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          A slide rule with a Wow!

The unusual slide rule of Hewley Mortimer Baines

The Baines hydrological slide rule
Click photo to enlarge

A lucky win

    I won this rule at the auction of Tom Wyman’s collection.
Tom Wyman (1927–2014) was one of the tribe elders of slide rule collecting as we know it today, and a co-founder of the Oughtred Society of which I am a proud Fellow. He had started collecting decades ago, and had ample opportunity to secure interesting rules. I had the pleasure of visiting him at his home in Palo Alto in his old age, and being shown some of the treasures he had collected. When he passed away in 2014, the Oughtred Society arranged an auction of Tom’s collection, and of the many historical items in it I was fortunate to carry home the Baines slide rule, shown below front and back.

The Baines slide rule - front
Click photo to enlarge
The Baines slide rule - back
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An amazing contraption

    The most adequate word to convey one’s first encounter with the Baines slide rule is “Wow!” (or, perhaps, “Whoa!!”). This is not exclaimed when you first see it in its folded state; then it seems like a nice old boxwood rule, at once solid and stolid. And then you pull on the little brass knobs at opposite corners of the rule, and – Whoa! – out pops a complicated pantograph structure of articulated metal rods that were hidden on the rule’s back side, and the four wooden scales spring into motion and extend this way and that. Wow!!...
    To illustrate the dynamic, here is a sequence showing a number of steps as you extend the rule.

The Baines slide rule  The Baines slide rule
The Baines slide rule  The Baines slide rule
The Baines slide rule  The Baines slide rule
The Baines slide rule  The Baines slide rule
Click a photo to enlarge
    In case you wonder how far the rule can extend itself, the answer is, VERY far: here it is pulled out all the way.
The Baines slide rule  The Baines slide rule
Click a photo to enlarge
    And of course, it can do the same trick in the other direction as well:
The Baines slide rule  The Baines slide rule
Click a photo to enlarge
    This rule implements the principle of forcing a dependent motion of the slides. It has no stator, or stock – the fixed part that the slide usually move in. Instead, it has four identically sized slides that all move relative to each other – under the constraints imposed by the pantograph in the back.
    You can see in the photo to the right how the four slides are kept sliding next to each other, using a variant of a sliding dovetail joint. The Baines slide rule - slide articulation
Click photo to enlarge
    A look at the edge of the rule tells us that it is the Baines slide rule No. 1, awarded the British patent number 25109/03 – the 03 stands for the year 1903. We also see the scratched imprint of one Elton, presumably the device’s owner, who also put his name on the back of one of the slides; and a number, 74, whose meaning is unclear.
The Baines slide rule - edge view
Click photo to enlarge

What it does

    The purpose of this unusual device is to do calculations involving the flow of water in iron pipes. This class of problems is complicated by friction in the water at the inner surface of the pipe, which causes a “head loss”. “Head” here means water pressure, and is expressed in length units – it is defined as the height of a column of water that would exert that pressure at its base. Thus, the energy loss to friction causes the pressure to drop, and the head loss measures this drop per length of pipe – in this slide rule it is specified in feet (of head) lost per 1000 feet (of pipe length).
    These problems are solved by using approximate formulas; well-known examples are the formula derived by Manning, and the one by Hazen and Williams (the basis for the Kally slide rule I describe here). Baines, however, uses Flamant’s formula for cast iron pipes, as stated on the topmost slide. The Flamant in question is Alfred Aimé Flamant (1839 –1915), a French hydrologist.

    The formula used, as shown on the rule, is

     V=76.28*d5/7*S4/7 ,


  • V = Velocity of the water in feet per second
  • S = Loss of head (due to friction) in feet per 1000 feet
  • d = Diameter of the pipe in inches

    These parameters are represented in the top three scales on the slide rule; a fourth parameter, shown on the last scale, is

  • Q = Discharge of water, in either gallons per minute or cubic feet per second

    While the Flamant formula does not include the discharge Q, the slide rule does calculate it from the velocity V and cross-sectional area A of the pipe: Q=A*V ,
where A is the area of the cross section of the pipe (itself derived from d).
The discharge in gallons per minute is proportional to that in cubic feet per second, and this conversion is done directly by the two juxtaposed scales on the bottom slide.

Usage example

    So – here is an example of a real calculation:

    Problem: Given a cast iron pipe of 12 inches diameter, carrying water at a head loss of 1 foot per 1000 feet. Calculate the velocity and the discharge (volume per unit time) of the water flowing through the pipe.

    Solution: Set the scales so the diameter and the loss of head are lined up vertically as in the following figure (blue-circled values). The remaining two scales will move into position (because of the pantograph) and the correct values will present themselves on the same vertical line (blue arrows): the velocity is 4.7 feet per second, and the discharge is 435 gallons per minute, or 1.16 cubic feet per second (we’re talking British, not US, gallons here).

The Baines slide rule - use example
    The nice thing is that once you align two parameters vertically, the other two truly “present themselves”: you don’t have to hunt them down elsewhere on the rule, they are right there on the same vertical line.

Who was Mr. Baines?

    The patent listed on the edge of the rule is accessible online, and from it we learn that this device was invented by Hewley Mortimer Baines. Further research informed me that Mr. Baines graduated in 1888 from the Royal Indian Engineering College. This college, located in Surrey, England, trained civil engineers for service in the Indian Public Works Department, and Baines had indeed shipped to India and spent two decades in the Punjab Public Works Department before retiring in 1910. Although the patent mentions the suitability of the idea to problems other than water flow, I find no reference anywhere to a similar device used for such a problem; perhaps Mr. Baines was happy to rest on his laurels after giving us the Slide Rule No. 1.

Exhibit provenance:
    I purchased this slide rule at the Oughtred Society’s auction of items from the Tom Wyman collection.
    I am indebted to the late Colin Barnes for much useful information about this device and its inventor; and to Stefan Heimann for guiding me through the relevant intricacies of hydraulic science and nomenclature.

More info:
    You can see Baines's patent, from 1903, here.
    See also “Baines and his Slide Rule”, by Colin Barnes and John Bolton, UKSRC Gazette, Issue 9 – Autumn 2008, p. 8. This article also reproduces an article in The Engineer, April 1, 1904.
    E. M. Horsburgh, Modern instruments and methods of calculation: a handbook of the Napier tercentenary exhibition (London : G. Bell and Sons, Ltd. ; The Royal Society of Edinburgh, 1914), has some of the math in page 162.
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