Curta

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A Curta II

The Curta is a mechanical calculating machine in the shape of a cylinder with a crank on top. The principle of operation is that of the double relay roller . It was developed by Curt Herzstark in the 1940s and produced from 1947 to 1970 by the Liechtenstein company Contina AG in a total of around 140,000. With a height of 85 mm and a diameter of 53 mm, the Curta I is the smallest mass-produced mechanical four-species calculating machine in the world.

Structure and variants

The Curta essentially consists of a central shaft, which takes on the function of the staggered roller and the crank is attached to the upper end, as well as the housing that carries the other elements.

The Curta staggered roller consists of a stack of sheets, with the individual sheets having different numbers of teeth and coding the digits in this way. Each sheet is available twice, so that you can subtract by moving the roller, which is done by adding the tens complement. Each digit is added to 9, the units digit to 10.

Example for a three-digit calculation: To subtract 173 from 451, add the complement 827 and get 1278. After removing the leading 1, you get the correct result.

The setting mechanism with eight or eleven setting handles is located on the side of the housing. The handles protrude inwards and move small gears there, which are loosely mounted on a second, further inner shaft. When the relay roller is rotated, the teeth turn the shafts of these wheels to different degrees depending on the position of the handle and the crank. In the rotatable upper part of the housing, the so-called “round car” (or “car” for short), there are the two result units, namely the 11 or 15-digit result counter (black background) and the 6 or 8-digit revolution counter (White background.) The transfer to these counters is done by small crown wheels at the upper end of the inner shafts. Additional gears and levers ensure the correct transfer of tens from point to point.

The round carriage is spring-loaded and can be moved after lifting it slightly, which is necessary for correct calculation. In the raised position, the extinguisher lever can also be operated, which with additional cogwheels sets all swept-over wheels of the result units to zero.

Curta I and II

Version I of the Curta could deliver up to eleven-digit results. It consists of 571 individual parts.

The later model Curta II delivers fifteen-digit results and consists of 719 individual parts. The side cover of the Curta II is painted black (model 1953) or gray (models 1958 and 1969).

service

The Curta masters the four basic arithmetic operations , whereby all calculations are based on additions and subtractions. For operation, you just have to keep in mind that every turn of the crank results in the correct addition of the setting mechanism (EW) to the result mechanism (RW) . As long as the EW is not adjusted, RW = EW * UW always applies after any number of rotations and displacements of the carriage.

The machine can be comfortably held in one hand and can also be operated largely with one hand. You can carry out all manipulations with your right hand (if you hold the machine in your left), but after some practice you will move the upper part (the "round car") and, if necessary, operate the EW with your left hand.

In the simplest case of addition, one of the two summands is entered via the adjusting slide on the outside of the cylinder (the digits can be read in small windows) and added to the result set by turning the crank. The crank revolutions are tracked with the correct sign in the rev counter (UW) . Then you set the second summand, turn the crank again and read off the result.

By repeating calculations with staggered positions (by lifting and moving the round car), multiplications can be carried out in a very similar way to written multiplication, i.e. one calculates one position after the other. If you pull the crank out a little bit in the axial direction, subtractions and - again in places - divisions can be calculated. A pawl prevents the crank from turning backwards, and a reset lever resets the total or revolution counter (or both). The crank, extinguishing lever and round carriage are locked against each other in such a way that only one of the operating elements can ever be outside of its basic position.

The "basic position" of the crank is clearly noticeable. If the crank is in this position, both counters are cleared, the adjustment handles are set to zero and the switch lever of the revolution counter is up, the machine is said to be "clear" .

Example: addition and subtraction

Exercise: 314.55 + 2135.30 - 875.92

The comma can be marked with the comma buttons if desired, but has no influence on the actual calculation.

  1. Make the machine clear.
  2. Set 31455, using handles 5 to 1.
  3. Make one turn of the crank. (31455 now appears in the result counter.)
  4. Set 213530, using handles 6 to 1.
  5. Make one turn of the crank. (The intermediate result 244985 appears in the result counter, which means 2449.85, but normally does not have to be taken into account.)
  6. Set 87592 using handles 5 to 1, do not forget to set handle 6 to zero.
  7. Optional : Move the switch lever of the revolution counter to the "counting in the opposite direction" position.
  8. Pull the crank out to the subtraction position and make a turn (this is abbreviated as "making a subtractive turn of the crank".)
  9. Read the result 1573.93. The revolution counter shows 3, the number of items, if you switched in step 7, otherwise 1.

Example: multiplication

Exercise: 4165.78 292.3

Again, commas are initially ignored. If you have the choice, it is advantageous to set the factor with more digits in the setting unit.

  1. Make the machine clear.
  2. Set 416578, using handles 6 to 1.
  3. Make three turns of the crank, check the revolution counter. Result counter shows 1249734.
  4. Raise the trolley and move it to position 2.
  5. Make two turns of the crank, check as above.
  6. Raise the trolley and move it to position 3.
  7. Make nine turns of the crank or - wiser - do a subtractive turn of the crank (290 = 300 - 10)
  8. Raise the trolley and move it to position 4.
  9. Make two or - if you have previously turned subtractively - three turns of the crank, control as above. The result counter shows 1217657494, which means 1217657.494, since the product must have as many decimal places as the two factors together, i.e. three here. Rough calculation in your head: 4000 * 300, 1.2 million, that's right. (As a rule, the result will be rounded off sensibly.)

With the same result, the calculation can be carried out "from above" - ​​that is, starting at carriage position 4 - or even in any order. The procedure is completely analogous to written multiplication, which is also done here and there.

For a further multiplication, the result usually has to be transferred back to the setting, which is only possible manually. (Larger table-top machines often offered an automatic system for this function.) For short numbers, however, there are tricks to carry out two multiplications "side by side" (see web link.)

Example: division

There are two methods for this: the "building up" and the "reducing" method. Both methods have in common that the divisor is located in the setting mechanism and the division result is read off in the revolution counter, which limits the number of digits to 6 or 8 (for the Curta II). In the incremental procedure, you start with zero in the result counter and build up the dividend here ; in the declining procedure, the dividend is already in the result set (advantageously as far to the left as possible, which may have to be taken into account beforehand), and the result is found by "cranking out ".

As a rule, a division does not open up; one then tries to find an approximate value that is as good as possible and rounds sensibly.

A. Building Procedure

Exercise: 310 / 4.68 =?

  1. Machine clear.
  2. Set 468 as far to the left as possible, i.e. handles 6 to 4 (Curta I) or 8 to 6 (Curta II). (The following example is calculated for the Curta I.)
  3. Car in position 6.
  4. Additive crank turns to get as close as possible to 310. After seven turns, 327.6 is reached, the 3 is no longer visible and must be added in front of the mind. (The comma tabs are excellently suited as a reminder, a fourth is placed between positions 9 and 10.) Alternatively, one could start a position lower; but this is at the expense of accuracy.
  5. Car in position 5.
  6. Subtractive crank turns (result mechanism is too high) to better approximate 310. After three turns 313.56 is reached, a fourth turn gives 308.88. The last value is closer (otherwise one would restore the last result by an additive rotation.)
  7. Car in position 4.
  8. Additive twists. After two rotations 309.816 is reached, a third rotation would lead to 310.284, both intermediate results are acceptable.
  9. Car in position 3.
  10. Further rotations towards 310. After four additive (if you have previously made two rotations) or six subtractive rotations you reach 310.0032. The revolution counter shows 662400, i.e. 66.24.
  11. Car in position 2.
  12. A subtractive rotation gives 309.99852.
  13. Car in position 1.
  14. Three additive rotations equals 309.999924, which is the closest approximation possible. The result 66.2393 can be read off in the revolution counter. The twelve-digit value would be 66.2393162393.

B. Degrading Process

Same as above, except that you start at the initial value and crank it to zero. To do this, the reversing lever must be moved to the "count in the opposite direction" position.

Exercise: 2040.3 / 17.26 =?

20403000000 is in the result unit, the revolution counter is cleared.

  1. Checking the reversing lever: lower position.
  2. Set 1726, as far to the left as possible, i.e. handles 6 to 3 (Curta I) or 8 to 5 (Curta II). (The following example is calculated for the Curta I.)
  3. Car in position 6.
  4. Make a subtractive twist.
  5. Car in position 5.
  6. Approach zero by further rotations; after two subtractive rotations, 99691000000 is reached, i.e. a negative number.
  7. Move the carriage to the other positions and crank further accordingly. You get 118.210, the twelve-digit value would be 118.209733488.

Example: Compound point calculation (rule of three)

This frequently occurring task can be calculated in abbreviated form with sufficiently "short" numbers or a sufficient number of digits by putting the numbers in the setting register at the ends. EW and RW are then mentally divided as far to the right as possible.

Task: 1980 * 395/144

  1. Machine clear.
  2. Set denominator 144 as far to the left as possible, i.e. handles 8 to 6 (again for Curta I.)
  3. Set the second factor 395 as far to the right as possible, i.e. handles 3 to 1. Three digits are required.
  4. Car in position 4, to the lowest point behind the mental division.
  5. By cranking you can now reach the first factor in the upper positions of the RW, i.e. one turn in position 4, four turns in position 3, two subtractive turns in position 2, and five more subtractive turns in position 1.
  6. RW now shows 19800543125, UW shows 001375. The end result is 5431.25 and the quotient of the first factor and denominator is 13.75.

Effectively, the EW and RW have been divided in the left part using the building-up method and in the right part the quotient has been multiplied by the second factor, here 395, in the same work step. If the division doesn't work out or the values ​​are too large, the numbers merge and you have to do the calculations individually.

Example: root

The determination of the square root is also possible with an iterative approximation method (see below). The prerequisite is that you know an approximate value, either from a table, by looking at the slide rule or by estimating.

Task: determine

The slide rule gives an approximation of 28.4. This value is first squared.

  1. Machine clear.
  2. Set 284, as far to the left as possible (taking into account the required doubling in step 9), i.e. handles 6 to 4 (Curta I) or 8 to 6 (Curta II). (The following example is calculated for the Curta I.)
  3. Car in position 6.
  4. Three additive rotations (as an 8 follows in the next lower digit); Check that the result counter does not overflow. (If this happens, you can start the calculation again one digit lower at the expense of accuracy, or you have to remember the dropped number as above and mentally add it to the front.)
  5. Car in position 5.
  6. Two subtractive turns. Checking the revolution counter, shows 280000.
  7. Car in position 4.
  8. Four additive twists. Checking the revolution counter shows 284000. The result counter shows 80656000000, ie 806.56, the estimate is a little too high, but very good.
  9. Set twice the original estimate in the same places in the setting. Here position 4 to “8”, position 5 to “6” and position 6 to “5”, taking into account the carryover of 2 * 284 = 568. With this setting you continue to crank until the result set shows a value as close as possible to the initial value.
  10. With the car still in position 4, a subtractive turn. You get 80088000000. (If the initial estimate was too bad, you can move the car back to higher positions.)
  11. Car in position 3.
  12. Crank until the required value is just reached or slightly exceeded. After six turns you get 80428800000.
  13. Car in position 2.
  14. Crank until the required value is approximately reached. After three subtractive rotations you get 80411760000.
  15. Car in position 1.
  16. Crank until the required value is approximately reached. After two additive rotations you get 80412896000. In the revolution counter you can read off: 283572, ie 28.3572, an improved approximation that is certainly accurate to four, probably to five digits. You will therefore normally stop here, but of course you can repeat the calculation with the improved value. (The twelve-digit value is 28.3571684059, so with a Curta I no further improvement is possible from a second approximation.)

Mathematical background ( Newton's method ): For the approximate value N, the unknown error E and the also unknown root R, and therefore . If one neglects the quadratic term in E, one recognizes that an estimate for E and thus a better value for N can be obtained by adding or subtracting 2N. As a rule, you can almost double the number of exact digits in each step. The method can also be used in a subtractive variant, analogous to division according to the degrading method. Finally, the method can also be used for third roots by using , cubing the estimated value and then using three times its square as a correction value.

Root: direct procedure

The direct root extraction succeeds with the "method of subtracting odd numbers" according to Töpler (see below). The radicand is divided into groups of two and you begin to subtract 1, 3, 5, etc., one after the other, at the units position of the highest group of two. As soon as an underflow occurs, the subtracted number is added up again, the next lower even number is set and the process is restarted at the next carriage position.

Task: determine

  1. Machine clear.
  2. Bring 27858575 as far to the left as possible into the result unit. Two steps are required for this, first the last two digits in position 4, then the remaining digits in position 8. (The example is calculated for the Curta I.)
  3. Clear revolution counter.
  4. Move the reversing lever to the “count in reverse” (lower) position.
  5. Set a 1 to the ones position of the highest group of two (ie the left “7”), all other levers to zero.
  6. Make a subtractive twist.
  7. Push the lever further to 3.
  8. Make a subtractive twist.
  9. Push the lever further to 5.
  10. Make a subtractive twist.
  11. Push the lever further to 7.
  12. Make a subtractive twist.
  13. Push the lever further to 9.
  14. Make a subtractive twist.
  15. Push the lever back to 1 and the next higher lever to 1 (corresponds to 11).
  16. Make a subtractive twist.
  17. An underflow occurs at this point. So you add the number again immediately.
  18. Lever 5 (the ones place) back one position (to 0, corresponding to 10).
  19. Car in position 5.
  20. Set lever 4 to 1.
  21. Make a subtractive twist.
  22. Push the lever further to 3.
  23. Make a subtractive twist.
  24. Push the lever further to 5.
  25. An underflow occurs again, so add the 5 again immediately.
  26. Lever back to 4.
  27. Car in position 4.
  28. Set lever 3 to 1.
  29. Make a subtractive twist.
  30. Push the lever further to 3.
  31. Make a subtractive twist.
  32. Push the lever further to 5.
  33. Make a subtractive twist.
  34. Push the lever further to 7.
  35. Make a subtractive twist.
  36. Push the lever further to 9.
  37. Make a subtractive twist.
  38. Slide lever back to 1 and move lever 4 from 4 to 5 to subtract 11.
  39. Make a subtractive twist.
  40. Push lever 3 further to 3.
  41. Make a subtractive twist.
  42. Slide lever 3 to 5 further.
  43. Make a subtractive twist.
  44. An underflow occurs again, so add the 5 (or 15) immediately again.
  45. Lever 3 back to 4.
  46. Car in position 3.
  47. Set lever 2 to 1.
  48. Subtract further in the same way, the underflow occurs after subtracting 17.
  49. Lever 2 back to 6.
  50. Car in position 2.
  51. Set lever 1 to 1.
  52. Subtract further in the same way, the underflow occurs after subtracting 3.
  53. At this point, there is no further lever available for the following positions, because you had to start at the ones position. Another position could be obtained by setting the front 2 only virtually and only evaluating the third underflow in the first steps, analogous to the procedure for division. The two values ​​that can be read in the revolution counter are 52.781 (before the subtraction of 3) and 52.782 (after the underflow). You can see from the result set that the exact value must be closer to the first, and could possibly gain further digits through interpolation. (The twelve-digit value is 52.781222987).

Development and sales

Inside view of the Curta
Curta I seen from above
Scope of delivery of a Curta II

The Curta is a cleverly constructed fine mechanical marvel. Its specialty is that, in contrast to the calculating machines that were otherwise common at the time, instead of one calculating gear, it only has one such gear per position, which is located on the central shaft and calculates the individual positions one after the other. This shaft is composed of segments that have different numbers of teeth and can therefore rotate the displays on the top to different degrees. All parts (on the Curta II over 700) are made of metal (exception: in machines from later production, the crank and extinguishing lever were made of plastic). Special deburring and selection processes ensure problem-free and silky smooth running.

It should be noted that this principle dates from the late 18th century; At that time, such calculating machines were known as "calculating mills". a. by Hahn, Schuster and Müller, and in the 19th century by Edmondson. A counterpart from a similar period is the “Gauß” machine from around 1905 by Christel Hamann . In its perfection and miniaturization, the Curta is unique.

It was designed by the Austrian office machine manufacturer Curt Herzstark . The construction plans in the Buchenwald concentration camp ended with a strong heart . The Curta only went into production after the war in Liechtenstein at the specially founded Contina AG. It was a technical sensation in its day; Although there were more powerful calculating machines, also with an electric drive and with extended functions such as transferring a result back to the setting mechanism, they were considerably larger and hardly transportable. Originally it was supposed to be called Liliput ; the name Curta is derived from the first name of the designer.

Production started in 1947. The 11-digit Type I was 85 mm high and had a diameter of 53 mm. It has a setting mechanism with eight digits and a result counter with eleven digits. From January 1954 the Type II was also produced with an 11-digit setting and 15-digit result counter. It is slightly larger at 90 × 65 mm, but its structure is basically the same.

A total of around 140,000 copies of the Curta were produced by 1971; however, exact production figures are not known. The price of version I in 1965 was 425 DM (864 € based on today's purchasing power), the larger Curta II cost 535 DM (1,087 €). In its time, the Curta was used in all areas in which pocket calculators can be found today, but in engineering the slide rule was often preferred (for point calculation ) , which delivers a more inaccurate result, but enables significantly faster calculations for this application. Due to its small size, it was very popular despite the comparatively high price, but was quickly supplanted by electronic computers ( HP-35 ) in the early 1970s . The Curtas have therefore disappeared from everyday use, but are a very popular collector's item, for which appropriate prices are paid, especially if they are in good condition.

For demonstration purposes there were also Curtas with openings in the housing and individual parts, also in large versions. A Curta with an electric drive (via battery) was rejected, as was a sample with trigonometric scales. The Vorarlberg engineer Elmar Maier dealt with the electrification of the Curta.

literature

  • C. Holub, U. Schröder, B. Schröder, H. Joss: Curt Herzstark - No gift for the leader - fate of a gifted inventor . Books on Demand, Norderstedt 2005, ISBN 3-8334-1136-8 .
  • Clifford Stoll: Calculating with the crank. In: Spektrum der Wissenschaft , No. 4, 2004, ISSN  0170-2971 , pp. 86ff.
  • Albert Rohrberg: Theory and Practice of Calculating Machines. BG Teubner, Stuttgart 1954. Page 14ff.
  • Friedrich L. Bauer, Karl Weinhart: Computer science - guide through the exhibition. Deutsches Museum, Munich 2004, ISBN 3-924183-92-9 .
  • Frank Thadeusz: Forgotten Master. Spiegel No. 27/2013, p. 106.
  • Curta - a adding machine and its story Podcast episode on Zeitsprung.fm

Video / DVD

  • Curta - a legend. The video shows the assembly of one of the last CURTAs.

Web links

Commons : Curta  - collection of images, videos and audio files