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Terms of the Cartesian Coordinates

Date: 12/29/2007 at 00:54:52
From: Monte
Subject: The name for the third coordinate in a Cartesian system

In looking for info on abscissa and ordinate I came across a 1998 
discussion of the name of the third coordinate. The conclusion 
was..., well, non-conclusive, methinks. However, on another website
(http://planetmath.org/encyclopedia/Abscissa.html) I found a reference
to the word 'applicate'. My question is, is this correct? A new 
discovery through the advent of the Internet? An urban legend?

My confusion comes with the plethora of information available on the 
Internet. Sometimes, when an answer is found, I can accept it 
unquestioningly. Other times, I'm just not sure. The 'applicate' 
answer looks good, but I haven't found anything else (yet) to 
corroborate the information.

I've conducted searches on many engines for 'abscissa' and 
'ordinate','Cartesian Coordinates' and a few other, more obscure

Date: 12/29/2007 at 23:25:36
From: Doctor Peterson
Subject: Re: The name for the third coordinate in a Cartesian system

Hi, Monte.

Here's what your link to PlanetMath says:

  The Cartesian coordinates of a point in R^3 for determining its
  place in three-dimensional space are the three real numbers x, y
  and z, which are called

    * x-coordinate or abscissa,
    * y-coordinate or ordinate,
    * z-coordinate or applicate.

  The last name “applicate” is rare in English, but its equivalents
  in continental European languages, as “die Applikate” in German
  and “aplikaat” in Estonian, are more known.

This very clearly states the truth: one hardly ever hears the term
used in English (I never have), but it is used in other languages, and
if you want a term in English this makes more sense than any other. 

I happen to have a German-English science/math dictionary, and it
lists "Applikate" as meaning "z-coordinate" (without offering any
other name for it in English); my (old) English math dictionary does
not list "applicate", nor do any of the usual sources.

Of course, I seldom have any reason to use the words abscissa and
ordinate, anyway, so there's little pressure to invent such a word.
That's a good reason to accept one ready-made from another language!

Thanks for contributing this!

- Doctor Peterson, The Math Forum

Date: 12/01/98 at 12:22:38
From: William
Subject: Cartesian Coordinate Names  

I have searched and searched for the third name of the Cartesian 
Coordinates. I know that the first two are the abcissa and the 
ordinate. I did find one name for it here at Dr. Math at


You labeled that third "z" coordinate the "altitude." My teacher did 
not accept this as the name, citing that it was simply a specific 
application name. What is this third name? Is it the altitude, or is 
there another name?

Date: 12/04/98 at 19:46:29
From: Doctor Mike
Subject: Re: Cartesian Coordinate Names  

Hi William,   

Yours is a very interesting question. I did some looking around. I saw 
the place where altitude was suggested. That, and similar words like 
"vertical," "depth," and "hill" show up in various documents I found. I 
agree that they are all too specific to be the general name.    
One way of being more general is to observe that the z coordinate is
very often given a value which is a function of x and y, or in the
customary mathematical notation z = f(x,y). This may be an altitude or
depth function, or temperature, or whatever. The set of points of the
form [x, y, f(x,y)] is often viewed as a surface. If someone were to 
try to make up a name for the z-axis, it might be productive to use the 
word function, or transformation, or surface, or something similar. 
My thoughts and investigations next turned to the history of how the 
two words abscissa and ordinate originated. In Jeff Miller's page on 
Earliest Known Uses of Some of the Words of Mathematics, at:

I found the following:  

   ABSCISSA was coined by Gottfried Wilhelm Leibniz (1646-1716) in 
   1692. According to Cajori (1919, page 175): The words "abscissa" 
   and "ordinate" were not used by Descartes. The technical use of 
   "abscissa" is observed in the eighteenth century by C. Wolf and 
   others. In the more general sense of a "distance" it was used 
   earlier by B. Cavalieri in his Indivisibles, by Stefano degli 
   Angeli (1623-1697), a professor of mathematics in Rome, and by 
   others. Abscissa appears in a 1692 manuscript by Sir Isaac Newton.  
This leads me to completely re-think the basic meaning of your 
question. The words abscissa and ordinate are words coined several 
hundred years ago, and I think we might just have to accept that Mr. 
Leibniz did not come up with a third word for the third direction.  
Based on your and my difficulties in finding such a word, it certainly 
is not in common use. 
If your teacher believes s/he knows such an historically accurate word, 
by all means write back and tell us what it is. If the teacher is just 
wondering what your search will turn up, my opinion is that it does not 
exist, but a clever mathematician/wordsmith with some familiarity with 
Latin could make a case for something coined in the 20th century. If 
that is your plan, better hurry up because there are only about 25 
months left!
I hope this helps some.
- Doctor Mike, The Math Forum
Associated Topics:
High School History/Biography

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