## On the Word “Abscissa”

In a (x,y) coordinate system, the abscissa is the x value. For example, the abscissa of (-3,5) is -3. (The y value is called the ordinate, which in the example would be 5.)*

This seems like a nice, simple word. Since it’s also in the As (before dictionary writers get tired) there shouldn’t be any ambiguity, issue, or controversy.

Right?

I. Ambiguous definitions

(in plane Cartesian coordinates) the x-coordinate of a point: its distance from the y-axis measured parallel to the x-axis.
— Random House Dictionary

The horizontal coordinate of a point in a plane Cartesian coordinate system obtained by measuring parallel to the x-axis.
— Merriam-Webster Dictionary

The horizontal coordinate of a point in a plane Cartesian coordinate system obtained by measuring parallel to the x-axis.
— The American Heritage Dictionary

Not sure where the confusion is yet? Try this picture from Merriam-Webster:

The definitions give the impression that the abscissa is the unsigned value of x. That is, (5,3) would give an abscissa of 5 and (-5,3) would also give an abscissa of 5. The impression is so strong to me I am left wondering if there the word has ever been used historically in such a way.

Just to nitpick, the abscissa also applies to oblique coordinates, not just Cartesian ones.

II. Even more ambiguity

But that’s not all! Try this version of the definition from MathWorld:

The x-(horizontal) coordinate of a point in a two dimensional coordinate system. Physicists and astronomers sometimes use the term to refer to the axis itself instead of the distance along it.

The definition avoids the confusion from the standard dictionaries (and even avoids the Cartesian nitpick), but tosses in an extra use: abscissa as the x-axis itself.

For example, here’s an excerpt from On the Relative Abundance of Bird Species:

For convenience, the abscissa is graduated logarithmically.

III. And other uses!

There’s also an abscissa of stability and an abscissa mapping and a spectral abscissa. None seem to have much of anything to do with the x-axis.

IV. Etymology

It’s Latin, from “linea abscissa”, meaning “a line cut off”. Going deeper, “ab-” means “off” or “away” (as in abnormal) and “scindere” means “to cut” (as in rescind).**

In other words: “a line cut off from another line”, rather like the Merriam-Webster picture above.

Nothing too strange there. The controversy is instead left for…

V. History

The Online Etymology Dictionary claims the first use of the word was 1698. Merriam-Webster claims the first use was in 1694. Pat Ballew claims 1692. Jeff Miller claims 1659. And a Professor Barney Hughes (from the Pat Ballew entry) claims 1220.

So, five different dates. Which is it?

For the later dates, the dictionaries might be aiming for the first use in English, even though the word is identical in Latin. Also, some dictionaries seem to be picking Leibniz as the first to use abscissa; however he was simply the first to popularize it.

Not quite the first

The 1659 date comes from this work by Stephano degli Angeli, entitled Miscellaneum Hyperbolicum, et Parabolicum.

Real first use of the word abscissa?

(An electronic version of the work is supposedly here at European Cultural Heritage Online but the document is on restricted access — if someone has that access, I would appreciate being able to take a look.)

However, the ultimate winner of the “earliest date” competition goes to none other than Leonardo of Pisa, aka Fibonacci:

Winner?

The use of the word isn’t from his most famous book, Liber Abaci (of the rabbits and the series that bears his name). It’s from the more obscure De Practica Geometrie.

But is his use of “abscissa” the same as ours? Here’s two excerpts:

Not to be overlooked is to show how to find the square on line eb called the residue, recisum, or abscissa. It is the difference between two lines commensurable only in their squares, such as between lines ae and ab. For example, let ae be the root of the rational number 720 and ab the number 10. Because line ae was divided into two parts at point b, the squares on lines ae and ab equal twice the product of ab by ae and the square on line eb, as was shown above. Therefore subtract twice the product of ab and ae from the squares on lines ae and ab; that is, subtract 20 times the root of 720 from 820. Now 20 roots of 720 equal the root of 288000, the number arising from the product of 400 the square of 20 and 720. The residue then is 820 less the root of 288000.

Whence if we take ec from ef, 12 less the root of 72 remains for line cf. (This is called the abscissa or recise or apotame since it consists of a number less a root.)

This doesn’t read at all to me like our use of abscissa, so (for now) I’ll have to hand the laurels back to Stephano degli Angeli.

VI. Why don’t we teach it anymore?

A quick scan of the current high school texts I have available indicate none of them mention the term (or the ordinate). It seems to have died out, and been replaced by talking about the x-coordinate and the y-coordinate.

I am wondering if that is entirely a good thing.

I can understand simplifying to have one less thing to memorize, but let me give a current example of a place this may be counterproductive: my trigonometry class is currently using the unit circle to find sines and cosines.

I would like them to remember that the x value of a point on the unit circle is the cosine and the y value is the sine. However, they almost invariably mix it up, because the letters “x” and “y” hardly solid pegs. If they had “abscissa” and “ordinate” pre-memorized, it might be easier to memorize the abscissa is the cosine and the ordinate is the sine; having an actual word to reference passes through different parts of the brain.

It’s a little like a magic trick I used to do where I pre-memorized objects matching the numbers one through ten (1 = pencil, 2 = swan, 3 = high-flying bird [looking like a 3], 4 = book, 5 = fishhook, 6 = table, 7 = cliff, 8 = hourglass, 9 = whistle, 10 = bat and ball). For the trick, someone in the audience would give me a list of 10 objects they decided. When they named the objects I would associate each one with the pre-memorized object (if their first object was a tomato, I would imagine a pencil pushed through a tomato). Afterwards I was able to immediately repeat back all the objects backwards, because I could easily go backwards through my pre-memorized objects and their associations.

I did the trick once at a summer camp (using 20 objects) and everyone afterwards thought I was some sort of memory genius, even after I explained the trick.

* Something not many people know: if there’s a z value it’s called the applicate.

** Quote from an essay on spelling reform: The “c”s in scissors and scythe are there because these sixteenth century etymologists wrongly believed these words came from scindere (to cut). In fact scissors comes from the Latin cisorium (cutting instrument) and scythe isn’t even Latin – it is an Old English word.

### 10 Responses

1. Teaching kids, as a mnemonic, that “x” comes before “y” alphabetically just as “cosine” comes before “sine” alphabetically is usually all many students need to help remember which is which – it seems that the words “abscissa” and “ordinate” would just clutter their minds much more. But every student is different, so perhaps it would be nice for books to at least present it as an option.

2. I’m really not sure — I’ve never tried teaching it (the words) before.

3. Just to add: this isn’t the only case where this kind of complaint comes up. For example, f(-x) flips the input, flipping the x-coordinates over the y-axis, and -f(x) flips the output, flipping the y-coordinates over the x-axis. This is terribly confusing even to teach and usually only the visuals get us through. It might be easier to say ‘abscissa’ instead of ‘x-coordinates’ and ‘ordinate’ instead of ‘y-coordinates’ — but only if the students already know the words.

It also can get confused with the ‘alternate definition’ tossed in there — flip the abscissa over the ordinate?

4. I was surprised to find that ” Pat Ballew claims 1692″ when My note, dating back to Prof. Hughes letter of 2006 has clearly given the credit to “I have a note that credits Leibniz with the orinin of the term in 1692, but in 2006 I received a note from Professor Barney Hughes that, ‘Fibonacci used the word in our meaning several times in his book, De practica geometrie. ‘. ”

I agree with you that his use seems not to describe the current usage, and will probably amend my references to include your comments (assuming of course you don’t file an injunction in the next thirty days)…
thanks for a nice historical update.
;zg

5. Sorry, missed that you had an actual person as a reference! I was simply referencing you as the compiler (as I am with Jeff Miller).

6. It seems to me like your example of pre-memorizing 10 objects is a very different cognitive trick than pre-memorizing an uncommon word. The object-memory trick taps into visual as well as linguistic memory. Visualizing the two objects interacting not only draws connections between the two concepts in a visual and linguistic way, but also gives you a single visual image to remember that you can unpack later. (Like mental data compression, maybe.)

Abscissa and ordinate don’t have visuals, and won’t have a wealth of prior usage to connect to outside a math class. To me they seem just as arbiitrary as x and y, and harder to spell.

7. It’s not a perfect analogy, no. It’s just some people are a lot better at words than numbers, so it would anchor in a different place. x and y by themselves seem to hit more the “number” part of the brain.

Also, of course the words seem arbitrary to us — they’d only be comfortable to someone who had early exposure where the connection became ingrained.

Abscissa being hard to spell (I can’t tell you how many times I had to check myself writing this post) might be why the words died out in the first place.

8. But now in our class we have the problem of plotting x vs t, and students have a problem if they have memorized that the horizontal axis is always the x-axis!

• Good point, although if I see t as using the abscissa it’s almost always f(t) as using the ordinate, not x.

9. Part of the reason why these words have fallen out of disuse is that Algebraic Geometry as a class is not common at the college level. In my experience, it is still common with students educated in foreign (to America) countries, and only people that are students of either history or Algebraic Geometry are aware of these words.

Of course, the math class I am teaching currently has these words in it, leading me to believe that the creator is probably older or studied in another country.