8.28 The Aver-O-Matic
Originally posted: 2024 Jul 09
A reader of Following the Rules reached out recently to inform me of a slide rule that his father had invented and sold in the 1960s. The Aver-O-Matic, by the Mims Company, was introduced by Powell O. Mims of Wheat Ridge, Colorado, in 1961. A specimen of the product was generously contributed by his son, Marc, and is shown in the figure below.
At first glance, this circular slide rule might appear to be a special “proportions” rule, with typical logarithmic scales that can be used to compute a basic ratio. And, indeed, this was one of the intended purposes of the rule, though with a slightly different take on the process. However, upon closer inspection, one can easily see that there is more going on here, as there are scales which include zero (not easy to include on a standard log scale) and other columns of numbers, located below the main scales at the top, which can be viewed through a window. And as the device has a system of plastic sheets of various opacity (one with a window), a cursor, and a strategically placed metal rivet, there is obviously a special procedure that is expected to be performed.
As the name might seem to imply, the primary intended use of this device is to perform an arithmetic mean, or average, of a collection of numbers. Marc Mims told me that his grandfather had an interest in greyhound racing, and according to Marc, “To place an informed bet, he liked to calculate the recent average times of each dog in a race. But that was hard work with pencil and paper. So, my father came up with an idea for a mechanical calculator to make that job fast and efficient.” And thus the idea for the Aver-O-Matic was born. Powell designed and enhanced his system and soon applied for a patent in 1961, when he was 26 years old. The patent, No. 3,095,144, was granted in 1963, and half assigned to Oscar B Mims – Powell’s father.
8.28.1 Using the Aver-O-Matic
The Aver-O-Matic (AoM) has two basic functions:
- Determine the average of \(N\) numbers, where each number is in the range from 0 to 100, for up to 10 numbers.
- Determine the percentage fraction of positive events \(c\) out of \(t\) trials, given the number of negative events \(m = t-c.\)
The first function is performed by a dedicated set of linear scales (linear in angle, that is), while the second function uses logarithmic scales to form an appropriate ratio.
8.28.1.1 Finding Averages
The first function to be performed – finding an average – requires a series of addition operations, which a standard slide rule is not designed to do. That is, we want to perform a summation of \(N\) values of the variable \(x\), and then divide the result by \(N\):
\[ \langle x \rangle = \frac{x_1 + x_2 + \ldots x_N }{N}. \]
The Aver-O-Matic utilizes a unique mechanical design following the geometry of some circular slide rules, including a rotating disc and a cursor. For the AoM, the values of the variable \(x\) are assumed to be in the range \(0\le x\le 100,\) and the number of observations \(N \le 10.\) As a circular device, the AoM can add distances by adding up angles, which can be designated along the arc of a circle. Scale A on the rule runs from 0 to 100, through a total angle of 70 degrees. Beginning with all scales lined up at zero, the user moves the cursor from its original position to the value of \(x\) on A. Then, by grasping both the the cursor and the largest outside disc, these are moved together to reset the cursor back to its original angular position, while thus registering the present status of the sum. (The sum is proportional to the total angle that the disc has been turned.) By repeating this \(N\) times, the angles corresponding to each number being summed are themselves added together.
One then needs to have this final sum divided by the number of samples, \(N.\) In the window of the outer plastic sheet, one sees a series of arcs which are labeled “1”, “2”, … “5” on the left side of the window. Each of these represent the number on an imaginary extended A scale, but divided by \(N\) = 1, 2, …, 5, respectively.
For example, suppose all 5 numbers being averaged are themselves “100”. Then, adding up 100 five times and dividing by five, the average is 100. Meanwhile, after performing our registration of the five numbers through the procedure above, the outside disc would be rotated by 350 degrees; and when we look at the smallest radius circle inside the window, labeled “5”, we see the result “100” sitting just 10 degrees away from the “0”. (See image above.)
The other three scales identified on the left side of the window are just scaled accordingly. For instance, on scale A (\(N\) = 1), the average of 100 is 100. But below 100 on A we find 50 = 100/2 on scale “2” (\(N\) = 2); below this is 33.3 on “3”, 25 on “4” and 20 on “5”. All other numbers on these scales are just divided linearly by angle from the zero position.
So, the A scale allows for up to 5 values to be averaged using the cursor/disc technique, with the result being found on the appropriate scale in the lower window. But by using the B scale (in red), which is half the angular expanse of scale A, the same procedure can be performed, but the total angular sums will be half as long. If we divide by values of \(N\) that are twice what was being used previously, then the arcs used to find final answers for values of \(N\) = 6, 8, and 10 are the same as the arcs used for \(N\) = 3, 4, and 5, respectively. Then, by adding new intermediate arcs of numbers for \(N\) = 7 and 9 we get a complete set of scales from which results can be read for any \(N\le 10.\)
8.28.1.2 Score Calculations
While the original motivation for the Aver-O-Matic might have been driven by Mims’ father’s interest in Greyhound racing, the final version of the AoM was likely marketed as a tool for the teaching profession. Teachers often need to find the average score for a series of exams, as an example, and the number of exams in a semester is generally fairly small. And, of course, the scoring of the exams themselves is also a common job. This second type of calculation is what constitutes the other function of the AoM. Its use in the education profession is also suggested by the wording at the bottom of the instructions that came with the slide rule.
Suppose an exam consisting of \(t\) questions is scored and found to have \(c\) correct answers. The score for the exam, in percent, can be defined simply as \(S = 100\times c/t.\) It was typical practice for a teacher to “mark” an exam and then count the number of missed questions. (And most teachers would hope that the typical student would get fewer wrong answers than right answers.) To find the final score, the teacher would then need to subtract the number missed (\(m\), say) from the total and find a ratio:
\[ S = 100\times \frac{c}{t} ~~=~~ 100\; \frac{t-m}{t} ~~=~~ 100\; \left(1- \frac{m}{t}\right). \]
Counting \(m\) and computing a score directly would be more desirable than having to subtract \(m\) from \(t\) before computing a score from a ratio. Our inventor of the Aver-O-Matic must have come to this realization as well.
The major scales labeled C and D on the AoM are used for this computation. The scale labeled C, on the “stock” of the slide rule, runs from right to left and is clearly logarithmic. The length of the C scale from 1 to 100 is about 9 inches, or a decade length of about 4.5 inches. The outer scale, on the “slide”, is labeled D. It also appears to have a logarithmic layout, but begins at 0 on the left and runs toward 100 on the right, with the spacing of numbers getting larger. In reality, the D scale is the scale for \(m\). Closer inspection shows that the scale ends at 99. If a standard log scale runs from \(x\) = 1 to 100, say, from right to left, then the logarithmic numbering on this D scale represents \(100-x\) and runs from 99 to 0, from right to left. In other words, the two scales C and D are both equal-length two-decade logarithmic scales running in the same direction, where the number written on D is equal to 100 minus the corresponding number on C when the two scales are aligned.
To use our two scales to do a division, it is convenient to look at writing our expression for \(S\) in terms of a ratio. For example,
\[ S= 100\;\left(1-\frac{m}{t}\right) ~~~~~~\longrightarrow ~~~~~~ 1-\left(\frac{S}{100}\right) = \frac{m}{t} ~~~~~~\longrightarrow ~~~~~~ \frac{100-S}{100} = \frac{m}{t}. \]
As scale D logarithmically represents the quantity \(100-x\), were \(x\) is the number shown on D, then we find that scales C and D can provide the following set of ratios:
\[ {\rm set} ~~\frac{C}{D} ~~~~~=~~~~~ \frac{m}{100-S} = \frac{t}{100-0} ~. \]
So to compute an exam score, \(t\) on C is set opposite \(0\) on D. Then opposite \(m\) on C will be \(S\) on D. As an example, the setting shown in the following image is for a total number of questions \(t\) = 75, where the number missed is \(m\) = 18, yielding a score of \(S\) = 76 (percent). With a single setting of the slide, the cursor can now be moved to \(m\) for each student’s exam and that student’s score \(S\) can be read directly and recorded, a great time saver for the instructor.
The clever use of a “\(100-x\)” logarithmic scale to compute a test score, and the development of a set of “average” scales using a circular layout makes the Aver-O-Matic a very unique calculating tool with potentially many specialized and every-day uses.
8.28.2 About Powell Mims
Having spent time grading exams and computing final grades, I can appreciate the effort that could have been saved through the use of an Aver-O-Matic in the 1960s. While the device looks like a rather simple circular slide rule at first glance, the thought that had to go into developing the scales just described and laying them out through a multi-disc apparatus with window and cursor was not a simple feat, and was clearly deserving of a patent. Powell manufactured the slide rules himself on an offset printing press.
But according to Marc, such displays of ingenuity were typical of his father. Powell Mims was born in 1935 and lived most of his life in Colorado, graduating high school in the town of Gypsum. After two years of college at the Colorado School of Mines, he left to join the US Navy where he served as a pilot during peacetime, soon after the Korean War. After his service, he moved to Sheridan, Wyoming, to operate a service station there that was owned by his father. This was also where he met and married his wife, Martha. After a short while, the young couple moved back to Colorado. It was about this time when Powell began working on the Aver-O-Matic slide rule design. By April of 1961, Marc – the first of four children – was born. One month later, the AoM patent was filed.
Powell worked for Martin Marietta as a technician, as well as for the Coors Brewing Company in Golden, Colorado, in their porcelain plant on projects funded by the US Department of Defense. He also was a technician on Coors’ first aluminum can project, which brought aluminum packaging to the world stage starting in 1959. Following a stint in California for a few years, where he worked as a surveyor for the Pacific Gas and Electric Company (PG&E), the family returned to Colorado and Powell finished his degree at the School of Mines, graduating with a B.S. in Mathematics in 1971. Working through college, he ran his own service business based upon his self-taught knowledge of electronics, repairing automated, unattended self-service gas pumps. A Denver area owner of several gas stations asked Powell if he could develop a solid-state controller to eliminate a problem with the stepper relays being used at the time. So, he did. And, he patented the product soon after college graduation.
After selling the patent rights to Tokheim International, Powell used his proceeds to start his own self-service gas station and convenience store near Hotchkiss, Colorado. Soon the business grew into a good number of stations throughout western Colorado, which led to a gasoline distribution arm of the company as well. Powell saw the need for his employees to be able to read gas pump purchase totals from inside the store, so he invented and produced an electronic remote readout system to do just that.
| As we can see, this is just how Powell Mims operated. He created solutions to practical problems all of his life, being interested more in solving problems than marketing solutions, and there are further interesting examples that could be discussed. According to Marc, “I have hundreds of his notes and drawings. For his whole adult life, he carried pencils, paper, and a pocket slide rule in his shirt pocket. If he was waiting to pick my mom up after work, in or the parking lot at the grocery store, or at home at the kitchen table, he was drawing, working math problems, inventing.” |  |
Powell Mims’ health failed him early. He died at 68 in 2003 following over two decades of heart-related issues. Said Marc about the last years of his dad’s life, “Dad spent his days reading, making notes and drawings, making cardboard mock-ups of ideas, and cooking. For him, cooking was not only part of his contribution to his and my mother’s household, it was an ever-expanding chemistry experiment.”
And the printing press remained in Powell’s possession until his death, never having been used since the production of the Aver-O-Matic.
As for addressing the needs of Marc Mims’ grandfather, the earliest version of Powell’s slide rule was for the Greyhound races, and evidently had a very limited distribution. According to Marc, the early “Greyhound” version of the Aver-O-Matic made by his father while in Sheridan, Wyoming, is still in Marc’s possession. Shown in the image below, it may be the only remaining specimen.
