Renard series

Renard series are a system of preferred numbers dividing an interval from 1 to 10 into 5, 10, 20, or 40 steps.[1] This set of preferred numbers was proposed in the 1877 by French army engineer Colonel Charles Renard.[2][3][4] His system was adopted by the ISO in 1949[5] to form the ISO Recommendation R3, first published in 1953[6] or 1954, which evolved into the international standard ISO 3.[1] Renard's system of preferred numbers divides the interval from 1 to 10 into 5, 10, 20, or 40 steps. The factor between two consecutive numbers in a Renard series is approximately constant (before rounding), namely the 5th, 10th, 20th, or 40th root of 10 (approximately 1.58, 1.26, 1.12, and 1.06, respectively), which leads to a geometric sequence. This way, the maximum relative error is minimized if an arbitrary number is replaced by the nearest Renard number multiplied by the appropriate power of 10. One application of the Renard series of numbers is to current rating of electric fuses.

Base series

The most basic R5 series consists of these five rounded numbers, which are powers of the fifth root of 10, rounded to two digits. Note that the Renard numbers are not always rounded to the closest three-digit number to the theoretical geometric sequence:

R5: 1.00 1.60 2.50 4.00 6.30


  • If some design constraints were assumed so that the two screws in the gadget should be placed between 32 mm and 55 mm apart, the resulting length would be 40 mm, because 4 is in the R5 series of preferred numbers.
  • If a set of nails with lengths between roughly 15 and 300 mm should be produced, then the application of the R5 series would lead to a product repertoire of 16 mm, 25 mm, 40 mm, 63 mm, 100 mm, 160 mm, and 250 mm long nails.
  • If traditional English wine cask sizes had been metricated, rundlet (68 liters), barrel (119 liters), tierce (159 liters), hogshead (239 liters), puncheon (318 liters), butt (477 liters) and tun (954 liters) could have become 63 (or 60 by R″5), 100, 160 (or 150), 250, 400, 630 (or 600) and 1000 liters, respectively.

Alternative series

If a finer resolution is needed, another five numbers are added to the series, one after each of the original R5 numbers, and one ends up with the R10 series. These are rounded to a multiple of 0.05. Where an even finer grading is needed, the R20, R40, and R80 series can be applied. The R20 series is usually rounded to a multiple of 0.05, and the R40 and R80 values interpolate between the R20 values, rather than being powers of the 80th root of 10 rounded correctly. In the table below, the additional R80 values are written to the right of the R40 values in the column named "R80 add'l". The R40 numbers 3.00 and 6.00 are higher than they "should" be by interpolation, in order to give rounder numbers.

In some applications more rounded values are desirable, either because the numbers from the normal series would imply an unrealistically high accuracy, or because an integer value is needed (e.g., the number of teeth in a gear). For these needs, more rounded versions of the Renard series have been defined in ISO 3. In the table below, rounded values that differ from their less rounded counterparts are shown in bold.

least rounded
R5 R10 R20 R40 R80 add'l
1.00 1.00 1.00 1.00 1.03
1.06 1.09
1.12 1.12 1.15
1.18 1.22
1.25 1.25 1.25 1.28
1.32 1.36
1.40 1.40 1.45
1.50 1.55
1.60 1.60 1.60 1.60 1.65
1.70 1.75
1.80 1.80 1.85
1.90 1.95
2.00 2.00 2.00 2.06
2.12 2.18
2.24 2.24 2.30
2.36 2.43
2.50 2.50 2.50 2.50 2.58
2.65 2.72
2.80 2.80 2.90
3.00 3.07
3.15 3.15 3.15 3.25
3.35 3.45
3.55 3.55 3.65
3.75 3.87
4.00 4.00 4.00 4.00 4.12
4.25 4.37
4.50 4.50 4.62
4.75 4.87
5.00 5.00 5.00 5.15
5.30 5.45
5.60 5.60 5.75
6.00 6.15
6.30 6.30 6.30 6.30 6.50
6.70 6.90
7.10 7.10 7.30
7.50 7.75
8.00 8.00 8.00 8.25
8.50 8.75
9.00 9.00 9.25
9.50 9.75
10.0 10.0 10.0 10.0
medium rounded
R′10 R′20 R′40
1.00 1.00 1.00
1.10 1.10
1.25 1.25 1.25
1.40 1.40
1.60 1.60 1.60
1.80 1.80
2.00 2.00 2.00
2.20 2.20
2.50 2.50 2.50
2.80 2.80
3.20 3.20 3.20
3.60 3.60
4.00 4.00 4.00
4.50 4.50
5.00 5.00 5.00
5.60 5.60
6.30 6.30 6.30
7.10 7.10
8.00 8.00 8.00
9.00 9.00
10.0 10.0 10.0
most rounded
R″5 R″10 R″20
1.0 1.0 1.0
1.2 1.2
1.5 1.5 1.6
2.0 2.0
2.5 2.5 2.5
3.0 3.0
4.0 4.0 4.0
5.0 5.0
6.0 6.0 6.0
8.0 8.0
10 10 10

As the Renard numbers repeat after every 10-fold change of the scale, they are particularly well-suited for use with SI units. It makes no difference whether the Renard numbers are used with metres or millimetres. But one would need to use an appropriate number base to avoid ending up with two incompatible sets of nicely spaced dimensions, if for instance they were applied with both inches and feet. In the case of inches and feet a root of 12 would be desirable, that is, n12 where n is the desired number of divisions within the major step size of twelve. Similarly a base of two, eight, or sixteen would fit nicely with the binary units commonly found in computer science.

Each of the Renard sequences can be reduced to a subset by taking every nth value in a series, which is designated by adding the number n after a slash.[4] For example, "R10″/3 (1…1000)" designates a series consisting of every third value in the R″10 series from 1 to 1000, that is, 1, 2, 4, 8, 15, 30, 60, 120, 250, 500, 1000.

See also


  1. ^ a b ISO 3:1973-04 - Preferred Numbers - Series of Preferred Numbers. International Standards Organization (ISO). April 1973. Retrieved 2016-12-18. (Replaced: ISO Recommendation R3-1954 - Preferred Numbers - Series of Preferred Numbers. July 1954. (July 1953))
  2. ^ Kienzle, Otto Helmut (2013-10-04) [1950]. Written at Hannover, Germany. Normungszahlen [Preferred numbers]. Wissenschaftliche Normung (in German). 2 (reprint of 1st ed.). Berlin / Göttingen / Heidelberg, Germany: Springer-Verlag OHG. ISBN 978-3-642-99831-7. Retrieved 2017-11-01. (340 pages)
  3. ^ Paulin, Eugen (2007-09-01). Logarithmen, Normzahlen, Dezibel, Neper, Phon - natürlich verwandt! [Logarithms, preferred numbers, decibel, neper, phon - naturally related!] (PDF) (in German). Archived (PDF) from the original on 2016-12-18. Retrieved 2016-12-18.
  4. ^ a b "preferred numbers". Sizes, Inc. 2014-06-10 [2000]. Archived from the original on 2017-11-01. Retrieved 2017-11-01.
  5. ^ ISO 17:1973-04 - Guide to the use of preferred numbers and of series of preferred numbers. International Standards Organization (ISO). April 1973. Archived from the original on 2017-11-02. Retrieved 2017-11-02. […] Preferred numbers were first utilized in France at the end of the nineteenth century. From 1877 to 1879, Captain Charles Renard, an officer in the engineer corps, made a rational study of the elements necessary in the construction of […] aircraft. He computed the specifications […] according to a grading system […]. Recognizing the advantage to be derived from the geometrical progression, he adopted […] a grading system […] that would yield a tenth multiple of the value […] after every fifth step of the series […] Renard's theory was to substitute […] more rounded but […] practical values […] as a power of 10, positive, nil or negative. He thus obtained […] 10 16 25 40 63 100 […] continued in both directions […] by the symbol R5 […] the R10, R20, R40 series were formed, each adopted ratio being the square root of the preceding one […] The first standardization drafts were drawn up on these bases in Germany by the Normenausschuss der Deutschen Industrie on 13 April 1920, and in France by the Commission permanente de standardisation in document X of 19 December 1921. […] the commission of standardization in the Netherlands proposed their unification […] reached in 1931 […] in June 1932, the International Federation of the National Standardizing Associations organized an international meeting in Milan, where the ISA Technical Committee 32, Preferred numbers, was set up and its Secretariat assigned to France. On 19 September 1934, the ISA Technical Committee 32 held a meeting in Stockholm; sixteen nations were represented: Austria, Belgium, Czechoslovakia, Denmark, Finland, France, Germany, Hungary, Italy, Netherlands, Norway, Poland, Spain, Sweden, Switzerland, U.S.S.R. With the exception of the Spanish, Hungarian and Italian […] the other delegations accepted the draft […] Japan communicated […] its approval […] the international recommendation was laid down in ISA Bulletin 11 (December 1935). […] After the Second World War, the work was resumed by ISO. The Technical Committee ISO/TC 19, Preferred numbers, was set up and France again held the Secretariat. This Committee at its first meeting […] in Paris in July 1949 […] recommended […] preferred numbers defined by […] ISA Bulletin 11, […] R5, R10, R20, R40. This meeting was attended by […] 19 […] nations: Austria, Belgium, Czechoslovakia, Denmark, Finland, France, Hungary, India, Israel, Italy, Netherlands, Norway, Poland, Portugal, Sweden, Switzerland, United Kingdom, U.S.A., U.S.S.R. During […] subsequent meetings in New York in 1952 and […] the Hague in 1953, […] attended also by Germany, […] series R80 was added […] The draft thus amended became ISO Recommendation R3. […] (Replaced: ISO Recommendation R17-1956 - Preferred Numbers - Guide to the Use of Preferred Numbers and of Series of Preferred Numbers. 1956. (1955) and ISO R17/A1-1966 - Amendment 1 to ISO Recommendation R17-1955. 1966.)
  6. ^ De Simone, Daniel V. (July 1971). U.S. Metric Study Interim Report - Engineering Standards (PDF). U.S. Government Printing Office. Washington, USA: The National Bureau of Standards (NBS). NBS Special Publication 345-11 (Code: XNBSA). Archived (PDF) from the original on 2017-11-03. Retrieved 2017-11-03.

Further reading

Circuit breaker

A circuit breaker is an automatically operated electrical switch designed to protect an electrical circuit from damage caused by excess current from an overload or short circuit. Its basic function is to interrupt current flow after a fault is detected. Unlike a fuse, which operates once and then must be replaced, a circuit breaker can be reset (either manually or automatically) to resume normal operation.

Circuit breakers are made in varying sizes, from small devices that protect low-current circuits or individual household appliance, up to large switchgear designed to protect high voltage circuits feeding an entire city. The generic function of a circuit breaker, RCD or a fuse, as an automatic means of removing power from a faulty system is often abbreviated as OCPD (Over Current Protection Device).

Drill bit sizes

Drill bits are the cutting tools of drilling machines. They can be made in any size to order, but standards organizations have defined sets of sizes that are produced routinely by drill bit manufacturers and stocked by distributors.

In the U.S., fractional inch and gauge drill bit sizes are in common use. In nearly all other countries, metric drill bit sizes are most common, and all others are anachronisms or are reserved for dealing with designs from the US. The British Standards on replacing gauge size drill bits with metric sizes in the UK was first published in 1959.

A comprehensive table for metric, fractional wire and tapping sizes can be found at the drill and tap size chart.

E series of preferred numbers

The E series is a system of preferred numbers (also called preferred values) derived for use in electronic components. It consists of the E1, E3, E6, E12, E24, E48, E96 and E192 series, where the number after the 'E' designates the quantity of value "steps" in each series. Although it is theoretically possible to produce components of any value, in practice the need for inventory simplification has led the industry to settle on the E series for resistors, capacitors, inductors, and zener diodes. Other types of electrical components are either specified by the Renard series (for example fuses) or are defined in relevant product standards (for example IEC 60228 for wires).

Indian 200-rupee note

The Indian 200-rupee note (₹200) is a denomination of the Indian rupee. Post 2016 Indian banknote demonetisation, the new currency notes have been announced by the Reserve Bank of India-- ₹2,000, ₹500, ₹200, ₹100, ₹50, ₹20 and ₹10.In order to determine currency denominations, the Reserve Bank of India follows a variation of the Renard series, called the 1-2-5 series, in which a ‘decade’ or a 1:10 ratio is covered in 3 steps, such as 1-, 2-, 5-, 10-, 20-, 50-, 100-, 200-, 500-, 1,000, etc. The Reserve Bank of India described the 200-rupee notes as the missing link in the Renard series. Besides the Indian Rupee, Euro and British Pound sterlings are two of the most notable currencies that are denominated in the 1-2-5 series. In March 2017, the decision to introduce ₹200 notes was taken by the Reserve Bank of India with the consultation of the Ministry of Finance. The currency is produced by printing units of the government-run Security Printing and Minting Corporation of India or at printing presses in Mysore and Salboni, managed by the Reserve Bank of India-owned Bharatiya Reserve Bank Note Mudran Private Limited, reportedly by Times of India. The Government of India (GOI) had examined the introduction of the ₹200 notes that would help citizens transact easily.

In June 2017, a photograph of a ₹200 banknote went viral on social media platforms like Facebook and WhatsApp. RBI announced the specifications of new the 200 rupee note in the Mahatma Gandhi New Series, bearing the signature of Dr. Urjit R. Patel, Governor of the Reserve Bank of India on 25 August 2017.

Mesh (scale)

Mesh is a measurement of particle size often used in determining the particle-size distribution of a granular material. For example, a sample from a truckload of peanuts may be placed atop a mesh with 5 mm openings. When the mesh is shaken, small broken pieces and dust pass through the mesh while whole peanuts are retained on the mesh. A commercial peanut buyer might use a test like this to determine if a batch of peanuts has too many broken pieces. This type of test is common in some industries, and, to facilitate uniform testing methods, several standardized mesh series have been established.

Metal surfaces mechanically polished are designated as having a mechanical finish related to the abrasive used.

Many mesh sizes were historically given in the number of holes per inch; due to the width of the wires in the mesh, mesh numbers did not correspond directly to fractional inch sizes, and several different systems standardized with slightly different mesh sizes for the same mesh numbers.


The Modulor is an anthropometric scale of proportions devised by the Swiss-born French architect Le Corbusier (1887–1965).

It was developed as a visual bridge between two incompatible scales, the imperial and the metric system. It is based on the height of a man with his arm raised.

It was used as a system to set out a number of Le Corbusier's buildings and was later codified into two books.

Preferred metric sizes

Preferred metric sizes are a set of international standards and de facto standards that are designed to make using the metric system easier and simpler, especially in engineering and construction practices. One of the methods used to arrive at these preferred sizes is the use of preferred numbers and convenient numbers such as the Renard series, the 1-2-5 series to limit the number of different sizes of components needed.

One of the largest benefits of such limits is an ensuing multiplicative or exponential reduction in the number of parts, tools and other items needed to support the installation and maintenance of the items built using these techniques. This occurs because eliminating one diameter fastener will typically allow the elimination of a large number of variations on that diameter (multiple thread pitches, multiple lengths, multiple tip types, multiple head types, multiple drive types, and the tools needed for installing each, including multiple drill bits (one for each different thread pitch, material, and fit combination).

Preferred number

In industrial design, preferred numbers (also called preferred values or preferred series) are standard guidelines for choosing exact product dimensions within a given set of constraints.

Product developers must choose numerous lengths, distances, diameters, volumes, and other characteristic quantities. While all of these choices are constrained by considerations of functionality, usability, compatibility, safety or cost, there usually remains considerable leeway in the exact choice for many dimensions.

Preferred numbers serve two purposes:

Using them increases the probability of compatibility between objects designed at different times by different people. In other words, it is one tactic among many in standardization, whether within a company or within an industry, and it is usually desirable in industrial contexts (unless the goal is vendor lock-in or planned obsolescence)

They are chosen such that when a product is manufactured in many different sizes, these will end up roughly equally spaced on a logarithmic scale. They therefore help to minimize the number of different sizes that need to be manufactured or kept in stock.Preferred numbers represent preferences of simple numbers (such as 1, 2, and 5) multiplied by the powers of a convenient basis, usually 10.

Prodromus Entomology

Prodromus Entomology is one of the earliest books about Australian natural history, and the first book about Australia containing plates engraved in Australia. The full title of the first edition is Prodromus Entomology. Natural History of Lepidopterous Insects of New South Wales, collected, engraved and faithfully painted after nature.

Sasha Roiz

Sasha Roiz (born October 21, 1973) is an Israeli-born Canadian-American actor. He is best known for his portrayals of Sam Adama in the science fiction television series Caprica and Captain Sean Renard in the American dark fantasy television series Grimm.

Ted Childs

Ted Childs commenced training as a programme director with ABC Television in 1962. He went on to produce and direct a wide variety of factual programmes and documentary films, including episodes of This Week, then the ITV current affairs flagship and also contributed to the acclaimed World At War Series.

He was one of the founding fathers of Euston Films, the film production company established by Thames Television in the early 1970s. Whilst there he produced The Sweeney, Special Branch and Quatermass Series, together with a number of theatrical and television films, as well as writing and/or directing films and series episodes for both ITV and the BBC.

In 1984, he was appointed Controller of Drama at Central Television and, subsequently, Managing Director of Central Films. In this dual role, he acted as executive producer on an extensive range of films and series for ITV. These included Inspector Morse, Soldier Soldier, Peak Practice, Sharpe, Brother Cadfael, Chancer, Gone to The Dogs, Kavanagh QC and Thieftakers.

Following the acquisition of Central Television by Carlton Communications, he opted to assume a freelance role. He continued to write and develop new television drama and films as an executive producer. As such, he was responsible for the Heat Of The Sun Series, the Morse film The Wench Is Dead, the television films: Into the Blue, Goodnight Mr. Tom and The Waiting Time, together with a Kavanagh QC film special, the Monsignor Renard Series and The Remorseful Day, the final episode in the Morse canon. In 2004/5 he was responsible for developing The Brief, a legal series. He acted as executive producer on two series of The Brief. He also acted as executive producer on Lewis, a television format derived from the Morse Series.

In 1991, The Production Guild honoured him with its annual Award of Merit. He was elected a Fellow of the Royal Television Society (RTS) in 1992. He was Chairman of the British Academy of Film and Television Arts (BAFTA) in 1994 and 1995. He received the RTS Award for outstanding creative contribution to British Television in 1995. In the same year he was awarded the RTS Baird Medal for outstanding contribution to British Regional Television. In the 1997 New Year’s Honours List, he was made an OBE for services to Broadcast Television. Also, in 1997, he was elected an Honorary Fellow of the Moving Image Society (formerly the British Kinematograph and Television Society.) In May 1998, he received the BAFTA Award for outstanding creative contribution to television. In 2000, he received the Broadcasting Press Guild’s Harvey Lee Award for outstanding contribution to broadcasting. In July, 2015, the University of Nottingham awarded him the honorary degree of doctor of letters.

The Tunnel (TV series)

The Tunnel (French: Tunnel) is a British-French crime drama television series adapted from the 2011 Danish-Swedish crime series The Bridge (Broen, Bron). The series began broadcasting on 16 October 2013 on Sky Atlantic in the UK, and on 11 November 2013 on Canal+ in France. The series stars Stephen Dillane and Clémence Poésy as British and French police detectives Karl Roebuck and Elise Wassermann. The plot follows the two detectives working together to find a serial killer who left the upper-half of a French politician and the lower-half of a British prostitute in the Channel Tunnel at the midpoint between France and the UK. The killer is nicknamed the "Truth Terrorist" and is on a moral crusade to highlight many social problems, terrorising both countries in the process. As the series progresses, the killer's true intention is revealed.

The Anglo-French adaptation of The Bridge was announced as a joint project between Sky and Canal+ in January 2013. Tunnel head writer Ben Richards worked with Hans Rosenfeldt, the creator of the original series. Due to the setting, the dialogue of the series is bilingual, a first for British and French television. Filming took place between February and August 2013 with a budget of £15 million, and was shot on location in Kent, England and Nord-Pas-de-Calais, France. It was produced with both British and French crew members. The premieres on both Sky Atlantic and Canal+ received strong ratings for the respective channels, with an initial consolidated figure of almost 900,000 in the UK and 1.3 million in France. Critical reception of the series has been generally positive, with Dillane and Poésy's acting being praised, as well as the plot's grittiness. Some reviewers made favourable comparisons with The Bridge, though others criticised The Tunnel for being identical. The producers admit that the first episode is a copy of the original.

On 16 February 2015, Canal+ and Sky Atlantic announced that a second series would begin production in March, set to air in early 2016, entitled The Tunnel: Sabotage, and consisting of eight episodes. Series 2 would focus on the crash of an airliner into the English Channel, with Dillane and Poésy returning; it premiered on Canal+ on 7 March 2016. The debut on Sky Atlantic was originally set for 5 April 2016 but was put off until a week later in deference to the Brussels terrorist attacks on 22 March 2016. It premiered in the UK on 12 April 2016 and was made available via Sky's On Demand service.The renewal for a third and final series was announced on 20 January 2017 entitled The Tunnel: Vengeance and consisting of six episodes. It began filming in March 2017 and premiered on Sky Atlantic on 14 December 2017, with all episodes released on the same day. Canal+ did not announce a corresponding date for France at the time of the UK release. Season 3 premiered on Canal+ on 4 June 2018.In the United States, the first season aired on many PBS stations from June through August 2016. The second season was broadcast from June through August 2017. The third season aired July through August 2018.

Volume and displacement indicators for an architectural structure

The volume (W) and displacement (Δ) indicators have been discovered by Philippe Samyn in 1997 to help the search for the optimal geometry of architectural structures.

ISO standards by standard number

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