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Fuller calculator

April 12, 2024

This article is about a type of slide rule. For the American slave, see Thomas Fuller (mental calculator).

The Fuller calculator, sometimes called Fuller's cylindrical slide rule, is a cylindrical slide rule with a helical main scale taking 50 turns around the cylinder. This creates an instrument of considerable precision – it is equivalent to a traditional slide rule 25.40 metres (1,000 inches) long. It was invented in 1878 by George Fuller, professor of engineering at Queen's University Belfast, and despite its size and price it remained on the market for nearly a century because it outperformed nearly all other slide rules.

Fuller calculator, Fuller-Bakewell model of 1928

As with other slide rules, the Fuller is limited to calculations based on multiplication and division with additional scales allowing for trigonometical and exponential functions. The mechanical calculators produced in the same era were generally restricted to addition and subtraction with only advanced versions, like the Arithmometer, able to multiply and divide. Even these advanced machines could not perform trigonometry or exponentiation and they were bigger, heavier and much more expensive than the Fuller. In the mid-twentieth century the handheld Curta mechanical calculator became available which also competed in convenience and price. However, for scientific calculations the Fuller remained viable until 1973 when it was made obsolete by the HP-35 handheld scientific electronic calculator.

Design edit

Model 1, the standard model edit

 
Fuller calculator, annotated

In essence, the calculator consists of three separate hollow cylindrical parts that can twist and slide over each other about a common axis without any tendency to slip. The following details describe the version made between 1921 and 1935. There is a papier-mâché cylinder (marked D in the annotated photograph) some 30 centimetres (12 inches) long and 6.2 centimetres (2.4 in) in diameter fastened to a mahogany handle. A second papier-mâché cylinder (marked C) – 16.3 centimetres (6.4 in) long and 8.1 centimetres (3.2 in) diameter – is a slide fit over the first. Both cylinders are covered in paper varnished with shellac. The second, outer, cylinder is printed with the slide rule's primary logarithmic scale in the form of a 50-turn helix 12.70 metres; 500 inches (41 ft 8 in) long with annotations on the scale going from 100 to 1000. A brass tube with a mahogany cap at the top is a slide fit into the first cylinder.[1][2][3][4]

A brass pointer with an engraved index marker at its tip (marked A) is attached to the handle so that it points to a place on the primary logarithmic scale, depending on the position to which the scale on cylinder C has been adjusted. A second brass pointer (marked B) is attached to the top cap pointing down over the logarithmic scale and it is positioned by rotating and sliding the cap at the top. This pointer has four index marks (marked B1, B2, B3, B4) such that whichever one is convenient may be used.[1][2] Printed on the inner cylinder D are simply tables of data for reference purposes.[5]

 
Fuller calculator in case

The calculator was sold in a hinged mahogany case 46 by 12 by 11 centimetres (18.1 in × 4.7 in × 4.3 in) which, if required, holds the instrument when in use by means a brass support that can be latched to the outer end of the case.[6][7] Out of its case the calculator weighs about 900 grams (32 oz).[8] For all except the earliest instruments the last two digits of the date and a serial number, believed to be consecutively allocated, are stamped at the top of pointer B.[9]

Other Fuller models edit

The calculator described above was called "Model No. 1" .[6] Model 2 had scales on the inner cylinder for calculating logs and sines. The "Fuller-Bakewell" model 3 had two scales of angles printed on the inner cylinder to calculate cosine² and sine⋅cosine[note 1] for use by engineers and surveyors for tacheometry calculations.[note 2][5][12] A smaller model with a 5.1 metres (200 in) scale was available for a short time but very few survive. In about 1935 the brass tube was replaced by one of phenolic resin and in about 1945 the mahogany was replaced by Bakelite.[13]

Included in Stanley's 1912 catalogue and continuing there until 1958 was Barnard's Coordinate calculator. It is very similar in construction to the Fuller instruments but its pointers have multiple indices so additional trigonometrical functions can be used. It cost slightly less than the Fuller-Bakewell and a 1919 example is held by the Science Museum, London.[14][15][16] In 1962 the Whythe-Fuller complex number calculator was introduced.[17][18] As well as being able to multiply and divide complex numbers it can convert between Cartesian and polar coordinates.[19]

Comparison with other slide rules and contemporaneous calculators edit

The calculator's unusual single-scale design[note 3] makes its 12.70-metre (500-inch) helical spiral equivalent to a scale twice this length on a traditional slide rule – 25.40 metres (1,000 inches) long. The scale can always be read to four significant figures and often to five.[21][22] In 1900 William Stanley, whose firm manufactured and sold scientific instruments including the Fuller calculator, described the slide rule as "possibly the highest refinement in this class of rules".[23]

When it was introduced the Fuller calculator had a much greater precision than other slide rules although the Thacher instrument became available a couple of years later. This was made in the United States and was comparable in size and precision but radically different in design.[24][25][26][27] However, both of these types of slide rule required some skill to operate accurately compared with mechanical calculators which manipulated exact numerical digits rather than using positioning and reading from a graduated scale. Mechanical calculators could only add and subtract (which the Fuller did not do at all) although models such as the Arithmometer could perform all four functions of elementary arithmetic.[26][28][29] No mechanical calculators could calculate transcendental functions, which slide rules could be designed to do, and they were bigger, heavier and much more expensive than any slide rule, including the Fuller.[26][28][30]

However, a revolutionary miniature mechanical calculator went on sale in the mid-twentieth century – while Curt Herzstark had been imprisoned in a Nazi concentration camp in World War II he had developed the design of the handheld Curta mechanical calculator. It was simple to use and, being digital, was completely accurate.[30] Because of these advantages and despite its somewhat higher price its total sales were 150,000 – over ten times more than the Fuller. Its range of mathematical calculations was seen as being adequate. However, for scientific calculations the Fuller remained viable until 1973 when, along with the Curta, it was made obsolete by the Hewlett-Packard HP-35 handheld scientific electronic calculator.[31][26][32]

Invention, sales and demise edit

 
Fuller's 1879 U.S. patent drawing

The calculator was invented by George Fuller (1829–1907[33]), professor of engineering at Queen's University Belfast (Queen's College at that time).[3] He patented it in Britain in 1878, described it in Nature in 1879 and in that year he also patented it the United States, depositing a patent model.[34][35]

Fuller's calculators were manufactured by the scientific instrument maker W.F. Stanley & Co. of London who made nearly 14,000 between 1878 and 1973.[8][36][37][5]

In Britain the prices charged by W.F. Stanley in 1900 were for model 1 £3 (equivalent to £345 in 2021) and for model 3 £4 10s.[38][note 4] The Whythe-Fuller model was advertised in a 1962 W.F. Stanley catalogue at £21 (£477 in 2021).[18] The calculator was still listed in Stanley's catalogue in 1976[note 5] when model 1 cost £60 (£459 in 2021) and model 2 was £61.25.[42]

In the United States the instrument was marketed by Keuffel and Esser who only supplied model 1. They described it as "Fuller's Spiral Slide Rule" and, over the period it was sold between 1895 and 1927, it rose in price from $28 to $42 (falling from $1025 to $737 in 2023 prices).[43][note 6]

From the time when serial numbers were first stamped (about 1900) to when production ceased in 1973 around 14,000 instruments were made.[note 7] Production was about 180 per year overall but it declined after about 1955.[9][45] In 1949 Encyclopædia Britannica, noting that the Fuller had been designed in 1878, reported that it "has been in considerable use up to the present time".[46]

In 1958 the mathematician and physicist Douglas Hartree[note 8] wrote that the Fuller "... is cheap compared with a desk machine[note 9] and may be found very useful in work for which its accuracy is adequate and in circumstances in which the cost of a desk machine is prohibitive. [...] With one of these slide-rules and an adding machine much useful numerical work can be done ...".[49] In 1968 the standard Fuller cost about $50 at a time when an electronic Hewlett-Packard HP 9100A desktop calculator (weighing 40 pounds (18 kg)) cost just under $5000.[50][51] But in 1972 Hewlett-Packard introduced the HP-35, the first handheld calculator with scientific functions, at $395 – the Fuller went out of production the next year.[52][31]

Operation edit

Multiplication and division edit

 
Multiplication procedure

The instrument operates on the principle that two pointers are set at an appropriate separation on the helical scale of the calculator. The relevant numbers are indexed by adjusting separately both the movable cylinder and the movable pointer. Since the scale is logarithmic the separation represents the ratio of the numbers. If the cylinder is then moved without altering the positions of the pointers, this same ratio applies between any other pair of numbers addressed.[53] In other words, it is a logarithmic Gunter's scale wound into a helix with Gunter's compass points being provided by pointers A and B.[54]

To multiply two numbers, p and q, cylinder C is rotated and shifted until pointer A points to p and pointer B is then moved so B1 points to 100. Next, cylinder C is moved so B1 points to q.[note 10] The product is then read from the pointer A. The decimal point is determined as with an ordinary slide rule. At the end of a calculation the slide rule is already positioned to continue with further multiplications (p x q x r ...).[1]

To divide p by q, cylinder C is rotated and shifted until pointer A points to p, B1 is brought to q, cylinder C is moved to bring 100 to B1 and the quotient is read from pointer A.[56] It turns out to be particularly efficient to alternate multiplication with division.[57]

Determining logarithms edit

 
Detail of scales for log calculation
( log10(1.1) = 0.04 + 0.0014 )

There are two other scales inscribed on the calculator which allow logarithms to be calculated and enabling such evaluations as pq and  .[58][53] The scales are linear and one is engraved along the length of pointer B and the other printed around the circumference of the top of cylinder C. Index B1 is set to the relevant value on cylinder C and then two readings are taken. The first reading is from the scale on pointer B where it crosses the topmost spiral of the helical scale on the cylinder. The second reading is from the scale at the top circumference of cylinder C where it crosses the left edge of pointer B. The sum of the readings provides the mantissa of the log of the value.[note 11][60]

Trigonometry and log functions edit

 
Model 3 Fuller-Bakewell use of sine.cosine[note 1]

For model 2 instruments with scales on the inner cylinder D, there is an index mark inscribed on both the top and bottom edges of cylinder C. As an example of use, when the lower index mark is set to an angle printed on the lower scale on cylinder D, pointer A points to the corresponding value of sine on cylinder C. The same approach apples for the log scale on the upper part of cylinder D.[note 12] The model 3 Fuller–Bakewell is used in the same way but its scales on cylinder D are for cosine² and sine⋅cosine[note 1][note 2](see photograph).[61]

Notes edit

  1. ^ a b c The ⋅ means multiplication. The ⋅ is the mid-line dot operator.
  2. ^ a b Stadiametric rangefinding: for an anallactic tacheometer with its vernier scale reading zero for a horizontal line of sight, the distance to a vertical stadia rod (d) and the height of the stadia rod above the tacheometer (v) may be calculated as follows. d = K.S cos2α and v = K.S sinα.cosα where K is a constant of the tacheometer (normally 100), S is the difference in height on the stadia rod as intercepted by the tacheometer's cross hairs (the stadia interval) and α is the vertical angle measured by the tacheometer.[10][11]
  3. ^ There is a single scale for multiplication and division. The other ancillary scales are for other purposes (transcendental functions).[20]
  4. ^ In 1907 and 1916 W.F. Stanley sold both model 1 and model 2 for £3 15s (for 1907 £422 in 2021, for 1916 £270 in 2021) and the Fuller-Bakewell model 3 for £4 10s (for 1907 £506 in 2021, for 1916 £325 in 2021).[39] The firm A.G. Thornton in Manchester made slide rules and also sold the Fuller – their 1916 pricelist shows £3 3s 6d (£229 in 2021) for model 1, £3 19s 0d for model 2 and £4 15s 0d for model 3.[40][41] Thornton's 1916 price for the Thacher was £7 18s 6d (£570 in 2021).[41]
  5. ^ Production stopped in 1973 but the calculators remained on sale.
  6. ^ In 1902 "Thacher’s calculating instrument" was priced at $35 (plus $10 for a reading glass).[44]
  7. ^ The ones sold in the U.S. were made in Britain and were numbered in the same way.[9]
  8. ^ Hartree had worked on ENIAC back in 1946.[47]
  9. ^ By "desk machine" Hartree meant a desktop calculator that could perform multiplication as well as addition and subtraction.[48]
  10. ^ Index mark B3 is used when B1 is off the scale – B1 and B3 are fixed on pointer B so their distance apart is the full length 100 – 1000 of the scale. Index marks B2 and B4 may be used if more convenient. B1 or B2 provide greater accuracy than B3 or B4.[3][55]
  11. ^ For model 2 calculators there is a better way of using logs (see "trigonometry and log functions").[59]
  12. ^ The angle is in degrees. For logs, the argument is pointed to on cylinder C and the resulting logarithm to the base 10 is at the upper index point. Note that pointer A and cylinder D are in a fixed orientation because both are rigidly attached to the handle.

References edit

Citations edit

  1. ^ a b c Larard & Golding (1907), p. 116.
  2. ^ a b Fuller (n.d.), pp. 2–3.
  3. ^ a b c Feely & Schure (1995), pp. 33–40.
  4. ^ Hopp (2003), p. 20.
  5. ^ a b c Hopp (2000), pp. 25–32.
  6. ^ a b Fuller (n.d.), p. 2.
  7. ^ Nichols (2009), pp. 3–8.
  8. ^ a b "Fuller's Cylindrical Slide Rule". University of Queensland Physics Museum. from the original on 7 June 2021. Retrieved 7 June 2021.
  9. ^ a b c Feely & Schure (1995), p. 35.
  10. ^ Fuller (n.d.), pp. 22–25.
  11. ^ "Tacheometric Surveying study notes" (PDF). Civil Engineering: B.Tech II Year II Semester. Annamacharya Institute of Technology & Sciences, Tirupati. pp. 7–8. (PDF) from the original on 27 January 2021. Retrieved 15 June 2021.
  12. ^ Fuller (n.d.), pp. 17–23.
  13. ^ De Cesaris (2011), pp. 196.
  14. ^ Hopp (2000), pp. 28–29.
  15. ^ W.F. Stanley & Co. Ltd (PDF). W.F. Stanley & Co. p. 79 (page 3 in pdf). (PDF) from the original on 7 April 2021. Retrieved 15 June 2021. accessed via "Catalogs, Brochures, Advertisements". www.sliderulemuseum.com. International Slide Rule Museum. from the original on 6 April 2021. Retrieved 15 June 2021.
  16. ^ "Barnard's Co-ordinate Calculator, spiral slide rule Science Museum Group Collection". collection.sciencemuseumgroup.org.uk. from the original on 16 June 2021. Retrieved 15 June 2021.
  17. ^ Whythe, D.J. (Spring 1999). "The Whythe Complex Calculator in Fuller Style". Journal of Oughtred Society. 8 (1): 15–17. from the original on 6 June 2021. Retrieved 6 June 2021.
  18. ^ a b Balilo, Erica. "Whythe-Fuller complex-number slide rule". collection.maas.museum. Museum of Applied Arts and Sciences. from the original on 6 June 2021. Retrieved 6 June 2021.
  19. ^ "Stanley Whythe Complex calculator". Oughtred Society Rarities Gallery. Oughtred Society. from the original on 6 June 2021. Retrieved 6 June 2021.
  20. ^ Feely & Schure (1995), p. 33.
  21. ^ Larard & Golding (1907), p. 115, (misprinted as page 11).
  22. ^ Fuller (1879a), p. 36.
  23. ^ Stanley (1900), p. 261, (page 275 online).
  24. ^ Lovett, Rod; Hume, Ted (eds.). "Thacher Slide Rule by K&E". Classic Slide Rules. Oughtred Society & United Kingdom Slide Rule Circle. from the original on 11 August 2020. Retrieved 15 June 2021.
  25. ^ Otnes, Bob (March 1993). "Thacher Notes" (PDF). Journal of the Oughtred Society. 2 (1): 21–24. (PDF) from the original on 16 June 2021. Retrieved 16 June 2021. quoting from a handout by Keuffel and Esser
  26. ^ a b c d Turner (1998), pp. 87–89.
  27. ^ Tympas (2017).
  28. ^ a b Girvan, Ray (1 May 2003). "The revealed grace of the mechanism: computing after Babbage". Scientific Computing World. from the original on 27 February 2021. Retrieved 15 June 2021.
  29. ^ Tout, Nigel. "Slide Rule". Vintage Calculator Collectors Museum. from the original on 13 January 2021. Retrieved 16 June 2021.
  30. ^ a b Szondy, David (11 October 2016). "Curta calculator: The mechanical marvel born in a Nazi death camp". New Atlas. from the original on 23 December 2020. Retrieved 15 June 2021.
  31. ^ a b "Hewlett-Packard-35 scientific calculator handheld scientific calculator, 1972". HP Virtual Museum. Hewlett-Packard. from the original on 6 June 2021. Retrieved 6 June 2021.
  32. ^ Flamm, Bruce. "The Amazing Curta". Vintage Calculators web Museun. from the original on 27 June 2020. Retrieved 16 June 2021.
  33. ^ "George Fuller". Science Museum Group Collection. Science Museum. from the original on 7 June 2021. Retrieved 7 June 2021.
    Newmann, Kate. "George Fuller". www.newulsterbiography.co.uk. Ulster History Circle. from the original on 16 June 2021. Retrieved 13 June 2021.
  34. ^ Fuller (1879a), pp. 36–37.
  35. ^ "Patent:GB187801044 - Rechnerlexikon". Die große Enzyklopädie des mechanischen Rechnens (in German). Rechnerlexikon. from the original on 6 June 2021. Retrieved 6 June 2021.
    Fuller, George (2 September 1879). "Improvement in calculators". Google Patents. from the original on 10 June 2021. Retrieved 10 June 2021.
    "Patent Model for Cylindrical Slide Rule Invented by George Fuller". National Museum of American History. from the original on 9 August 2020. Retrieved 13 June 2021.
  36. ^ Fuller (n.d.), p. 1.
  37. ^ De Cesaris (2011), pp. 195–215.
  38. ^ Stanley (1900), p. 365, item 261, page 378 online).
  39. ^ De Cesaris (2011), p. 205.
  40. ^ "P.I.C./ British Thornton Slide Rules". www.sliderulemuseum.com. International Slide Rule Museum. from the original on 5 June 2021. Retrieved 10 June 2021.
  41. ^ a b Thornton, A.G. (1916). "Drawing, Surveying & Scientific Instruments" (PDF). Slide Rule Museum. p. 370. (PDF) from the original on 7 April 2021. Retrieved 10 June 2021.
  42. ^ "U.K. Price List - February, 1976" (PDF). drawing-instruments.groups.io. W.F. Stanley and company. p. 5. (PDF) from the original on 12 June 2021. Retrieved 10 June 2021.
  43. ^ McCoy, Clark. "Collection of pages from K&E catalogues for the 4015 Fuller's family of Slide Rules". Keuffel & Esser catalogs. from the original on 7 April 2021. Retrieved 10 June 2021.
  44. ^ Tympas (2017), p. 25.
  45. ^ Nichols & Hopp (2009), p. 38.
  46. ^ Baxandall, David; Eckert, Wallace John (1949). "Calculating machines". In Yust, Walter (ed.). Encyclopædia Britannica (revised 14th ed.). University of Chicago. p. 553.
  47. ^ Ceruzzi (1983), Chapter 5.
  48. ^ Hartree (1958), p. 11, (page 30 online).
  49. ^ Hartree (1958), p. 24, (page 44 online).
  50. ^ Tympas (2017), p. 31.
  51. ^ "History of the 9100A desktop calculator, 1968". HP Virtual Museum. Hewlett-Packard. from the original on 11 March 2010. Retrieved 6 June 2021.
  52. ^ Stoll, Cliff (May 2006). "When Slide Rules Ruled". Scientific American. 294 (5): 80–87. Bibcode:2006SciAm.294e..80S. doi:10.1038/scientificamerican0506-80. PMID 16708492.
  53. ^ a b Pickworth (1900), pp. 85–86, (pages 91–92 in pdf).
  54. ^ Walker, David. "Slide rules: The quest for precision and accuracy – making the scales longer". www.microscopy-uk.org.uk. Microscopy Magazine. Retrieved 11 June 2021.
  55. ^ Fuller (n.d.), p. 5.
  56. ^ Larard & Golding (1907), pp. 116–117.
  57. ^ Fuller (n.d.), p. 7.
  58. ^ Larard & Golding (1907), pp. 117–119.
  59. ^ Fuller (n.d.), p. 12.
  60. ^ Fuller (n.d.), p. 11.
  61. ^ Fuller (n.d.), pp. 16–25.

Works cited edit

  • Ceruzzi, Paul E. (1983). "5 Faster, Faster: The ENIAC". Reckoners: The Prehistory of the Digital Computer, from Relays to the Stored Program Concept, 1935-1945. Greenwood Press. ISBN 0-313-23382-9. – author granted permission to display on website.[1]
  • De Cesaris, Robert G. (September 2011). "The Fuller Calculator Revisited". Rod Lovett's Slide Rules. UK Slide Rule Circle, Oughtred Society.
  • Feely, Wayne; Schure, Conrad (March 1995). "The Fuller Calculating Instrument". Journal of the Oughtred Society. 4 (1): 33–40.
  • Fuller, George (8 May 1879). "Spiral Slide Rule". Nature. 20 (2). London, Macmillan: 36–37. Bibcode:1879Natur..20...36.. doi:10.1038/020036a0. S2CID 4079378.
  • Fuller, George (n.d.). Instructions for the use of the Fuller Calculator. London: W.F. Stanley & Co. from the original on 10 June 2017.
  • Hartree, Douglas R. (1958). Numerical Analysis. Oxford University Press.
  • Hopp, Peter M. (Autumn 2003). "How many Fullers make 5?". Sliderule Gazette. 4.
  • Hopp, M. (Autumn 2000). "Fuller Style Calculators". Slide Rule Gazette. 1: 25–32. updated at Nichols & Hopp (2009)
  • Larard, Charles E.; Golding, Henry A. (1907). Practical Calculations for Engineers. Charles Griffin. pp. 115–119.
  • Nichols, David (Autumn 2009). "Fuller Calculator Boxes - Differing Styles". Sliderule Gazette. 10: 3–8.
  • Nichols, David; Hopp, Peter M. (Autumn 2009). "Fuller Calculator Transition Points: An Update". Slide Rule Gazette. 10: 38.
  • Pickworth, Charles N. (1900). "Long-scale slide rules: Fuller's Calculating Rule". The Slide Rule: A Practical Manual (PDF) (6 ed.). Emmot & Pickworth.
  • Stanley, William Ford (1900). Mathematical Drawing and Measuring Instruments (Seventh ed.). London: E. & F.N. Spon.
  • Turner, Gerard L'Estrange (1998). Gilbert, John; Ayers, Tim (eds.). Scientific instruments, 1500-1900: an introduction. London: Philip Wilson. pp. 87–89. ISBN 9780520217287.
  • Tympas, Aristotle (2017). Calculation and computation in the pre-electronic era : the mechanical and electrical ages. History of Computing. New York: Springer. doi:10.1007/978-1-84882-742-4. ISBN 978-1-84882-741-7. S2CID 29384955.

Further reading edit

 
Wikimedia Commons has media related to Professor Fuller's Calculating Slide Rule.
  • "Fuller's Spiral Cylindrical Slide Rule". National Museum of American History. Smithsonian Museum. – description of Model 1
  • Chamberlain, Edwin J. (Spring 1999). "Long-Scale Slide Rules". Journal of the Oughtred Society. 8 (1): 24–35.
  • Stanley, William Ford (1901). Surveying and Leveling Instruments : Third Edition. London: E. & F.N. Spon. pp. 542–543.
  • Pflugfelder, Bob (29 October 2021). Mother of all Slide Rules: The Fuller Calculator (video). Northern Michigan: ResearchFlatMoon. Retrieved 4 November 2021. Fuller calculator instructional video

April 12, 2024
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This article is about a type of slide rule For the American slave see Thomas Fuller mental calculator The Fuller calculator sometimes called Fuller s cylindrical slide rule is a cylindrical slide rule with a helical main scale taking 50 turns around the cylinder This creates an instrument of considerable precision it is equivalent to a traditional slide rule 25 40 metres 1 000 inches long It was invented in 1878 by George Fuller professor of engineering at Queen s University Belfast and despite its size and price it remained on the market for nearly a century because it outperformed nearly all other slide rules Fuller calculator Fuller Bakewell model of 1928As with other slide rules the Fuller is limited to calculations based on multiplication and division with additional scales allowing for trigonometical and exponential functions The mechanical calculators produced in the same era were generally restricted to addition and subtraction with only advanced versions like the Arithmometer able to multiply and divide Even these advanced machines could not perform trigonometry or exponentiation and they were bigger heavier and much more expensive than the Fuller In the mid twentieth century the handheld Curta mechanical calculator became available which also competed in convenience and price However for scientific calculations the Fuller remained viable until 1973 when it was made obsolete by the HP 35 handheld scientific electronic calculator Contents 1 Design 1 1 Model 1 the standard model 1 2 Other Fuller models 1 3 Comparison with other slide rules and contemporaneous calculators 2 Invention sales and demise 3 Operation 3 1 Multiplication and division 3 2 Determining logarithms 3 3 Trigonometry and log functions 4 Notes 5 References 5 1 Citations 5 2 Works cited 6 Further readingDesign editModel 1 the standard model edit nbsp Fuller calculator annotatedIn essence the calculator consists of three separate hollow cylindrical parts that can twist and slide over each other about a common axis without any tendency to slip The following details describe the version made between 1921 and 1935 There is a papier mache cylinder marked D in the annotated photograph some 30 centimetres 12 inches long and 6 2 centimetres 2 4 in in diameter fastened to a mahogany handle A second papier mache cylinder marked C 16 3 centimetres 6 4 in long and 8 1 centimetres 3 2 in diameter is a slide fit over the first Both cylinders are covered in paper varnished with shellac The second outer cylinder is printed with the slide rule s primary logarithmic scale in the form of a 50 turn helix 12 70 metres 500 inches 41 ft 8 in long with annotations on the scale going from 100 to 1000 A brass tube with a mahogany cap at the top is a slide fit into the first cylinder 1 2 3 4 A brass pointer with an engraved index marker at its tip marked A is attached to the handle so that it points to a place on the primary logarithmic scale depending on the position to which the scale on cylinder C has been adjusted A second brass pointer marked B is attached to the top cap pointing down over the logarithmic scale and it is positioned by rotating and sliding the cap at the top This pointer has four index marks marked B1 B2 B3 B4 such that whichever one is convenient may be used 1 2 Printed on the inner cylinder D are simply tables of data for reference purposes 5 nbsp Fuller calculator in caseThe calculator was sold in a hinged mahogany case 46 by 12 by 11 centimetres 18 1 in 4 7 in 4 3 in which if required holds the instrument when in use by means a brass support that can be latched to the outer end of the case 6 7 Out of its case the calculator weighs about 900 grams 32 oz 8 For all except the earliest instruments the last two digits of the date and a serial number believed to be consecutively allocated are stamped at the top of pointer B 9 Other Fuller models edit The calculator described above was called Model No 1 6 Model 2 had scales on the inner cylinder for calculating logs and sines The Fuller Bakewell model 3 had two scales of angles printed on the inner cylinder to calculate cosine and sine cosine note 1 for use by engineers and surveyors for tacheometry calculations note 2 5 12 A smaller model with a 5 1 metres 200 in scale was available for a short time but very few survive In about 1935 the brass tube was replaced by one of phenolic resin and in about 1945 the mahogany was replaced by Bakelite 13 Included in Stanley s 1912 catalogue and continuing there until 1958 was Barnard s Coordinate calculator It is very similar in construction to the Fuller instruments but its pointers have multiple indices so additional trigonometrical functions can be used It cost slightly less than the Fuller Bakewell and a 1919 example is held by the Science Museum London 14 15 16 In 1962 the Whythe Fuller complex number calculator was introduced 17 18 As well as being able to multiply and divide complex numbers it can convert between Cartesian and polar coordinates 19 Comparison with other slide rules and contemporaneous calculators edit The calculator s unusual single scale design note 3 makes its 12 70 metre 500 inch helical spiral equivalent to a scale twice this length on a traditional slide rule 25 40 metres 1 000 inches long The scale can always be read to four significant figures and often to five 21 22 In 1900 William Stanley whose firm manufactured and sold scientific instruments including the Fuller calculator described the slide rule as possibly the highest refinement in this class of rules 23 When it was introduced the Fuller calculator had a much greater precision than other slide rules although the Thacher instrument became available a couple of years later This was made in the United States and was comparable in size and precision but radically different in design 24 25 26 27 However both of these types of slide rule required some skill to operate accurately compared with mechanical calculators which manipulated exact numerical digits rather than using positioning and reading from a graduated scale Mechanical calculators could only add and subtract which the Fuller did not do at all although models such as the Arithmometer could perform all four functions of elementary arithmetic 26 28 29 No mechanical calculators could calculate transcendental functions which slide rules could be designed to do and they were bigger heavier and much more expensive than any slide rule including the Fuller 26 28 30 However a revolutionary miniature mechanical calculator went on sale in the mid twentieth century while Curt Herzstark had been imprisoned in a Nazi concentration camp in World War II he had developed the design of the handheld Curta mechanical calculator It was simple to use and being digital was completely accurate 30 Because of these advantages and despite its somewhat higher price its total sales were 150 000 over ten times more than the Fuller Its range of mathematical calculations was seen as being adequate However for scientific calculations the Fuller remained viable until 1973 when along with the Curta it was made obsolete by the Hewlett Packard HP 35 handheld scientific electronic calculator 31 26 32 Invention sales and demise edit nbsp Fuller s 1879 U S patent drawingThe calculator was invented by George Fuller 1829 1907 33 professor of engineering at Queen s University Belfast Queen s College at that time 3 He patented it in Britain in 1878 described it in Nature in 1879 and in that year he also patented it the United States depositing a patent model 34 35 Fuller s calculators were manufactured by the scientific instrument maker W F Stanley amp Co of London who made nearly 14 000 between 1878 and 1973 8 36 37 5 In Britain the prices charged by W F Stanley in 1900 were for model 1 3 equivalent to 345 in 2021 and for model 3 4 10s 38 note 4 The Whythe Fuller model was advertised in a 1962 W F Stanley catalogue at 21 477 in 2021 18 The calculator was still listed in Stanley s catalogue in 1976 note 5 when model 1 cost 60 459 in 2021 and model 2 was 61 25 42 In the United States the instrument was marketed by Keuffel and Esser who only supplied model 1 They described it as Fuller s Spiral Slide Rule and over the period it was sold between 1895 and 1927 it rose in price from 28 to 42 falling from 1025 to 737 in 2023 prices 43 note 6 From the time when serial numbers were first stamped about 1900 to when production ceased in 1973 around 14 000 instruments were made note 7 Production was about 180 per year overall but it declined after about 1955 9 45 In 1949 Encyclopaedia Britannica noting that the Fuller had been designed in 1878 reported that it has been in considerable use up to the present time 46 In 1958 the mathematician and physicist Douglas Hartree note 8 wrote that the Fuller is cheap compared with a desk machine note 9 and may be found very useful in work for which its accuracy is adequate and in circumstances in which the cost of a desk machine is prohibitive With one of these slide rules and an adding machine much useful numerical work can be done 49 In 1968 the standard Fuller cost about 50 at a time when an electronic Hewlett Packard HP 9100A desktop calculator weighing 40 pounds 18 kg cost just under 5000 50 51 But in 1972 Hewlett Packard introduced the HP 35 the first handheld calculator with scientific functions at 395 the Fuller went out of production the next year 52 31 Operation editMultiplication and division edit nbsp Multiplication procedureThe instrument operates on the principle that two pointers are set at an appropriate separation on the helical scale of the calculator The relevant numbers are indexed by adjusting separately both the movable cylinder and the movable pointer Since the scale is logarithmic the separation represents the ratio of the numbers If the cylinder is then moved without altering the positions of the pointers this same ratio applies between any other pair of numbers addressed 53 In other words it is a logarithmic Gunter s scale wound into a helix with Gunter s compass points being provided by pointers A and B 54 To multiply two numbers p and q cylinder C is rotated and shifted until pointer A points to p and pointer B is then moved so B1 points to 100 Next cylinder C is moved so B1 points to q note 10 The product is then read from the pointer A The decimal point is determined as with an ordinary slide rule At the end of a calculation the slide rule is already positioned to continue with further multiplications p x q x r 1 To divide p by q cylinder C is rotated and shifted until pointer A points to p B1 is brought to q cylinder C is moved to bring 100 to B1 and the quotient is read from pointer A 56 It turns out to be particularly efficient to alternate multiplication with division 57 Determining logarithms edit nbsp Detail of scales for log calculation log10 1 1 0 04 0 0014 There are two other scales inscribed on the calculator which allow logarithms to be calculated and enabling such evaluations as pq and pq displaystyle sqrt q p nbsp 58 53 The scales are linear and one is engraved along the length of pointer B and the other printed around the circumference of the top of cylinder C Index B1 is set to the relevant value on cylinder C and then two readings are taken The first reading is from the scale on pointer B where it crosses the topmost spiral of the helical scale on the cylinder The second reading is from the scale at the top circumference of cylinder C where it crosses the left edge of pointer B The sum of the readings provides the mantissa of the log of the value note 11 60 Trigonometry and log functions edit nbsp Model 3 Fuller Bakewell use of sine cosine note 1 For model 2 instruments with scales on the inner cylinder D there is an index mark inscribed on both the top and bottom edges of cylinder C As an example of use when the lower index mark is set to an angle printed on the lower scale on cylinder D pointer A points to the corresponding value of sine on cylinder C The same approach apples for the log scale on the upper part of cylinder D note 12 The model 3 Fuller Bakewell is used in the same way but its scales on cylinder D are for cosine and sine cosine note 1 note 2 see photograph 61 Notes edit a b c The means multiplication The is the mid line dot operator a b Stadiametric rangefinding for an anallactic tacheometer with its vernier scale reading zero for a horizontal line of sight the distance to a vertical stadia rod d and the height of the stadia rod above the tacheometer v may be calculated as follows d K S cos2a and v K S sina cosa where K is a constant of the tacheometer normally 100 S is the difference in height on the stadia rod as intercepted by the tacheometer s cross hairs the stadia interval and a is the vertical angle measured by the tacheometer 10 11 There is a single scale for multiplication and division The other ancillary scales are for other purposes transcendental functions 20 In 1907 and 1916 W F Stanley sold both model 1 and model 2 for 3 15s for 1907 422 in 2021 for 1916 270 in 2021 and the Fuller Bakewell model 3 for 4 10s for 1907 506 in 2021 for 1916 325 in 2021 39 The firm A G Thornton in Manchester made slide rules and also sold the Fuller their 1916 pricelist shows 3 3s 6d 229 in 2021 for model 1 3 19s 0d for model 2 and 4 15s 0d for model 3 40 41 Thornton s 1916 price for the Thacher was 7 18s 6d 570 in 2021 41 Production stopped in 1973 but the calculators remained on sale In 1902 Thacher s calculating instrument was priced at 35 plus 10 for a reading glass 44 The ones sold in the U S were made in Britain and were numbered in the same way 9 Hartree had worked on ENIAC back in 1946 47 By desk machine Hartree meant a desktop calculator that could perform multiplication as well as addition and subtraction 48 Index mark B3 is used when B1 is off the scale B1 and B3 are fixed on pointer B so their distance apart is the full length 100 1000 of the scale Index marks B2 and B4 may be used if more convenient B1 or B2 provide greater accuracy than B3 or B4 3 55 For model 2 calculators there is a better way of using logs see trigonometry and log functions 59 The angle is in degrees For logs the argument is pointed to on cylinder C and the resulting logarithm to the base 10 is at the upper index point Note that pointer A and cylinder D are in a fixed orientation because both are rigidly attached to the handle References editCitations edit a b c Larard amp Golding 1907 p 116 a b Fuller n d pp 2 3 a b c Feely amp Schure 1995 pp 33 40 Hopp 2003 p 20 a b c Hopp 2000 pp 25 32 a b Fuller n d p 2 Nichols 2009 pp 3 8 a b Fuller s Cylindrical Slide Rule University of Queensland Physics Museum Archived from the original on 7 June 2021 Retrieved 7 June 2021 a b c Feely amp Schure 1995 p 35 Fuller n d pp 22 25 Tacheometric Surveying study notes PDF Civil Engineering B Tech II Year II Semester Annamacharya Institute of Technology amp Sciences Tirupati pp 7 8 Archived PDF from the original on 27 January 2021 Retrieved 15 June 2021 Fuller n d pp 17 23 De Cesaris 2011 pp 196 Hopp 2000 pp 28 29 W F Stanley amp Co Ltd PDF W F Stanley amp Co p 79 page 3 in pdf Archived PDF from the original on 7 April 2021 Retrieved 15 June 2021 accessed via Catalogs Brochures Advertisements www sliderulemuseum com International Slide Rule Museum Archived from the original on 6 April 2021 Retrieved 15 June 2021 Barnard s Co ordinate Calculator spiral slide rule Science Museum Group Collection collection sciencemuseumgroup org uk Archived from the original on 16 June 2021 Retrieved 15 June 2021 Whythe D J Spring 1999 The Whythe Complex Calculator in Fuller Style Journal of Oughtred Society 8 1 15 17 Archived from the original on 6 June 2021 Retrieved 6 June 2021 a b Balilo Erica Whythe Fuller complex number slide rule collection maas museum Museum of Applied Arts and Sciences Archived from the original on 6 June 2021 Retrieved 6 June 2021 Stanley Whythe Complex calculator Oughtred Society Rarities Gallery Oughtred Society Archived from the original on 6 June 2021 Retrieved 6 June 2021 Feely amp Schure 1995 p 33 Larard amp Golding 1907 p 115 misprinted as page 11 Fuller 1879a p 36 Stanley 1900 p 261 page 275 online Lovett Rod Hume Ted eds Thacher Slide Rule by K amp E Classic Slide Rules Oughtred Society amp United Kingdom Slide Rule Circle Archived from the original on 11 August 2020 Retrieved 15 June 2021 Otnes Bob March 1993 Thacher Notes PDF Journal of the Oughtred Society 2 1 21 24 Archived PDF from the original on 16 June 2021 Retrieved 16 June 2021 quoting from a handout by Keuffel and Esser a b c d Turner 1998 pp 87 89 Tympas 2017 a b Girvan Ray 1 May 2003 The revealed grace of the mechanism computing after Babbage Scientific Computing World Archived from the original on 27 February 2021 Retrieved 15 June 2021 Tout Nigel Slide Rule Vintage Calculator Collectors Museum Archived from the original on 13 January 2021 Retrieved 16 June 2021 a b Szondy David 11 October 2016 Curta calculator The mechanical marvel born in a Nazi death camp New Atlas Archived from the original on 23 December 2020 Retrieved 15 June 2021 a b Hewlett Packard 35 scientific calculator handheld scientific calculator 1972 HP Virtual Museum Hewlett Packard Archived from the original on 6 June 2021 Retrieved 6 June 2021 Flamm Bruce The Amazing Curta Vintage Calculators web Museun Archived from the original on 27 June 2020 Retrieved 16 June 2021 George Fuller Science Museum Group Collection Science Museum Archived from the original on 7 June 2021 Retrieved 7 June 2021 Newmann Kate George Fuller www newulsterbiography co uk Ulster History Circle Archived from the original on 16 June 2021 Retrieved 13 June 2021 Fuller 1879a pp 36 37 Patent GB187801044 Rechnerlexikon Die grosse Enzyklopadie des mechanischen Rechnens in German Rechnerlexikon Archived from the original on 6 June 2021 Retrieved 6 June 2021 Fuller George 2 September 1879 Improvement in calculators Google Patents Archived from the original on 10 June 2021 Retrieved 10 June 2021 Patent Model for Cylindrical Slide Rule Invented by George Fuller National Museum of American History Archived from the original on 9 August 2020 Retrieved 13 June 2021 Fuller n d p 1 De Cesaris 2011 pp 195 215 Stanley 1900 p 365 item 261 page 378 online De Cesaris 2011 p 205 P I C British Thornton Slide Rules www sliderulemuseum com International Slide Rule Museum Archived from the original on 5 June 2021 Retrieved 10 June 2021 a b Thornton A G 1916 Drawing Surveying amp Scientific Instruments PDF Slide Rule Museum p 370 Archived PDF from the original on 7 April 2021 Retrieved 10 June 2021 U K Price List February 1976 PDF drawing instruments groups io W F Stanley and company p 5 Archived PDF from the original on 12 June 2021 Retrieved 10 June 2021 McCoy Clark Collection of pages from K amp E catalogues for the 4015 Fuller s family of Slide Rules Keuffel amp Esser catalogs Archived from the original on 7 April 2021 Retrieved 10 June 2021 Tympas 2017 p 25 Nichols amp Hopp 2009 p 38 Baxandall David Eckert Wallace John 1949 Calculating machines In Yust Walter ed Encyclopaedia Britannica revised 14th ed University of Chicago p 553 Ceruzzi 1983 Chapter 5 Hartree 1958 p 11 page 30 online Hartree 1958 p 24 page 44 online Tympas 2017 p 31 History of the 9100A desktop calculator 1968 HP Virtual Museum Hewlett Packard Archived from the original on 11 March 2010 Retrieved 6 June 2021 Stoll Cliff May 2006 When Slide Rules Ruled Scientific American 294 5 80 87 Bibcode 2006SciAm 294e 80S doi 10 1038 scientificamerican0506 80 PMID 16708492 a b Pickworth 1900 pp 85 86 pages 91 92 in pdf Walker David Slide rules The quest for precision and accuracy making the scales longer www microscopy uk org uk Microscopy Magazine Retrieved 11 June 2021 Fuller n d p 5 Larard amp Golding 1907 pp 116 117 Fuller n d p 7 Larard amp Golding 1907 pp 117 119 Fuller n d p 12 Fuller n d p 11 Fuller n d pp 16 25 Works cited edit Ceruzzi Paul E 1983 5 Faster Faster The ENIAC Reckoners The Prehistory of the Digital Computer from Relays to the Stored Program Concept 1935 1945 Greenwood Press ISBN 0 313 23382 9 author granted permission to display on website 1 De Cesaris Robert G September 2011 The Fuller Calculator Revisited Rod Lovett s Slide Rules UK Slide Rule Circle Oughtred Society Feely Wayne Schure Conrad March 1995 The Fuller Calculating Instrument Journal of the Oughtred Society 4 1 33 40 Fuller George 8 May 1879 Spiral Slide Rule Nature 20 2 London Macmillan 36 37 Bibcode 1879Natur 20 36 doi 10 1038 020036a0 S2CID 4079378 Fuller George n d Instructions for the use of the Fuller Calculator London W F Stanley amp Co Archived from the original on 10 June 2017 Hartree Douglas R 1958 Numerical Analysis Oxford University Press Hopp Peter M Autumn 2003 How many Fullers make 5 Sliderule Gazette 4 Hopp M Autumn 2000 Fuller Style Calculators Slide Rule Gazette 1 25 32 updated at Nichols amp Hopp 2009 Larard Charles E Golding Henry A 1907 Practical Calculations for Engineers Charles Griffin pp 115 119 Nichols David Autumn 2009 Fuller Calculator Boxes Differing Styles Sliderule Gazette 10 3 8 Nichols David Hopp Peter M Autumn 2009 Fuller Calculator Transition Points An Update Slide Rule Gazette 10 38 Pickworth Charles N 1900 Long scale slide rules Fuller s Calculating Rule The Slide Rule A Practical Manual PDF 6 ed Emmot amp Pickworth Stanley William Ford 1900 Mathematical Drawing and Measuring Instruments Seventh ed London E amp F N Spon Turner Gerard L Estrange 1998 Gilbert John Ayers Tim eds Scientific instruments 1500 1900 an introduction London Philip Wilson pp 87 89 ISBN 9780520217287 Tympas Aristotle 2017 Calculation and computation in the pre electronic era the mechanical and electrical ages History of Computing New York Springer doi 10 1007 978 1 84882 742 4 ISBN 978 1 84882 741 7 S2CID 29384955 Further reading edit nbsp Wikimedia Commons has media related to Professor Fuller s Calculating Slide Rule Fuller s Spiral Cylindrical Slide Rule National Museum of American History Smithsonian Museum description of Model 1 Chamberlain Edwin J Spring 1999 Long Scale Slide Rules Journal of the Oughtred Society 8 1 24 35 Stanley William Ford 1901 Surveying and Leveling Instruments Third Edition London E amp F N Spon pp 542 543 Pflugfelder Bob 29 October 2021 Mother of all Slide Rules The Fuller Calculator video Northern Michigan ResearchFlatMoon Retrieved 4 November 2021 Fuller calculator instructional video Retrieved from https en wikipedia org w index php title Fuller calculator amp oldid 1200652339, wikipedia, wiki, book, books, library,

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