Whenever you see an English word beginning with “bi-“, you can be fairly sure that it has something to do with the number two. “Bi-“ comes to English from Latin, although its roots are even older; in Ancient Greek it was “δι-“, and in Sanskrit it was “dvi-“. Even in Latin, “bi-” served the same purpose as in English; to form compound words that mean two of something.
English acquired “bi-” around the 1300s, by way of French (which is most often the case when you see a date of first usage around that time). Oddly enough, one of the first recorded “bi-” words in English was “bigami” (having two spouses) — today we spell it “bigamy.” Over the next century or so the number of “bi-” words began to expand pretty quickly, a period when so you see words appearing such as “biceps,” “bifer” (a plant that flowers twice every year), “bicorn” (an animal with two horns), and so forth.
Even though (ahem) binary choices seem like they should be pretty straightforward — yes or no, chocolate or vanilla, one or zero — there are a few “bi-” words into which a bit of uncertainty has crept. These all have to do with periods of time, such as “biennial” (having to do with years; “biyearly” is also sometimes used), “bimonthly,” “biweekly,” and a couple of more obscure ones: “bi-hourly” and “bi-diurnal” (having to do with days) and its counterpart “bi-nocturnal” (nights). The hyphens in the last three are, in fact, optional (more or less) in all of the time-related “bi-” words. British usage tends to include hyphens, while US usage omits them. But nobody is going to look at “bi-weekly” and complain that they WOULD know what it meant except for that the hyphen in the middle.
The vagueness around “biweekly” and its relatives comes from a basic question of mathematics; while everybody understands that the “bi-“ indicates two, is the operation multiplication or division? If it’s multiplication, then something “biweekly” happens every two weeks. If it’s division, then it happens twice per week. It’s the same with the other terms; “bimonthly” might mean twice in one month or once every two months.
As far as anybody knows, the “once every two [weeks/months/etc]” meaning came first. But really this might just be due to the slower pace of life a couple thousand years ago. At today’s pace we might be concerned with events that happen bi-hourly (which means every two…that is, twice per… never mind), but back then, come on, what could be so urgent that it happened twice in one measley week?
This being English, there are plenty of other prefixes and words ready to step in to solve the confusion reigning in our bicameral minds. Take “semi-,” for example. It would be much clearer to use “semiweekly” for something happening twice per week, and “biweekly” for every two weeks. “Semi-”, as a prefix, means “half” — although it’s not used as precisely as “bi-,” so a “semi-” word usually means “less than whole” rather than a binary bifurcation of a given quantity. Another way to express the frequency “twice-per-[period]” is to use “half-” as in “half-weekly.” That’s used in British English, but hasn’t ever caught on in the US. And of course there’s the word I’ve been using repeatedly because there’s no ambiguity about it at all: twice. If you just say “twice per week” and “every two weeks”, you’ve eliminated all the vagueness in one go. The key to clarity in writing, after all, is to forestall any possibility of bialternate interpretations to your message.
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