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Date: 06/12/2003 at 14:20:01
From: Ed
Subject: Nomographs

What are nomographs used for?  How does one use them?  Are they still in 
use?  Last but not least, how are they constructed? 

Date: 06/13/2003 at 17:09:44
From: Doctor Peterson
Subject: Re: Nomographs

Hi, Ed.

I have been interested in these for a long time, but have never found 
a good source of information on their design and use. Here is the 
American Heritage Dictionary's definition of "nomograph" (with the 
alternative form "nomogram"):

  1. A graph consisting of three coplanar curves, each graduated
  for a different variable so that a straight line cutting all
  three curves intersects the related values of each variable.
  2. A chart representing numerical relationships

That is, they have three (or more) scales (usually straight lines, but 
sometimes curved) arranged so that, if you know the values of two of 
the variables, placing a straightedge across those values on their 
scales yields the corresponding value of the third variable on its 

Here is an interesting history I found:

   Blood, Dirt, and Nomograms. A Particular History of Graphs, by 
   Thomas L. Hankins, History of Science Society Distinguished Lecture

Among other things, it says

  The nomogram has several great advantages. First, it allows for
  great economy of expression: the diagram is much less cluttered
  than a graph with Cartesian coordinates. Second, the same nomogram
  can express all the parameters of a formula and can handle many
  more variables. And third, one reads off the values on a nomogram
  by stretching a thread or laying a straightedge across the scales
  and noting where it intercepts a third scale, thereby greatly
  increasing the speed and accuracy of reading the graph. There is
  no need to follow a line from the abcissa to the curve to the
  ordinate, as is the case with Cartesian coordinates. Wherever
  speed is more important than precision nomograms have an
  advantage. For us, in the age of supersonic missiles and
  electronic computation, it is hard to believe that gunners in
  World War I used nomograms to direct antiaircraft fire. Nomograms
  spread quickly, first in military and civil engineering and then
  in a wide variety of scientific applications. They are very neat
  devices. I had never heard of them until last year, which is
  either a confession of personal inadequacy or a comment on the
  narrowness of our field; I am not sure which. Some of them are
  quite beautiful. As aesthetic expressions of mathematical laws
  they are the "fractals" of the early twentieth century. 

Although I think of them as old-fashioned, and for many purposes 
replaced by the computer, a search for the word "nomograph" or 
"nomogram" finds many references, including examples of nomograms that 
apparently are used currently for calculations in many fields. Here 
are just a few I turned up, that illustrate several styles of nomogram 
and where they are used. First, "nomograms":

  The number needed to treat: a clinically useful nomogram
  in its proper context 

  Nomograms for the determination of the body surface area 

  Body surface area 

  Fagan nomogram (probabaility of a disease) 

  Percent Body Fat

Interestingly, almost all of the examples I found with this name seem 
to be in medicine. But under "nomograph," I found examples from many 
other fields in engineering, technology, and agriculture:

  Relative Centrifugal Force (PDF file)


  Sprayer Calibration Nomograph (PDF file) 

  Grounding Nomograph (PDF file) 

  Color Temperature Nomograph 

  Nomograph for Calculation of Boiling Points Under Vacuum 

Here is a brief discussion of the (former) use of nomograms in 

  Engines of our Ingenuity: Nomograms 

The following page makes a nomogram to add, subtract, multiply, or 
divide two quantities with any scaling you want:

  An Interactive Graphic Calculator for creating custom nomograms 

This site shows how to make the simplest kind of nomograph for school 

  Signed Number Nomograph 

Many of the nomographs I found are of this very basic type, which 
just adds and scales; when the scales are logarithmic, as on a slide 
rule, it multiplies. These use three parallel lines:

          : \
          :  \
          :    \
          :     \
          :      \
          :       \
          :        \
          :         \
          :          \

We locate X and Y, draw line XY, and read off the location of Z. How 
is Z related to X and Y?

By similar triangles, if XV/XU = k, then VZ/UY = k, so

  CZ - AX = VZ = k*UY = k*(EY - AX)


  CZ = AX + k*EY - k*AX = k*EY + (1 - k)AX

So the location of Z indicates the sum of AX and EY, scaled by 1 - k 
and k respectively. Commonly k = 1/2 (that is, the lines are 
equidistant), so both scales are the same.

There are many more complicated configurations, some requiring curves 
(sometimes computer-generated) rather than lines, but the most 
interesting involve different arrangements of lines. For example, 
consider the case where the middle line is not parallel, but 
intersects the others:

     \               /
        \           /
           \       /
              \   /
                /   \
               /       \
              /           \
             /               \
            /                   \
           /                       \
          /                           \
         /                               \

Now the ratio of the similar triangles is no longer constant, but 
varies with the position of Z:

  CY/AX = CZ/AZ = (AC - AZ)/AZ

so that


  AZ(CY + AX) = AX * AC

  AZ = AC * AX / (AX + CY) = AC / (1 + CY/AX)

With the right markings, Z represents the quotient CY/AX.

A great variety of equations relating three variables can be 
represented geometrically; and as you see in some of the example 
links above, additional tricks (like using several alternative lines 
to represent different multipliers, and using a pivot line to 
represent an intermediate result) allow more variables can be 

I still need to find a reference (other than the old books listed on 
one of the sites I found) that really goes into detail on the 
different kinds of nomographs that were invented in their heyday. The 
following site is the best I have found so far:

  The Analysis of Observations with applications in atmospheric 
  science - William A. Cooper, NCAR Advanced Study Program

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum 

Date: 06/14/2003 at 11:31:52
From: Ed
Subject: Thank you (Nomographs)

Thank you for helping me.  This gives me a better idea of what a 
nomograph is and their various uses.  I appreciate the other Web sites 
listed.  You are a great source for all kinds of math questions.  

Much appreciated.  Be well, 

Date: 09/30/2004 at 11:18:35
From: Kirk
Subject: Supplemental Answer to Nomographs

In the nomographs question, Dr. Peterson writes:  "I still need to
find a reference (other than the old books listed on one of the sites
I found) that really goes into detail on the different kinds of 
nomographs that were invented in their heyday."

I heartily recommend "Nomography and Empirical Equations" by Lee H.
Johnson.  Yes, it's an "old" book (1978).  It's still the best 
reference on the subject I've found.

Date: 09/30/2004 at 12:46:33
From: Doctor Peterson
Subject: Re: Supplemental Answer to Nomographs

Hi, Kirk.

Thanks for writing. Unfortunately, a search of all the libraries I 
have access to doesn't turn up that book, but several others with 
dates like 1947 or 1965 (by Kulmann, Davis, Adams, Douglass, 
Levens). If you happen to know any of those older books enough to 
recommend the best of that crop, I just might be interested in doing 
some more study! Of course, what I was doing in the page you refer 
to is looking for on-line information to help people without access 
to big libraries, but a good bibliography may help some readers.

- Doctor Peterson, The Math Forum

Date: 09/30/2004 at 13:01:21
From: Kirk
Subject: Supplemental Answer to Nomographs

Ah.  Given the desire for an online source - one with more meat - I'll
direct your attention to:

Winchell Chung (the author) has a few hobbies, one of which is a
passionate love of slide-rules and other nomographs.  I think you'll
find this site shows that interest quite well.  Worth noting is HIS
bibliography on the subject, which may well overlap the books you have
already found.

And the Lee book is not one of Winch's - instead, it was suggested to
me by yet another wargame hobbyist/designer, one Tony Valle.  His game
makes outstanding use of a few nomographs for modern air combat
maneuver, which in turn meant I took his recommendation of "best
single work" to heart.

I hope this will be found useful,


Date: 09/30/2004 at 14:55:16
From: Doctor Peterson
Subject: Re: Supplemental Answer to Nomographs

Hi, Kirk.

Thanks. Since my main interest is the math behind nomographs, I was 
a little disappointed when I found the following in the site:

  DISCLAIMER: My mathematical grounding is rather shaky. Any or all
  of the information in this site may be inaccurate. This is why
  the site will concentrate on the methods of construction while
  ignoring the theory of why they work (i.e., in many cases I don't
  know why they work either). A reading list will be supplied for
  those who want to go further. I would supply links to web sites
  about advanced nomography but there don't seem to be any.

But his bibliography, as he says, looks very helpful; it does 
include several of the books available to me. And I'm sure there's 
good material on making and using nomographs on the site, as well. 
I'll be looking around, and maybe looking for suggestions to make 
for improving its mathematical content!

- Doctor Peterson, The Math Forum

Associated Topics:
High School Definitions
High School Equations, Graphs, Translations
Middle School Definitions
Middle School Graphing Equations

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