "a hundred times slower" is Consummate English (= a hundred times less speed), if subtly ambiguous given that either of (3a1) or (3a2) might apply (in VHM's case, the former)… but if this discussion is any indication, we didn't.

62. ### Michael Rank said,

October 25, 2012 @ 2:32 pm

Do Americans say/write hundreth when we Brits say/write hundredth?

63. ### Ellen K. said,

October 25, 2012 @ 8:11 pm

Actually, judging from Google ngrams, it looks like Brits are more likely to write (in edited writing) hundreth than Americans. Not a huge difference in the present, with hundredth much more common in both cases, but historically it was a lot more common in British writing than American.

64. ### W. Kiernan said,

October 27, 2012 @ 7:45 pm

Brian T:

If the units of fastness are miles per hour, then the corresponding units of slowness would be hours per mile. Does this sound far fetched? Then consider the units of American fuel mileage ratings, in miles per gallon or distance traveled per fuel volume, as compared with European mileage ratings in liters per hundred kilometers, or fuel volume per distance traveled.

65. ### richwarm said,

October 28, 2012 @ 10:52 pm

"VHM: … conceptually it's actually very much like the jiāyǐ 加以 ("increasingly") construction in Chinese, where you can — seemingly paradoxically — have expressions like this: jiāyǐ jiǎnshǎo 加以减少 ("increasingly fewer"). Those types of expressions … are still very common today."

If I can ask a question as a Chinese learner, could you give an example where 加以减少 means "increasingly fewer"? Because I can only find examples where it seems to simply mean "to reduce" (or, perhaps more literally, "to effect a reduction").

e.g.
我们可采用一些手段加以减少。
如何在車主離開後可以加以減少偷竊發生
这种变形是可以用改进操作加以减少和避免的。
学校已将学生的人数加以减少，只有去年的80%。

66. ### Jukka Kohonen said,

October 29, 2012 @ 4:44 am

"Observation" writes,

"I think it is true that 'two times more than X' really means 'three times as many as X', at least in Chinese."

I can't make sense of this. 'two times more than X' is not Chinese, neither is 'three times as many as X'. They are English. What does it mean that an English phrase "really means" some other English phrase "in Chinese"?

"Observation" then presents the following, presumably made-up example:
"高鐵比普通鐵路快X倍。(=high-speed trains are X times faster than normal trains)"

I can't claim much fluency in Chinese, but trying to decipher, roughly,

高鐵 (high-speed trains)
比 (bi3, comparison)
普通 (ordinary)
鐵路 (railway)
快 (kuai4, fast)
X
倍 (bei4, times)

Now this indeed has some similar structural elements as "X times faster"; namely 比 (bi3, comparison, you might relate it to the English comparative suffix "-er" but there are structural differences) and 倍 (bei4, times). But how do you support the claim that this "really means X+1 times as fast"?

Can you cite actual usage (for example, in a newspaper, or a physics textbook) and demonstrate the intended meaning? Perhaps someone more fluent in Chinese might help here. If the expression is idiomatic, it shouldn't be too difficult to find samples (but you probably need to understand Chinese to analyze them).

67. ### Jukka Kohonen said,

January 14, 2013 @ 6:49 pm

Above, I mentioned Micrologus (circa A.D. 1026). But we can go earlier with the "DUPLO MAIOR" construction. Pliny the Elder's "Naturalis Historia" (circa A.D. 77) contains instances such as:

"Silvestre rapum in arvis maxime nascitur, fruticosum, semine candido, DUPLO MAIORE quam papaver."

("Wild rape is mostly found growing in the fields; it has a tufted top, with a white seed, twice as large as that of the poppy." John Bostock's translation.)

I would venture to guess that other instances can be found in Classical Latin texts, so the folks who insist that "twice larger" is a modern corruption may have about 2000 years of modern corruption to deal with.