# The Evolution of Weights and Measures

At long last, I’ve exhausted my curiosity in mathematical etymologies. Many word histories have been explored in the previous three installments:

This time around, I want to look at some of the words we use for measurements. There are a few interesting histories in the metric system (SI), but most of the fun comes from the English Imperial system.

The Roman Empire provided us with the primary pre-SI system of measurement in Europe, from which many of the medieval systems were derived. The Latin word mille gives us two important words today: million (related to “thousand”, as detailed in a previous post), and mile. As Roman legions marched across the Mediterranean world, they measured their distances according to paces, with a thousand paces being milia passuum. A pace is the distance traveled in two full steps, and is about 58-62 inches (depending, obviously, on an individual’s height). Using this reckoning, the Roman definition of a mile clocks in at 4,833-5,167 feet.

When the Roman Empire fractured in the West, their uniform measurement system fractured as well, occasionally with hilarious consequences. Later, by the 18th century, the Roman mile had evolved from one definition to many: there were Scots miles, English miles, German miles, and so on. The German mile was 24,000-some feet (at least according to Wikipedia), compared to the English mile’s comparably-paltry 5,280 feet. (Go check that Wikipedia reference, too—there are many more variants!)

But before I get too distracted by the history of the mile, let’s move on to some other length measurements.

• Inch — this is a fun one. The word comes from the Latin uncia, which basically means “unit”. The strange thing is that an uncia was a unit of weight rather than length—it was 1/12th of a Roman pound. While the English inch is still 1/12th of its parent measure, the ounce somehow became 1/16th of a pound.
• Furlong — rather simply, it’s a combination of furrow and long, with a furrow being the length of a ten-acre farm field. This makes it about 1/8th of a mile.
• Yard and Rod — these two have an intertwined history. Today, a yard is 3 feet long, and a rod is 16.5 feet long. The word yard comes from Old English gierd, meaning “rod” or “stick.” Rod comes from the Old Norse rudda, meaning “club”. According to Schwartzman, the rod and the yard were used somewhat interchangeably during the Medieval period, and only later did they settle on 3 and 16.5 feet (or thereabouts)—the “short” and the “long” yard.
• Fathom — originating from the Old English fæðm (“faythm”), meaning “arms” or “grasp”. It was the length of a person’s outstretched arms, and is defined as 6 feet today. Perhaps, given its nautical use, a fathom was the distance you could fall off the boat while still being rescued by someone on board?

While there are lots of other words I could choose from, here are two in particular that have a surprising connection.

• Pound — comes from the Latin pondus, meaning “a weight.” The abbreviation lb. comes from the Latin word libra, meaning “pound” or “balance.” In most markets, merchants would assess the value of precious metals offered for payment using a balance scale (still with us in the popular imagination today). Indeed, one of the signs of the Zodiac is a balance scale. Of course, you’d need to balance the payment against a set of known weights. Over time, then, the word for the weights themselves came to be the English pound, while the word for the scale itself (libra) evolved into its abbreviation.
• Liter — comes from the Greek litra, which was a unit of weight. Yes, libra and litra have a common origin! Schwartzman notes that lytre and pound were used interchangeably in England as late as the 17th century. When France adopted a decimal system (the precursor to modern SI units), they borrowed the word litron, changing it from a unit of weight to a unit of volume.

There are many, many more words that I didn’t have the time or energy to write up! But hopefully it’s kept your interest throughout the whole series of posts. Get a copy of Schwartzman’s The Words of Mathematics if you want to learn more.

# The Evolution of Plane Curves

This is the third post in the “Evolution Of…” series; the first and second posts can be found here and here.

This time around, we’ll explore some of the words that have come to be used for various plane curves. First of all, a disclaimer: often, the names of the curves existed many centuries before the development of modern algebra and the Cartesian coordinate system. As a consequence, the original names for the curves are more geometric in origin (imagine one of the ancient Greeks saying “umm, well, it looks like a flower… so let’s call it the flower curve“).

While reviewing the curves listed in Schwartzman’s book, I noticed that most of them can be classified into four major groups: the conics, the chrones, the trixes, and the oids.

1. Conics. You’ve probably heard of them—circle, ellipse, parabola, and hyperbola. The first one has its origins in the Latin word circus, which means “ring” or “hoop.” The other three are Greek, with their original meanings reflecting the Greeks’ use of conic sectionsEllipse comes from en (meaning “in”) and leipein (meaning “to leave out”). For the other two, note that –bola comes from ballein which means “to throw” or “to cast”. So hyperbola means “to cast over” and parabola means “to cast alongside”. (If you check out this image from Wikipedia, it may start to make more sense.)
2. Chrones. The two curves I have in mind here are brachistochrone and tautochrone. In Greek, chrone means “time”. The prefixes come from brakhus and tauto-, which mean “short” and “same”, respectively. So these curves’ names are really “short time” and “same time.” Naturally enough, the brachistochrone is the curve on which a ball will take the least amount of time to roll down, while the tautochrone is the curve on which the time to roll down is the same regardless of the ball’s starting point. Finding equations for these curves occupied the time of many scientists and mathematicians in the 17th century, including a controversy between the brothers Jakob and Johann Bernoulli. You could say they had a “chronic” case of sibling rivalry.
3. Trixes. I am a fan of the feminine suffix –trix because it also provides us with the modern word obstetrics (literally, “the woman who gets in the way”—i.e., a midwife). We don’t use this suffix very much anymore, though aviatrix comes to mind. The algebraic curves in this category are trisectrix (“cut into three”) and tractrix (“the one that pulls”), along with the parabola-related term directrix (“the one that directs”). Interestingly, the masculine form of tractrix gives us the English word tractor.
4. Oids. These were the most fun for me. The suffix is Greek, originating in oeides, which means “form” (though in modern English, “like” might be more appropriate). Here are a some examples: astroidcardioidcissoidcochleoidcycloidramphoid, strophoid. Here are their original Greek/Latin meanings: “star-like”, “heart-like”, “ivy-like”, “snail-like”, “circle-like”, “like a bird’s beak”, “(having the) form of turning”. I’ve provided images of each one below—see if you can match the name to the curve!

Don’t worry, there are plenty more word origins coming later! However, I’ll need a break to recharge my etymology batteries. Expect an “intermission” post in the next few weeks.

# The Evolution of Arithmetic

This post is the second in a series; if you haven’t read the first post, on the evolution of English counting words, I’d recommend reading that one first.

As promised, this post looks at the origins of the English words for arithmetic operations. Read on, friend!

• Plus and Minus. These two are fairly straightforward—they’re the Latin words for “more” and “less”, respectively. The symbols, though, are less clear. It appears that the letters p and m were used (sometimes appearing as p and m) during the 1400s—Wikipedia claims that these first appeared in Luca Pacioli’s Summa de Arithmetica, though I’ve been unable to find a satisfactory example. In the 1500s, the modern + and – signs began to appear; Schwartzman attributes the + to an abbreviation of the Latin “et” (taking the t only) and the – to the bar from m.
• Multiply. This word comes from the Latin multiplicare, meaning “to increase.” Breaking it down a little further, we have the prefix multi– (“many”) and the suffix -plex (“fold”) so that the compound word multiplex means “many folds.” (We still use “fold” language today—when we speak of a “threefold increase,” we mean that something had been multiplied by three.) The x symbol for multiplication is attributed to William Oughtred, while Schwartzman gives credit for the dot • to Gottfried Wilhelm Leibniz.
• Divide. This word comes from Latin as well, with the origin being dividere, meaning “to separate.” (As a side note, the root videre means “to see” and gives us the modern word video, which means “I see”.) Putting di– and videre together, I suppose this means that division is literally “to see in two.”

Notice that all four of these words originate in a description of the operation itself. It turns out that exponents and roots are a little more metaphorical in their meaning:

• Exponent. Once again, we have a Latin origin: the prefix ex– and the verb ponere, roughly meaning “to put out.” Unlike the four arithmetic operations, though, the original meaning is typographical—the exponent is the number that is “put out” above and to the right of the base. In part, it’s because the exponent is a relatively new development; Schwartzman attributes the notation to Descartes, specifically La Géométrie (1637).
• Root. Finally, a non-Latin word! The word rot means “cause” or “origin”, which makes sense when you consider that since 8 = 23, its “origin” is 2. If you trace the word further back, the Proto-Indo-European root (see what I did there?) is wrad-. Thus, the Latin-based words radical and radish come from a source similar to root.

And there you have it! In the next installment, I’ll get a little more geometric and explore some words we’ve come to use for algebraic curves.