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:

mwpicture

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

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?

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?

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:

abpic2

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.

abpic3

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.

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My Favorite Math Magic Trick (as a puzzle)

I was just reading about a magic trick invented by Charles Peirce and described by Martin Gardner and I wanted to mention my own favorite in math magic tricks (also described by Martin Gardner). I’ve done it 8 times and it has received quite a reception each time. I have modified it slightly from the original to improve the presentation.

Supplies: deck of cards, calculator, envelope. Blindfold optional.

Turn your back to the audience, eyes closed. (The blindfold goes here if you have one.)

Have an audience member pick any row, column, or diagonal on the calculator.


789
456
123

Have them type those three digits in any order (so it could be for example 879, 528, or 195). Have them hit the “times” button on the calculator.

Have another audience member pick a different row, column or diagonal and type those three digits in any order, then hit equals.

Get a third audience member to then look at that number, say, 447795, and find cards from the deck that match the number. If a digit is repeated there should be multiple cards; so the number above means the audience member would find two 4s, two 7s, one 9, and one 5. Any zeros should be ignored (so for 56088 it should be one 5, one 6, and two 8s). The remaining deck of cards should be set aside.

Using the cards selected from the deck, have an audience member choose one at random and seal it in the envelope. Have the envelope handed to another person at the far corner of the room, and have the audience member now read the remaining cards to you.

You can now name the missing card and ask for the envelope to be opened. Remove the blindfold, turn, and wait for the cries of disbelief.

(No, really, you get cries of disbelief.)

So, how does the trick work?