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IP over Avian Carriers

In computer networking, IP over Avian Carriers (IPoAC) is a joke proposal to carry Internet Protocol (IP) traffic by birds such as homing pigeons. IP over Avian Carriers was initially described in issued by the Internet Engineering Task Force, written by David Waitzman, and released on April 1, 1990. It is one of several April Fools' Day Request for Comments.

Waitzman described an improvement of his protocol in, IP over Avian Carriers with Quality of Service (1 April 1999). Later, in —released on 1 April 2011, and 13 years after the introduction of IPv6—Brian Carpenter and Robert Hinden published Adaptation of RFC 1149 for IPv6.

IPoAC has been successfully implemented, but for only nine packets of data, with a packet loss ratio of 55% (due to operator error), and a response time ranging from 3000 sec to over 6000 sec. Thus, this technology suffers from high latency.

Real-life implementation
On 28 April 2001, IPoAC was implemented by the Bergen Linux user group, under the name CPIP (for Carrier Pigeon Internet Protocol). They sent nine packets over a distance of approximately 5 km, each carried by an individual pigeon and containing one ping (ICMP echo request), and received four responses.

Script started on Sat Apr 28 11:24:09 2001 $ /sbin/ifconfig tun0 tun0     Link encap:Point-to-Point Protocol inet addr:10.0.3.2 P-t-P:10.0.3.1  Mask:255.255.255.255 UP POINTOPOINT RUNNING NOARP MULTICAST MTU:150  Metric:1 RX packets:1 errors:0 dropped:0 overruns:0 frame:0 TX packets:2 errors:0 dropped:0 overruns:0 carrier:0 collisions:0 RX bytes:88 (88.0 b) TX bytes:168 (168.0 b)

$ ping -c 9 -i 900 10.0.3.1 PING 10.0.3.1 (10.0.3.1): 56 data bytes 64 bytes from 10.0.3.1: icmp_seq=0 ttl=255 time=6165731.1 ms 64 bytes from 10.0.3.1: icmp_seq=4 ttl=255 time=3211900.8 ms 64 bytes from 10.0.3.1: icmp_seq=2 ttl=255 time=5124922.8 ms 64 bytes from 10.0.3.1: icmp_seq=1 ttl=255 time=6388671.9 ms

--- 10.0.3.1 ping statistics --- 9 packets transmitted, 4 packets received, 55% packet loss round-trip min/avg/max = 3211900.8/5222806.6/6388671.9 ms

Script done on Sat Apr 28 14:14:28 2001

This real life implementation was mentioned by the French member of parliament Martine Billard in the French National Assembly, during debates about HADOPI.

Risks
In December 2005, a Gartner report on bird flu that concluded "A pandemic wouldn't affect IT systems directly" was humorously criticized for neglecting to consider RFC 1149 and RFC 2549 in its analysis.

Known risks to the protocol include:
 * Carriers being attacked by birds of prey. RFC2549: "Unintentional encapsulation in hawks has been known to occur, with decapsulation being messy and the packets mangled."
 * Carriers being blown off course. RFC1149: "While broadcasting is not specified, storms can cause data loss."
 * The absence of viable local carriers. RFC6214: "In some locations, such as New Zealand, a significant proportion of carriers are only able to execute short hops, and only at times when the background level of photon emission is extremely low." This describes the flightless and nocturnal nature of kiwi.
 * Loss of availability of species, such as the extinction of the passenger pigeon.
 * Disease affecting the carriers. RFC6214: "There is a known risk of infection by the so-called H5N1 virus."
 * The network topologies supported for multicast communication are limited by the homing abilities of carriers. RFC6214: "... [carriers] prove to have no talent for multihoming, and in fact enter a routing loop whenever multihoming is attempted."

Other avian data transfer methods
Rafting photographers already use pigeons as a sneakernet to transport digital photos on flash media from the camera to the tour operator. Over a 30 mi distance, a single pigeon may be able to carry tens of gigabytes of data in around an hour, which on an average bandwidth basis compares very favorably to current ADSL standards, even when accounting for lost drives.

On March 12, 2004, Yossi Vardi, Ami Ben-Bassat, and Guy Vardi sent three homing pigeons a distance of 100 km, "each carrying 20–22 tiny memory cards containing 1.3 GB, amounting in total of 4 GB of data." An effective throughput of 2.27 Mbps was achieved. The purpose of the test was to measure and confirm an improvement over RFC 2549. Since the developers used flash memory instead of paper notes as specified by RFC 2549, the experiment was widely criticized as an example in which an optimized implementation breaks an official standard.

Inspired by RFC 2549, on 9 September 2009, the marketing team of The Unlimited, a regional company in South Africa, decided to host a tongue-in-cheek pigeon race between their pet pigeon Winston and local telecom company Telkom SA. The race was to send 4 gigabytes of data from Howick to Hillcrest, approximately 60 km apart. The pigeon carried a microSD card and competed against a Telkom ADSL line. Winston beat the data transfer over Telkom's ADSL line, with a total time of two hours, six minutes and 57 seconds from uploading data on the microSD card to completion of download from the card. At the time of Winston's victory, the ADSL transfer was just under 4% complete.

In November 2009, the Australian comedy/current-affairs television program Hungry Beast repeated this experiment. The Hungry Beast team took up the challenge after a fiery parliament session wherein the government of the time blasted the opposition for not supporting telecommunications investments, saying that if the opposition had their way, Australians would be doing data transfer over carrier pigeons. The Hungry Beast team had read about the South African experiment and assumed that, as a developed western country, Australia would have higher speeds. The experiment had the team transfer a 700 MB file via three delivery methods to determine which was the fastest: a carrier pigeon with a microSD card, a car carrying a USB stick, and a Telstra (Australia's largest telecom provider) ADSL line. The data was to be transferred from Tarana in rural New South Wales to the western-Sydney suburb of Prospect, New South Wales, a distance of 132 km by road. Approximately halfway through the race, the internet connection unexpectedly dropped and the transfer had to be restarted. The pigeon won the race with a time of approximately 1 hour 5 minutes, the car came in second at 2 hours 10 minutes, while the internet transfer did not finish, having dropped out a second time and not come back. The estimated time to upload completion at one point was as high as 9 hours, and at no point did the estimated upload time fall below 4 hours.

A similar pigeon race was conducted in September 2010 by tech blogger (trefor.net) and ISP Timico CTO Trefor Davies with farmer Michelle Brumfield in rural Yorkshire, England: delivering a five-minute video to a BBC correspondent 75 miles away in Skegness. The pigeon (carrying a memory card with a 300 MB HD video of Davies having a haircut) was pitted against an upload to YouTube via British Telecom broadband; the pigeon was released at 11:05 am and arrived in the loft one hour and fifteen minutes later while the upload was still incomplete, having failed once in the interim.

title: IP over Avian Carriers


text:
Alphonse de Polignac

Alphonse de Polignac (1826–1863) was a French mathematician and aristocrat. He his known for Polignac's Conjecture.

Biography
His father, Jules de Polignac (1780-1847) was prime minister of Charles X until the Bourbon dynasty was overthrown in the July Revolution of 1830. Alphonse was born in London during his father's time as ambassador to the United Kingdom. In 1849 he was admitted to Polytechnique and went onto serve in the Crimean War as an artillery officer, achieving the rank of Captain.

He was also a historian, a poet, a musician, and authored a translation of the play Faust by Goethe.

His work in mathematics mainly focused on Number Theory and he specifically worked with prime numbers.

Polignac's Conjecture
In his first year at Polytechnique Polignac formulated his eponymous conjecture, which states that: "For every positive integer k, there are infinitely many prime gaps of size 2k."

Other work in Mathematics
Polignac also conjectured Romanov's Theorem, which states that: "Every odd number larger than 3 can be written as the sum of an odd prime and a power of 2"The same year that he formulated his two most famous conjectures, Polignac also formulated a false conjecture which he presented to the Academy of Sciences. It incorrectly asserted that any odd number was equal to the power of 2 plus a prime number.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Alphonse de Polignac


text:
Phosgene

Phosgene is an organic chemical compound with the formula COCl2. It is a toxic, colorless gas; in low concentrations, its musty odor resembles that of freshly cut hay or grass. It can be thought of chemically as the double acyl chloride analog of carbonic acid, or structurally as formaldehyde with the hydrogen atoms replaced by chlorine atoms. Phosgene is a valued and important industrial building block, especially for the production of precursors of polyurethanes and polycarbonate plastics.

Phosgene is extremely poisonous and was used as a chemical weapon during World War I, where it was responsible for 85,000 deaths. It is a highly potent pulmonary irritant and quickly filled enemy trenches due to it being a heavy gas.

It is classified as a Schedule 3 substance under the Chemical Weapons Convention. In addition to its industrial production, small amounts occur from the breakdown and the combustion of organochlorine compounds, such as chloroform.

Structure and basic properties
Phosgene is a planar molecule as predicted by VSEPR theory. The C=O distance is 1.18 Å, the C−Cl distance is 1.74 Å and the Cl−C−Cl angle is 111.8°. Phosgene is a carbon oxohalide and it can be considered one of the simplest acyl chlorides, being formally derived from carbonic acid.

Production
Industrially, phosgene is produced by passing purified carbon monoxide and chlorine gas through a bed of porous activated carbon, which serves as a catalyst:
 * CO + Cl2 → COCl2 (ΔHrxn = −107.6 kJ/mol)

This reaction is exothermic and is typically performed between 50 and 150 °C. Above 200 °C, phosgene reverts to carbon monoxide and chlorine, Keq(300 K) = 0.05. World production of this compound was estimated to be 2.74 million tonnes in 1989.

Phosgene is fairly simple to produce, but is listed as a Schedule 3 substance under the Chemical Weapons Convention. As such, it is usually considered too dangerous to transport in bulk quantities. Instead, phosgene is usually produced and consumed within the same plant, as part of an "on demand" process. This involves maintaining equivalent rates of production and consumption, which keeps the amount of phosgene in the system at any one time fairly low, reducing the risks in the event of an accident. Some batch production does still take place, but efforts are made to reduce the amount of phosgene stored.

Atmospheric chemistry
Simple organochlorides slowly convert into phosgene when exposed to ultraviolet (UV) irradiation in the presence of oxygen. Before the discovery of the Ozone hole in the late 1970s large quantities of organochlorides were routinely used by industry, which inevitably led to them entering the atmosphere. In the 1970-80s phosgene levels in the troposphere were around 20-30 pptv (peak 60 pptv). However, these levels had not decreased significantly nearly 30 years later, despite organochloride production becoming restricted under the Montreal Protocol.

Phosgene in the troposphere can last up to about 70 days and is removed primarily by hydrolysis with ambient humidity or cloudwater. Less than 1% makes it to the stratosphere, where it is expected to have a lifetime of several years, since this layer is much drier and phosgene decomposes slowly through UV photolysis. Consequently, it does play a minor part in ozone depletion.

Combustion
Carbon tetrachloride (CCl4) can turn into phosgene when exposed to heat in air. This was a problem as carbon tetrachloride is an effective fire suppressant and was formerly in widespread use in fire extinguishers. There are reports of fatalities caused by its use to fight fires in confined spaces. Carbon tetrachloride's generation of phosgene and its own toxicity mean it is no longer used for this purpose.

Biologically
Phosgene is also formed as a metabolite of chloroform, likely via the action of cytochrome P-450.

History
Phosgene was synthesized by the Cornish chemist John Davy (1790–1868) in 1812 by exposing a mixture of carbon monoxide and chlorine to sunlight. He named it "phosgene" from Greek φῶς (phos, light) and γεννάω (gennaō, to give birth) in reference of the use of light to promote the reaction. It gradually became important in the chemical industry as the 19th century progressed, particularly in dye manufacturing.

Reactions and uses
The reaction of an organic substrate with phosgene is called phosgenation. Phosgenation of diols give carbonates (R = H, alkyl, aryl), which can be either linear or cyclic:
 * n HO\sCR2\sX\sCR2\sOH + n COCl2 → [\sO\sCR2\sX\sCR2\sO\sC(\dO)\s]_{n} + 2n HCl

An example is the reaction of phosgene with bisphenol A to form polycarbonates. Phosgenation of diamines gives di-isocyanates, like toluene diisocyanate (TDI), methylene diphenyl diisocyanate (MDI), hexamethylene diisocyanate (HDI), and isophorone diisocyanate (IPDI). In these conversions, phosgene is used in excess to increase yield and minimize side reactions. The phosgene excess is separated during the work-up of resulting end products and recycled into the process, with any remaining phosgene decomposed in water using activated carbon as the catalyst. Diisocyanates are precursors to polyurethanes. More than 90% of the phosgene is used in these processes, with the biggest production units located in the United States (Texas and Louisiana), Germany, Shanghai, Japan, and South Korea. The most important producers are Dow Chemical, Covestro, and BASF. Phosgene is also used to produce monoisocyanates, used as pesticide precursors (e.g. methyl isocyanate (MIC).

Aside from the widely used reactions described above, phosgene is also used to produce acyl chlorides from carboxylic acids:
 * R\sC(\dO)\sOH + COCl2 → R\sC(\dO)\sCl + HCl + CO2

For this application, thionyl chloride is commonly used instead of phosgene.

Laboratory uses
The synthesis of isocyanates from amines illustrates the electrophilic character of this reagent and its use in introducing the equivalent synthon "CO2+":
 * R\sNH2 + COCl2 → R\sN\dC\dO + 2 HCl, where R = alkyl, aryl

Such reactions are conducted on laboratory scale in the presence of a base such as pyridine that neutralizes the hydrogen chloride side-product.

Phosgene is used to produce chloroformates such as benzyl chloroformate:
 * R\sOH + COCl2 → R\sO\sC(\dO)\sCl + HCl

In these syntheses, phosgene is used in excess to prevent formation of the corresponding carbonate ester.

With amino acids, phosgene (or its trimer) reacts to give amino acid N-carboxyanhydrides. More generally, phosgene acts to link two nucleophiles by a carbonyl group. For this purpose, alternatives to phosgene such as carbonyldiimidazole (CDI) are safer, albeit expensive. CDI itself is prepared by reacting phosgene with imidazole.

Phosgene is stored in metal cylinders. In the US, the cylinder valve outlet is a tapered thread known as "CGA 160" that is used only for phosgene.

Alternatives to phosgene
In the research laboratory, due to safety concerns phosgene nowadays finds limited use in organic synthesis. A variety of substitutes have been developed, notably trichloromethyl chloroformate ("diphosgene"), a liquid at room temperature, and bis(trichloromethyl) carbonate ("triphosgene"), a crystalline substance.

Other reactions
Phosgene reacts with water to release hydrogen chloride and carbon dioxide:
 * COCl2 + H2O → CO2 + 2 HCl

Analogously, upon contact with ammonia, it converts to urea:
 * COCl2 + 4 NH3 → CO(NH2)2 + 2 [NH4]Cl

Halide exchange with nitrogen trifluoride and aluminium tribromide gives COF2 and COBr2, respectively.

Chemical warfare

It is listed on Schedule 3 of the Chemical Weapons Convention: All production sites manufacturing more than 30 tonnes per year must be declared to the OPCW. Although less toxic than many other chemical weapons such as sarin, phosgene is still regarded as a viable chemical warfare agent because of its simpler manufacturing requirements when compared to that of more technically advanced chemical weapons such as tabun, a first-generation nerve agent.

Phosgene was first deployed as a chemical weapon by the French in 1915 in World War I. It was also used in a mixture with an equal volume of chlorine, with the chlorine helping to spread the denser phosgene. Phosgene was more potent than chlorine, though some symptoms took 24 hours or more to manifest.

Following the extensive use of phosgene during World War I, it was stockpiled by various countries.

Phosgene was then only infrequently used by the Imperial Japanese Army against the Chinese during the Second Sino-Japanese War. Gas weapons, such as phosgene, were produced by the IJA's Unit 731.

Toxicology and safety
Phosgene is an insidious poison as the odor may not be noticed and symptoms may be slow to appear.

Phosgene at low concentrations, may have a pleasant odor of freshly mown hay or green corn, but has also been described as sweet, like rotten banana peels. The odor detection threshold for phosgene is 0.4 ppm, four times the threshold limit value (time weighted average). Its high toxicity arises from the action of the phosgene on the \sOH, \sNH2 and \sSH groups of the proteins in pulmonary alveoli (the site of gas exchange), respectively forming ester, amide and thioester functional groups in accord with the reactions discussed above. This results in disruption of the blood–air barrier, eventually causing pulmonary edema. The extent of damage in the alveoli does not primarily depend on phosgene concentration in the inhaled air, with the dose (amount of inhaled phosgene) being the critical factor. Dose can be approximately calculated as "concentration" × "duration of exposure". Therefore, persons in workplaces where there exists risk of accidental phosgene release usually wear indicator badges close to the nose and mouth. Such badges indicate the approximate inhaled dose, which allows for immediate treatment if the monitored dose rises above safe limits.

In case of low or moderate quantities of inhaled phosgene, the exposed person is to be monitored and subjected to precautionary therapy, then released after several hours. For higher doses of inhaled phosgene (above 150 ppm × min) a pulmonary edema often develops which can be detected by X-ray imaging and regressive blood oxygen concentration. Inhalation of such high doses can eventually result in fatality within hours up to 2–3 days of the exposure.

The risk connected to a phosgene inhalation is based not so much on its toxicity (which is much lower in comparison to modern chemical weapons like sarin or tabun) but rather on its typical effects: the affected person may not develop any symptoms for hours until an edema appears, at which point it could be too late for medical treatment to assist. Nearly all fatalities as a result of accidental releases from the industrial handling of phosgene occurred in this fashion. On the other hand, pulmonary edemas treated in a timely manner usually heal in the mid- and longterm, without major consequences once some days or weeks after exposure have passed. Nonetheless, the detrimental health effects on pulmonary function from untreated, chronic low-level exposure to phosgene should not be ignored; although not exposed to concentrations high enough to immediately cause an edema, many synthetic chemists (e.g. Leonidas Zervas) working with the compound were reported to experience chronic respiratory health issues and eventual respiratory failure from continuous low-level exposure.

If accidental release of phosgene occurs in an industrial or laboratory setting, it can be mitigated with ammonia gas; in the case of liquid spills (e.g. of diphosgene or phosgene solutions) an absorbent and sodium carbonate can be applied.

Accidents

 * The first major phosgene-related incident happened in May 1928 when eleven tons of phosgene escaped from a war surplus store in central Hamburg. Three hundred people were poisoned, of whom ten died.
 * In the second half of 20th century several fatal incidents implicating phosgene occurred in Europe, Asia and the US. Most of them have been investigated by authorities and the outcome made accessible to the public. For example, phosgene was initially blamed for the Bhopal disaster, but investigations proved methyl isocyanate to be responsible for the numerous poisonings and fatalities.
 * Recent major incidents happened in January 2010 and May 2016. An accidental release of phosgene gas at a DuPont facility in West Virginia killed one employee in 2010. The US Chemical Safety Board released a video detailing the accident. Six years later, a phosgene leak occurred in a BASF plant in South Korea, where a contractor inhaled a lethal dose of phosgene.
 * 2023 Ohio train derailment: A freight train carrying vinyl chloride derailed and burned in East Palestine, Ohio, releasing phosgene and hydrogen chloride into the air and contaminating the Ohio River.

title: Phosgene


text:
Archibald Leach (disambiguation)

Archibald Leach is the birth name of film actor Cary Grant (1904–1986).

Archibald Leach may also refer to:

 * Archie Leach, a fictional character played by John Cleese in the film A Fish Called Wanda
 * Archie Leach, a fictional character played by Richard Chamberlain in the TV series Leverage

title: Archibald Leach (disambiguation)


text:
Talk:Continued fraction

Ordinary arithmetic operations on continued fraction representation
While the representation of numbers as continued fractions is very pretty, I wonder how we do basic arithmetic operations like addition, subtraction, multiplication, and division directly in this form, without having to resort to converting them to ordinary fractions. I feel that the usefulness of this article can be greatly enhanced by this inclusion. Manoguru (talk) 18:22, 17 December 2019 (UTC)

"Complete convergent"?
The paper patz5.pdf cited under Pell's equation says on page 2:
 * ... If $$x_n = (P_n + \sqrt{D}) / Q_n$$ is the n-th complete convergent of the simple continued fraction for $$\omega = (P_0 + \sqrt{D}) / Q_0$$, ...

but I can't find any definition of complete convergent. Is it just another name for a complete quotient? If it is, perhaps someone sufficiently knowledgeable could mention it in that page. Hv (talk) 13:46, 22 May 2021 (UTC)

Requires citations and clarifications
This is a long article; and I jumped in to the "Examples" section. The table completely mystified me, in terms of "how do I get from the 'stuff' in the table to a continued fraction?" What I'm getting at is that the column and row headings are neither descriptive nor described -- at least a quick definition of terms or a "For example, using the table able, we can generate pi with the following continued fraction. Notice how the values correspond to the table entries ."

The citation that I'd like is one in the "History" section, where it states, "300 BCE Euclid's Elements contains an algorithm for the greatest common divisor which generates a continued fraction as a by-product" -- unfortunately, neither hotlink "Euclid's Elements" nor "greatest common divisor" leads me to text that deals with the "greatest common divisor which generates a continued fraction as a by-product" goal that I was looking for. Each leads me to its respective "narrow" definition, but nothing leads me to the overall concept about how GCD relates to continued fractions. 198.84.205.118 (talk) 00:03, 29 May 2021 (UTC)
 * I have edited the first item of section "History", and added in section "Motivation and notation": "The sequence of the integers that occur in this representation [of a rational numbers by a continued fraction] is the sequence of the successive quotients that are computed by the Euclidean algorithm". D.Lazard (talk) 08:32, 29 May 2021 (UTC)

Too hard formula represented on article
Correct formula for programming conversion $$a/b$$ to continued fraction is very simple and like this within loop:

an1 := an0 / bn0; // integer div (you need this $$a_n$$) bn1 := an0 % bn0; // integer mod an0 := bn0; bn0 := bn1;

Mvitaminus (talk) 23:28, 2 June 2021 (UTC)

Division by zero
The article currently indicates "where $a_{i}$ and $b_{i}$ can be any complex numbers." Before we required $b_{i}$ to be 1 -- because we were defining a simple continued fraction before -- and we also required that $a_{i}$ be a positive integer. As a consequence, this prevented division by zero in every convergent. The current definition has no such safeguard. Should it? — Q uantling (talk | contribs) 17:21, 2 May 2022 (UTC)

Reciprocal of 1
The article suggest that the reciprocal of a number is given by adding/removing a zero at the begining. It is not true for 1 or -1 since their reciprocals are themselves. 2601:648:8601:93A0:AC29:71C5:7EE0:6ABA (talk) 06:55, 13 December 2022 (UTC)

 * Did you work out the value of the continued fraction obtained when you add a zero at the beginning for these numbers? Remember that continued fraction representations are not always unique. —David Eppstein (talk) 07:01, 13 December 2022 (UTC)

The golden ratio is not the most irrational number.
If you use square roots, instead of fractions, you can get something more irrational than the golden ratio, because many square roots already become irrational with 1 iteration, which does not apply with fractions.

example: 3+sqrt(7+sqrt(15+sqrt(1+sqrt(292+sqrt(... 84.151.244.169 (talk) 15:07, 8 May 2023 (UTC)

 * I don't think that there is a precise meaning for "most irrational number". The point in the article is to highlight that the continued fraction expansion of the golden ratio $φ$ is all ones.  It turns out that this means that regardless of the fraction $p/q$, it is the case that the value $(|φ − p/q|) × q2$ is large in comparison to what can be achieved with rational approximations for other irrational numbers.  — Q uantling (talk | contribs) 16:20, 8 May 2023 (UTC)

Revamping the page
Hello!

I just wanted to check in with anyone that might care whether or not they'd take issue with me trying to reorganise the page. I've become quite interested in continued fractions not just as a weird means of 'calculating numbers', but as an alternate means of representing numbers between the integers to that of the usual decimal 'negative power series', and I'd really like to try to do justice to them through updating the page.

Some notes I've taken thus far as as follows:

- Motivation and Notation section spends a lot of time explaining how to calculate the continued fraction form from usual decimal negative power series form, not enough time talking about actual motivations/history/desirability, and the notation. Either I can change the name and collate related information or remove from this section and write a clearer explanation of the method elsewhere, preserving Pier4r's request above.

- Whole separate Notation section exists, which actually discusses 'alternative notations' to the ones presented in the 'motivation and notations section'; I think this heading should be changed and it should be subsumed by a broader section on notation.

- Repeated references throughout to the effect of "sqrt(2) actually equals 1.41421..., so you can calculate this from its continued fraction form [1;2,2,2,..] by doing so and so." Seems to be a neglect for the consideration of a continued fraction representation of a number as equally 'valid' as the power series representation, probably due to unfamiliarity and the somewhat cumbersome but necessary notation. To be clear, I think there's little reason to not switch the notation such that for example pi = 3.7(15)1(292)111213... (in continued fraction form) = [3;1,4,1,5,9,2,6,..] (in power series form) -- now imagine analogously saying that " pi actually = 3.7(15)1(292)111213..., so you can calculate this from its power series form via... ". I personally think it's reductive and unnecessary, so I wonder what you guys might think of this point in particular.

- The continued fraction notation version of a bunch of mathematical constants in the Motivation and Notation section seems really helpful to me for familiarising the reader with this perspective on these numbers, and I would like to preserve something similar, but when you look for its context you see that all this space is actually serving to elucidate infinite continued fractions, which is off-topic from the heading. I'd like to flesh out some of these kinds of examples with more numbers that aren't just infinite cfs, and reserve discussion of infinite cfs for maybe the section 'Infinite continued fractions and convergents'.

- Notice that there's no mention nor use of the 'repeating' notation usually seem with 'decimal' notation of numbers like 1/3 or 1/7, only ellipsis like sqrt(3) = [1;1,2,1,2,1, 2,...]. I'd like to explicitly incorporate that.

- Having read the page quite a few times, I'm confused as to whether it's about cfs in the canonical form or the generalised form. Given that a page exists solely for the generalised form, I'd be inclined to dedicate this to the canonical form, but I also feel that that would be too specific and might mislead people given the name. The diagram in the introduction shows it in the canonical form, mention of definition as 'the reciprocal of another number' somewhat suggests the understanding that it's about the canonical form, the intro makes the delineation between the two and suggests a prioritisation of the canonical form, yet the section on Basic Formula immediately jumps into the generalised form, despite that formula being mirrored on the generalised cf's page. That formula is then repeated in the later section titled 'generalized continued fraction(s?)', which I feel again is redundant and I'd like to remove and possibly move over any interesting information to its respective article if it's not there already.

There's more I'd like to add, including some interesting patterns I've found myself, some restructuring to be done, and more I need to study in order to be able to really speak on some topics. I'd like to ensure I'm factoring in the suggestions that others have already made too, and in particular I'd love to be able to address Manoguru's concerns about the natural operations of numbers in continued fraction form, but that's all I can speak on for now.

Please do let me know your thoughts on my potential changes, thank you for reading if you made it this far! CallumMScott (talk) 14:17, 14 August 2023 (UTC)

how "larger term" better approximation
I don't like the sentence
 * The larger a term is in the continued fraction, the closer the corresponding convergent is to the irrational number being approximated.

It goes on to explain that the golden ratio is the hardest to approximate because all terms of its continued fraction are "1".

I think what the sentence should say is something like "the larger a term is, the more that one term improves the approximation." But then I want to natter on about percentage reduction in absolute error.

I would like to hear from someone who understands the article before I try to "improve" it.

Jmichael ll (talk) 20:41, 8 November 2023 (UTC)

 * Which is closer to 4: 4$1/7$ or 4$1/3$?
 * The greater a partial quotient is, the less effect it and its successors have on the number; in other words, the more accurate the fraction already is. —Tamfang (talk) 05:32, 15 November 2023 (UTC)
 * Perhaps I'm mistaken in this concern, but isn't there a problem about what we mean by "the corresponding convergent"? Would it be an improvement to add a "next", as in
 * The larger the next term in the continued fraction is, the closer a convergent is to the irrational number being approximated.
 * I think both are true and mean slightly different things. Dhrm77 (talk) 15:40, 15 November 2023 (UTC)
 * I think both are true and mean slightly different things. Dhrm77 (talk) 15:40, 15 November 2023 (UTC)

Constructive observation
A real number $$x$$ has an infinite continued fraction expansion iff it is apart from all rationals: $$\forall q:\Q.\exists m:\N +. |x-q| \geq (1/m)$$. This is constructively stronger than being irrational (not rational).

46.33.143.125 (talk) 15:52, 28 January 2024 (UTC)

 * What does "apart from all rationals" mean, if not "irrational"? —Tamfang (talk) 17:28, 27 March 2024 (UTC)
 * Constructively, "apart" has a stronger meaning than "not equal". Two numbers are "apart" if they differ by at least some $1/n$. 46.33.143.125 (talk) 19:04, 13 April 2024 (UTC)
 * Can you give an example of an irrational number which is not "apart" from the rationals? –jacobolus (t) 19:37, 13 April 2024 (UTC)
 * What is an example of an irrational that has a finite neighborhood containing no rationals?? —Tamfang (talk) 21:07, 13 April 2024 (UTC)
 * The definition in the top comment here lets you pick a different neighborhood excluding each rational number. But I don't understand what's different about it than the concept of "irrational" per se. –jacobolus (t) 22:22, 13 April 2024 (UTC)

title: Talk:Continued fraction


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User:Kip~enwiki

Just a System Administrator for the Ann Arbor District Library in Michigan, USA.

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User talk:Kip~enwiki

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October 2007
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Darius the Great

Darius I (Δαρεῖος ; c. 550 – 486 BCE), commonly known as Darius the Great, was a Persian ruler who served as the third King of Kings of the Achaemenid Empire, reigning from 522 BCE until his death in 486 BCE. He ruled the empire at its territorial peak, when it included much of Western Asia, parts of the Balkans (Thrace–Macedonia and Paeonia) and the Caucasus, most of the Black Sea's coastal regions, Central Asia, the Indus Valley in the far east, and portions of North Africa and Northeast Africa including Egypt (Mudrâya), eastern Libya, and coastal Sudan.

Darius ascended the throne by overthrowing the Achaemenid monarch Bardiya (or Smerdis), who he claimed was in fact an imposter named Gaumata. The new king met with rebellions throughout the empire but quelled each of them; a major event in Darius's life was his expedition to subjugate Greece and punish Athens and Eretria for their participation in the Ionian Revolt. Although his campaign ultimately resulted in failure at the Battle of Marathon, he succeeded in the re-subjugation of Thrace and expanded the Achaemenid Empire through his conquests of Macedonia, the Cyclades, and the island of Naxos.

Darius organized the empire by dividing it into administrative provinces, each governed by a satrap. He organized Achaemenid coinage as a new uniform monetary system, and he made Aramaic a co-official language of the empire alongside Persian. He also put the empire in better standing by building roads and introducing standard weights and measures. Through these changes, the Achaemenid Empire became centralized and unified. Darius undertook other construction projects throughout his realm, primarily focusing on Susa, Pasargadae, Persepolis, Babylon, and Egypt. He had an inscription carved upon a cliff-face of Mount Behistun to record his conquests, which would later become important evidence of the Old Persian language.

Darius is mentioned in the books of Haggai, Zechariah, Daniel, and Ezra–Nehemiah of the Hebrew Bible (the Christian Old Testament).

Etymology
Dārīus and Dārēus are the Latin forms of the Greek Dareîos (Δαρεῖος), itself from Old Persian Dārayauš (𐎭𐎠𐎼𐎹𐎢𐏁, ; which is a shortened form of Dārayavaʰuš . The longer Persian form is reflected in the Elamite Da-ri-(y)a-ma-u-iš, Babylonian Da-(a-)ri-ia-(a-)muš, and Aramaic drywhwš (𐡃𐡓𐡉𐡅𐡄𐡅𐡔) forms, and possibly in the longer Greek form, Dareiaîos (Δαρειαῖος). The name in nominative form means "he who holds firm the good(ness)", which can be seen by the first part dāraya, meaning "holder", and the adverb vau, meaning "goodness".

Early life
Darius was the eldest of five sons to Hystaspes. The identity of his mother is uncertain. According to the modern historian Alireza Shapour Shahbazi (1994), Darius's mother was thought to have been a woman named Rhodogune. However, according to Lloyd Llewellyn-Jones (2013), recently uncovered texts in Persepolis indicate that his mother was Irdabama, an affluent landowner descended from a family of local Elamite rulers. Richard Stoneman likewise refers to Irdabama as the mother of Darius. The Behistun Inscription of Darius states that his father was satrap of Bactria in 522 BCE. According to Herodotus (III.139), Darius, prior to seizing power and "of no consequence at the time", had served as a spearman (doryphoros) in the Egyptian campaign (528–525 BCE) of Cambyses II, then the Persian Great King; this is often interpreted to mean he was the king's personal spear-carrier, an important role. Hystaspes was an officer in Cyrus's army and a noble of his court.

Before Cyrus and his army crossed the river Araxes to battle with the Armenians, he installed his son Cambyses II as king in case he should not return from battle. However, once Cyrus had crossed the Aras River, he had a vision in which Darius had wings atop his shoulders and stood upon the confines of Europe and Asia (the known world). When Cyrus awoke from the dream, he inferred it as a great danger to the future security of the empire, as it meant that Darius would one day rule the whole world. However, his son Cambyses was the heir to the throne, not Darius, causing Cyrus to wonder if Darius was forming treasonable and ambitious designs. This led Cyrus to order Hystaspes to go back to Persis and watch over his son strictly, until Cyrus himself returned.

Accession
There are different accounts of the rise of Darius to the throne from both Darius himself and Greek historians. The oldest records report a convoluted sequence of events in which Cambyses II lost his mind, murdered his brother Bardiya, and was killed by an infected leg wound. After this, Darius and a group of six nobles traveled to Sikayauvati to kill an usurper, Gaumata, who had taken the throne by pretending to be Bardiya during the true king's absence.

Darius's account, written at the Behistun Inscription, states that Cambyses II killed his own brother Bardiya, but that this murder was not known among the Iranian people. A would-be usurper named Gaumata came and lied to the people, stating that he was Bardiya. The Iranians had grown rebellious against Cambyses's rule and, on 11 March 522 BCE, a revolt against Cambyses broke out in his absence. On 1 July, the Iranian people chose to be under the leadership of Gaumata, as "Bardiya". No member of the Achaemenid family would rise against Gaumata for the safety of their own life. Darius, who had served Cambyses as his lance-bearer until the deposed ruler's death, prayed for aid and, in September 522 BCE, along with Otanes, Intaphrenes, Gobryas, Hydarnes, Megabyzus and Aspathines, killed Gaumata in the fortress of Sikayauvati.

Herodotus provides a dubious account of Darius's ascension: Several days after Gaumata had been assassinated, Darius and the other six nobles discussed the fate of the empire. At first, the seven discussed the form of government: A democratic republic (Isonomia) was strongly pushed by Otanes, an oligarchy was pushed by Megabyzus, while Darius pushed for a monarchy. After stating that a republic would lead to corruption and internal fighting, while a monarchy would be led with a single-mindedness not possible in other governments, Darius was able to convince the other nobles.

To decide who would become the monarch, six of them decided on a test, with Otanes abstaining, as he had no interest in being king. They were to gather outside the palace, mounted on their horses at sunrise, and the man whose horse neighed first in recognition of the rising sun would become king. According to Herodotus, Darius had a slave, Oebares, who rubbed his hand over the genitals of a mare that Darius's horse favored. When the six gathered, Oebares placed his hands beside the nostrils of Darius's horse, who became excited at the scent and neighed. This was followed by lightning and thunder, leading the others to dismount and kneel before Darius in recognition of his apparent divine providence. In this account, Darius himself claimed that he achieved the throne not through fraud, but cunning, even erecting a statue of himself mounted on his neighing horse with the inscription: "Darius, son of Hystaspes, obtained the sovereignty of Persia by the sagacity of his horse and the ingenious contrivance of Oebares, his groom."

According to the accounts of Greek historians, Cambyses II had left Patizeithes in charge of the kingdom when he headed for Egypt. He later sent Prexaspes to murder Bardiya. After the killing, Patizeithes put his brother Gaumata, a Magian who resembled Bardiya, on the throne and declared him the Great King. Otanes discovered that Gaumata was an impostor, and along with six other Iranian nobles, including Darius, created a plan to oust the pseudo-Bardiya. After killing the impostor along with his brother Patizeithes and other Magians, Darius was crowned king the following morning.

The details regarding Darius's rise to power is generally acknowledged as forgery and was in reality used as a concealment of his overthrow and murder of Cyrus's rightful successor, Bardiya. To legitimize his rule, Darius had a common origin fabricated between himself and Cyrus by designating Achaemenes as the eponymous founder of their dynasty. In reality, Darius was not from the same house as Cyrus and his forebears, the rulers of Anshan.

Early revolts

Following his coronation at Pasargadae, Darius moved to Ecbatana. He soon learned that support for Bardiya was strong, and revolts in Elam and Babylonia had broken out. Darius ended the Elamite revolt when the revolutionary leader Aschina was captured and executed in Susa. After three months the revolt in Babylonia had ended. While in Babylonia, Darius learned a revolution had broken out in Bactria, a satrapy which had always been in favour of Darius, and had initially volunteered an army of soldiers to quell revolts. Following this, revolts broke out in Persis, the homeland of the Persians and Darius and then in Elam and Babylonia, followed by in Media, Parthia, Assyria, and Egypt.

By 522 BCE, there were revolts against Darius in most parts of the Achaemenid Empire leaving the empire in turmoil. Even though Darius did not seem to have the support of the populace, Darius had a loyal army, led by close confidants and nobles (including the six nobles who had helped him remove Gaumata). With their support, Darius was able to suppress and quell all revolts within a year. In Darius's words, he had killed a total of nine "lying kings" through the quelling of revolutions. Darius left a detailed account of these revolutions in the Behistun Inscription.

Elimination of Intaphernes
One of the significant events of Darius's early reign was the slaying of Intaphernes, one of the seven noblemen who had deposed the previous ruler and installed Darius as the new monarch. The seven had made an agreement that they could all visit the new king whenever they pleased, except when he was with a woman. One evening, Intaphernes went to the palace to meet Darius, but was stopped by two officers who stated that Darius was with a woman. Becoming enraged and insulted, Intaphernes drew his sword and cut off the ears and noses of the two officers. While leaving the palace, he took the bridle from his horse, and tied the two officers together.

The officers went to the king and showed him what Intaphernes had done to them. Darius began to fear for his own safety; he thought that all seven noblemen had banded together to rebel against him and that the attack against his officers was the first sign of revolt. He sent a messenger to each of the noblemen, asking them if they approved of Intaphernes's actions. They denied and disavowed any connection with Intaphernes's actions, stating that they stood by their decision to appoint Darius as King of Kings. Darius's choice to ask the noblemen indicates that he was not yet completely sure of his authority.

Taking precautions against further resistance, Darius sent soldiers to seize Intaphernes, along with his son, family members, relatives and any friends who were capable of arming themselves. Darius believed that Intaphernes was planning a rebellion, but when he was brought to the court, there was no proof of any such plan. Nonetheless, Darius killed Intaphernes's entire family, excluding his wife's brother and son. She was asked to choose between her brother and son. She chose her brother to live. Her reasoning for doing so was that she could have another husband and another son, but she would always have but one brother. Darius was impressed by her response and spared both her brother's and her son's life.

Egyptian campaign
After securing his authority over the entire empire, Darius embarked on a campaign to Egypt where he defeated the armies of the Pharaoh and secured the lands that Cambyses had conquered while incorporating a large portion of Egypt into the Achaemenid Empire.

Through another series of campaigns, Darius I would eventually reign over the territorial apex of the empire, when it stretched from parts of the Balkans (Thrace-Macedonia, Bulgaria-Paeonia) in the west, to the Indus Valley in the east.

Invasion of the Indus Valley

In 516 BCE, Darius embarked on a campaign to Central Asia, Aria and Bactria and then marched into Afghanistan to Taxila in modern-day Pakistan. Darius spent the winter of 516–515 BCE in Gandhara, preparing to conquer the Indus Valley. Darius conquered the lands surrounding the Indus River in 515 BCE. Darius I controlled the Indus Valley from Gandhara to modern Karachi and appointed the Greek Scylax of Caryanda to explore the Indian Ocean from the mouth of the Indus to Suez. Darius then marched through the Bolan Pass and returned through Arachosia and Drangiana back to Persia.

Babylonian revolt
After Bardiya was murdered, widespread revolts occurred throughout the empire, especially on the eastern side. Darius asserted his position as king by force, taking his armies throughout the empire, suppressing each revolt individually. The most notable of all these revolts was the Babylonian revolt which was led by Nebuchadnezzar III. This revolt occurred when Otanes withdrew much of the army from Babylon to aid Darius in suppressing other revolts. Darius felt that the Babylonian people had taken advantage of him and deceived him, which resulted in Darius gathering a large army and marching to Babylon. At Babylon, Darius was met with closed gates and a series of defences to keep him and his armies out.

Darius encountered mockery and taunting from the rebels, including the famous saying "Oh yes, you will capture our city, when mules shall have foals." For a year and a half, Darius and his armies were unable to retake the city, though he attempted many tricks and strategies—even copying that which Cyrus the Great had employed when he captured Babylon. However, the situation changed in Darius's favour when, according to the story, a mule owned by Zopyrus, a high-ranking soldier, foaled. Following this, a plan was hatched for Zopyrus to pretend to be a deserter, enter the Babylonian camp, and gain the trust of the Babylonians. The plan was successful and Darius's army eventually surrounded the city and overcame the rebels.

During this revolt, Scythian nomads took advantage of the disorder and chaos and invaded Persia. Darius first finished defeating the rebels in Elam, Assyria, and Babylon and then attacked the Scythian invaders. He pursued the invaders, who led him to a marsh; there he found no known enemies but an enigmatic Scythian tribe.

European Scythian campaign
The Scythians were a group of north Iranian nomadic tribes, speaking an Eastern Iranian language (Scythian languages) who had invaded Media, killed Cyrus in battle, revolted against Darius and threatened to disrupt trade between Central Asia and the shores of the Black Sea as they lived between the Danube River, River Don and the Black Sea.

Darius crossed the Black Sea at the Bosphorus Straits using a bridge of boats. Darius conquered large portions of Eastern Europe, even crossing the Danube to wage war on the Scythians. Darius invaded European Scythia in 513 BCE, where the Scythians evaded Darius's army, using feints and retreating eastwards while laying waste to the countryside, by blocking wells, intercepting convoys, destroying pastures and continuous skirmishes against Darius's army. Seeking to fight with the Scythians, Darius's army chased the Scythian army deep into Scythian lands, where there were no cities to conquer and no supplies to forage. In frustration Darius sent a letter to the Scythian ruler Idanthyrsus to fight or surrender. The ruler replied that he would not stand and fight with Darius until they found the graves of their fathers and tried to destroy them. Until then, they would continue their strategy as they had no cities or cultivated lands to lose.

Despite the evading tactics of the Scythians, Darius's campaign was so far relatively successful. As presented by Herodotus, the tactics used by the Scythians resulted in the loss of their best lands and of damage to their loyal allies. This gave Darius the initiative. As he moved eastwards in the cultivated lands of the Scythians in Eastern Europe proper, he remained resupplied by his fleet and lived to an extent off the land. While moving eastwards in the European Scythian lands, he captured the large fortified city of the Budini, one of the allies of the Scythians, and burnt it.

Darius eventually ordered a halt at the banks of Oarus, where he built "eight great forts, some 8 mi distant from each other", no doubt as a frontier defence. In his Histories, Herodotus states that the ruins of the forts were still standing in his day. After chasing the Scythians for a month, Darius's army was suffering losses due to fatigue, privation and sickness. Concerned about losing more of his troops, Darius halted the march at the banks of the Volga River and headed towards Thrace. He had conquered enough Scythian territory to force the Scythians to respect the Persian forces.

Persian invasion of Greece

Darius's European expedition was a major event in his reign, which began with the invasion of Thrace. Darius also conquered many cities of the northern Aegean, Paeonia, while Macedonia submitted voluntarily, after the demand of earth and water, becoming a vassal kingdom. He then left Megabyzus to conquer Thrace, returning to Sardis to spend the winter. The Greeks living in Asia Minor and some of the Greek islands had submitted to Persian rule already by 510 BCE. Nonetheless, there were certain Greeks who were pro-Persian, although these were largely based in Athens. To improve Greek-Persian relations, Darius opened his court and treasuries to those Greeks who wanted to serve him. These Greeks served as soldiers, artisans, statesmen and mariners for Darius. However, the increasing concerns amongst the Greeks over the strength of Darius's kingdom along with the constant interference by the Greeks in Ionia and Lydia were stepping stones towards the conflict that was yet to come between Persia and certain of the leading Greek city states.

When Aristagoras organized the Ionian Revolt, Eretria and Athens supported him by sending ships and troops to Ionia and by burning Sardis. Persian military and naval operations to quell the revolt ended in the Persian reoccupation of Ionian and Greek islands, as well as the re-subjugation of Thrace and the conquering of Macedonia in 492 BCE under Mardonius. Macedon had been a vassal kingdom of the Persians since the late 6th century BCE, but retained autonomy. Mardonius's 492 campaign made it a fully subordinate part of the Persian kingdom. These military actions, coming as a direct response to the revolt in Ionia, were the beginning of the First Persian invasion of (mainland) Greece. At the same time, anti-Persian parties gained more power in Athens, and pro-Persian aristocrats were exiled from Athens and Sparta.

Darius responded by sending troops led by his son-in-law across the Hellespont. However, a violent storm and harassment by the Thracians forced the troops to return to Persia. Seeking revenge on Athens and Eretria, Darius assembled another army of 20,000 men under his Admiral, Datis, and his nephew Artaphernes, who met success when they captured Eretria and advanced to Marathon. In 490 BCE, at the Battle of Marathon, the Persian army was defeated by a heavily armed Athenian army, with 9,000 men who were supported by 600 Plataeans and 10,000 lightly armed soldiers led by Miltiades. The defeat at Marathon marked the end of the first Persian invasion of Greece. Darius began preparations for a second force which he would command, instead of his generals; however, before the preparations were complete, Darius died, thus leaving the task to his son Xerxes.

Family
Darius was the son of Hystaspes and the grandson of Arsames. Darius married Atossa, daughter of Cyrus, with whom he had four sons: Xerxes, Achaemenes, Masistes and Hystaspes. He also married Artystone, another daughter of Cyrus, with whom he had two sons, Arsames and Gobryas. Darius married Parmys, the daughter of Bardiya, with whom he had a son, Ariomardus. Furthermore, Darius married his niece Phratagune, with whom he had two sons, Abrokomas and Hyperantes. He also married another woman of the nobility, Phaidyme, the daughter of Otanes. It is unknown if he had any children with her. Before these royal marriages, Darius had married an unknown daughter of his good friend and lance carrier Gobryas from an early marriage, with whom he had three sons, Artobazanes, Ariabignes and Arsamenes. Any daughters he had with her are not known. Although Artobazanes was Darius's first-born, Xerxes became heir and the next king through the influence of Atossa; she had great authority in the kingdom as Darius loved her the most of all his wives.

Death and succession
After becoming aware of the Persian defeat at the Battle of Marathon, Darius began planning another expedition against the Greek city-states; this time, he, not Datis, would command the imperial armies. Darius had spent three years preparing men and ships for war when a revolt broke out in Egypt. This revolt in Egypt worsened his failing health and prevented the possibility of his leading another army. Soon afterwards, Darius died, after thirty days of suffering through an unidentified illness, partially due to his part in crushing the revolt, at about sixty-four years old. In October 486 BCE, his body was embalmed and entombed in the rock-cut tomb at Naqsh-e Rostam, which he had been preparing. An inscription on his tomb introduces him as "Great King, King of Kings, King of countries containing all kinds of men, King in this great earth far and wide, son of Hystaspes, an Achaemenian, a Persian, son of a Persian, an Aryan, having Aryan lineage." A relief under his tomb portraying equestrian combat was later carved during the reign of the Sasanian King of Kings, Bahram II ((r. 274 – 293)).

Xerxes, the eldest son of Darius and Atossa, succeeded to the throne as Xerxes I; before his accession, he had contested the succession with his elder half-brother Artobarzanes, Darius's eldest son, who was born to his first wife before Darius rose to power. With Xerxes's accession, the empire was again ruled by a member of the house of Cyrus.

Organization

Early in his reign, Darius wanted to reorganize the structure of the empire and reform the system of taxation he inherited from Cyrus and Cambyses. To do this, Darius created twenty provinces called satrapies (or archi) which were each assigned to a satrap (archon) and specified fixed tributes that the satrapies were required to pay. A complete list is preserved in the catalogue of Herodotus, beginning with Ionia and listing the other satrapies from west to east excluding Persis, which was the land of the Persians and the only province which was not a conquered land. Tributes were paid in both silver and gold talents. Tributes in silver from each satrap were measured with the Babylonian talent. Those paid in gold were measured with the Euboic talent. The total tribute from the satraps came to an amount less than 15,000 silver talents.

The majority of the satraps were of Persian origin and were members of the royal house or the six great noble families. These satraps were personally picked by Darius to monitor these provinces. Each of the provinces was divided into sub-provinces, each having its own governor, who was chosen either by the royal court or by the satrap. To assess tributes, a commission evaluated the expenses and revenues of each satrap. To ensure that one person did not gain too much power, each satrap had a secretary, who observed the affairs of the state and communicated with Darius; a treasurer, who safeguarded provincial revenues; and a garrison commander, who was responsible for the troops. Additionally, royal inspectors, who were the "eyes and ears" of Darius, completed further checks on each satrap.

The imperial administration was coordinated by the chancery with headquarters at Persepolis, Susa, and Babylon with Bactria, Ecbatana, Sardis, Dascylium and Memphis having branches. Darius kept Aramaic as the common language, which soon spread throughout the empire. However, Darius gathered a group of scholars to create a separate language system only used for Persis and the Persians, which was called Aryan script and was only used for official inscriptions. Before this, the accomplishments of the king were addressed in Persian solely through narration and hymns and through the "masters of memory". Indeed, oral history continued to play an important role throughout the history of Iran.

Economy

Darius introduced a new universal currency, the daric, sometime before 500 BCE. Darius used the coinage system as a transnational currency to regulate trade and commerce throughout his empire. The Daric was also recognized beyond the borders of the empire, in places such as Celtic Central Europe and Eastern Europe. There were two types of darics, a gold daric and a silver daric. Only the king could mint gold darics. Important generals and satraps minted silver darics, the latter usually to recruit Greek mercenaries in Anatolia. The daric was a major boost to international trade. Trade goods such as textiles, carpets, tools and metal objects began to travel throughout Asia, Europe and Africa. To further improve trade, Darius built the Royal Road, a postal system and Phoenician-based commercial shipping.

The daric also improved government revenues as the introduction of the daric made it easier to collect new taxes on land, livestock and marketplaces. This led to the registration of land which was measured and then taxed. The increased government revenues helped maintain and improve existing infrastructure and helped fund irrigation projects in dry lands. This new tax system also led to the formation of state banking and the creation of banking firms. One of the most famous banking firms was Murashu Sons, based in the Babylonian city of Nippur. These banking firms provided loans and credit to clients.

In an effort to further improve trade, Darius built canals, underground waterways and a powerful navy. According to Herodotus, qanat irrigation technology was introduced to Egypt, which is supported by the historian Albert T. Olmstead. He further improved and expanded the network of roads and way stations throughout the empire, so that there was a system of travel authorization for the King, satraps and other high officials, which entitled the traveller to draw provisions at daily stopping places.

Religion
While there is no general consensus in scholarship whether Darius and his predecessors had been influenced by Zoroastrianism, it is well established that Darius was a firm believer in Ahura Mazda, whom he saw as the supreme deity. However, Ahura Mazda was also worshipped by adherents of the (Indo-)Iranian religious tradition. As can be seen at the Behistun Inscription, Darius believed that Ahura Mazda had appointed him to rule the Achaemenid Empire.

Darius had dualistic philosophical convictions and believed that each rebellion in his kingdom was the work of druj, the enemy of Asha. Darius believed that because he lived righteously by Asha, Ahura Mazda supported him. In many cuneiform inscriptions denoting his achievements, he presents himself as a devout believer, perhaps even convinced that he had a divine right to rule over the world. However, his relationship with the deity was far more complex: in one inscription he writes "Ahura Mazda is mine, I am Ahura Mazda's"

In the lands that were conquered by his empire, Darius followed the same Achaemenid tolerance that Cyrus had shown and later Achaemenid kings would show. He supported faiths and religions that were "alien" as long as the adherents were "submissive and peaceable", sometimes giving them grants from his treasury for their purposes. He had funded the restoration of the Israelite temple which had originally been decreed by Cyrus, was supportive towards Greek cults which can be seen in his letter to Gadatas, and supported Elamite priests. He had also observed Egyptian religious rites related to kingship and had built the temple for the Egyptian god, Amun.

Building projects

During Darius's Greek expedition, he had begun construction projects in Susa, Egypt and Persepolis. He had linked the Red Sea to the river Nile by building a canal (Darius Canal) which ran from modern Zaqāzīq to modern Suez. To open this canal, he travelled to Egypt in 497 BCE, where the inauguration was carried out with great fanfare and celebration. Darius also built a canal to connect the Red Sea and Mediterranean. On this visit to Egypt he erected monuments and executed Aryandes on the charge of treason. When Darius returned to Persis, he found that the codification of Egyptian law had been finished.

Additionally, Darius sponsored large construction projects in Susa, Babylon, Egypt, and Persepolis. In Susa, Darius built a new palace complex in the north of the city. An inscription states that the palace was destroyed during the reign of Artaxerxes I, but was rebuilt. Today only glazed bricks of the palace remain, the majority of them in the Louvre. In Pasargadae, Darius finished all incomplete construction projects from the reign of Cyrus the Great. A palace was also built during the reign of Darius, with an inscription in the name of Cyrus the Great. It was previously believed that Cyrus had constructed this building, however due to the cuneiform script being used, the palace is believed to have been constructed by Darius.

In Egypt, Darius built many temples and restored those that had previously been destroyed. Even though Darius was a believer of Ahura Mazda, he built temples dedicated to the Gods of the Ancient Egyptian religion. Several temples found were dedicated to Ptah and Nekhbet. Darius also created several roads and routes in Egypt. The monuments that Darius built were often inscribed in the official languages of the Persian Empire, Old Persian, Elamite and Babylonian and Egyptian hieroglyphs. To construct these monuments, Darius employed a large number of workers and artisans of diverse nationalities. Several of these workers were deportees who had been employed specifically for these projects. These deportees enhanced the empire's economy and improved inter-cultural relations. At the time of Darius's death construction projects were still under way. Xerxes completed these works and in some cases expanded his father's projects by erecting new buildings of his own.

title: Darius the Great


text:
Darius II

Darius II ( Dārayavaʰuš; Δαρεῖος Dareios), also known by his given name Ochus (Greek: Ὦχος Ochos), was King of Kings of the Achaemenid Empire from 423 BC to 405 or 404 BC.

Artaxerxes I, who died in 424 BC, was followed by his son Xerxes II. After a month and half Xerxes II was murdered by his brother Sogdianus. His illegitimate brother, Ochus, satrap of Hyrcania, rebelled against Sogdianus, and after a short fight killed him, and suppressed by treachery the attempt of his own brother Arsites to imitate his example. Ochus adopted the name Darius (Greek sources often call him Darius Nothos, "Bastard"). Neither the names Xerxes II nor Sogdianus occur in the dates of the numerous Babylonian tablets from Nippur; here effectively the reign of Darius II follows immediately after that of Artaxerxes I. Historians know little about Darius II's reign. A rebellion by the Medes in 409 BC is mentioned by Xenophon. It does seem that Darius II was quite dependent on his wife Parysatis. In excerpts from Ctesias some harem intrigues are recorded, in which he played a disreputable part. The Elephantine papyri mention Darius II as a contemporary of the high priest Johanan of Ezra 10:6.

Conflict with Athens
As long as the power of Athens remained intact he did not meddle in Greek affairs. When in 413 BC, Athens supported the rebel Amorges in Caria, Darius II would not have responded had not the Athenian power been broken in the same year at Syracuse. As a result of that event, Darius II gave orders to his satraps in Asia Minor, Tissaphernes and Pharnabazus, to send in the overdue tribute of the Greek towns and to begin a war with Athens. To support the war with Athens, the Persian satraps entered into an alliance with Sparta. In 408 BC he sent his son Cyrus to Asia Minor, to carry on the war with greater energy.

Darius II may have expelled various Greek dynasts who had been ruling cities in Ionia: Pausanias wrote that the sons of Themistocles, which include Archeptolis, Governor of Magnesia, "appear to have returned to Athens", and that they dedicated a painting of Themistocles in the Parthenon and erected a bronze statue to Artemis Leucophryene, the goddess of Magnesia, on the Acropolis. They may have returned from Asia Minor in old age, after 412 BC, when the Achaemenids took again firm control of the Greek cities of Asia, and they may have been expelled by the Achaemenid satrap Tissaphernes sometime between 412 and 399 BC. In effect, from 414 BC, Darius II had started to resent increasing Athenian power in the Aegean and had Tissaphernes enter into an alliance with Sparta against Athens, which in 412 BC led to the Persian conquest of the greater part of Ionia.

Darius is said to have received the visit of Greek athlete and Olympic champion Polydamas of Skotoussa, who made a demonstration of his strength by killing three Immortals in front of the Persian ruler. A sculpture representing the scene is visible in the Museum of the History of the Olympic Games of antiquity.

Darius II died in 404 BC, in the nineteenth year of his reign, and was followed as Persian king by Artaxerxes II.

Issue
Prior to his accession, Darius II was married to the daughter of Gobryas. With the daughter of Gobryas, Darius II had four sons, one of whom fathered Artabazanes, who served as King of Media Atropatene in the second half of the 3rd century BC.

 * By Parysatis
 * Artaxerxes II
 * Cyrus the Younger
 * Oxathres or Oxendares or Oxendras
 * Artoxexes
 * Ostanes
 * Amestris wife of Teritouchmes & then Artaxerxes II
 * & seven other unnamed children

 * By other wives
 * Artostes
 * The unnamed satrap of Media at 401 B.C.

title: Darius II


text:
Darius III

Darius III ( Dārayavaʰuš; Δαρεῖος Dareios; c. 380 – 330 BC) was the last Achaemenid King of Kings of Persia, reigning from 336 BC to his death in 330 BC.

Contrary to his predecessor Artaxerxes IV Arses, Darius was a distant member of the Achaemenid dynasty. During his early career, he was reportedly an obscure figure among his peers and first rose to prominence during the Cadusian expedition of Artaxerxes III in the 350s BC. As a reward for his bravery, he was given the Satrapy of Armenia. Around 340 BC, he was placed in charge of the royal "postal service," a high-ranking position. In 338 BC, Artaxerxes III met an abrupt end after being poisoned by the court eunuch and chiliarch (hazahrapatish) Bagoas, who installed Artaxerxes' youngest son Arses on the throne. He only reigned for a few years, until Bagoas had him poisoned as well. Darius was subsequently installed on the throne and soon forced Bagoas to drink his poison after discovering that the eunuch had planned to poison him as well.

In 334 BC, Alexander the Great began his invasion of the Persian Empire and subsequently defeated the Persians in several battles before looting and destroying their capital, Persepolis, by fire in 330 BC. With the Persian Empire now effectively under Alexander's control, Alexander then decided to pursue Darius. Before Alexander reached him, however, Darius was killed by his relative Bessus, who was also the satrap of Bactria.

Darius is partially remembered in Iranian traditions as Dara II, the last king of the mythological Kayanian dynasty, which reflected memories of the Achaemenids.

Name
Before his accession, Darius bore the name of Artashata (Old Persian: *Artašiyāta, "Happy in Arta"). The 2nd-century Roman historian Justin is the only historian to refer Darius as Codomannus, a name he supposedly bore before he rose to prominence. It may have been his nickname, or possibly a third name. Its etymology is uncertain. Badian has suggested that the name was of Western Semitic origin, most likely from the Aramaic qdmwn ("from the East, Easterner"). Having two names was not unusual; there are several recorded instances of Babylonian figures having two names, often a Babylonian and Aramaic name. One Persian magnate is also attested with a Babylonian and Iranian name. However, this practice seems to have fallen out of favour during Darius' lifetime. Badian suggests that his birth name was the Aramaic Codomannus, which he later dropped in favour of Artashata when he rose in ranks. He first adopted the regnal name of Darius (Old Persian: Daraya-vahauš, "he who holds firm the good(ness)") when he ascended the Achaemenid throne in 336 BC.

Historiography
The last century of the Achaemenid era is notable for lacking in sources, especially during the reign of Darius III. He is not attested in any Persian sources and is almost completely only known from the reports of Greek historians, who portray his career as a contradiction to that of the successful Alexander the Great.

Background
Darius was born in c. 380 BC. He was a distant member of the royal Achaemenid dynasty. He was the son of a certain Arsames, and grandson of Ostanes, whose father Darius II ruled the Achaemenid Empire from 424 BC to 405 BC. His mother was Sisygambis, a woman of obscure origins. She was probably of Achaemenid descent, although it is unknown to which branch she belonged. She may have been the daughter of Ostanes, and thus the sister of Arsames. Darius had a brother and sister; Oxyathres and Stateira I, respectively.

Early life
Artashata is first attested during the Cadusian expedition of Artaxerxes III ((r. 358 – 338)) in the 350s BC. During a battle, Artashata distinguished himself by slaying a warrior in single combat. His exploit was noticed by Artaxerxes III, who sent him gifts and gave him the Satrapy of Armenia. It was probably after this promotion that Artashata married for the first time, to an unknown noblewoman, who bore him two daughters, and possibly a son named Ariobarzanes. His first marriage has suggested being the terminus ad quem for when he dropped his Aramaic name, Codomannus. Greek writers report that Artashata later became the "courier" and "slave" of the king, which some modern historians believe was written to belittle Artashata. However, the terms are in reality a Greek translation of the Old Persian bandaka, which did not mean slave, but "henchman, (loyal) servant, vassal."

On the Behistun inscription, Darius the Great ((r. 522 – 486)) uses the term to refer to his high-ranking officers. It has been deduced that Artashata was probably in charge of the royal "postal service," a high-ranking position. He may have held the same Persepolis sector headed by the prominent Persian officer Pharnaces (d. 497 BC). It is uncertain when this happened, it has been suggested this promotion took place around 340 BC, when Artashata married for a second time, with his sister Stateira I. In 339 BC, they had a son named Ochus. This demonstrates that Artaxerxes III, who was highly alert on any possible relative to challenge the throne, did not consider Artashata to pose the slightest threat to him or his son Arses.

In late August/late September 338 BC, the court eunuch and chiliarch (hazahrapatish) Bagoas orchestrated the poisoning and subsequent death of Artaxerxes III through the latter's physician. The majority of Artaxerxes III's sons, except for Arses and Bisthanes, were also murdered by Bagoas. Artaxerxes III's early death proved to be a problematic issue for Persia. Several modern historians argue that his sudden death paved the way for the fall of the Achaemenid Empire. Bagoas, acting as kingmaker, put the young Arses (Artaxerxes IV) on the throne. Arses was determined on trying to free himself from Bagoas' authority and influence; he made an unsuccessful effort to have the latter poisoned, only to be poisoned himself along with the rest of his family by Bagoas, who put Artashata on the throne in 336 BC. Darius III's reputation for bravery, his probable royal descent, and the support he enjoyed from Artaxerxes III all helped him gain acceptance amongst the aristocracy. Macedonian propaganda, made to legitimize the conquests of Alexander the Great a few years later, accused Darius III of playing a key role in the murder of Arses, who was portrayed as the last king of the Achaemenid royal house.

Philip's campaign
Bagoas soon attempted to poison Darius III as well, but his plans were discovered. Darius III summoned Bagoas and asked him to drink a toast for him, giving him his cup which was filled with poison. Bagoas was forced to drink the cup, resulting in his death. This happened around the time of Alexander's accession to the Macedonian throne in the autumn of 336 BC. At the start of Darius III's reign, Egypt and Babylon may have briefly been plunged into rebellion. Regardless, they were seemingly not of heavy significance, as reports about the events quickly disappear.

In 336 BC Philip II of Macedon was authorized by the League of Corinth as its Hegemon to initiate a sacred war of vengeance against the Persians for desecrating and burning the Athenian temples during the Second Persian War, over a century before. He sent an advance force into Asia Minor under the command of his generals Parmenion and Attalus to liberate the Greeks living under Persian control. After they took the Greek cities of Asia from Troy to the Maiandros river, Philip was assassinated and his campaign was suspended while his heir consolidated his control of Macedonia and the rest of Greece.

Alexander's campaign

In the spring of 334 BC, Philip's heir, Alexander, who had himself been confirmed as Hegemon by the League of Corinth, invaded Asia Minor at the head of an army of Macedonian and other Greek soldiers. This invasion, which marked the beginning of the Wars of Alexander the Great, was followed almost immediately by the victory of Alexander over the Persians at Battle of the Granicus. Darius never showed up for the battle, because there was no reason for him to suppose that Alexander intended to conquer the whole of Asia, and Darius may well have supposed that the satraps of the 'lower' satrapies could deal with the crisis, so he instead decided to remain at home in Persepolis and let his satraps handle it. In the previous invasion of Asia Minor by the Spartan king Agesilaus II, the Persians had pinned him in Asia Minor while fomenting rebellion in Greece. Darius attempted to employ the same strategy, with the Spartans rebelling against the Macedonians, but the Spartans were defeated at Megalopolis.

Darius did not take the field against Alexander's army until a year and a half after Granicus, at the Battle of Issus in 333 BC. His forces outnumbered Alexander's soldiers by at least a 2 to 1 ratio, but Darius was still outflanked, defeated, and forced to flee. It is told by Arrian that at the Battle of Issus the moment the Persian left went to pieces under Alexander's attack and Darius, in his war-chariot, saw that it was cut off, he incontinently fled – indeed, he led the race for safety. On the way, he left behind his chariot, his bow, and his royal mantle, all of which were later picked up by Alexander. Greek sources such as Diodorus Siculus' Library of History and Justin's Epitoma Historiarum Philippicarum recount that Darius fled out of fear at the Battle of Issus and again two years later at the Battle of Gaugamela despite commanding a larger force in a defensive position each time. At the Battle of Issus, Darius III even caught Alexander by surprise and failed to defeat Alexander's forces. Darius fled so far so fast that Alexander was able to capture Darius' headquarters and take Darius' family as prisoners in the process. Darius petitioned to Alexander through letters several times to get his family back, but Alexander refused to do so unless Darius would acknowledge him as the new emperor of Persia.

Circumstances were more in Darius' favor at the Battle of Gaugamela in 331 BC. He had a good number of troops who had been organized on the battlefield properly, he had the support of the armies of several of his satraps, and the ground on the battlefield was almost perfectly even, so as not to impede movement of his scythed chariots. Despite all these beneficial factors, he still fled the battle before any victor had been decided and deserted his experienced commanders as well as one of the largest armies ever assembled. Another source accounts that when Darius perceived the fierce attack of Alexander, as at Issus he turned his chariot around, and was the first to flee, once again abandoning all of his soldiers and his property to be taken by Alexander. Many Persian soldiers died that day, so many in fact that after the battle the casualties of the enemy ensured that Darius would never again raise an imperial army. Darius then fled to Ecbatana and attempted to raise a third army, while Alexander took possession of Babylon, Susa, and the Persian capital at Persepolis. Darius reportedly offered all of his empire west of the Euphrates River to Alexander in exchange for peace several times, each time denied by Alexander against the advice of his senior commanders. Alexander could have declared victory after the capture of Persepolis, but he instead decided to pursue Darius.

The Battle of Gaugamela, in which Alexander the Great defeated Darius III of Persia in 331 BC, took place approximately 100 kilometres (62 mi) west of Erbil, Iraq. After the battle, Darius managed to flee to the city. However, somewhat inaccurately, the confrontation is sometimes known as the "Battle of Arbela."

Flight, imprisonment, and death
Darius did attempt to restore his once-great army after his defeat at the hands of Alexander, but he failed to raise a force comparable to that which had fought at Gaugamela, partly because the defeat had undermined his authority, and also because Alexander's liberal policy, for instance in Babylonia and Persis, offered an acceptable alternative to Persian policies.

When at Ecbatana, Darius learned of Alexander's approaching army, he decided to retreat to Bactria where he could better use his cavalry and mercenary forces on the more even ground of the plains of Asia. He led his army through the Caspian Gates, the main road through the mountains that would work to slow the following army. Persian forces became increasingly demoralized with the constant threat of a surprise attack from Alexander, leading to many desertions and eventually a coup led by Bessus, a satrap, and Nabarzanes, who managed all audiences with the King and was in charge of the palace guard. The two men suggested to Darius that the army regroup under Bessus and that power would be transferred back to the King once Alexander was defeated. Darius did not accept this plan and his conspirators became more anxious to remove him for his successive failures against Alexander and his forces. Patron, a Greek mercenary, encouraged Darius to accept a bodyguard of Greek mercenaries rather than his usual Persian guard to protect him from Bessus and Nabarzanes, but the King could not accept for political reasons and grew accustomed to his fate. Bessus and Nabarzanes eventually bound Darius and threw him in an ox-cart while they ordered the Persian forces to continue. According to Curtius' History of Alexander, at this point Alexander and a small, mobile force arrived and threw the Persians into a panic, leading Bessus and two other conspirators, Satibarzanes and Barsaentes, to wound the king with their javelins and leave him to die.

A Macedonian soldier found Darius either dead or dying in the wagon shortly thereafter—a disappointment to Alexander, who wanted to capture Darius alive. Alexander saw Darius’ dead body in the wagon and took the signet ring off the dead king's finger. Afterwards, he sent Darius's body back to Persepolis, gave him a magnificent funeral and ordered that he be buried, like all his royal predecessors, in the royal tombs. Darius's tomb has not yet been discovered. Alexander eventually married Darius' daughter Stateira at Susa in 324 BC.

With the old king defeated and given a proper burial, Alexander's rulership of Persia became official. This led to Darius being regarded by some historians as cowardly and inefficient, as under his rulership, the entirety of the Persian Empire fell to a foreign invader. After killing Darius, Bessus took the regal name Artaxerxes V and began calling himself the King of Asia. He was subsequently captured by Alexander, tortured, and executed. Another of Darius' generals ingratiated himself to Alexander by giving the conqueror Darius' favored companion, Bagoas.

Ancient works

 * Diodorus Siculus, Bibliotheca historica.

title: Darius III


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Henry John Stephen Smith

Prof Henry John Stephen Smith FRS FRSE FRAS LLD (2 November 1826 – 9 February 1883) was an Irish mathematician and amateur astronomer remembered for his work in elementary divisors, quadratic forms, and Smith–Minkowski–Siegel mass formula in number theory. In matrix theory he is visible today in having his name on the Smith normal form of a matrix. Smith was also first to discover the Cantor set.

Life
Smith was born in Dublin, Ireland, the fourth child of John Smith (1792–1828), a barrister, who died when Henry was two. His mother, Mary Murphy (d.1857) from Bantry Bay, very soon afterwards moved the family to England. He had thirteen siblings, including Eleanor Smith, who became a prominent educational activist. He lived in several places in England as a boy. His mother did not send him to school but educated him herself until age 11, at which point she hired private tutors. At age 15 Smith was admitted in 1841 to Rugby School in Warwickshire, where Thomas Arnold was the school's headmaster. This came about because his tutor Henry Highton took up a housemaster position there.

At 19 he won an entrance scholarship to Balliol College, Oxford. He graduated in 1849 with high honours in both mathematics and classics. Smith was fluent in French having spent holidays in France, and he took classes in mathematics at the Sorbonne in Paris during the 1846–7 academic year. He was unmarried and lived with his mother until her death in 1857. He then brought his sister, Eleanor Smith, to live with him as housekeeper at St Giles.

Smith remained at Balliol College as a mathematics tutor following his graduation in 1849 and was soon promoted to Fellow status.

In 1861, he was promoted to the Savilian Chair of Geometry at Oxford. In 1873, he was made the beneficiary of a fellowship at Corpus Christi College, Oxford, and gave up teaching at Balliol.

In 1874 he became Keeper of the University Museum and moved (with his sister) to the Keeper's House on South Parks Road in Oxford.

On account of his ability as a man of affairs, Smith was in demand for academic administrative and committee work: he was Keeper of the Oxford University Museum; a Mathematical Examiner for the University of London; a member of a Royal Commission to review scientific education practice; a member of the commission to reform University of Oxford governance; chairman of the committee of scientists overseeing the Meteorological Office; twice president of the London Mathematical Society; etc.

He died in Oxford on 9 February 1883. He is buried in St Sepulchre's Cemetery in Oxford.

Researches in number theory
''An overview of Smith's mathematics contained in a lengthy obituary published in a professional journal in 1884 is reproduced at NumberTheory.Org. The following is an extract from it.''

Smith's two earliest mathematical papers were on geometrical subjects, but the third concerned the theory of numbers. Following the example of Gauss, he wrote his first paper on the theory of numbers in Latin: "De compositione numerorum primorum formæ $$4n+1$$ ex duobus quadratis." In it he proves in an original manner the theorem of Fermat---"That every prime number of the form $$4n+1$$ ($$n$$ being an integer) is the sum of two square numbers." In his second paper he gives an introduction to the theory of numbers.

In 1858, Smith was selected by the British Association to prepare a report upon the Theory of Numbers. It was prepared in five parts, extending over the years 1859–1865. It is neither a history nor a treatise, but something intermediate. The author analyzes with remarkable clearness and order the works of mathematicians for the preceding century upon the theory of congruences, and upon that of binary quadratic forms. He returns to the original sources, indicates the principle and sketches the course of the demonstrations, and states the result, often adding something of his own.

During the preparation of the Report, and as a logical consequence of the researches connected therewith, Smith published several original contributions to the higher arithmetic. Some were in complete form and appeared in the Philosophical Transactions of the Royal Society of London; others were incomplete, giving only the results without the extended demonstrations, and appeared in the Proceedings of that Society. One of the latter, entitled "On the orders and genera of quadratic forms containing more than three indeterminates," enunciates certain general principles by means of which he solves a problem proposed by Eisenstein, namely, the decomposition of integer numbers into the sum of five squares; and further, the analogous problem for seven squares. It was also indicated that the four, six, and eight-square theorems of Jacobi, Eisenstein and Liouville were deducible from the principles set forth.

In 1868, Smith returned to the geometrical researches which had first occupied his attention. For a memoir on "Certain cubic and biquadratic problems" the Royal Academy of Sciences of Berlin awarded him the Steiner prize.

In February, 1882, Smith was surprised to see in the Comptes rendus that the subject proposed by the Paris Academy of Science for the Grand prix des sciences mathématiques was the theory of the decomposition of integer numbers into a sum of five squares; and that the attention of competitors was directed to the results announced without demonstration by Eisenstein, whereas nothing was said about his papers dealing with the same subject in the Proceedings of the Royal Society. He wrote to M. Hermite calling his attention to what he had published; in reply he was assured that the members of the commission did not know of the existence of his papers, and he was advised to complete his demonstrations and submit the memoir according to the rules of the competition. According to the rules each manuscript bears a motto, and the corresponding envelope containing the name of the successful author is opened. There were still three months before the closing of the concours (1 June 1882) and Smith set to work, prepared the memoir and despatched it in time.

Two months after Smith's death, the Paris Academy made their award. Two of the three memoirs sent in were judged worthy of the prize. When the envelopes were opened, the authors were found to be Smith and Minkowski, a young mathematician of Königsberg, Prussia. No notice was taken of Smith's previous publication on the subject, and M. Hermite on being written to, said that he forgot to bring the matter to the notice of the commission.

Work on the Riemann integral
In 1875 Smith published the important paper on the integrability of discontinuous functions in Riemann's sense. In this work, while giving a rigorous definition of the Riemann integral as well as explicit rigorous proofs of many of the results published by Riemann, he also gave an example of a meagre set which is not negligible in the sense of measure theory, since its measure is not zero: a function which is everywhere continuous except on this set is not Riemann integrable. Smith's example shows that the proof of sufficient condition for the Riemann integrability of a discontinuous function given earlier by Hermann Hankel was incorrect and the result does not hold: however, his result remained unnoticed until much later, having no influence on successive developments. In an 1875 paper, he discussed a nowhere-dense set of positive measure on the real line, an early version of the Cantor set, now known as the Smith–Volterra–Cantor set.

title: Henry John Stephen Smith