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
scale.

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
http://www.journals.uchicago.edu/Isis/journal/demo/v000n000/000000/000000.text.html

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
http://www.hbroussais.fr/Broussais/InforMed/Nomogram/Nomo.html

Nomograms for the determination of the body surface area
http://www.bioscience.org/atlases/clinical/nomogram/nomogram.htm

Body surface area
http://www.smm.org/heart/lessons/nomogram_adult.htm

Fagan nomogram (probabaility of a disease)
http://www.cmh.edu/stats/definitions/fagan.htm

Percent Body Fat
http://www.insitefitness.com.au/lessons/fitness%20testing/Anthropometry/nomogram.html

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)
http://www.bdbiosciences.com/discovery_labware/technical_resources/labtools/p43.pdf

NOMOGRAPH - MAX OPERATING CURRENT VS. WIRE GAUGE
http://www.kepcopower.com/nomomax.htm

Sprayer Calibration Nomograph (PDF file)
http://www.oznet.ksu.edu/library/ageng2/MF415.PDF

Grounding Nomograph (PDF file)
http://www.dranetz-bmi.com/pdf/nomograph.pdf

Color Temperature Nomograph
http://www.mic-d.com/java/nomograph/

Nomograph for Calculation of Boiling Points Under Vacuum
http://www.rhodium.ws/chemistry/equipment/nomograph.html

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

Engines of our Ingenuity: Nomograms
http://www.uh.edu/engines/epi1664.htm

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
http://www.ece.rochester.edu:8080/~jones/NOMO_FILES/nomoMain.html

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

Signed Number Nomograph
http://www.mathnstuff.com/papers/nomogrf/nomo.htm

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:

A-------X---------------------------------B
:\
: \
: \
C-------V---Z-----------------------------D
: \
: \
: \
: \
: \
: \
: \
E-------U-----------Y---------------------F

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)

and

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:

A-------------------X---------------------B
\ /
\ /
\ /
\ /
Z
/ \
/ \
/ \
/ \
/ \
/ \
/ \
/ \
D-----Y-----------------------------------C

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*AC - AX*AZ

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
introduced.

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
Nomograms
http://www.asp.ucar.edu/colloquium/1992/notes/part1/node102.html

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





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,
Ed




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.




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.





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

http://www.projectrho.com/nomogram/

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,

Kirk




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!