Dead Reckonings

Nomography, truly a forgotten art, is the graphical representation of mathematical relationships or laws (the Greek word for law is nomos). These graphs are variously called nomograms (the term used here), nomographs, alignment charts, and abacs. This area of practical and theoretical mathematics was invented in 1880 by Philbert Maurice d’Ocagne (1862-1938) and used extensively for many years to provide engineers with fast graphical calculations of complicated formulas to a practical precision.

Along with the mathematics involved, a great deal of ingenuity went into the design of these nomograms to increase their utility as well as their precision. Many books were written on nomography and then driven out of print with the spread of computers and calculators, and it can be difficult to find these books today even in libraries. Every once in a while a nomogram appears in a modern setting, and it seems odd and strangely old-fashioned—the multi-faceted Smith Chart for transmission line calculations is still sometimes observed in the wild. The theory of nomograms “draws on every aspect of analytic, descriptive, and projective geometries, the several fields of algebra, and other mathematical fields” [Douglass].

This essay is an overview of how nomograms work and how they are constructed from scratch. Part I of this essay is concerned with straight-scale designs, Part II additionally addresses nomograms having one or more curved scales, and Part III describes how nomograms can be transformed into different shapes, the status of nomograms today, and the nomographic references I consulted.

The simplest form of nomogram is a scale such as a Fahrenheit vs. Celsius scale seen on an analog thermometer or a conversion chart. Linear spacing can be replaced with logarithmic spacing to handle conversions involving powers. Slide rules also technically qualify as nomograms but are not considered here. A slide rule is designed to provide basic arithmetic operations so it can solve a wide variety of equations with a sequence of steps, while the traditional nomogram is designed to solve a specific equation in one step. It’s interesting to note that the nomogram has outlived the slide rule.

Most of the nomograms presented here are the classic forms consisting of three or more straight or curved scales, each representing a function of a single variable appearing in an equation. A straightedge, called an index line or isopleth, is placed across these scales at known values of these variables, and the value of an unknown variable is found at the point crossed on that scale. This provides an analog means of calculating the solution of an equation involving one unknown, and for finding one variable in terms of two others it is much easier than trying to read a 3-D surface plot. We will see later that it is sometimes possible to overlay scales so the number of scale lines can be reduced.

The Geometry of Nomograms

We can design nomograms composed of straight scales by analyzing their geometric properties, and a variety of interesting nomograms can be constructed from these derivations. Certainly these seem to be the most prevalent types of nomograms.

The figure on the right shows the basic parallel scale nomogram for calculating a value f₃(w) as the sum of two functions f₁(u) and f₂(v):

          f₁(u)  +  f₂(v) =  f₃(w)

Each function plotted on a vertical scale using a corresponding scaling factor (sometimes called a scale modulus) m1, m2 or m3 that provides a conveniently sized nomogram. The spacing of the lines is shown here as a and b. Now by similar triangles, [m₁f₁(u) – m₃f₃(w)] / a =  [m₃f₃(w) – m₂f₂(v)] / b. This can be rearranged as:

          m₁f₁(u) +  (a/b) m₂f₂(v)   =  (1 + a/b) m₃f₃(w)

So to arrive at the original equation f₁(u)  +  f₂(v)  =  f₃(w), we have to cancel out all the terms involving m, a and b, which is accomplished by setting m₁ = (a/b) m₂ = (1 + a/b)m₃.  The left half of this relationship provides the relative scaling of the two outer scales and the outer parts provide the scaling of the middle scale:

          m₁/ m₂  =  a / b             m₃ = m₁m₂ / (m₁+ m₂)

Also note that the baseline does not have to be perpendicular to the scales for the similar triangle proportion to be valid. Now a = b for the case where the middle scale is located halfway between the outer scales, and in this case m₁ = m₂ and m₃ = ½m₁. For a smaller range and greater accuracy of an outer scale, we can change its scale m and move the middle line away from it and toward the other outer scale. In fact, if the unknown scale w has a very small range it can be moved outside the two other scales to widen the scale. Additions to u, v or w simply shift the scale values up or down. Multipliers of u, v and w multiply the value when drawing the scales (they are not included in the values of m in the above calculations). Subtracting a value simply reverses the up/down direction of the scale, and if two values are negative their scales can simply be swapped. The example on the right shows a parallel-scale nomogram for the equation (u–425) – 2(v–120) = w designed for ranges 530<u<590 and 120<v<180.

So this looks like a lot of work to solve a simple linear equation. But in fact plotting logarithmic rather than linear scales expands the use of parallel scale nomograms to very complicated equations! The use of logarithms allows multiplications to be represented by additions and powers to be represented by multiplications according to the following rules:

          log(cd)  =  log c  +  log d           log c^d  =  d log c

So if we have an equation such as f₁(u) x f₂(v) = f₃(w), we can replace it with

          log [f₁(u) x f₂(v)]  =  log f₃(w)

or

          log f₁(u) + log f₂(v) = log f₃(w)

and we have converted the original equation into one without multiplication of variables. And note that there is actually no need to solve symbolically for the variable (we just plot these logs on the scales), a significant advantage when we come to more complicated equations.

Let’s create a nomogram for the engineering equation (u + 0.64)^0.58(0.74v) = w as given in Douglass. We assume that the engineering ranges we are interested in are 1.0<u<3.5 and 1.0<v<2.0.

          0.58 log (u + 0.64) + log (0.74v) = log w

          0.58 log (u + 0.64) + log (0.74) + log v = log w

          0.58 log (u + 0.64) + log v = log w –  log (0.74)

We will directly plot the three components here as our u, v and w scales. To find the scaling factors we divide the final desired height of the u and v scales (say, 6 inches for both) by the ranges (maximum – minimum) of u and v:

          m₁ =  6 / [0.58 log (3.5 + 0.64) – 0.58 log (1 + 0.64) ]  =  25.72

          m₂ =  6 / [log 2.0 – log 1.0]  =  19.93

          m₃ = m₁m₂ / (m₁+ m₂)  =  11.23

Let’s set the width of the chart to 3 inches:

          a / b  =  m₁ / m₂  = 1.29    so  a = 1.29b

          a + b  =  3   so  1.29b + b  =  3  yielding  b = 1.31 inches and a = 1.69 inches

We draw the u-scale on the left marked off from u = 1.0 to u = 3.5.  To do this we mark a baseline value of 1.0 and place tick marks spaced out as 25.72 [0.58 log (u + 0.64) – 0.58 log (1.0 – 0.64)] which will result in a 6 inch high line. Then 3 inches to the right of it we draw the v-scale with a baseline value of 1.0 and tick marks spaced out as 19.93 (log v – log 1). Finally, 1.69 inches to the right of the u-scale we draw the w-scale with a baseline of (1.0 + 0.64)^0.58(0.74)(1) = 0.98 and tick marks spaced out as 11.23 (log w –  log 0.74). And we arrive at the nomogram on the right, where a straightedge connecting values of u and v crosses the middle scale at the correct solution for w, and in fact any two of the variables will generate the third. Flexibility in arranging terms of the equation into different scales provides a means of optimizing the ranges and accuracies of the nomogram. A larger scale and finer tick marks can produce a quite accurate parallel scale nomogram that is deceptively simple in appearance, and one that can be manufactured and re-used indefinitely for this engineering equation.

It is also possible to create a circular nomogram to solve a 3-variable equation. Details on doing this from geometrical derivations are given in Douglass.

N or Z Charts

A nomogram like that shown in the figure on the right is called an “N Chart” or more commonly a “Z Chart” because of its shape. The slanting middle scale joins the baseline values of the two outer scales (which are now plotted in opposition). The middle line can slant in either direction by flipping the diagram, and it can be just a partial section anchored at one end or floating in the middle if the entire scale isn’t needed in the problem, thus appearing, as Douglass puts it,  “rather more spectacular” to the casual observer. A Z chart can be used to solve a 3-variable equation involving a division:

          f₃(w) = f₁(u) / f₂(v)

By similar triangles, m₁f₁(u) / m₂f₂(v) =  Z / [L – Z]. Substituting f₃(w) for f₁(u) / f₂(v) and rearranging terms yields the distance along Z for tick marks corresponding to f₃(w):

          Z =  L f₃(w) /  [(m₂/m₁) + f₃(w)]

The f₃(w) scale does not have a uniform scaling factor m₃ as before. We could have used a parallel scale chart with logarithmic scales to plot this division, but the Z chart performs this with linear scales for u and v and it was once a real chore to calculate logarithms. But further, the linear scales of the Z Chart are much more suitable for combining a division with an addition or subtraction than compound parallel scales with their logarithmic scales. And of course if the scale for the unknown variable is an outside one, we have a Z chart for multiplication.

An example of a Z chart is shown here for the equation Q² = (8R+4) / (P-3). To create this, the desired height of the nomogram and the ranges of P and R provide their scaling factors m₁ and m₂ as done earlier. The desired width of the chart along with this height defines the length L needed for the Q-scale (L² = W² + H²). The tick marks for Q are located a distance from the end calculated from the formula for Z above, where f₃(w) is replaced with Q².  It is also possible to slide the outer scales up or down without changing the tick mark spacing of the Z-scale as it also rotates due to its endpoints (because similar triangles still result), yielding in a nomogram with a perpendicular Z-scale as shown in an example in the second part of this essay.

Proportional Charts

The proportional chart solves an equation in four unknowns of the type

          f₁(u) / f₂(v)  =   f₃(w) / f₄(t)

If we take our Z chart diagram and a second isopleth that intersects the Z line at the same point as the first, we have by similar triangles:

          m₁f₁(u) / m₂f₂(v)  =  m₃f₃(w) / m₄f₄(t)

which matches our equation above if we choose the scaling of the outer scales such that

          m₁ / m₂ = m₃ / m₄

We then overlay two variables on each outer scale with this ratio of scaling factors, as shown in the nomogram to the right from Josephs for the approximate pitch of flange rivets in a plate girder, where p is the rivet pitch in inches, R is the rivet value in lbs, h is the effective depth of the girder in inches, and V is the total vertical shear in lbs: p = Rh/V.

Another type of proportional chart uses crossed lines within a boxed area, as shown below. Again, the scaling factors for the four variables are given by m₁ / m₂ = m₃ / m₄ where these are related as before to the u, v, w and t scales, respectively. (Actually, similar triangles still exist and the ratios still hold for any parallelogram, not just a rectangle.)

But there are other types of proportional charts as shown below. In the ones labeled Type 3 an isopleth is drawn between two scale variables, then moved parallel until it spans the third variable value and the fourth unknown variable. The flange rivet example done in this manner is shown here. In the Type 4 nomogram the second isopleth is drawn perpendicular rather than parallel to the first one; it’s actually easier to draw a perpendicular than a parallel line if you have a drafting square or even a rectangular sheet of paper.

Concurrent Scale Charts

The concurrent chart solves an equation of the type

          1/f₁(u)  +  1/f₂(v) =  1/f₃(w)

The effective resistance of two parallel resistors is given by this equation, and a concurrent scale nomogram for this is shown on the right.

The derivation is somewhat involved, but in the end the scaling factors m must meet the following conditions:

          m₁ =  m₂  =  m₃ / (2 cos A)

where A is the angle between the u-scale and the v-scale, and also the angle between the v-scale and the w-scale. The scaling factor m₃ corresponds to the w-scale. The zeros of the scales must meet at the vertex. If the angle A is chosen to be 60°, then 2 cos A = 1 and the three scaling factors are identical, as is the case in this figure.

To solve the 4-variable equation 1/f₁(u)  +  1/f₂(v)  +  1/f₄(t)  =  1/f₃(w), the equation is first re-arranged as 1/f₁(u)  +  1/f₂(v)  =  1/f₃(w)  –  1/f₄(t). Then the two halves are set equal to an intermediate value f(q). A compound concurrent chart is then created in a similar way to other compound charts as shown in this figure (here A is chosen to be less than 60°).

4-Variable Charts

A 4-variable equation with one unknown can be represented as a combination of two separate charts of any type.  The first step is to break the equation into two parts in three variables that are equal to one another. For f₁(u)  +  f₂(v) +  f₃(w) = f₄(t) and t unknown, we can re-arrange the equation into f₁(u)  +  f₂(v) = f₄(t) –  f₃(w) and create a new variable k to equal this sum. Then a blank scale for k is created such that a parallel scale nomogram for f₁(u)  +  f₂(v) = k marks a pivot point on the k-scale, then a second straightedge alignment from this point is used for a parallel-scale nomogram for f₄(t) –  f₃(w) = k to find f₄(t). The scaling for u, v and w and the position chosen for the k-scale can be optimized to minimize errors at the pivot point for small errors in the straightedge alignment. The figure on the right shows a compound parallel scale nomogram. Below are examples from Levens of compound nomograms of Z charts and concurrent and proportional charts. A key often provides instructions on the use of a compound nomogram as shown in the first figure. Of course, this concept can be extended to equations with additional variables, where color coding would be helpful.

Curved Scale Charts

It is possible to geometrically derive relationships for nomograms that have one or more curved scales, but the design of these more complicated nomograms is so much easier using determinants. Designing nomograms with determinants is the subject of Part II of this essay.

                                                                                                                           >>> Go to Part II of this essay

Printer-friendly PDF file of Nomography Parts I-III.

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