Gerber's Equameter

Elaborating on Christopher Staecker's description with a simulation.

The Gerber Variable Scale with its elastic scale is well known among mathematical instrument collectors.[1] 

Figure 1: Gerber Variable Scale.[2] 

Another instrument from the Gerber Scientific Instrument Company that includes an elastic scale as well was sold at the IM2022 auction: a Gerber GraphAnalogue. This instrument also contains a number of fixed scales: logarithmic, reciprocal, trigonomic scales and scales for powers and probabilities. It is used for analog solving a wide variety of technical problems that are discussed in detail in the instruction manual.[3] 


Figure 2: Gerber GraphAnalogue.[4] 

The elastic scale also occurs in an even more complicated instrument: the Equameter.[5]  The Equameter is used to determine a polynomial that describes a given graph. Instead of a polynomial, a Fourier series can also be determined.

Heinz Joseph Gerber's 1956 patent describes a complex device,[6]  the size of a drawing board (Figure 3)

Figure 3: Gerber Equameter[7] 

Figure 4: Elements of Gerber Equameter, US patent 2,918,213. The Variable Scale is highlighted.

Chris Staecker, a mathematician from Fairfield University, has published a simplified version of the Equameter and described it in a YouTube video.[7]  Chris's other YouTube videos are also recommended.

Chris's website contains PDF files with which you can make a simplified Equameter yourself. I have created a web page that simulates Chris' Equameter.[8] ]

Figure 5: Log Curve (a) and Scale (b) of the simplified Gerber Equameter[7]

The log curve in figure 5 has y as the coordinate and x as the ordinate.

Let's illustrate how the Equameter works with a simple example.

Take an unknown curve (Figure 6):

Figure 6: Unknown curve[9]

Choose which polynomial (or order of Fourier series) should be fitted to it. Of course we choose a parabola.
 Take the corresponding worksheet (Figure 7)

How is this worksheet structured? The curve must be measured at three points: x=0, x=1 and x=2. Higher degree polynomials require more points. The Gerber Variable Scale can be used to find the three equidistant x-values on a scale associated with the curve.

For the parabola we have three equations:
  y(x=0) = y0 = a + b*0 + c*0*0
   y(x=1) = y1 = a + b*1 + c*1*1
   y(x=2) = y2 = a + b*2 + c*2*2

Figure 7: Parabola worksheet

so
   a = y0
   b = (−3y0 + 4y1 − y2)/2 = −1.5y0 + 2y1 – 0.5y2
  c = (y0 − 2y1 + y2)/2= 0.5y0 − y1 + 0.5y2

If we now change the numbers in the factors by letters (A=1, B=2, ...) we get

  a = A y0
   b = −A.E y0 + B y1 – .E y2
  c = .E y0 – A y1 + _.E y2

And this exactly matches the capital letters in columns a, b and c in the table.

Figure 8: Log curve intersects our curve at x=0

We now place the Log curve from Figure 5 over our curve and shift it horizontally so that the Log curve intersects our curve at x=0. This shift is proportional to − log(y0)  See Figure 8.

Then we place the Scale from Figure 5 over the entire stack so that the brown hairline falls over the intersection.


Figure 9: Hairline of the Scale at x=0

Read the values on the Scale that correspond to the capital letters in the row for x=0 in the worksheet, so A:1.5 for columns a and b, and E:7.5 for columns b and c. The values found are entered in the x=0 row of the worksheet.

The slide rule collector will notice that for A we add the values of log(y0) and log(1), and for E we add the values of log(y0)  and log(5) and thus the products y0*1 and y0*5  are calculated.
This is exactly what we need in the solution of the system of equations. And just like with a slide rule, we have to pay close attention to the position of the decimal point.

Figure 10: Worksheet with row for x=0 completed.

In the next step, the Logcurve is shifted so that our curve is cut at x=1. The hairline of the Scale is then shifted over that point again. The values at B and A are read and written in the row for x=1. 

This is repeated for x=2, where we only need to read E.

We ultimately add up the values found column by column and then get the required values for a, b and c.

Gerber could have added an Addiator to the Equameter for this addition, but one can assume that someone who has already come this far can add these numbers by heart.

According to the Equameter, the curve is described by the parabola y = 1.5 + 0.6*x + 0.49*x*x

Figure 11: Completed worksheet

For polynomials of a higher degree and for Fourier series, more points must be measured and many more “capital letters” must be read. Inaccuracies in the setting of the Log curve and hairline and in the reading of the Scale are added up and will lead to a less than optimal fit of the polynomial to the curve. Gerber therefore cut the Log curve into pieces in his design of the Equameter, see the blue curves in the left part of Figure 4. This meant that the Scale also had to be cut into pieces, and that reading the capital letters was much more complex. .

The Gerber Scientific Instrument Company put the Equameter into production, and several models were made.[10]  
The Model GEQ-450 sold in 1959 for $550.[11]
It is not known how many Equameters were made and how many still exist.

Acknowledgements

Thanks to Chris Staecker for drawing attention to the Equameter, and for discussions about the simulation.

References

A Dutch version of this paper is published in MIR 89, November 2022