Alert Ready

The National Public Alerting System (), branded as Alert Ready (), is the national warning system in Canada. The system consists of infrastructure and standards for the presentation and distribution of public alerts issued by government authorities (including Environment and Climate Change Canada and other provincial public safety agencies), such as weather emergencies, AMBER Alerts, and other emergency notifications, by all broadcasters and last mile distributors in the affected region, including television stations, radio stations, television providers, and mobile networks in the affected region.
The system is based upon the Common Alerting Protocol (CAP), while wireless alerts (Wireless Public Alerting System) use a Canadian variant of the Wireless Emergency Alerts standard adopted in the United States. Alert Ready officially launched on March 31, 2015; under Canadian Radio-television and Telecommunications Commission (CRTC) orders, all terrestrial radio and television broadcasters, digital multichannel television providers are required to relay emergency messages issued through NAAD. Wireless providers were required to implement the system in April 2018. Pelmorex Media (owner of The Weather Network) operates its backend — the "National Alert Aggregation and Dissemination" (NAAD) system — which distributes alerts to broadcasters and other parties. Pelmorex is also responsible for public marketing of the system.
Alert Ready has faced criticism, particularly due to wireless alerts being handled under a blanket category with no opt-outs (as opposed to the severity-based system used by the U.S. WEA system), and criticism over AMBER Alerts (particularly in Ontario) being issued province-wide rather than geographically targeted to relevant regions. In April 2020, the RCMP faced criticism for not cooperating with local officials in use of the system to warn of an active gunman, which had attacked multiple locations in the province of Nova Scotia.
Environment and Climate Change Canada also maintains Weatheradio Canada, which transmits weather information and hazard alerts. Unlike Alert Ready, it utilizes Specific Area Message Encoding (SAME)—the same protocol used by NOAA Weather Radio and the Emergency Alert System in the United States.
History.
Proposals.
Various attempts had been made in the 21st century to establish a public alert system in Canada, by both departments of government and by television broadcasters. In 2001, Pelmorex, owners of The Weather Network and its French language counterpart MétéoMédia, applied to the CRTC for an amendment to their licenses to encompass a mandatory "All Channel Alert" system, requiring all television providers to relay emergency messages on behalf of governments across all of their channels. The service would have used proprietary hardware developed by Pelmorex, and would have been funded primarily by a $0.13 increase in carriage fees for the two channels. However, its initial proposal was denied by the CRTC, citing the need for consultation with broadcasters, television providers, and other parties on how the system would be designed, along with its costs. There were also concerns over the means in which the system was to be implemented technologically, and that the system was not inclusive to the visually impaired because Pelmorex only specified use of a text crawl, without an audio component.
Establishment of such a system in a voluntary form was hampered by CRTC rules at the time, which required television providers to obtain consent from broadcasters before they could overlay emergency notifications onto their programming. In 2005, CRTC called for proposals regarding a national alerting system; Pelmorex and the Canadian Broadcasting Corporation submitted proposals for a national system, while Bell ExpressVu submitted a proposal requesting the removal of the consent requirement. The CBC's proposal would have fed alerts via satellite to decoders installed at local CBC Radio transmitters. It would have allowed television providers to participate on a voluntary basis by installing decoders of their own, if the aforementioned consent requirement were removed. All three applicants promised to adopt the standardized alerting specifications that were developed by the government-backed CANALERT initiative.
While broadcasters and governments supported the proposals for a national alerting system, CTV, Canwest Global and CHUM Limited showed concerns surrounding the Pelmorex proposal, as it would be operated by a for-profit venture that would have the power to override their signals with third-party content, and be redundant to alerts already provided as a public service by some broadcasters. Shaw and Rogers argued that the Pelmorex system was less cost-efficient than the CBC's proposal. It was also disputed whether the CRTC could even order the mandatory distribution of the service, as the CRTC does not regulate alphanumeric content because it is not considered television programming. In 2007, the CRTC removed the consent requirement to ease the adoption of voluntary alerting by broadcast distribution undertakings, but stated that it would reconsider the possibility of a mandatory alert system in the future.
Development.
In 2009, the CRTC approved a proposal by Pelmorex to grant must-carry status to The Weather Network and MétéoMédia on digital television services. As part of the proposal, Pelmorex committed to developing a "national aggregator and distributor" (NAAD) of localized emergency alert messages compliant with the Common Alerting Protocol. Pelmorex established a governance council for the system, including representatives of the broadcasting industry, federal government, members of the Senior Officials Responsible for Emergency Management (SOREM), and the Canadian Association for Public Alerting and Notification, to oversee its operations. NAAD would become operational in 2010.
In a 2011 renewal of the must-carry status, the CRTC praised Pelmorex's work, considering the NAAD system to be "an essential element of a national public alerting system", and expressed an expectation for all broadcasters to voluntarily participate in distributing its alerts. However, as a condition of the renewal, the CRTC ordered Pelmorex to reach agreements with all federal, provincial and territorial emergency management officials to allow them to transmit messages through the system, implement the capability of processing "broadcast intrusive alerts" through the system, and develop a public awareness campaign surrounding the alert system with a budget of at least $1 million per-year. In June of the same year, the province of Alberta launched a new alerting system of its own, Alberta Emergency Alert, which distributes alerts on radio and television, as well as online and via social networking services. This replaced the previous Emergency Public Warning System which had been used since 1992.
Environment Canada, the Canadian Council of Emergency Management Organizations, and the provinces of Manitoba and New Brunswick endorsed the potential use of the NAAD framework as a backend for a mandatory public alerting system. On May 26, 2013, SOREM published a "Common Look and Feel" specification for alerts. Developed with guidance from the broadcasting industry, it describes how and when alerts are to be distributed and presented to the public. As part of a license renewal in May 2013, CBC Radio agreed to implement an alerting system using NAAD by December 31, 2014.
On February 27, 2014, the CRTC issued a proposal to mandate participation in the national alert system by all broadcasters. The commission felt that owing to the importance of the endeavour, broadcasters had displayed an inconsistent level of commitment to implement it voluntarily.
Launch.
On August 29, 2014, the CRTC ruled that all Canadian broadcasters, including over-the-air television broadcasters, radio broadcasters, and broadcast distribution undertakings, must begin participating in the National Public Alerting System by March 31, 2015. Community, campus, and aboriginal broadcasters were given an extended deadline of March 31, 2016 to implement the system. The National Public Alerting System was launched on March 31, 2015 under the public-facing brand Alert Ready (French: En Alerte).
Bell Satellite TV, MTS, Shaw Direct, and Sogetel do not fully participate in the system, as some of their customers utilize legacy set-top boxes that cannot be updated to support the display of public alerts. They were granted a six-month extension for the implementation deadline by the CRTC in order to address these issues, under the condition that they inform customers that they cannot receive public alerts unless they update their hardware, and must present bi-weekly progress reports to the CRTC. CRTC chairman Jean-Pierre Blais criticized the providers for their inability to properly implement the system, stating that the commission "will not hide our disappointment that certain television service providers are not ready, despite having been given more than enough time [to implement it.]"
In December 2015, the CRTC granted an indefinite extension of the exceptions and reporting guidelines to Bell, Shaw Direct, and MTS until they completely phase out hardware that is not compatible with the NPAS. The CRTC felt that the providers had made a good-faith effort in informing customers of their inability to receive public alerts and offering hardware replacements. Bell reported that some customers had declined their offering of a free set-top box replacement as they did not want to participate in receiving alerts. MTS discontinued its legacy "Classic TV" service due to its inability to display alerts, and provided incentives for its remaining customers to upgrade to its current "Ultimate TV" service.
Provincial tests began to improve public awareness of the new system, such as in Manitoba, and Quebec. One of the tests in the province on May 19, 2015, simulating a tornado emergency in the Centre-du-Québec region, surprised many people tuned in to radio or TV in the region, leading some to believe that there was an actual tornado emergency.
Mobile alerting.
On April 6, 2017, the CRTC required all wireless carriers in Canada to begin relaying public alerts over their LTE and post-LTE wireless networks by April 6, 2018, using Cell Broadcast-based standards by ATIS similar to the U.S. Wireless Emergency Alerts system, collaborating with the SOREM Common Look and Feel guidelines. The CRTC required at least half of devices offered by wireless carriers to support wireless public alerts (including at least one device available at no-charge on-contract, and one "accessible" device) by the implementation date. As of April 2019, all devices sold by wireless carriers must support wireless public alerts. The mandate does not apply to networks older than LTE; the CRTC chose to exclude older network technologies (such as 3G) from the mandate, citing wide deployment of LTE networks across the country, and that government funding has been used to support its deployment in underserved areas.
On January 29, 2018, the CRTC informed the Standing Senate Committee on National Security and Defence of concerns surrounding the possibility of false positives (such as a false missile alert in Hawaii earlier that month), and ensuring that the industry meets its goal of at least 50% of devices being capable of receiving mobile alerts by that date.
Operation.
Alert Ready alerts are broadcast to last-mile distributors using the Anik F1R satellite over C-band on virtual channel 206 (with virtual channel 550 as a backup), and prior to September 1, 2019, using the Anik F2 satellite over Ku-band. The Ku-band signals were terminated August 31, 2019 after consultation with the Last Mile Distributors determined that it was underutilized and sufficient alternate services exist. Alerts are also distributed over the internet from web servers based in Oakville and Montreal on TCP port 8080. An RSS feed of past alerts is also available.
The presentation of alerts is dictated by the NPAS Common Look and Feel Guidance. Messages are formatted using the Canadian Profile of the Common Alerting Protocol (CAP-CP), and are provided in at least one of Canada's official languages (either English, French, or both, as determined by local policies and laws). Alerts can contain text and audio components, and contain information designating the region that an alert applies to.
Messages marked with "Broadcast Immediately" flags are used to designate alerts that present "an imminent or unexpected threat to life, that alerting officials wish to be distributed and presented to the public as soon as possible, even if it means disrupting the programming of last mile distributors." On television and radio, relevant alerts marked with this flag are immediately presented to viewers when they are received, interrupting programming to facilitate their display. These alerts are prefaced by the Canadian Alerting Attention Signal. The attention signal is an 8-second sequence of alternating half-second duration complex tones, the first being a combination of tones at frequencies of 932.33 Hz, 1046.5 Hz and 3135.96 Hz, and the second at 440 Hz, 659.26 Hz and 3135.96 Hz (the same signal that is used by Alberta Emergency Alert). The attention signal is followed by the audio of the alert where applicable or supported by hardware (in the absence of audio, the alert may be read using a text to speech system, or a generic message played).
The Common Look and Feel Guidance prescribes that, on television, the text of alerts be displayed on either a crawler, or as a full screen notice that covers programming, in white text on a red background in both cases. Crawlers inserted by television channels are positioned at the centre of the screen out of respect for those inserted by broadcast distribution undertakings at the bottom of the screen. The guidelines note that "automated broadcast interruption need not be used if a person can present the text of an audience alert message verbally and visually mindful of the other guidance found in [the guidelines]."
A separate "Wireless Immediately" flag is used to require Cell Broadcast distribution as well. Wireless alerts on devices specifically supporting Cell Broadcast alerts utilize the same attention signal sound and a specific vibration pattern mimicking this sound. The Common Look and Feel Guidance specifies that the phone's software must display these alerts with a capitalized, bilingual heading reading "EMERGENCY ALERT / ALERTE D'URGENCE", and a limit of 600 characters due to device limitations. Devices may not be configured to allow users to opt out of alerts: they are coded using the same priority level as U.S. presidential alerts—the only U.S. category which may not be disabled. If the device's software is primarily designed for the U.S. implementation of WEA, it will typically identify Canadian wireless emergency alerts as a presidential alert.
Mobile apps are available that distribute alerts as push notifications to mobile devices such as smartphones.
Public awareness testing.
Public awareness tests are held twice per-year, in which a 30-second test message (60 seconds in provinces where bilingual messages are issued) is distributed to radio and television outlets, and a Cell Broadcast message is sent to wireless phones. One is held on a Wednesday in May during Public Safety Canada's Emergency Preparedness Week, and the second is held in November.
Initially, four, quarterly tests were scheduled annually, on the third Wednesday of every third month of the year, along with the Emergency Preparedness Week test. After the introduction of wireless alerts in May 2018, however, SOREM quietly ceased all quarterly tests, citing concerns that tests only directed to broadcast media could cause confusion to the public over the effectiveness and correct operation of the extended system. The CRTC supported this position, officially requiring wireless service providers to participate in two public-visible alerts per year, distributed on all platforms.
There is precedent for alert tests to be postponed or cancelled on a regional basis if there is an ongoing threat of an actual emergency in an area, such as for instance, Spring flooding in parts of Ontario and Quebec in May 2019.
Participating agencies.
Alerts distributed by NAAD originate from specially-designated provincial and federal agencies and ministries, such as Environment and Climate Change Canada, and provincial alerting agencies.
As the system was designed primarily to handle domestic situations such as weather, Environment and Climate Change Canada is currently the only federal agency that is capable of issuing alerts. Following the false alarm in Hawaii in January 2018, discussions began on how the possibility of an actual missile attack threat would be handled, and the Canadian Press reported in December 2018 that they were "finalizing" a protocol for the handling of such events.
Reception.
An activation of the system in Manitoba for a tornado spotting led to criticism over the quality of the text-to-speech system used by Manitoba's implementation of the alert system, with viewers reporting a "garbled" message and mispronunciations of community names.
On March 6, 2016, Alert Ready was used to distribute an AMBER Alert in Ontario relating to an alleged kidnapping of a child in Orillia (the child's father was actually picking up the child after they had run from home). Viewers felt that the frequent notifications (especially as it occurred during the U.S. airing of the series finale of "Downton Abbey", seen via PBS stations carried in Canada) with full-screen messages and alarm sounds as being disruptive, in comparison to the previous, voluntary practice of Canadian broadcasters displaying AMBER Alert messages on tickers. At the same time, the move was praised for providing a higher degree of prominence to the alert; Orillia Ontario Provincial Police commander Patrick Morris defended its use, stating that "while I will apologize for any inconvenience this may have caused, we won't apologize for using all of the tools available to us to find a missing child."
"National Post" columnist Matt Gurney provided similar praise, but noted that the system's operation hindered its ability to disseminate information quickly. He explained that while the purpose of such a system is to "[get] information to the public as rapidly and as clearly as possible", on his television provider's set-top box "the text was arriving on my screen incredibly slowly. Several minutes into the alert, we were just starting to get the description of the child and the suspect vehicle. It was embarrassing — when seconds count, the province needed minutes to deliver incredibly basic, utterly crucial facts about the emergency. The contrast between the urgently screeching buzz of the alarm and the text crawling up the screen in ultra-slow motion seemed designed for comedic effect." He also, similarly, noted the poor quality of the text-to-speech systems.
Wireless alerting.
The first public awareness test to include Cell Broadcast transmission occurred in Ontario and Quebec on May 7, 2018, with the remaining provinces following suit on May 9, in observance of Emergency Preparedness Week. In Quebec, a syntax error in the test message's XML file caused the wireless alert to fail, while users in Ontario reported mixed results. Testing in the remaining provinces and territories, excluding Nunavut, occurred May 9, 2018, again to mixed results, with people reporting having not gotten an alert despite their phones meeting the requirements. Though the test was reported to have worked in Nova Scotia and PEI, reports from Alberta, Manitoba, the Northwest Territories, Newfoundland and Labrador, and Saskatchewan saw people failing to get alerts (although the crown telecom SaskTel deemed it a success). Following the first test, Bell Mobility and Telus acknowledged issues preventing receipt of the messages.
In September 2018, Scott Shortliffe, chief consumer officer of the CRTC, acknowledged that the effectiveness of the system was being affected by technical problems, following a province-wide AMBER Alert in North Battleford, Saskatchewan where the wireless cell broadcast was delayed by several hours (as well as complaints by Manitoba residents over the cross-provincial extension of the alert), and mixed reports during tornadoes in the Ottawa-Gatineau region (including some users not receiving any message, and users in Quebec only receiving an English-language message). On October 3, 2018, CRTC Secretary General Claude Doucet issued a letter ordering service providers to participate in another visible all-channel alert test that was being scheduled in November (scheduled for November 28). SOREM also requested that the CRTC mandate biannual tests of wireless alerts rather than just annual.
Criticism emerged following an AMBER Alert in Ontario on May 14, 2018 for a child allegedly kidnapped in Thunder Bay, Ontario—the first to have included mobile alerts. Three different emergency messages were sent, the first contained only English text, but was resent merely a half hour later in order to add a French-language translation, followed by one more alert message stating that the AMBER Alert had been cancelled, and the child found safe. Criticism was directed primarily towards the intrusive presentation of the message, and that all AMBER Alerts are sent across the entire province rather than regionalized to only cover a vicinity around the city (the alert was received as far as Toronto, which is at least away from Thunder Bay), and the fact that all wireless alerts are mandatory and cannot be disabled (unlike in the U.S., where AMBER Alerts are one of three categories of alerts users may opt out from, while "Presidential" level alerts are mandatory and may not be disabled). There were concerns that such excessive use could result in alarm fatigue and desensitization to actual local emergencies.
In a 2019 follow-up to his previous criticisms of the system, Gurney noted that several instances of AMBER Alerts issued during the early-morning hours in Ontario had prompted residents to call 9-1-1 to complain. He argued that such alerts were only of immediate relevance to recipients that were awake, attentive, and "out-and-about" (who could be targeted via radio and television rather than cell broadcast), but were being given the same intrusive presentation as alerts for imminent threats to public safety that affect a larger population. Gurney thus argued that in such a situation, those who were not awake could learn the details of the alert in the morning instead. He also acknowledged similar opinions presented by Philip Cross in an editorial for its sister publication, the "Financial Post"; Cross felt that AMBER Alerts on phones should be optional as in the U.S., comparing the disruption of sleep caused by late-night alerts to health effects associated with daylight saving time, and noting that the public would be "more likely to arrive at better solutions if we can discuss the issue sanely, rather than have all complaints dismissed as callous disregard for children's safety." In response to an Ontario AMBER Alert that concluded with the victim found murdered, a "Maclean's" editor had deemed all critics of the system to be "horrible" and expressing entitlement for their refusal to participate in the civic duty of locating a missing child in imminent danger.
In October 2019, a 70-year old resident of Hamilton, Ontario was charged with mischief for disrupting emergency services, after repeatedly calling 9-1-1 to complain about an AMBER Alert broadcast.
False alarms.
On January 12, 2020 at 7:24 a.m. ET, an emergency alert was issued for the entire province of Ontario, advising that an incident had been reported at the Pickering Nuclear Generating Station. The alert stated that "there has been no abnormal release of radioactivity from the station", staff was responding to the situation, and that no immediate protective actions were required. Approximately 40 minutes later, Ontario Power Generation issued a statement on its Twitter account indicating that the alert had been sent in error. At around 9:10 a.m., a second emergency alert message was issued, containing a similar retraction. Solicitor General Sylvia Jones stated that the erroneous alert was the result of a mistake during a "routine training exercise" by Ontario's emergency operations centre.
In its aftermath, Dave Ryan and John Tory—the mayors of Pickering and Toronto, made calls for a full investigation into the incident. MPP and Ontario NDP energy and climate change critic Peter Tabuns also responded, stating that "confidence in the accuracy of the [alert] system is essential for public safety." U.S. media outlets drew comparisons between the incident and the 2018 Hawaii false missile alert (coincidentally, the Pickering incident occurred on the eve of the Hawaii incident's second anniversary).
On February 27, 2020, the results of a formal investigation were published. It found that the alert was triggered by a duty officer of Ontario's emergency operations centre at the beginning of his shift, who had accidentally remained logged into the live alert system when performing a regular internal test. It was considered common practice for officers to log into the live system first to ensure it is functioning before switching to a secondary, internal instance of the system designed specifically for testing purposes. The officer had realized his error, but the alert retracting the previous message was delayed due to an inability to receive clear instructions from Emergency Management Ontario supervisors on how to make an alert broadcast-intrusive, or whether the second alert should have been at all.
Testing errors.
On May 12, 2021, Emergency Management British Columbia accidentally re-issued a test alert that had been used for the biannual provincial test one week prior.
Non-usage during the 2020 Nova Scotia attacks.
The Royal Canadian Mounted Police was criticized following the April 19, 2020 killing spree in Nova Scotia, for not having used the alert system to warn of the active gunman at-large. Premier Stephen McNeil stated that he had not formally received any request for the issuance of an emergency alert, and explained that "I can tell you, I'm not going to second-guess why someone or the organization did what they did or didn't do at this moment in time. This was an active environment, I can tell you. Deaths, gunfire. Let's give them an opportunity as an organization to explain that to you." RCMP Chief Superintendent Chris Leather stated that they had primarily relied on Twitter to issue timely updates on the situation due to its fluidity. However, the areas where the shootings occurred had poor internet access, and a large population of seniors (who may not necessarily be active users of social media).
On April 22, Premier McNeil stated that at 10:15 a.m. AT, provincial emergency management officials had attempted to contact RCMP officials "a number of times" for permission and cooperation in issuing an alert. However, the RCMP did not respond; the suspect was shot and killed by police just over an hour later at 11:26 a.m.
The RCMP does not currently have a national strategy or policy for use of emergency alerts during such situations, but some law enforcement agencies began to express interest in evaluating future use of the system as part of their communications strategies.


CTM Festival

The annual CTM Festival is a music and visual arts event held in Berlin, Germany. Founded in 1999, the festival originally focused on electronic music, but has since evolved to cover a wide range of genres under the banner "Festival for Adventurous Music and Art".
Changing through various shapes and formats over the years, the festival currently takes place as a 10-day long event in which the music program is supplemented by an extensive daytime program of workshops, art installations, panel discussions, screenings and presentations that illustrate the latest artistic, technological and economic developments in music and media cultures.
Distinguishing the festival from many others in its field is the fact that CTM spotlights music’s social role in electronic and digital culture. Through the festival, as well as various events curated by CTM throughout the year, the organization reflects the latest musical currents against a backdrop of new technologies, modern art, historical perspective, and social issues.
Books.
GENDERTRONICS – DER KÖRPER IN DER ELEKTRONISCHEN MUSIK
Edited by CTM and Meike Jansen
When, in the early 50s, electronic music appeared on the scene with the promise of abandoning all physical limits of music-making this was – like much besides – a Promethian male fantasy. Indeed, this music subsequently led to everything but disembodiment. From the psychedelic trances of the 60s and Kraftwerk robotics of the 70s, through to Techno ecstasies, gender-political interventions in the 90s and laptop performance – the questions as to how, from whom, to what ends and in which contexts electronics and the human body might be cable-linked have continually had to be addressed anew.
“Gendertronics” is an in-depth study of questions raised by last year’s festival theme, "Performing Sound", edited by CTM and Meike Jansen and published by Edition Suhrkamp Verlag. With contributions from Olaf Arndt, Claudia Basrawi, Jochen Bonz & Thomas Meinecke, Mariola Brillowska, Kurt Dahlke, Diedrich Diederichsen, Harald Fricke, Tom Holert, Miss Kittin, Pinky Rose, Birgit Richard, Terre Thaemlitz, Marc Weiser and 17 black and white drawings by Jan Rohlf. – In German language only.


276 Adelheid

Adelheid (minor planet designation: 276 Adelheid) is a dark Alauda asteroid from the outer region of the asteroid belt, approximately 121 kilometers in diameter. It was discovered by Austrian astronomer Johann Palisa at Vienna Observatory on 17 April 1888. The meaning of the asteroids's name is unknown.
Classification.
"Adelheid" is a member of the Alauda family (), a large family of typically bright carbonaceous asteroids and named after its parent body, 702 Alauda.
Physical characteristics.
Photometric observations in 1992 gave a lightcurve with a period of 6.328 ± 0.012 hours and a brightness variation of 0.10 ± 0.02 in magnitude. The curve is regular with two maxima and minima.
In the Tholen classification, its spectrum has been characterized as that of an X-type asteroid, while polarimetric observations refined its classification to a primitive P-type.
Naming.
Any reference of this minor planet's name to a person or occurrence is unknown.
Unknown meaning.
Among the many thousands of named minor planets, "Adelheid" is one of 120 asteroids, for which no official naming citation has been published. All of these low-numbered asteroids have numbers between and and were discovered between 1876 and the 1930s, predominantly by astronomers Auguste Charlois, Johann Palisa, Max Wolf and Karl Reinmuth "(also see )".


Karate (software)

Karate is an open-source general-purpose test-automation framework that can script calls to HTTP end-points and assert that the JSON or XML responses are as expected. Karate also has support for service-virtualization where it can bring up "mock" (or stub) servers which can substitute for web-services that need to participate in an integration-test. Karate's capabilities include being able to run tests in parallel, HTML reports and compatibility with Continuous Integration tools.
The additional capability to re-use functional tests as performance-tests via integration with the Gatling tool was released in July 2018 The project also added the capability to perform web-UI automation in 2019 which was declared out of RC (release-candidate) status in 2020.
Karate is implemented in Java but test-scripts are written in Gherkin since Karate was originally an extension of the Cucumber framework. It was built within Intuit and released under the MIT license.
History.
Karate was created by Peter Thomas, an Intuit employee.
Basic usage.
This example shows what a simple Karate test script looks like and how it is based on the Gherkin syntax.
Feature: karate 'hello world' example
Scenario: create and retrieve a cat
Given url 'http://myhost.com/v1/cats'
When method post
Then status 201
Given path response.id
When method get
Then status 200
This actually makes two calls, first an HTTP codice_1 to 'http://myhost.com/v1/cats' and then a codice_2 to the same URL but with the value of codice_3 appended as a REST-ful path parameter. The codice_4 keyword is used for asserting that a given payload is as expected. The use of the codice_5 "fuzzy match" token takes care of "ignoring" the actual value since it is dynamic, as it is a server-side auto-generated identifier.
Reception.
Karate was featured as one of the top 5 open-source API testing tools within six months of its release. It was also mentioned as one of the 10 API testing tools to try in 2017.
Karate was first listed in the ThoughtWorks Technology Radar in 2019 with a rating of "Assess". One year later it moved into the "Trial" category in May 2020.


Mono No Aware (organization)

Mono No Aware is a cinema-arts non-profit organization. Founded in 2007, its mission entails the advancement of “connectivity through the cinematic experience.” The organization is named after the Japanese concept mono no aware (物の哀れ), which expresses the impermanence of being and beauty.
Mono No Aware is based in Brooklyn, New York and organizes artist screenings, analog filmmaking workshops, equipment rentals, and film stock distribution. Film artists Bill Brand and Leslie Thornton serve on its Advisory Board. Since its founding, the organization has hosted an annual film festival, exhibiting expanded cinema works from around the world that utilize both analog technologies and live performance. Following the completion of the tenth festival, a month-long series of exhibitions, Mono No Aware undertook the construction of the first non-profit motion picture laboratory to operate in the United States.
Annual film festival.
In 2007, the first Mono No Aware film festival was held in the Galapagos Art Space in Brooklyn. It was free to attend and free to submit and the same holds true today. In 2016, the organization, in celebration of its tenth anniversary, organized MONO X, a 21-event film festival spanning three boroughs in New York City. It presented the work of 150 artists to an audience of 5,000. Works included sculpture, expanded cinema, and installation art incorporating the moving image on film and/or video. Presenting sponsors included Kodak, Brooklyn Brewery, and Technicolor.
Educational Initiatives.
Mono No Aware runs workshops in Brooklyn year-round, and collaborates with institutions throughout North America, to teach students analog filmmaking techniques. Workshops include non-toxic film processing with coffee and beer, hand-making photographic emulsion, 2D and 3D analog animation, optical printing and contact printing, and Super 8mm, 16mm, and 35mm filmmaking.


Sokha (unit)

Sokha (big plow) – a unit of land measure in Russia in 13th-17th centuries. Sokha was used as unit of taxation for worked land owned by someone else or the State. The term originated when Tatars took rent from the actual plows - the land working tool which used 2-3 horses.
At the end of 15th century Novgorod sokha equalled 3 обжs. A Moscow sokha was equal to 10 Novgorod ones and was paid according to the different sizes in various regions of the state.
In 1550 a new land-measure unit was established which kept the old name "sokha". The name was then used for a certain amount of exploited land. A sokha was divided into quarters.
There were several types of sokha:
Here, "government" meant someone who was working in government sectors, and "black" meant an average person ("the masses"), "quarter" or "quart" meant an area of land that is planted with a quarter of rye. The land was planted in such a way that the area of land seeded with two "quarters" was equal to a (an area of land approximately equal to one hectare).
Because of the different types of sokha, land owned by lower classes was worth less towards the tax, and most of the tax was paid by the peasants. Given the same amount of land, they paid twice as much as government officials, and 1.5 times as much as church officials.
In order to find out how many sokhas were in the state, a census measured and counted the rented lands. Quarters of land that were planted for personal use were free from payment. Until the beginning of the 17th century, a peasant who lived and worked on the land owned by a "government" employee, just like a peasant living on their own land or state land, paid equally for each exploited quarter of land. This section of land was called "living". At the beginning of the 17th century, the term "living quarter" was used not for the actual quarter of worked land, but for several whole farms in which more than a single quarter was planted.
This was done in the interest of government officials - their lands were used to establish "living quarters" which constituted 10-16 yards. This means that if every yard was divided into 4 quarters (two desyatinas), then such a "living quarter" was then 40 to 64 quarters of "actual" quarters of utilized land. Therefore, 40 to 64 quarters of government official land paid the same amount of tax as one peasant paid from the single quarter that he lived on (whether self- or state-owned).
In 1678–1679 the sokha was replaced with the "yard count", meaning taxes were to be paid for the number of peasant yards, rather than the area of the land occupied by the yards.
Taxes from small villages was also collected in sokhas. The term sokha remained but now it referred to a certain number of yards/houses. There were several types of sokha again: "best", "average", "lesser", and "worst" villagers. Sokha of the best villages was equal to 40 yards, average - 80, lesser - 160, and worst - 320 yards. Every sokha was charged the same amount of tax. Therefore, best villages paid the same amount for half the number of farms than the average ones. If there were not enough lesser or worst sokhas, then the tax was charged from the people that lived in average or best sokhas.


Just diatonic semitone


2045 Peking


Feddan

A feddan () is a unit of area. It is used in Egypt, Sudan, Syria and the Sultanate of Oman. In Classical Arabic, the word means 'a yoke of oxen': implying the area of ground that could be tilled by them in a certain time. In Egypt the feddan is the only non-metric unit which remained in use following the switch to the metric system. A feddan is divided into 24 kirat (, "qīrāt") in which one kirat equals 175 square metres.
Equivalent units.
1 feddan = 24 kirat = 60 metre × 70 metre = 4200 square metres (m²) = 0.420 hectares = 1.037 acres
In Syria, the feddan ranges from 2295 square metres (m²) to 3443 square metres (m²).


Malagasy units of measurement

A number of units of measurement were used in Madagascar to measure length, mass, capacity, etc. The Metric system was introduced in Madagascar in 1897.
System before metric system.
Several units were used.
Length.
One rahf was equal to 46.46 in (1.18 m) (the value was double in the northern part of Madagascar).
Mass.
For gold and silver.
Units included:
1 vari = 3 nanki
1 sompi = 2 vari = 60 grain.
Capacity.
One bambou was equal to 0.0568 bushel.


Double drop D tuning

Double drop D tuning: DADGBD, also known simply as double drop D, is an alternative guitar tuning: E strings are tuned down ("dropped") one whole step (2 frets) to D rather than E as in standard tuning (EADGBE).
Uses of double dropped D tuning.
The main use for double dropped D is so that guitarists can play intervals of a fifth with one finger on the bass strings, and play the treble side of a barre chord. Some recordings that make effective use of this tuning are "Black Water" by The Doobie Brothers, "The Loner," "Cortez the Killer," "Don't Let It Bring You Down," "Ohio" and "Cinnamon Girl" by Neil Young, "Choctaw Bingo" and "We Can't Make It Here" by James McMurtry, "Find the Cost of Freedom" by Crosby, Stills, Nash, and Young, "Going to California" by Led Zeppelin, "The End" by The Doors, "Devils & Dust" by Bruce Springsteen, "Satellite" by Elliott Smith, "Overkill" by Men At Work, "Bryter Layter" by Nick Drake and "Nobody's Fault but My Own" by Beck.
Examples of chords in dropped D tuning.
Chords in double dropped D tuning are formed as they are in standard tuning, with the exception of the first and sixth strings, which are either omitted or fretted one whole step higher:
Note that these chords are not the power chords commonly played in double drop D tuning. Power chords generally mute the higher notes rather than the lower notes:
For purposes of making the table easier to read, spaces are provided between each number when the fret number becomes a double digit.
Some examples of the simplified barre chords are shown below.
Chords involving the minor third may be more difficult to fret.


Universal testing machine

A universal testing machine (UTM), also known as a universal tester, materials testing machine or materials test frame, is used to test the tensile strength and compressive strength of materials. An earlier name for a tensile testing machine is a tensometer. The "universal" part of the name reflects that it can perform many standard tensile and compression tests on materials, components, and structures (in other words, that it is versatile).
Components.
Several variations are in use. Common components include:
Use.
The set-up and usage are detailed in a test method, often published by a standards organization. This specifies the sample preparation, fixturing, gauge length (the length which is under study or observation), analysis, etc.
The specimen is placed in the machine between the grips and an extensometer if required can automatically record the change in gauge length during the test. If an extensometer is not fitted, the machine itself can record the displacement between its cross heads on which the specimen is held. However, this method not only records the change in length of the specimen but also all other extending / elastic components of the testing machine and its drive systems including any slipping of the specimen in the grips.
Once the machine is started it begins to apply an increasing load on specimen. Throughout the tests the control system and its associated software record the load and extension or compression of the specimen.
Machines range from very small table top systems to ones with over 53 MN (12 million lbf) capacity.


TSE (examination)

The Test of Spoken English (TSE) was an oral test developed by Educational Testing Service which measured the ability of nonnative English speakers to communicate effectively. As of March 2010, the TSE has been superseded by the speaking portion of the TOEFL iBT as well as by the SPEAK test.
Before its retirement, TSE scores were used by academic institutions, corporations, government agencies, health care systems, and other organizations to guide their decisions regarding graduate assistantships in teaching and research, hiring new workers, and licensing and certifying existing employees.
Scoring.
The TSE score consisted of a single score of communicative language ability, which was reported on a scale of 20 to 60. Assigned score levels were averaged across items and raters, and the scores were reported in increments of five (i.e., 20, 25, 30, 35, 40, 45, 50, 55, and 60).


Atom interferometer

An atom interferometer is an interferometer which uses the wave character of atoms. Similar to optical interferometers, atom interferometers measure the difference in phase between atomic matter waves along different paths. Atom interferometers have many uses in fundamental physics including measurements of the gravitational constant, the fine-structure constant, the universality of free fall, and have been proposed as a method to detect gravitational waves. They also have applied uses as accelerometers, rotation sensors, and gravity gradiometers.
Overview.
Interferometry inherently depends on the wave nature of the object. As pointed out by de Broglie in his PhD thesis, particles, including atoms, can behave like waves (the so-called wave–particle duality, according to the general framework of quantum mechanics). More and more high precision experiments now employ atom interferometers due to their short de Broglie wavelength. Some experiments are now even using molecules to obtain even shorter de Broglie wavelengths and to search for the limits of quantum mechanics. In many experiments with atoms, the roles of matter and light are reversed compared to the laser based interferometers, i.e. the beam splitter and mirrors are lasers while the source instead emits matter waves (the atoms).
Interferometer types.
While the use of atoms offers easy access to higher frequencies (and thus accuracies) than light, atoms are affected much more strongly by gravity. In some apparatuses, the atoms are ejected upwards and the interferometry takes place while the atoms are in flight, or while falling in free flight. In other experiments gravitational effects by free acceleration are not negated; additional forces are used to compensate for gravity. While these guided systems in principle can provide arbitrary amounts of measurement time, their quantum coherence is still under discussion. Recent theoretical studies indicate that coherence is indeed preserved in the guided systems, but this has yet to be experimentally confirmed.
The early atom interferometers deployed slits or wires for the beam splitters and mirrors. Later systems, especially the guided ones, used light forces for splitting and reflecting of
the matter wave.
History.
The separation of matter wave packets from complete atoms was first observed by Esterman and Stern in 1930, when a Na beam was diffracted off a surface of NaCl. The first modern atom interferometer reported was a Young's-type double-slit experiment with metastable helium atoms and a microfabricated double slit by Carnal and Mlynek in 1991, and an interferometer using three microfabricated diffraction gratings and Na atoms in the group around Pritchard at MIT. Shortly afterwards, an optical version of a Ramsey spectrometer typically used in atomic clocks was recognized also as an atom interferometer at the PTB in Braunschweig, Germany. The largest physical separation between the partial wave packets of atoms was achieved using laser cooling techniques and stimulated Raman transitions by S. Chu and coworkers in Stanford. More recently atom interferometers have begun moving out of laboratory conditions and have begun to address a variety of applications in real word environments.
Inertial navigation.
The first team to make a working model, Pritchard's, which included D.W. Keith, prompted Keith to leave atomic physics after achieving success, in part because one of the most obvious applications for atom interferometry was in highly accurate gyroscopes for submarines carrying ballistic missiles. AIGs (atomic interferometer gyroscopes) and ASGs (atomic spin gyroscopes) use atomic interferometer to sense rotation or in the latter case, uses atomic spin to sense rotation with both having compact size, high precision, and the possibility of being made on a chip-scale. "AI gyros" may compete, along with ASGs, with the established ring laser gyroscope, fiber optic gyroscope and hemispherical resonator gyroscope in future inertial guidance applications.


Staatsexamen Nederlands als tweede taal

Staatsexamen Nederlands als tweede taal (State Exams Dutch as a Second Language or State Examination of Dutch as a Second Language), often abbreviated as Staatsexamen NT2, is a standardised examination of Dutch language for those who are not native speakers of Dutch. By passing the State Examinations NT2 candidates can show that they have sufficient command of the Dutch language to be able to work or study in the Netherlands.
Organiser.
The Staatsexamen NT2 is organised by Dienst Uitvoering Onderwijs (DUO) (Office of Education) on behalf of the College voor Examens (CvE) (Board of Examinations). CvE is nonetheless still responsible for the organisation, development of the State Examinations NT2, and provision of sample papers.
The actual execution of the examination is done by , a testing and measurement company with international reputation, and Bureau ICE. While Cito is responsible for the listening and speaking exam, Bureau ICE is responsible for the reading and writing exam. Therefore, Cito also provides further training for markers for the speaking exam and facilitates the entire process of grading for this part of the exam.
Level of examinations.
As different levels of language competence are needed for working and for studying, there are two levels of examination, which could be chosen by the candidates. Programme 1 should be taken by candidates who need to work or be under vocational training in a Dutch environment. Programme 2 certifies that the candidates who pass the test have mastered enough Dutch for employment or education in college or university. These exams are set according to the Common European Framework of Reference (CEFR), where Programme 1 corresponds to the B1 level and Programme 2 corresponds to the B2 level.
Structure.
The examination consists of four sections: reading, listening, writing and speaking. A candidate gets the Diploma NT2, only if they pass all four parts.
Candidates complete the listening and speaking tests on a computer, and finish the writing tests partly on the computer and partly on an answer sheet. For the reading test, the reading texts would be printed on a booklet and the candidates should mark their answers on the computer.
Registration.
The exam is registered through the website of DUO. The exam could be taken at Amsterdam, Eindhoven, Delft, Rotterdam, Utrecht and Zwolle almost every week.
Preparation.
Sample papers for the exam are available. Candidates may also decide to enrol in preparation courses offered by various language schools.
Results.
The result for the exam is marked either as "Satisfactory" or "Not satisfactory". The result is announced about five weeks after the exam.
Recognition.
A pass in Programme 2 can fulfil the language requirement for enrolment in college or university in the Netherlands. According to the Dutch nationality law and Integration law for immigrants to the Netherlands, a pass in Programme 1 can exempt candidates from taking the language test in the process of immigration and naturalisation.


Apollonius of Perga

Apollonius of Perga ( "Apollonios o Pergeos"; ; ) was an Ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the contributions of Euclid and Archimedes on the topic, he brought them to the state prior to the invention of analytic geometry. His definitions of the terms ellipse, parabola, and hyperbola are the ones in use today.
Apollonius worked on numerous other topics, including astronomy. Most of this work has not survived, where exceptions are typically fragments referenced by other authors. His hypothesis of eccentric orbits to explain the apparently aberrant motion of the planets, commonly believed until the Middle Ages, was superseded during the Renaissance. The Apollonius crater on the Moon is named in his honor.
Life.
For such an important contributor to the field of mathematics, scant biographical information remains. The 6th century Greek commentator, Eutocius of Ascalon, on Apollonius’ major work, "Conics", states:
Apollonius, the geometrician, ... came from Perga in Pamphylia in the times of Ptolemy Euergetes, so records Herakleios the biographer of Archimedes ...
Perga at the time was a Hellenized city of Pamphylia in Anatolia. The ruins of the city yet stand. It was a center of Hellenistic culture. Euergetes, "benefactor", identifies Ptolemy III Euergetes, third Greek dynast of Egypt in the diadochi succession. Presumably, his “times” are his regnum, 246-222/221 BC. Times are always recorded by ruler or officiating magistrate, so that if Apollonius was born earlier than 246, it would have been the “times” of Euergetes’ father. The identity of Herakleios is uncertain. The approximate times of Apollonius are thus certain, but no exact dates can be given. The figure Specific birth and death years stated by the various scholars are only speculative.
Eutocius appears to associate Perga with the Ptolemaic dynasty of Egypt. Never under Egypt, Perga in 246 BC belonged to the Seleucid Empire, an independent diadochi state ruled by the Seleucid dynasty. During the last half of the 3rd century BC, Perga changed hands a number of times, being alternatively under the Seleucids and under the Kingdom of Pergamon to the north, ruled by the Attalid dynasty. Someone designated “of Perga” might well be expected to have lived and worked there. To the contrary, if Apollonius was later identified with Perga, it was not on the basis of his residence. The remaining autobiographical material implies that he lived, studied and wrote in Alexandria.
A letter by the Greek mathematician and astronomer Hypsicles was originally part of the supplement taken from Euclid's Book XIV, part of the thirteen books of Euclid's Elements.
The times of Apollonius.
Apollonius lived toward the end of a historical period now termed the Hellenistic Period, characterized by the superposition of Hellenic culture over extensive non-Hellenic regions to various depths, radical in some places, hardly at all in others. The change was initiated by Philip II of Macedon and his son, Alexander the Great, who, subjecting all of Greece in a series of stunning victories, went on to conquer the Persian Empire, which ruled territories from Egypt to Pakistan. Philip was assassinated in 336 BC. Alexander went on to fulfill his plan by conquering the vast Persian empire.
The short autobiography of Apollonius.
The material is located in the surviving false “Prefaces” of the books of his "Conics." These are letters delivered to influential friends of Apollonius asking them to review the book enclosed with the letter. The Preface to Book I, addressed to one Eudemus, reminds him that "Conics" was initially requested by a house guest at Alexandria, the geometer, Naucrates, otherwise unknown to history. Naucrates had the first draft of all eight books in his hands by the end of the visit. Apollonius refers to them as being “without a thorough purgation” ("ou diakatharantes" in Greek, "ea non perpurgaremus" in Latin). He intended to verify and emend the books, releasing each one as it was completed.
Hearing of this plan from Apollonius himself on a subsequent visit of the latter to Pergamon, Eudemus had insisted Apollonius send him each book before release. The circumstances imply that at this stage Apollonius was a young geometer seeking the company and advice of established professionals. Pappus states that he was with the students of Euclid at Alexandria. Euclid was long gone. This stay had been, perhaps, the final stage of Apollonius’ education. Eudemus was perhaps a senior figure in his earlier education at Pergamon; in any case, there is reason to believe that he was or became the head of the Library and Research Center (Museum) of Pergamon. Apollonius goes on to state that the first four books were concerned with the development of elements while the last four were concerned with special topics.
There is something of a gap between Prefaces I and II. Apollonius has sent his son, also Apollonius, to deliver II. He speaks with more confidence, suggesting that Eudemus use the book in special study groups, which implies that Eudemus was a senior figure, if not the headmaster, in the research center. Research in such institutions, which followed the model of the Lycaeum of Aristotle at Athens, due to the residency of Alexander the Great and his companions in its northern branch, was part of the educational effort, to which the library and museum were adjunct. There was only one such school in the state. Owned by the king, it was under royal patronage, which was typically jealous, enthusiastic, and participatory. The kings bought, begged, borrowed and stole the precious books whenever and wherever they could. Books were of the highest value, affordable only to wealthy patrons. Collecting them was a royal obligation. Pergamon was known for its parchment industry, whence “parchment” is derived from “Pergamon.”
Apollonius brings to mind Philonides of Laodicea, a geometer whom he introduced to Eudemus in Ephesus. Philonides became Eudemus' student. He lived mainly in Syria during the 1st half of the 2nd century BC. Whether the meeting indicates that Apollonius now lived in Ephesus is unresolved. The intellectual community of the Mediterranean was international in culture. Scholars were mobile in seeking employment. They all communicated via some sort of postal service, public or private. Surviving letters are abundant. They visited each other, read each other's works, made suggestions to each other, recommended students and accumulated a tradition termed by some “the golden age of mathematics.”
Preface III is missing. During the interval Eudemus passed away, says Apollonius in IV, again supporting a view that Eudemus was senior over Apollonius. Prefaces IV–VII are more formal, omitting personal information and concentrating on summarizing the books. They are all addressed to a mysterious Attalus, a choice made “because”, as Apollonius writes to Attalus, “of your earnest desire to possess my works.” By that time a good many people at Pergamum had such a desire. Presumably, this Attalus was someone special, receiving copies of Apollonius’ masterpiece fresh from the author's hand. One strong theory is that Attalus is Attalus II Philadelphus, 220-138 BC, general and defender of his brother's kingdom (Eumenes II), co-regent on the latter's illness in 160 BC, and heir to his throne and his widow in 158 BC. He and his brother were great patrons of the arts, expanding the library into international magnificence. The dates are consonant with those of Philonides, while Apollonius’ motive is consonant with Attalus' book-collecting initiative.
Apollonius sent to Attalus Prefaces V–VII. In Preface VII he describes Book VIII as “an appendix” ... “which I will take care to send you as speedily as possible.” There is no record that it was ever sent or ever completed. It may be missing from history because it was never in history, Apollonius having died before its completion. Pappus of Alexandria, however, provided lemmas for it, so at the very least some edition of it must once have been in circulation.
Documented works of Apollonius.
Apollonius was a prolific geometer, turning out a large number of works. Only one survives, "Conics". It is a dense and extensive reference work on the topic, even by today's standards, serving as a repository of now little known geometric propositions as well as a vehicle for some new ones devised by Apollonius. Its audience was not the general population, which could not read or write. It was always intended for savants of mathematics and their small number of educated readers associated with the state schools and their associated libraries. It always was, in other words, a library reference work. Its basic definitions have become an important mathematical heritage. For the most part its methods and conclusions have been superseded by Analytic Geometry.
Of its eight books, only the first four have a credible claim to descent from the original texts of Apollonius. Books 5-7 have been translated from the Arabic into Latin. The original Greek has been lost. The status of Book VIII is unknown. A first draft existed. Whether the final draft was ever produced is not known. A "reconstruction" of it by Edmond Halley exists in Latin. There is no way to know how much of it, if any, is verisimilar to Apollonius. Halley also reconstructed "De Rationis Sectione" and "De Spatii Sectione". Beyond these works, except for a handful of fragments, documentation that might in any way be interpreted as descending from Apollonius ends.
Many of the lost works are described or mentioned by commentators. In addition are ideas attributed to Apollonius by other authors without documentation. Credible or not, they are hearsay. Some authors identify Apollonius as the author of certain ideas, consequently named after him. Others attempt to express Apollonius in modern notation or phraseology with indeterminate degrees of fidelity.
"Conics".
The Greek text of "Conics" uses the Euclidean arrangement of definitions, figures and their parts; i.e., the “givens,” followed by propositions “to be proved.” Books I-VII present 387 propositions. This type of arrangement can be seen in any modern geometry textbook of the traditional subject matter. As in any course of mathematics, the material is very dense and consideration of it, necessarily slow. Apollonius had a plan for each book, which is partly described in the "Prefaces". The headings, or pointers to the plan, are somewhat in deficit, Apollonius having depended more on the logical flow of the topics.
An intellectual niche is thus created for the commentators of the ages. Each must present Apollonius in the most lucid and relevant way for his own times. They use a variety of methods: annotation, extensive prefatory material, different formats, additional drawings, superficial reorganization by the addition of capita, and so on. There are subtle variations in interpretation. The modern English speaker encounters a lack of material in English due to the preference for New Latin by English scholars. Such intellectual English giants as Edmund Halley and Isaac Newton, the proper descendants of the Hellenistic tradition of mathematics and astronomy, can only be read and interpreted in translation by populations of English speakers unacquainted with the classical languages; that is, most of them.
Presentations written entirely in native English begin in the late 19th century. Of special note is Heath's "Treatise on Conic Sections". His extensive prefatory commentary includes such items as a lexicon of Apollonian geometric terms giving the Greek, the meanings, and usage. Commenting that “the apparently portentious bulk of the treatise has deterred many from attempting to make its acquaintance,” he promises to add headings, changing the organization superficially, and to clarify the text with modern notation. His work thus references two systems of organization, his own and Apollonius’, to which concordances are given in parentheses.
Heath's work is indispensable. He taught throughout the early 20th century, passing away in 1940, but meanwhile another point of view was developing. St. John's College (Annapolis/Santa Fe), which had been a military school since colonial times, preceding the United States Naval Academy at Annapolis, Maryland, to which it is adjacent, in 1936 lost its accreditation and was on the brink of bankruptcy. In desperation the board summoned Stringfellow Barr and Scott Buchanan from the University of Chicago, where they had been developing a new theoretical program for instruction of the Classics. Leaping at the opportunity, in 1937 they instituted the “new program” at St. John's, later dubbed the Great Books program, a fixed curriculum that would teach the works of select key contributors to the culture of western civilization. At St. John's, Apollonius came to be taught as himself, not as some adjunct to analytic geometry.
The “tutor” of Apollonius was R. Catesby Taliaferro, a new PhD in 1937 from the University of Virginia. He tutored until 1942 and then later for one year in 1948, supplying the English translations by himself, translating Ptolemy's "Almagest" and Apollonius’ "Conics". These translations became part of the Encyclopædia Britannica's Great Books of the Western World series. Only Books I-III are included, with an appendix for special topics. Unlike Heath, Taliaferro did not attempt to reorganize Apollonius, even superficially, or to rewrite him. His translation into modern English follows the Greek fairly closely. He does use modern geometric notation to some degree.
Contemporaneously with Taliaferro's work, Ivor Thomas an Oxford don of the World War II era, was taking an intense interest in Greek mathematics. He planned a compendium of selections, which came to fruition during his military service as an officer in the Royal Norfolk Regiment. After the war it found a home in the Loeb Classical Library, where it occupies two volumes, all translated by Thomas, with the Greek on one side of the page and the English on the other, as is customary for the Loeb series. Thomas' work has served as a handbook for the golden age of Greek mathematics. For Apollonius he only includes mainly those portions of Book I that define the sections.
Heath, Taliaferro, and Thomas satisfied the public demand for Apollonius in translation for most of the 20th century. The subject moves on. More recent translations and studies incorporate new information and points of view as well as examine the old.
Book I.
Book I presents 58 propositions. Its most salient content is all the basic definitions concerning cones and conic sections. These definitions are not exactly the same as the modern ones of the same words. Etymologically the modern words derive from the ancient, but the etymon often differs in meaning from its reflex.
A conical surface is generated by a line segment rotated about a bisector point such that the end points trace circles, each in its own plane. A cone, one branch of the double conical surface, is the surface with the point (apex or vertex), the circle (base), and the axis, a line joining vertex and center of base.
A “section” (Latin sectio, Greek tome) is an imaginary “cutting” of a cone by a plane.
The Greek geometers were interested in laying out select figures from their inventory in various applications of engineering and architecture, as the great inventors, such as Archimedes, were accustomed to doing. A demand for conic sections existed then and exists now. The development of mathematical characterization had moved geometry in the direction of Greek geometric algebra, which visually features such algebraic fundamentals as assigning values to line segments as variables. They used a coordinate system intermediate between a grid of measurements and the Cartesian coordinate system. The theories of proportion and application of areas allowed the development of visual equations. (See below under Methods of Apollonius).
The “application of areas” implicitly asks, given an area and a line segment, does this area apply; that is, is it equal to, the square on the segment? If yes, an applicability (parabole) has been established. Apollonius followed Euclid in asking if a rectangle on the abscissa of any point on the section applies to the square of the ordinate. If it does, his word-equation is the equivalent of formula_1 which is one modern form of the equation for a parabola. The rectangle has sides formula_2 and formula_3. It was he who accordingly named the figure, parabola, “application.”
The “no applicability” case is further divided into two possibilities. Given a function, formula_4, such that, in the applicability case, formula_5, in the no applicability case, either formula_6 or formula_7. In the former, formula_8 falls short of formula_9 by a quantity termed the ellipsis, ”deficit.” In the latter, formula_8 overshoots by a quantity termed the hyperbole, "surfeit."
Applicability could be achieved by adding the deficit, formula_11, or subtracting the surfeit, formula_12. The figure compensating for a deficit was named an ellipse; for a surfeit, a hyperbola. The terms of the modern equation depend on the translation and rotation of the figure from the origin, but the general equation for an ellipse,
can be placed in the form
formula_13
where C/B is the d, while an equation for the hyperbola,
becomes
formula_14
where C/B is the s.
Book II.
Book II contains 53 propositions. Apollonius says that he intended to cover "the properties having to do with the diameters and axes and also the asymptotes and other things ... for limits of possibility." His definition of "diameter" is different from the traditional, as he finds it necessary to refer the intended recipient of the letter to his work for a definition. The elements mentioned are those that specify the shape and generation of the figures. Tangents are covered at the end of the book.
Book III.
Book III contains 56 propositions. Apollonius claims original discovery for theorems "of use for the construction of solid loci ... the three-line and four-line locus ..." The locus of a conic section is the section. The three-line locus problem (as stated by Taliafero's appendix to Book III) finds "the locus of points whose distances from three given fixed straight lines ... are such that the square of one of the distances is always in a constant ratio to the rectangle contained by the other two distances." This is the proof of the application of areas resulting in the parabola. The four-line problem results in the ellipse and hyperbola. Analytic geometry derives the same loci from simpler criteria supported by algebra, rather than geometry, for which Descartes was highly praised. He supersedes Apollonius in his methods.
Book IV.
Book IV contains 57 propositions. The first sent to Attalus, rather than to Eudemus, it thus represents his more mature geometric thought. The topic is rather specialized: "the greatest number of points at which sections of a cone can meet one another, or meet a circumference of a circle, ..." Nevertheless, he speaks with enthusiasm, labeling them "of considerable use" in solving problems (Preface 4).
Book V.
Book V, known only through translation from the Arabic, contains 77 propositions, the most of any book. They cover the ellipse (50 propositions), the parabola (22), and the hyperbola (28). These are not explicitly the topic, which in Prefaces I and V Apollonius states to be maximum and minimum lines. These terms are not explained. In contrast to Book I, Book V contains no definitions and no explanation.
The ambiguity has served as a magnet to exegetes of Apollonius, who must interpret without sure knowledge of the meaning of the book's major terms. Until recently Heath's view prevailed: the lines are to be treated as normals to the sections. A normal in this case is the perpendicular to a curve at a tangent point sometimes called the foot. If a section is plotted according to Apollonius’ coordinate system (see below under Methods of Apollonius), with the diameter (translated by Heath as the axis) on the x-axis and the vertex at the origin on the left, the phraseology of the propositions indicates that the minima/maxima are to be found between the section and the axis. Heath is led into his view by consideration of a fixed point p on the section serving both as tangent point and as one end of the line. The minimum distance between p and some point g on the axis must then be the normal from p.
In modern mathematics, normals to curves are known for being the location of the center of curvature of that small part of the curve located around the foot. The distance from the foot to the center is the radius of curvature. The latter is the radius of a circle, but for other than circular curves, the small arc can be approximated by a circular arc. The curvature of non-circular curves; e.g., the conic sections, must change over the section. A map of the center of curvature; i.e., its locus, as the foot moves over the section, is termed the evolute of the section. Such a figure, the edge of the successive positions of a line, is termed an envelope today. Heath believed that in Book V we are seeing Apollonius establish the logical foundation of a theory of normals, evolutes, and envelopes.
Heath's was accepted as the authoritative interpretation of Book V for the entire 20th century, but the changing of the century brought with it a change of view. In 2001, Apollonius scholars Fried & Unguru, granting all due respect to other Heath chapters, balked at the historicity of Heath's analysis of Book V, asserting that he “reworks the original to make it more congenial to a modern mathematician ... this is the kind of thing that makes Heath’s work of dubious value for the historian, revealing more of Heath’s mind than that of Apollonius.” Some of his arguments are in summary as follows. There is no mention of maxima/minima being per se normals in either the prefaces or the books proper. Out of Heath's selection of 50 propositions said to cover normals, only 7, Book V: 27-33, state or imply maximum/minimum lines being perpendicular to the tangents. These 7 Fried classifies as isolated, unrelated to the main propositions of the book. They do not in any way imply that maxima/minima in general are normals. In his extensive investigation of the other 43 propositions, Fried proves that many cannot be.
Fried and Unguru counter by portraying Apollonius as a continuation of the past rather than a foreshadowing of the future. First is a complete philological study of all references to minimum and maximum lines, which uncovers a standard phraseology. There are three groups of 20-25 propositions each. The first group contains the phrase “from a point on the axis to the section,” which is exactly the opposite of a hypothetical “from a point on the section to the axis.” The former does not have to be normal to anything, although it might be. Given a fixed point on the axis, of all the lines connecting it to all the points of the section, one will be longest (maximum) and one shortest (minimum). Other phrases are “in a section,” “drawn from a section,” “cut off between the section and its axis,” cut off by the axis,” all referring to the same image.
In the view of Fried and Unguru, the topic of Book V is exactly what Apollonius says it is, maximum and minimum lines. These are not code words for future concepts, but refer to ancient concepts then in use. The authors cite Euclid, Elements, Book III, which concerns itself with circles, and maximum and minimum distances from interior points to the circumference. Without admitting to any specific generality they use terms such as “like” or “the analog of.” They are known for innovating the term “neusis-like.” A neusis construction was a method of fitting a given segment between two given curves. Given a point P, and a ruler with the segment marked off on it. one rotates the ruler around P cutting the two curves until the segment is fitted between them. In Book V, P is the point on the axis. Rotating a ruler around it, one discovers the distances to the section, from which the minimum and maximum can be discerned. The technique is not applied to the situation, so it is not neusis. The authors use neusis-like, seeing an archetypal similarity to the ancient method.
Book VI.
Book VI, known only through translation from the Arabic, contains 33 propositions, the least of any book. It also has large lacunae, or gaps in the text, due to damage or corruption in the previous texts.
The topic is relatively clear and uncontroversial. Preface 1 states that it is “equal and similar sections of cones.” Apollonius extends the concepts of congruence and similarity presented by Euclid for more elementary figures, such as triangles, quadrilaterals, to conic sections. Preface 6 mentions “sections and segments” that are “equal and unequal” as well as “similar and dissimilar,” and adds some constructional information.
Book VI features a return to the basic definitions at the front of the book. “Equality” is determined by an application of areas. If one figure; that is, a section or a segment, is “applied” to another (Halley's "si applicari possit altera super alteram"), they are “equal” (Halley's "aequales") if they coincide and no line of one crosses any line of the other. This is obviously a standard of congruence following Euclid, Book I, Common Notions, 4: “and things coinciding ("epharmazanta") with one another are equal ("isa").” Coincidence and equality overlap, but they are not the same: the application of areas used to define the sections depends on quantitative equality of areas but they can belong to different figures.
Between instances that are the same (homos), being equal to each other, and those that are different, or unequal, are figures that are “same-ish” (hom-oios), or similar. They are neither entirely the same nor different, but share aspects that are the same and do not share aspects that are different. Intuitively the geometricians had scale in mind; e.g., a map is similar to a topographic region. Thus figures could have larger or smaller versions of themselves.
The aspects that are the same in similar figures depend on the figure. Book 6 of Euclid's Elements presents similar triangles as those that have the same corresponding angles. A triangle can thus have miniatures as small as you please, or giant versions, and still be “the same” triangle as the original.
In Apollonius' definitions at the beginning of Book VI, similar right cones have similar axial triangles. Similar sections and segments of sections are first of all in similar cones. In addition, for every abscissa of one must exist an abscissa in the other at the desired scale. Finally, abscissa and ordinate of one must be matched by coordinates of the same ratio of ordinate to abscissa as the other. The total effect is as though the section or segment were moved up and down the cone to achieve a different scale.
Book VII.
Book VII, also a translation from the Arabic, contains 51 Propositions. These are the last that Heath considers in his 1896 edition. In Preface I, Apollonius does not mention them, implying that, at the time of the first draft, they may not have existed in sufficiently coherent form to describe. Apollonius uses obscure language, that they are “peri dioristikon theorematon”, which Halley translated as “de theorematis ad determinationem pertinentibus,” and Heath as “theorems involving determinations of limits.” This is the language of definition, but no definitions are forthcoming. Whether the reference might be to a specific kind of definition is a consideration but to date nothing credible has been proposed. The topic of Book VII, completed toward the end of Apollonius’ life and career, is stated in Preface VII to be diameters and “the figures described upon them,” which must include conjugate diameters, as he relies heavily on them. In what way the term “limits” or “determinations” might apply is not mentioned.
Diameters and their conjugates are defined in Book I (Definitions 4-6). Not every diameter has a conjugate. The topography of a diameter (Greek diametros) requires a regular curved figure. Irregularly-shaped areas, addressed in modern times, are not in the ancient game plan. Apollonius has in mind, of course, the conic sections, which he describes in often convolute language: “a curve in the same plane” is a circle, ellipse or parabola, while “two curves in the same plane” is a hyperbola. A chord is a straight line whose two end points are on the figure; i.e., it cuts the figure in two places. If a grid of parallel chords is imposed on the figure, then the diameter is defined as the line bisecting all the chords, reaching the curve itself at a point called the vertex. There is no requirement for a closed figure; e.g., a parabola has a diameter.
A parabola has symmetry in one dimension. If you imagine it folded on its one diameter, the two halves are congruent, or fit over each other. The same may be said of one branch of a hyperbola. Conjugate diameters (Greek suzugeis diametroi, where suzugeis is “yoked together”), however, are symmetric in two dimensions. The figures to which they apply require also an areal center (Greek kentron), today called a centroid, serving as a center of symmetry in two directions. These figures are the circle, ellipse, and two-branched hyperbola. There is only one centroid, which must not be confused with the foci. A diameter is a chord passing through the centroid, which always bisects it.
For the circle and ellipse, let a grid of parallel chords be superimposed over the figure such that the longest is a diameter and the others are successively shorter until the last is not a chord, but is a tangent point. The tangent must be parallel to the diameter. A conjugate diameter bisects the chords, being placed between the centroid and the tangent point. Moreover, both diameters are conjugate to each other, being called a conjugate pair. It is obvious that any conjugate pair of a circle are perpendicular to each other, but in an ellipse, only the major and minor axes are, the elongation destroying the perpendicularity in all other cases.
Conjugates are defined for the two branches of a hyperbola resulting from the cutting of a double cone by a single plane. They are called conjugate branches. They have the same diameter. Its centroid bisects the segment between vertices. There is room for one more diameter-like line: let a grid of lines parallel to the diameter cut both branches of the hyperbola. These lines are chord-like except that they do not terminate on the same continuous curve. A conjugate diameter can be drawn from the centroid to bisect the chord-like lines.
These concepts mainly from Book I get us started on the 51 propositions of Book VII defining in detail the relationships between sections, diameters, and conjugate diameters. As with some of Apollonius other specialized topics, their utility today compared to Analytic Geometry remains to be seen, although he affirms in Preface VII that they are both useful and innovative; i.e., he takes the credit for them.
Lost and reconstructed works described by Pappus.
Pappus mentions other treatises of Apollonius:
Each of these was divided into two books, and—with the "Data", the "Porisms", and "Surface-Loci" of Euclid and the "Conics" of Apollonius—were, according to Pappus, included in the body of the ancient analysis. Descriptions follow of the six works mentioned above.
"De Rationis Sectione".
"De Rationis Sectione" sought to resolve a simple problem: Given two straight lines and a point in each, draw through a third given point a straight line cutting the two fixed lines such that the parts intercepted between the given points in them and the points of intersection with this third line may have a given ratio.
"De Spatii Sectione".
"De Spatii Sectione" discussed a similar problem requiring the rectangle contained by the two intercepts to be equal to a given rectangle.
In the late 17th century, Edward Bernard discovered a version of "De Rationis Sectione" in the Bodleian Library. Although he began a translation, it was Halley who finished it and included it in a 1706 volume with his restoration of "De Spatii Sectione".
"De Sectione Determinata".
"De Sectione Determinata" deals with problems in a manner that may be called an analytic geometry of one dimension; with the question of finding points on a line that were in a ratio to the others. The specific problems are: Given two, three or four points on a straight line, find another point on it such that its distances from the given points satisfy the condition that the square on one or the rectangle contained by two has a given ratio either (1) to the square on the remaining one or the rectangle contained by the remaining two or (2) to the rectangle contained by the remaining one and another given straight line. Several have tried to restore the text to discover Apollonius's solution, among them Snellius (Willebrord Snell, Leiden, 1698); Alexander Anderson of Aberdeen, in the supplement to his "Apollonius Redivivus" (Paris, 1612); and Robert Simson in his "Opera quaedam reliqua" (Glasgow, 1776), by far the best attempt.
"De Tactionibus".
"De Tactionibus" embraced the following general problem: Given three things (points, straight lines, or circles) in position, describe a circle passing through the given points and touching the given straight lines or circles. The most difficult and historically interesting case arises when the three given things are circles. In the 16th century, Vieta presented this problem (sometimes known as the Apollonian Problem) to Adrianus Romanus, who solved it with a hyperbola. Vieta thereupon proposed a simpler solution, eventually leading him to restore the whole of Apollonius's treatise in the small work "Apollonius Gallus" (Paris, 1600). The history of the problem is explored in fascinating detail in the preface to J. W. Camerer's brief "Apollonii Pergaei quae supersunt, ac maxime Lemmata Pappi in hos Libras, cum Observationibus, &c" (Gothae, 1795, 8vo).
"De Inclinationibus".
The object of "De Inclinationibus" was to demonstrate how a straight line of a given length, tending towards a given point, could be inserted between two given (straight or circular) lines. Though Marin Getaldić and Hugo d'Omerique ("Geometrical Analysis", Cadiz, 1698) attempted restorations, the best is by Samuel Horsley (1770).
"De Locis Planis".
"De Locis Planis" is a collection of propositions relating to loci that are either straight lines or circles. Since Pappus gives somewhat full particulars of its propositions, this text has also seen efforts to restore it, not only by P. Fermat ("Oeuvres", i., 1891, pp. 3–51) and F. Schooten (Leiden, 1656) but also, most successfully of all, by R. Simson (Glasgow, 1749).
Lost works mentioned by other ancient writers.
Ancient writers refer to other works of Apollonius that are no longer extant:
Early printed editions.
The early printed editions began for the most part in the 16th century. At that time, scholarly books were expected to be in Latin, today's New Latin. As almost no manuscripts were in Latin, the editors of the early printed works translated from the Greek or Arabic to Latin. The Greek and Latin were typically juxtaposed, but only the Greek is original, or else was restored by the editor to what he thought was original. Critical apparatuses were in Latin. The ancient commentaries, however, were in ancient or medieval Greek. Only in the 18th and 19th centuries did modern languages begin to appear. A representative list of early printed editions is given below. The originals of these printings are rare and expensive. For modern editions in modern languages see the references.
Ideas attributed to Apollonius by other writers.
Apollonius' contribution to astronomy.
The equivalence of two descriptions of planet motions, one using excentrics and another deferent and epicycles, is attributed to him. Ptolemy describes this equivalence as Apollonius' theorem in the "Almagest" XII.1.
Methods of Apollonius.
According to Heath, “The Methods of Apollonius” were not his and were not personal. Whatever influence he had on later theorists was that of geometry, not of his own innovation of technique. Heath says,
As a preliminary to the consideration in detail of the methods employed in the Conics, it may be stated generally that they follow steadily the accepted principles of geometrical investigation which found their definitive expression in the Elements of Euclid.
With regard to moderns speaking of golden age geometers, the term "method" means specifically the visual, reconstructive way in which the geometer unknowingly produces the same result as an algebraic method used today. As a simple example, algebra finds the area of a square by squaring its side. The geometric method of accomplishing the same result is to construct a visual square. Geometric methods in the golden age could produce most of the results of elementary algebra.
Geometrical algebra.
Heath goes on to use the term geometrical algebra for the methods of the entire golden age. The term is “not inappropriately” called that, he says. Today the term has been resurrected for use in other senses (see under geometric algebra). Heath was using it as it had been defined by Henry Burchard Fine in 1890 or before. Fine applies it to La Géométrie of René Descartes, the first full-blown work of analytic geometry. Establishing as a precondition that “two algebras are formally identical whose fundamental operations are formally the same,” Fine says that Descartes’ work “is not ... mere numerical algebra, but what may for want of a better name be called the algebra of line segments. Its symbolism is the same as that of numerical algebra; ...”
For example, in Apollonius a line segment AB (the line between Point A and Point B) is also the numerical length of the segment. It can have any length. AB therefore becomes the same as an algebraic variable, such as "x" (the unknown), to which any value might be assigned; e.g., "x"=3.
Variables are defined in Apollonius by such word statements as “let AB be the distance from any point on the section to the diameter,” a practice that continues in algebra today. Every student of basic algebra must learn to convert “word problems” to algebraic variables and equations, to which the rules of algebra apply in solving for "x". Apollonius had no such rules. His solutions are geometric.
Relationships not readily amenable to pictorial solutions were beyond his grasp; however, his repertory of pictorial solutions came from a pool of complex geometric solutions generally not known (or required) today. One well-known exception is the indispensable Pythagorean Theorem, even now represented by a right triangle with squares on its sides illustrating an expression such as a2 + b2 = c2. The Greek geometers called those terms “the square on AB,” etc. Similarly, the area of a rectangle formed by AB and CD was "the rectangle on AB and CD."
These concepts gave the Greek geometers algebraic access to linear functions and quadratic functions, which latter the conic sections are. They contain powers of 1 or 2 respectively. Apollonius had not much use for cubes (featured in solid geometry), even though a cone is a solid. His interest was in conic sections, which are plane figures. Powers of 4 and up were beyond visualization, requiring a degree of abstraction not available in geometry, but ready at hand in algebra.
The coordinate system of Apollonius.
All ordinary measurement of length in public units, such as inches, using standard public devices, such as a ruler, implies public recognition of a Cartesian grid; that is, a surface divided into unit squares, such as one square inch, and a space divided into unit cubes, such as one cubic inch. The ancient Greek units of measurement had provided such a grid to Greek mathematicians since the Bronze Age.
Prior to Apollonius, Menaechmus and Archimedes had already started locating their figures on an implied window of the common grid by referring to distances conceived to be measured from a left-hand vertical line marking a low measure and a bottom horizontal line marking a low measure, the directions being rectilinear, or perpendicular to one another. These edges of the window become, in the Cartesian coordinate system, the axes. One specifies the rectilinear distances of any point from the axes as the coordinates. The ancient Greeks did not have that convention. They simply referred to distances.
Apollonius does have a standard window in which he places his figures. Vertical measurement is from a horizontal line he calls the “diameter.” The word is the same in Greek as it is in English, but the Greek is somewhat wider in its comprehension. If the figure of the conic section is cut by a grid of parallel lines, the diameter bisects all the line segments included between the branches of the figure. It must pass through the vertex (koruphe, "crown"). A diameter thus comprises open figures such as a parabola as well as closed, such as a circle. There is no specification that the diameter must be perpendicular to the parallel lines, but Apollonius uses only rectilinear ones.
The rectilinear distance from a point on the section to the diameter is termed tetagmenos in Greek, etymologically simply “extended.” As it is only ever extended “down” (kata-) or “up” (ana-), the translators interpret it as ordinate. In that case the diameter becomes the x-axis and the vertex the origin. The y-axis then becomes a tangent to the curve at the vertex. The abscissa is then defined as the segment of the diameter between the ordinate and the vertex.
Using his version of a coordinate system, Apollonius manages to develop in pictorial form the geometric equivalents of the equations for the conic sections, which raises the question of whether his coordinate system can be considered Cartesian. There are some differences. The Cartesian system is to be regarded as universal, covering all figures in all space applied before any calculation is done. It has four quadrants divided by the two crossed axes. Three of the quadrants include negative coordinates meaning directions opposite the reference axes of zero.
Apollonius has no negative numbers, does not explicitly have a number for zero, and does not develop the coordinate system independently of the conic sections. He works essentially only in Quadrant 1, all positive coordinates. Carl Boyer, a modern historian of mathematics, therefore says:
However, Greek geometric algebra did not provide for negative magnitudes; moreover, the coordinate system was in every case superimposed "a posteriori" upon a given curve in order to study its properties ... Apollonius, the greatest geometer of antiquity, failed to develop analytic geometry...
No one denies, however, that Apollonius occupies some sort of intermediate niche between the grid system of conventional measurement and the fully developed Cartesian Coordinate System of Analytic Geometry. In reading Apollonius, one must take care not to assume modern meanings for his terms.
The theory of proportions.
Apollonius uses the "Theory of Proportions" as expressed in Euclid’s "Elements", Books 5 and 6. Devised by Eudoxus of Cnidus, the theory is intermediate between purely graphic methods and modern number theory. A standard decimal number system is lacking, as is a standard treatment of fractions. The propositions, however, express in words rules for manipulating fractions in arithmetic. Heath proposes that they stand in place of multiplication and division.
By the term “magnitude” Eudoxus hoped to go beyond numbers to a general sense of size, a meaning it still retains. With regard to the figures of Euclid, it most often means numbers, which was the Pythagorean approach. Pythagoras believed the universe could be characterized by quantities, which belief has become the current scientific dogma. Book V of Euclid begins by insisting that a magnitude (megethos, “size”) must be divisible evenly into units (meros, “part”). A magnitude is thus a multiple of units. They do not have to be standard measurement units, such as meters or feet. One unit can be any designated line segment.
There follows perhaps the most useful fundamental definition ever devised in science: the ratio (Greek logos, meaning roughly “explanation.”) is a statement of relative magnitude. Given two magnitudes, say of segments AB and CD. the ratio of AB to CD, where CD is considered unit, is the number of CD in AB; for example, 3 parts of 4, or 60 parts per million, where ppm still uses the “parts” terminology. The ratio is the basis of the modern fraction, which also still means “part,” or “fragment”, from the same Latin root as fracture.
The ratio is the basis of mathematical prediction in the logical structure called a “proportion” (Greek analogos). The proportion states that if two segments, AB and CD, have the same ratio as two others, EF and GH, then AB and CD are proportional to EF and GH, or, as would be said in Euclid, AB is to CD as EF is to GH.
Algebra reduces this general concept to the expression AB/CD = EF/GH. Given any three of the terms, one can calculate the fourth as an unknown. Rearranging the above equation, one obtains AB = (CD/GH)•EF, in which, expressed as y = kx, the CD/GH is known as the “constant of proportionality.” The Greeks had little difficulty with taking multiples (Greek pollaplasiein), probably by successive addition.
Apollonius uses ratios almost exclusively of line segments and areas, which are designated by squares and rectangles. The translators have undertaken to use the colon notation introduced by Gottfried Wilhelm Leibniz in "Acta Eruditorum", 1684. Here is an example from "Conics", Book I, on Proposition 11:
External links.
Many of the popular sites in the history of mathematics linked below reference or analyze concepts attributed to Apollonius in modern notations and concepts. Since much of Apollonius is subject to interpretation, and he does not per se use modern vocabulary or concepts, the analyses below may not be optimal or accurate. They represent the historical theories of their authors.