Willis and Lippincott planimeters.

Last updated 10 April 2020, David Green

1. Introduction.

In 1855 Jacob Amsler revolutionised planimeter design by introducing the polar planimeter. The planimeter measures area by recording the rotation of a wheel, ignoring the distance the wheel is dragged sideways.

So polar planimeters were well established in 1894 when Edward Willis patented a revised design of the polar planimeter (Section #3) recording the distance the wheel is dragged sideways and ignoring the rotation of the wheel. There is something of a mystery here, however, because Alpheus Lippincott was making planimeters using this principle in 1892, and very probably earlier (Section #2).

Willis produced at least three variations of his planimeter before introducing his "Improved" planimeter in about 1901. The first of these (Section #4) seems to predate his 1894 patent. Two other designs (Section #5) more closely match the patent.

The "Improved" planimeter went through four versions (Section #7). Since I have been monitoring the Willis planimeters offered on Ebay (1999) (spreadsheet), I have noted the following occurrences of each type:

  1. Section 4   3  (pre-patent)
  2. Section 5   2  ("Willis Planimeter", brace)
  3. Section 5   2  ("Willis Planimeter", no brace)
  4. Section 8   2  ("Improved Planimeter", 'square' hinge, long spindle, screw connection)
  5. Section 8   4  ("Improved Planimeter", 'square' hinge, long spindle, latch connection)
  6. Section 8  88  ("Improved Planimeter", 'square' hinge, short spindle, latch connection)
  7. Section 8  25  ("Improved Planimeter", 'round' hinge , short spindle, latch connection)

Willis continued to tinker with the design (Section #8), and was awarded a third patent in 1922. He also produced a model of a "two wheel" planimeter which is now in the National Museum of American History.

Lippincott was also active during the period. As mentioned earlier, it seems to have been his idea to measure the slippage of the planimeter wheel against a ruler, and, in his 1896 patent, the improvement to avoid having to turn over the instrument when adjusting it to the width of an indicator diagram (Section #9).

Besides the New Planimeter, he introduced three polar models, first with glass scales, then his "Improved Planimeter" with boxwood scales and the "Simplex Planimeter", a cheaper version with separate scales. Some time later, the Trill Indicator Company acquired the rights to make his Improved Planimeter (Section #11).

The Willis Improved Planimeter was the most successful of the models. I believe about 7,000 were sold (Section #13), compared with perhaps 4,000 Lippincott models.

2. The New Planimeter.

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This planimeter was manufactured by The Mechanical Specialties Mfg Co. "The Mechanical Specialties Manufacturing Company was known for its 'Arc' steam engine indicator, which it started to sell in 1888. The firm was apparently short-lived. By 1900, the Boston Journal of Commerce and Textile Industries was published from its address." [NMAH web page]

Carpenter, in 1892, stated: "A planimeter, termed the New Planimeter, is manufactured by the Mechanical Specialties Co. of Boston, Mass. This instrument consists of a single arm, one end of which moves in a fixed groove; the other end carries a tracing-point which is moved around the diagram to be evaluated, and in this respect is similar to the Coffin Planimeter. The perpendicular motion of the arm is measured by the slipping of a sharp-edged wheel on a graduated axis perpendicular to the arm." [Ref #2 page 36]

So the New Planimeter was in production in 1892 and possibly as early as 1888. It was a linear planimeter and its structure closely resembles Figure 6 from Lippincott's 1896 patent [Ref #3].

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The New Planimeter was also made by Hine and Robertson, but whether before, after, or perhaps concurrently with Mechanical Specialties is not known. Hine & Robertson of New York City made steam-engine indicators and sold planimeters from the 1880s until 31 December 1896, when the firm was renamed James L. Robertson & Sons [archivingindustry.com].

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3. Willis's Patent, 1894.

The first of the patents refering to this type of planimeter was Willis's Patent No. 529,008, dated November 13,1894 [Ref #5].

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Willis's aim was "to simplify and improve the construction and operation [of polar planimeters] so as to enable the user to read the mean effective steam pressure or horse power direct from a scale attached to the instrument, without calculation of any kind and without application of measurements to the diagram." Further, as indicators could be used with different springs, scales matching these springs were to be easily attached and detached from the planimeter.

"Prior to my invention planimeters have employed measuring wheels having rotary movement, the indications given thereby being merely the height of the line denoting the mean effective pressure on the diagram. So far as I am aware there has been no provision made, heretofore, for reading the mean effective pressure in pounds and fractions of a pound, direct from the scale of the instrument. Neither am I aware of any instance in which the instrument has been so organised as to permit the use of interchangeable scales, or the reading of horse powers directly from the instrument."

Importantly, Willis patented a better way of mounting his scales than Lippincott, whose scales were inscribed on the axis along which the wheel slid and were read by the edge of the hub of the wheel.

4. A Prototype Willis planimeter?

This planimeter showed up on Ebay in June 2019.

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It is nicely made and barely used. There is no serial number, maker's name or patent number engraved in the metal. The only identifier is the name "Theo Alteneder & Sons Philadelphia" on the rule.

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It is undoubtedly an early "unimproved" Willis model, although it does not exactly match the 1894 patent or the other "early" Willis planimeters I have seen (see Section 5). It appears to be an even earlier version.

The planimeter fits snugly into a smart, lined presentation case. It rather suggests that Willis commissioned an instrument maker to make the planimeter. Possibly it was made by Alteneder, as their name is on the rule. Their 1892 catalogue shows that they would have had the capability. But they sold their scales retail, so any maker could have used one.

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Points of difference from the patent drawing include:

  1. The use of a triangular rule providing six scales, rather than the single scale shown in the patent. But it has this in common with all the other examples I have seen. It seems Willis discarded the idea of a single scale very early on.
  2. The scales are on the right of the shaft, rather than on the left.
  3. The method of connecting the tracer-bar 2 to the rectangular housing 1 (using the numbering system in the patent). In the patent, and in all Willis planimeters I have seen, early or improved, the tracer-bar passes through a longitudinal hole in the housing. This can be seen clearly in Hawkins' drawing of the planimeter, Figure 111 shown below. In the "prototype" it drops from above into a slot in the housing. Because of this, and to keep the line from the "small conical point 6" to the "nipple 15" parallel to the tracer-bar, the tip of the tracer-bar is bent.
  4. The shaft 7 is made of metal. The patent simply mentions that "numeral 7 indicates a fixed shaft ... upon [which] is placed the measuring wheel 8, accurately fitting the shaft, but capable of turning freely and moving longitudinally from end to end thereof". No mention is made of the material to be used but the production standard seems to have been glass. Hawkins mentions that "The wheel is mounted upon a glass rod, and needs no lubrication." The instruction sheet accompanying the first Improved Willis planimeters states "The glass rod of the Willis Planimeter and the glass tube of the Lippincott have been avoided, thus rendering the instrument thoroughly portable, and not liable to breakage".

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An additional difference between this planimeter and all the other Willis planimeters I have seen is the use of a plain boxwood triangular scale rather than a white edged triangular scale.

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What can we say about this planimeter? A first thought was that it might have been a patent model. However, the patent (#529,008, dated November 13, 1894), mentions: "Application filed August 2, 1894 ... No model."

Rather, it seems to have been a prototype.

A major selling point of the Willis Planimeter was its ability to read the mean effective pressure of an engine directly from an indicator diagram, without calculation. To do this the tracer arm had to be set equal in length to the width of the diagram, as shown in Figure 112. This was a clumsy operation involving turning the planimeter up-side-down, and loosening and tightening a retaining screw.

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With the "prototype" this operation is even more clumsy because the arm is likely to fall out of the frame.

Two other such planimeters have been offered on Ebay. In February 2018, one was mistakenly offered as an Improved Planimeter because it was accompanied by a brochure for the improved model. The scale rule shows the slogan "S & H. WATERPROOF INDIA INK" which appears on Alteneder rules.

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More recently a third was offered on Ebay. Sadly it is in poor condition, missing the tracer-bar and rather rusty. Again, the only identification on the planimeter is the name "Theo Alteneder & Sons Philadelphia" on the scale rule.

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The three planimeters appear to be identical. It seems likely that they pre-date the 1894 patent and that Willis had a batch of them made, perhaps for sale, perhaps to circulate amongst his colleagues to get their reactions.

5. The "Willis Planimeter".

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To my knowledge only four "Willis Planimeters" have been offered on eBay since 1999. Call them A, B, C and D for convenience.

(A) The first was sold in May 2001.

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(B) The second was offered in November 2008.

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(C) The third was sold in July 2018.

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(D) The fourth was sold in September 2019.

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Hawkins' description of the planimeter was that "The wheel is mounted upon a glass rod and needs no lubrication. The scale is of box wood, finely graduated, and the rest of the instrument is of brass (nickel plated) and steel. It is supplied by its inventor, E. J. Willis, M. E., 211 East Franklin Street, Richmond, Va." [Hawkins 1898 page 139]

Unfortunately the photos accompanying the two earlier eBay lots are few and small and offer little detail. The four look substantially the same and none of them exactly matches the 1894 patent.

  1. In particular, they all use a triangular rule providing six scales, rather than the single scale shown in the patent.
  2. All four were provided with a horse power attachment.
  3. They all have a transparent shaft along which the wheel slides. This seems to have been standard because Hawkins mentions "The wheel is mounted upon a glass rod ..."
  4. The serial number of (C) was #20. Presumably the others were also numbered but their numbers are not known.

However, on closer inspection there is a minor variation. For A and D the connection of the fulcrum-bar 4 to the frame 1 is braced, as shown below, while for B and C this brace appears to be absent.

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I can find no record that Willis manufactured his own planimeters. The Improved Planimeters were all made by Jas. L. Robertson & Sons, initially of 204 Fulton Street, New York. There is no indication whether Willis made the earlier planimeters himself or had them made to order.

6. Willis's 1895 patent.

Willis was awarded a second patent, No. 542,511, on 9th July 1895.

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This differed from Patent 529,008 in that:

  1. The fulcrum-bar is replaced by a guide-strip (12), that is, it is a linear planimeter. I have never seen an example of this version. It would have been more bulky and cumbersome than the polar model. I suspect it was never a commercial proposition.
  2. Rather than the single detachable scales, Willis "prefers to use a revolvable scale-bar ... of polygonal form in cross-section, and preferably triangular, its three faces representing the sides of an equilateral triangle". This is the only form of scale-bar I have seen.

Much of the patent is devoted to explaining his horse power attachment.

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The horse power attachment was provided with the Willis Planimeter but later was sold separately from the Willis Improved Planimeter. They were numbered and the highest serial number I have seen was #343. It seems there was little demand for them and that just a few hundred were made.

This isn't really surprising. As Willis himself explained in a paper in Detroit in June 1895, the horse power is calculated as the mean effective pressure, read from the indicator diagram, multiplied by a constant for the engine. That is:
horse power = mep x (piston area x stroke x revolutions per minute)/ 33,000 = mep x constant.

So the horse power attachment was an alternative to doing a simple multiplication. This is not a big deal even if done by hand and, as Zoller has pointed out, is trivial if using a slide rule. For $20, the same cost as a planimeter, it just wasn't good value.

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The patent was reissued on 22nd September 1896 with hardly discernible changes.

7. The Improved Willis Planimeter.

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Most of the Willis planimeters one sees are the "Improved Willis Planimeter". The major improvements were:

1. a better way of adjusting the tracer arm to the width of the indicator diagram. This was actually an innovation proposed by Lippincott in his October 1896 patent.

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Lippincott explains: "In planimeters the length of the diagram controls the distance between the pivot of the frame and the tracing-point of the tracing-bar. It has been customary to mount pointed lugs over these two points, and they are brought into coincidence with the length of the diagram to be measured by turning the entire instrument over, or partly so, in making the necessary measurements. A source of inaccuracy is introduced from the fact that these two points may not be exactly the same distance apart as the two measuring-points above which they are located. To overcome this, I provide the frame of the planimeter with a depressible pin the point of which is preferably directly beneath or in alignment with the pivot of the frame. The pin is normally held out of the way when not in use by means of a spring, but can be forced down while the instrument is being adjusted for length, the point of this pin and the tracing-point corresponding with the pointed lugs heretofore used. In this way it is not necessary to turn the instrument over to ascertain the length of the card."

2. reduction of the friction between the measuring wheel and the shaft.

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The Willis patent says: " ... the measuring-wheel 11 comprises a blade or disk 17 and a collar or sleeve 18, which passes through the hub of the disk or blade, ball-bearings 20 being interposed between the sleeve and the hub of the disk, the latter being free to rotate upon the sleeve, and the balls serving to reduce the friction to a minimum. The sleeve 18 slides upon the spindle 12 and is free to move lengthwise thereof, and for the purpose of reducing the friction between the sleeve and the spindle the opposite ends of the sleeve are each provided with brackets 22, in which are journaled the antifriction rollers or wheels 23, which contact the spindle 12, so as to reduce the friction in the sliding movement of the sleeve and its disk along the spindle."

Not satisfied with these improvements, Willis continued to tweak the design of the Improved Planimeter and there appears to have been four versions.

For the two earliest planimeters (serial numbers #560 and #588) the frame was attached to the pole arm by means of screws. Thereafter, latches were used to hold the two parts together. The purpose, if any, of the two large holes in the housing is unknown.

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At about the same time, Willis slightly extended the length of the tracer bar.

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And for a short time, for which there have been six examples (with serial numbers from #560 to #783), Willis used an intermediate design to reduce the friction of the wheel on the axle, typified by planimeter #676, shown below.

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This version was also illustrated in the earliest instruction booklets.

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The design utilised rollers held by brackets attached to the frame (R and S). The wheel was rigidly fixed to a cylindrical axle that was long enough to allow the wheel to register against all of the scale rule. The improved method for setting the length of the tracer-bar is already in use (E).

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Some time between planimeter #783 and planimeter #807 the axle arrangement was changed to the design specified in the 1901 patent, with the wheel constructed as below.

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This design remained unaltered until planimeter #6083 and accounts for 88 of the 126 planimeters in my records. At some time between then and the next planimeter recorded (#7137) a change was made to the hinge. It is not clear why this change was needed but it is very distinctive. On the left is the early hinge (#688), on the right is the later circular hinge (#162). The final 25 planimeters in my records had this hinge.

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The information for 120 Improved Planimeters from various sources, mainly eBay sales between 1999 and 2019, can be summarised as follows:

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The serial numbers are rather confusing. The two "earliest" numbers recorded (#108 and #162) were actually the last two planimeters in the list to have been made.

A few planimeters still had dated packing slips in their boxes. But the planimeters were not packed in any sort of order so the slips are not much of a guide to the manufacturing dates.

Curiously the planimeter accompanying the following instruction brochure had no serial number.

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8. Later Willis Planimeter designs.

8.1 1922 Patent.

The 1922 patent introduced a yet more complicated "means of supporting the measuring wheel on the spindle, whereby end play of the wheel is entirely eliminated and a very accurate construction is produced ... producing an instrument which will retain its accuracy even after long use." It also made provision for a magnifying glass to more conveniently read the scale. I wonder if these improvements were overkill and the extra construction costs could not be justified. Such a model may have been made. I have never seen one.

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8.2 A Prototype Two-wheel Planimeter.

The National Museum of American History has in its collection a "Two Wheel Planimeter" constructed by Willis. The date on the box is the date of the 1922 patent, but it is nothing like the patent drawings. The pair of wheels suggests it was intended to be a linear planimeter, while the triangular metal plate with three prickers appears to be intended as the pole of a polar planimeter. It beats me how this model was supposed to function.

Hyman Schwartz mentions this planimeter in his article about Willis. The museum has seven interesting pictures of the model on its website.

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9. Lippincott 1896 Patent.

The events during the last decade of the nineteenth century are rather mysterious.

In 1894 Willis patented the idea of using the slippage of a wheel to measure the area of a diagram, even though, for at least two years before 1894, Lippincott was selling linear planimeters that did just that.

He also patented the idea of providing interchangeable scales to calculate mean effective pressure for different springs, even though Robertson was quite possibly making the Lippincott planimeters with interchangeable glass tubes by then.

Lippincott appears to be playing catch up in his 1896 patent. He finally patents the linear planimeter he was making in 1892 or earlier. He also patents the interchangeable glass tubes and the depressible pin for setting the length of the tracer bar to the width of the indicator diagram. In doing so he appears to be violating parts of Willis's patents.

I am not a patent attorney but I suspect that you are not allowed to "invent" something someone else is already making, either deliberately or innocently. Perhaps this is why Lippincott was allowed to make these belated claims?

The upshot of all this seems to have been that Willis was drawn into the Robertson orbit and a compromise agreed in which Robertson made the Willis Improved Planimeter, using the Willis scales and the Lippincott depressible pin. Perhaps both parties got royalties from this arrangement.

And this left Lippincott free to make his own planimeters.

10. Lippincott models.

10.1 Lippincott with Glass Scales.

Since 1999, I have noted 23 of these planimeters on Ebay, and the Computer History Museum has one on it's website. The planimeters were numbered and range from #190 to #1086. I speculate that about 1,200 were made. Only half of the 24 survivors are complete. The wheel was stored separately in the box and three wheels have been lost. The glass tubes were susceptible to breakage and ten of the planimeters have less than the full complement of three tubes.

14 examples (#190 to #818) are marked Hine & Robertson and 10 examples (#909 to #1086) are marked James L. Robertson & Sons. As the Hine and Robertson name ceased on 31 December 1896, about 850 were made before then. But I have been unable to determine when their production started.

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10.2 Lippincott Improved.

Clearly, Lippincott wanted to continue to make planimeters, but he had to do so without his plunger and needing to mount scales in a way that didn't violate Willis's patent. Which leads us to this interesting design.

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This drawing is from Wakeman's book, copyrighted in 1900 and published in 1901. It may be a drawing of an actual planimeter but one like this has not appeared on Ebay since 1999, so far as I know. However, some may have been made because the same drawing appears in The Electrical World and Engineer, dated 28 July 1900, and with it is what appears to be a photo of one in a box.

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More information, but no more clarity, is provided by an advertising pamphlet entitled "Preliminary Circular and Instruction for Using the Improved Lippincott Planimeter" issued by the "Engineering & Power Co." (General Manager A. C. Lippincott) (page 1, page 2). It is not dated but the illustrations in Wakeman's book and The Electrical World and Engineer dated 28 July 1900 are clearly copies from this circular, so the circular must date to late 1899 or early 1900.

The circular contains some interesting remarks:

  1. "The original [Lippincott Planimeter] has been in use for over five years, and had a larger sale than all others combined".
  2. "... the testimonials of over 2000 users."
  3. "The wheel is mounted on a tool-steel shaft, which slides under roller bearings practically eliminating friction."
  4. "Special scales to read horse-power will be graduated to order, but as the reading needs but to be multiplied by the 'constant' for the engine, an operation so rapid and simple, that the horse-power reading seems unnecessary."
  5. "With the unprotected [boxwood] triangular scale, the edges soon become chipped, and the surface so soiled that legibility is destroyed."
  6. "The depressible pin ... for which Mr Lippincott was granted patents, is not found necessary, as the folding bar admits of the use of a stationary point."
  7. ... manufacturing in large numbers we are enabled to fix a list price of $18.00 and will make a special introductory discount for 60 days."

Claim (a) suggests the glass-tube planimeters were introduced in 1894.

(b) looks like advertising hyperbole. Since 1999, 21 of these planimeters have been sold on Ebay, with serial numbers only from #199 to #1086.

(c) is intriguing. The Willis patent in 1901 is the first mention of rollers to eliminate friction, but attached to the wheel itself. It seems Lippincott actually introduced rollers in 1900, but attached to the frame. And early versions of the Willis Improved Planimeter also used this arrangement (for serial numbers before about #807).

I am still struggling to understand what (f) means. As noted below, Lippincott later replaced the depressible pin with a lever.

Something happened to interrupt the sales offered in (g). I have never seen an actual example in over 20 years of looking and I believe few, if any, were made.

The spike where the plunger would have been looks like an occupational health hazard. Lippincott modified the design significantly, replacing the flat scales with a pair of triangular scales (setting aside his comments in (e)) and replacing the health hazard with a lever. The result was this production model.

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The 1896 Willis patent was for "A planimeter comprising ... a measuring wheel having ... a longitudinal movement [on its axis] perpendicular to the tracer bar, and a scale along which the wheel moves ...". Lippincott avoided this by locating the scales on the other side of the tracer bar and attaching a pointer to the end of the spindle, which obviously moves the same distance as the wheel.

The lever seems to me a nicer idea than the original plunger. When up, it is out of the way. When down, the foot rests exactly below the center line of the tracer bar and exactly adjacent to the center pin of the pole bar.

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The planimeters were numbered but the number was tucked away at the end of the rectangular frame and below the tracer arm. It isn't captured in most photos and I have been able to record only three. Curiously they are the close numbers #765, #780 and #788. This doesn't help in estimating how many were made. A tenuous guess is about 1750, which is based on the following:

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The planimeter was not without its faults. To adjust the scales involves lining up four little holes with four little pins. I can attest that this is very awkward. Later the latch was replaced by two nuts. I have no idea whether this made life any easier.

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One other variation emerged in 2005. It appears to have an octagonal metal scale but I know nothing more about it.

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10.3 Lippincott Simplex.

Another way that Lippincott got round the problem of mounting scales on his planimeters was to do away with them. This allowed for a simpler instrument that could be sold more cheaply.

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Lippincott provided a simple instruction sheet with the simplex planimeters (front, back). It notes that the planimeter was made in two grades, retailing for $7.50 or $10.00. Both had the same accuracy and the difference in cost was accounted for by a more careful finish and a more expensive box. For $10.00 the box was velvet lined and boxwood scales were supplied. For $7.50 you got fibre scales and a cheaper lining in the box. Sadly, the document is not dated.

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The Simplex Planimeter was modestly successful. Eighteen have appeared on Ebay since 1999. Using the same rationale as earlier, this would suggest production of about 900.

These fibre scales mention that Lippincott is General Manager of "Lippincott Steam Speciality and Supply Co." and also of "Indicator Instruction Co.", both of Newark, NJ.

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Judging from labels in the Simplex boxes Lippincott had some difficulty deciding what to call his Company.

10.4 A modified Lippincott Simplex?

An unusual Lippincott planimeter was sold on eBay in May 2007. It was in a box marked: "LIPPINCOTT PLANIMETER - PATENTED DECEMBER 31, 1901- LIPPINCOTT S. S & S CO - NEWARK. NJ". As the design would contravene the Willis patent I suspect it is simply a private, DIY modification to a Simplex planimeter.

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11. Trill.

The Trill Indicator Company of Corry, Pennsylvania, U.S.A. was founded by William L. Trill in 1901. This metalworking business made a selection of internal- and external-spring indicators. Trade is said to have declined after Trill's death in 1943, though employees continued to work until the end of the Second World War. The moribund Trill business was then purchased by the Corry Instrument Company of Corry, Pennsylvania, U.S.A. in 1945 [archivingindustry.com].

Trill made a planimeter that was a copy of the earlier Lippincott design.

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12. Eugene Scoville Patent, 2nd January 1906.

As a footnote to the Lippincott/Willis wars, Eugene Scoville patented another version of this type of planimeter in 1906 [Ref #15]. This combined a linear planimeter that measured mean effective pressure using the distance the planimeter wheel was dragged and a specialised slide rule for calculating the horse power from this value of m.e.p. It is a neat looking instrument that would be very collectible, but I have seen no evidence that they were sold commercially.

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13. The Serial Numbers of Willis Planimeters.

The serial numbers of the Improved Planimeters are a puzzle. The two "earliest" numbers recorded (#108 and #162) were actually the last two planimeters in the list to have been made; effectively #10108 and #10162. It seems the company started to renumber the planimeters (from #001?) at about the time it transitioned from James. L. Robertson & Sons to James. L. Robertson 3rd.

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One might expect the planimeters to be numbered sequentially and to be equally likely to survive, in which case they should approximately follow the straight red line shown below. Instead, there is a distinctly nonlinear pattern with three runs and four possible gaps.

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In this graph the circles are replaced by the black cumulative curve. The red line is the continuous uniform distribution. The maximum distance between the two curves is d = 0.185, shown in green. One statistical test of goodness of fit between the uniform distribution and the observed serial numbers uses this distance d (Ref 12). This particular value of d indicates that there is very much less than one chance in a hundred that there were 10200 planimeters which were numbered sequentially.

A plausible explanation for the gap in the numbers between #8734 and #(10)108 is that James. L. Robertson 3rd started renumbering the planimeters when he took over James. L. Robertson & Sons. The old numbers probably ended at about #8800. If we arbitrarily renumber the last two planimeters as #8908 and #8962 we get the following graph.

It is also unlikely that 13 planimeters between #560 and #985 and none between #1 and #559 would survive if the serial numbers of the Improved Planimeter started at #1. At least one earlier Willis Planimeter was numbered (#20) so I believe they all were and that Robertson continued the number sequence when the improved version was introduced. Those four planimeters are included in the next graph.

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The value d=0.116 is smaller than the 95% limit (0.122) so there is no statistical justification to assume these serial numbers are not a random sample from the uniform sequence. However, they still look odd to me.

What might have caused gaps in the numbering? Originally, Robertson's company occupied 204 Fulton Street, NY. The addresses on some of their brochures show that they moved premises twice, first to 48 Warren Street NY and then to 78-80 Murray St NY. There is no guide to the dates of these moves. It may be a coincidence but the two moves may coincide with the two hiccups in the serial numbers for some reason. It is also notable that #6083, before the gap, is the last planimeter with a "square" hinge, and #7137 has the "round" hinge. Perhaps numbering was restarted at #7000 for the new design.

Be that as it may, the next graph shows the sequence resulting from removing the gaps from #2188 to #3520 and from #6083 to #7042. As before, the red line is the uniform distribution but now it represents my model of events: serial numbers started from #1 and included the early Willis Planimeter; they were sequential but the sequence was interrupted three times for reasons only the Robertson company knew.

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For interest, the blue line is the linear regression fitting the adjusted serial numbers. My conclusion is that although the serial numbers run up to (10)162, only about 7000 Willis planimeters were actually made. But the above does not constitute proof.

13. References.

[1] N Hawkins, "Hawkin's Indicator Catechism" published: New York, Theo. Audel & Co., 1898
available from https://archive.org/

[2] Rolla C. Carpenter, "A text-book of experimental engineering. For engineers and for students in engineering laboratories." published: New York, J. Wiley & Sons, 1892.

also Rolla C. Carpenter, "A text-book of experimental engineering. For engineers and for students in engineering laboratories." published: New York, J. Wiley & Sons, 1893. p36

[3] Lippincott's Patent No. 569,107, dated October 6, 1896.
To download a copy, enter 569107 in "Query" in http://patft.uspto.gov/netahtml/PTO/srchnum.htm
then click "Images"

[4] Lippincott's Patent No. 689,978, dated December 31, 1901.
To download a copy, enter 689978 in "Query" in http://patft.uspto.gov/netahtml/PTO/srchnum.htm
then click "Images"

[5] Willis's Patent No. 529,008, dated November 13, 1894
To download a copy, enter 529008 in "Query" in http://patft.uspto.gov/netahtml/PTO/srchnum.htm
then click "Images"

[6] Willis's Patent No. 542,511, dated July 9, 1895
To download a copy, enter 542511 in "Query" in http://patft.uspto.gov/netahtml/PTO/srchnum.htm
then click "Images"

[7] Willis's Patent No. 11,568, dated September 22, 1896
To download a copy, enter RE11568 in "Query" in http://patft.uspto.gov/netahtml/PTO/srchnum.htm
then click "Images"

[8] W H Wakeman, "Engineering Practice and Theory, for Steam Engineers" published: New Haven, Conn., U.S.A. by the author, 1901. [but copyrighted 15 September 1900]
available from https://archive.org/

[9] Theo Alteneder & Sons Philadelphia, "Alteneder Drawing Instruments Catalogue", 1892
available from https://archive.org/

[10] Hyman A. Schwartz, "The Willis Planimeter", Rittenhouse, Vol 7, No 2, 1993.

[11] The Electrical World and Engineer. v.36 (1900 mo.JUL-DEC, a weekly journal) 28 July 1900

[12] S Siegel & N J Castellan, "Nonparametric Statistics for the Behavioral Sciences", 2nd edition, McGraw-Hill Book Company, 1988, page 51.
or W J Conover, "Practical Nonparametric Statistics", John Wiley & Sons, 1971, page 293.

[13] Paul Zoller, "Instruments and Methods for the Evaluation of Indicator Diagrams." Bulletin of the Scientific Instrument Society, No 75, December 2002, pages 11-17.

[14] E. J. Willis, "A Horse Power Planimeter" Transactions of ASME. v.16(1894-1895).
can be read at https://babel.hathitrust.org/cgi/pt?id=uiug.30112007864108&view=1up&seq=1209

[15] Scoville's Patent No. 809,019, dated January 2, 1906
To download a copy, enter 809019 in "Query" in http://patft.uspto.gov/netahtml/PTO/srchnum.htm
then click "Images"