A nomogram is a graphical device for solving equations by reading values from graduated scales. In an alignment nomogram, a straight line intersects scales whose readings satisfy a specified equation. Brodetsky (1920, pp. xi-xii) attributes
the development of alignment nomograms to d'Ocagne and also describes earlier contributions
by Lalanne and Massau.
with realcoefficients and uses two parallel vertical scales
and a graduated hyperbola (Brodetsky 1920, pp. 75-78;
Trott 2004, Ch. 1, Exercise 19 and its solution). Let be the horizontal separation of the scales. Mark
at the point on the left scale and at on the right scale. The line
through these points is
by , for . Substitution into the line equation
gives .
Therefore, the labels at its intersections with
the curve give the real roots
other than .
The excluded root occurs iff and must be checked separately. A repeated root
other than
makes the linetangent to
the curve.
The illustration uses ,
, and . The blue line meets the curve
at the labels
and . Only finite
portions of the hyperbola are shown, so the plotted
ranges must be extended to read roots outside them.
and 
uses two parallel vertical scales
and a graduated hyperbola (Brodetsky 1920, pp. 75-78;
Trott 2004, Ch. 1, Exercise 19 and its solution). Let 
be the horizontal separation of the scales. Mark
at the point 
on the left scale and 
at 
on the right scale. The line
through these points is
, for 
. Substitution into the line equation
gives 
.
Therefore, the labels at its intersections with
the curve give the real roots
other than 
.
The excluded root 
occurs iff 
and must be checked separately. A repeated root
other than 
makes the line tangent to
the curve.
,
, and 
. The blue line meets the curve
at the labels 
and 
. Only finite
portions of the hyperbola are shown, so the plotted
ranges must be extended to read roots outside them.
