Signals intelligence (SIGINT) is the act and field of intelligence-gathering by interception of signals, whether communications between people (communications intelligence—abbreviated to COMINT) or from electronic signals not directly used in communication (electronic intelligence—abbreviated to ELINT). As classified and sensitive information is usually encrypted, signals intelligence may necessarily involve cryptanalysis (to decipher the messages). Traffic analysis—the study of who is signaling to whom and in what quantity—is also used to integrate information, and it may complement cryptanalysis. == History == === Origins === Electronic interceptions appeared as early as 1900, during the Boer War of 1899–1902. The British Royal Navy had installed wireless sets produced by Marconi on board their ships in the late 1890s, and the British Army used some limited wireless signalling. The Boers captured some wireless sets and used them to make vital transmissions. Since the British were the only people transmitting at the time, the British did not need special interpretation of the signals that they were. The birth of signals intelligence in a modern sense dates from the Russo-Japanese War of 1904–1905. As the Russian fleet prepared for conflict with Japan in 1904, the British ship HMS Diana stationed in the Suez Canal intercepted Russian naval wireless signals being sent out for the mobilization of the fleet, for the first time in history. === Development in World War I === Over the course of the First World War, a new method of signals intelligence reached maturity. Russia's failure to properly protect its communications fatally compromised the Russian Army's advance early in World War I and led to their disastrous defeat by the Germans under Ludendorff and Hindenburg at the Battle of Tannenberg. In 1918, French intercept personnel captured a message written in the new ADFGVX cipher, which was cryptanalyzed by Georges Painvin. This gave the Allies advance warning of the German 1918 Spring Offensive. The British in particular, built up great expertise in the newly emerging field of signals intelligence and codebreaking (synonymous with cryptanalysis). On the declaration of war, Britain cut all German undersea cables. This forced the Germans to communicate exclusively via either (A) a telegraph line that connected through the British network and thus could be tapped; or (B) through radio which the British could then intercept. Rear Admiral Henry Oliver appointed Sir Alfred Ewing to establish an interception and decryption service at the Admiralty; Room 40. An interception service known as 'Y' service, together with the post office and Marconi stations, grew rapidly to the point where the British could intercept almost all official German messages. The German fleet was in the habit each day of wirelessing the exact position of each ship and giving regular position reports when at sea. It was possible to build up a precise picture of the normal operation of the High Seas Fleet, to infer from the routes they chose where defensive minefields had been placed and where it was safe for ships to operate. Whenever a change to the normal pattern was seen, it immediately signalled that some operation was about to take place, and a warning could be given. Detailed information about submarine movements was also available. The use of radio-receiving equipment to pinpoint the location of any single transmitter was also developed during the war. Captain H.J. Round, working for Marconi, began carrying out experiments with direction-finding radio equipment for the army in France in 1915. By May 1915, the Admiralty was able to track German submarines crossing the North Sea. Some of these stations also acted as 'Y' stations to collect German messages, but a new section was created within Room 40 to plot the positions of ships from the directional reports. Room 40 played an important role in several naval engagements during the war, notably in detecting major German sorties into the North Sea. The battle of Dogger Bank was won in no small part due to the intercepts that allowed the Navy to position its ships in the right place. It played a vital role in subsequent naval clashes, including at the Battle of Jutland as the British fleet was sent out to intercept them. The direction-finding capability allowed for the tracking and location of German ships, submarines, and Zeppelins. The system was so successful that by the end of the war, over 80 million words, comprising the totality of German wireless transmission over the course of the war, had been intercepted by the operators of the Y-stations and decrypted. However, its most astonishing success was in decrypting the Zimmermann Telegram, a telegram from the German Foreign Office sent via Washington to its ambassador Heinrich von Eckardt in Mexico. === Postwar consolidation === With the importance of interception and decryption firmly established by the wartime experience, countries established permanent agencies dedicated to this task in the interwar period. In 1919, the British Cabinet's Secret Service Committee, chaired by Lord Curzon, recommended that a peace-time codebreaking agency should be created. The Government Code and Cypher School (GC&CS) was the first peace-time codebreaking agency, with a public function "to advise as to the security of codes and cyphers used by all Government departments and to assist in their provision", but also with a secret directive to "study the methods of cypher communications used by foreign powers". GC&CS officially formed on 1 November 1919, and produced its first decrypt on 19 October. By 1940, GC&CS was working on the diplomatic codes and ciphers of 26 countries, tackling over 150 diplomatic cryptosystems. The US Cipher Bureau was established in 1919 and achieved some success at the Washington Naval Conference in 1921, through cryptanalysis by Herbert Yardley. Secretary of War Henry L. Stimson closed the US Cipher Bureau in 1929 with the words "Gentlemen do not read each other's mail." === World War II === The use of SIGINT had even greater implications during World War II. The combined effort of intercepts and cryptanalysis for the whole of the British forces in World War II came under the code name "Ultra", managed from Government Code and Cypher School at Bletchley Park. Properly used, the German Enigma and Lorenz ciphers should have been virtually unbreakable, but flaws in German cryptographic procedures, and poor discipline among the personnel carrying them out, created vulnerabilities which made Bletchley's attacks feasible. Bletchley's work was essential to defeating the U-boats in the Battle of the Atlantic, and to the British naval victories in the Battle of Cape Matapan and the Battle of North Cape. In 1941, Ultra exerted a powerful effect on the North African desert campaign against German forces under General Erwin Rommel. General Sir Claude Auchinleck wrote that were it not for Ultra, "Rommel would have certainly got through to Cairo". Ultra decrypts featured prominently in the story of Operation SALAM, László Almásy's mission across the desert behind Allied lines in 1942. Prior to the Normandy landings on D-Day in June 1944, the Allies knew the locations of all but two of Germany's fifty-eight Western Front divisions. Winston Churchill was reported to have told King George VI: "It is thanks to the secret weapon of General Menzies, put into use on all the fronts, that we won the war!" Supreme Allied Commander, Dwight D. Eisenhower, at the end of the war, described Ultra as having been "decisive" to Allied victory. Official historian of British Intelligence in World War II Sir Harry Hinsley argued that Ultra shortened the war "by not less than two years and probably by four years"; and that, in the absence of Ultra, it is uncertain how the war would have ended. At a lower level, German cryptanalysis, direction finding, and traffic analysis were vital to Rommel's early successes in the Western Desert Campaign until British forces tightened their communications discipline and Australian raiders destroyed his principal SIGINT Company. == Technical definitions == The United States Department of Defense has defined the term "signals intelligence" as: A category of intelligence comprising either individually or in combination all communications intelligence (COMINT), electronic intelligence (ELINT), and foreign instrumentation signals intelligence (FISINT), however transmitted. Intelligence derived from communications, electronic, and foreign instrumentation signals. Being a broad field, SIGINT has many sub-disciplines. The two main ones are communications intelligence (COMINT) and electronic intelligence (ELINT). == Disciplines shared across the branches == === Targeting === A collection system has to know to look for a particular signal. "System", in this context, has several nuances. Targeting is the process of developing collection requirements: "1. A
Sprayprinter
SprayPrinter is a device that attaches to aerosol paint cans whereby users can print images via Bluetooth from a smartphone onto a wall or almost any surface. == History == The technology behind SprayPrinter was developed by Mihkel Joala. He explained in a 2016 interview with New Atlas that his idea was inspired by the modern car engine and the Nintendo Wii console. "Engines nowadays use extremely fast valves to spray fuel to [the] combustion chamber," says Joala. "I realized I can use them to shoot paint with pinpoint accuracy." As of December 2021, the company appears to be no longer selling products. == Awards and Recognitions == In 2015, SprayPrinter received €8,000 from the Estonian prototyping contest Prototron for its initial prototype. In 2016, the SprayPrinter team won the grand prize of €30,000 from the televised pitching competition Ajujaht.
Predictor–corrector method
In numerical analysis, predictor–corrector methods belong to a class of algorithms designed to integrate ordinary differential equations – to find an unknown function that satisfies a given differential equation. All such algorithms proceed in two steps: The initial, "prediction" step, starts from a function fitted to the function-values and derivative-values at a preceding set of points to extrapolate ("anticipate") this function's value at a subsequent, new point. The next, "corrector" step refines the initial approximation by using the predicted value of the function and another method to interpolate that unknown function's value at the same subsequent point. == Predictor–corrector methods for solving ODEs == When considering the numerical solution of ordinary differential equations (ODEs), a predictor–corrector method typically uses an explicit method for the predictor step and an implicit method for the corrector step. === Example: Euler method with the trapezoidal rule === A simple predictor–corrector method (known as Heun's method) can be constructed from the Euler method (an explicit method) and the trapezoidal rule (an implicit method). Consider the differential equation y ′ = f ( t , y ) , y ( t 0 ) = y 0 , {\displaystyle y'=f(t,y),\quad y(t_{0})=y_{0},} and denote the step size by h {\displaystyle h} . First, the predictor step: starting from the current value y i {\displaystyle y_{i}} , calculate an initial guess value y ~ i + 1 {\displaystyle {\tilde {y}}_{i+1}} via the Euler method, y ~ i + 1 = y i + h f ( t i , y i ) . {\displaystyle {\tilde {y}}_{i+1}=y_{i}+hf(t_{i},y_{i}).} Next, the corrector step: improve the initial guess using trapezoidal rule, y i + 1 = y i + 1 2 h ( f ( t i , y i ) + f ( t i + 1 , y ~ i + 1 ) ) . {\displaystyle y_{i+1}=y_{i}+{\tfrac {1}{2}}h{\bigl (}f(t_{i},y_{i})+f(t_{i+1},{\tilde {y}}_{i+1}){\bigr )}.} That value is used as the next step. === PEC mode and PECE mode === There are different variants of a predictor–corrector method, depending on how often the corrector method is applied. The Predict–Evaluate–Correct–Evaluate (PECE) mode refers to the variant in the above example: y ~ i + 1 = y i + h f ( t i , y i ) , y i + 1 = y i + 1 2 h ( f ( t i , y i ) + f ( t i + 1 , y ~ i + 1 ) ) . {\displaystyle {\begin{aligned}{\tilde {y}}_{i+1}&=y_{i}+hf(t_{i},y_{i}),\\y_{i+1}&=y_{i}+{\tfrac {1}{2}}h{\bigl (}f(t_{i},y_{i})+f(t_{i+1},{\tilde {y}}_{i+1}){\bigr )}.\end{aligned}}} It is also possible to evaluate the function f only once per step by using the method in Predict–Evaluate–Correct (PEC) mode: y ~ i + 1 = y i + h f ( t i , y ~ i ) , y i + 1 = y i + 1 2 h ( f ( t i , y ~ i ) + f ( t i + 1 , y ~ i + 1 ) ) . {\displaystyle {\begin{aligned}{\tilde {y}}_{i+1}&=y_{i}+hf(t_{i},{\tilde {y}}_{i}),\\y_{i+1}&=y_{i}+{\tfrac {1}{2}}h{\bigl (}f(t_{i},{\tilde {y}}_{i})+f(t_{i+1},{\tilde {y}}_{i+1}){\bigr )}.\end{aligned}}} Additionally, the corrector step can be repeated in the hope that this achieves an even better approximation to the true solution. If the corrector method is run twice, this yields the PECECE mode: y ~ i + 1 = y i + h f ( t i , y i ) , y ^ i + 1 = y i + 1 2 h ( f ( t i , y i ) + f ( t i + 1 , y ~ i + 1 ) ) , y i + 1 = y i + 1 2 h ( f ( t i , y i ) + f ( t i + 1 , y ^ i + 1 ) ) . {\displaystyle {\begin{aligned}{\tilde {y}}_{i+1}&=y_{i}+hf(t_{i},y_{i}),\\{\hat {y}}_{i+1}&=y_{i}+{\tfrac {1}{2}}h{\bigl (}f(t_{i},y_{i})+f(t_{i+1},{\tilde {y}}_{i+1}){\bigr )},\\y_{i+1}&=y_{i}+{\tfrac {1}{2}}h{\bigl (}f(t_{i},y_{i})+f(t_{i+1},{\hat {y}}_{i+1}){\bigr )}.\end{aligned}}} The PECEC mode has one fewer function evaluation than PECECE mode. More generally, if the corrector is run k times, the method is in P(EC)k or P(EC)kE mode. If the corrector method is iterated until it converges, this could be called PE(CE)∞.
BRS/Search
BRS/Search is a full-text database and information retrieval system. BRS/Search uses a fully inverted indexing system to store, locate, and retrieve unstructured data. It was the search engine that in 1977 powered Bibliographic Retrieval Services (BRS) commercial operations with 20 databases (including the first national commercial availability of MEDLINE); it has changed ownership several times during its development and is currently sold as Livelink ECM Discovery Server by Open Text Corporation. == Early development == Development on what was to become BRS began as Biomedical Communications Network (BCN) at the State University of New York at Albany (SUNY). BCN, which went online in 1968, provided on-line access to nine databases, including MEDLINE and BIOSIS Previews, to large universities and medical schools primarily in the Northeast of the USA. State funding for the project was withdrawn in 1975, and Bibliographic Retrieval Services (BRS) was formed as a non-profit concern the following year. It was incorporated in May 1976 as a for-profit corporation with Ron Quake as president, Jan Egeland as vice president in charge of marketing and training, and Lloyd Palmer as vice president of systems. == BRS commercial operations == In December 1976, the First BRS User Meeting was held in Syracuse, New York, and by January 1977 BRS started commercial operations with 20 databases (including the first national commercial availability of MEDLINE) and 9 million records, using modified IBM STAIRS (STorage And Information Retrieval System) software, Telenet for telecommunications, and timesharing mainframe computers of Carrier Corporation. In October 1980 BRS was sold by Egeland and Quake to Indian Head, Inc., a subsidiary of the Dutch company Thyssen-Bornemisza Group. == 1989–1993 == In 1989 Robert Maxwell acquired BRS and the BRS/Search software; he announced the planned incorporation of the ORBIT Search Service and BRS Information Technologies and renamed the whole group Maxwell Online, Inc. At that time BRS Information Technologies was serving the medical and academic library marketplace with over 150 databases. Maxwell later bought the publishing company Macmillan and put Maxwell Online under Macmillan. In the same year BRS/LINK (hypertext connection of databases; first application delivering full text) was announced. The initial BRS/LINK application "relates the citation in a bibliographic database to its full-text article in a second database," and "eliminates the need to re-execute a search strategy in the second database in order to find the corresponding full-text article." Initially BRS/LINK supported linking only selected bibliographic databases: MEDLINE, Health Planning and Administration, and MEDLINE References on AIDS to the full-text Comprehensive Core Medical Library. At the time of Robert Maxwell’s death in 1991, Macmillan brought in Andrew Gregory to represent the company during the 2 years that Maxwell’s affairs were being settled and to prepare Maxwell Online to be able to sell the components. Maxwell Online shortly thereafter underwent yet another name change, this time to InfoPro Technologies. == Dataware Technologies ownership of BRS/SEARCH == Early in 1994, InfoPro Technologies, a subsidiary of MHC Inc. (holding company for Macmillan Inc.), the former Maxwell Online service, sold off all its subsidiaries. ORBIT Search Services went to the French-owned Questel, the dial-up BRS Search Services to CD Plus Technologies (later to become OVID), and BRS Software Products (including BRS/SEARCH) to Dataware Technologies. Almost up to the end of InfoPro Technologies, BRS Software had been the fastest growing segment of the company. At the 14th BRS North American Users Group Conference in 1999, Dave Schubmehl of Dataware Technologies presented a paper in which he stated "The purpose of this presentation is to update BRS users on upcoming releases of BRS/Search, NetAnswer, and other Dataware products. BRS/Search 7.0 will include features specifically requested by customers, as well as other enhancements. Earlier this year, Dataware acquired Sovereign Hill Software, makers of InQuery. In light of that acquisition, and Dataware's other development projects, we'll look at Dataware's plans for all products, including BRS/Search and NetAnswer." == Open Text acquisition of BRS/Search == In 2001 BRS/Search was acquired by Open Text and became LiveLink ECM Discovery Server. It is now referred to as Open Text Discovery Server. Open Text still supports both BRS/Search and NetAnswer. The core BRS/Search technology in the Open Text portfolio was augmented with other capabilities through various acquisitions. For example, Dataware's acquisition of Sovereign-Hill brought InQuery, “a probabilistic information retrieval system using an inference network”, which was developed by the University of Massachusetts Amherst Center for Intelligent Information Retrieval] out of the UMass CIIR and into the marketplace. A product re-branding table shows the range of products, their old names and their new names. InQuery is a concept search engine that uses noun phrases, parts of speech and other co-occurrence relationships in overlapping passages of text rather than single term inverted indexes of single words in documents. Open Text's portfolio has grown to include Hummingbird Content Management, and has always included BASIS. == 2003 == BRS/Search North America User's Group (BRSNAUG) website with a June 8, 2003 date listed the following features for BRS/Search. The BRSNAUG also disincorporated in 2003. Cross-references to BRS/Search on the World Wide Web point to Open Text Livelink. Engine features include: Rapid query response time. Numerical data handling and elementary statistical processing (sum, avg, min, max) Search results weighting and relevancy ranking Left- and right-truncation and expansion of search terms Superior data compression – loaded databases typically use only about 1.5 times the input stream size in disk space Large capacity databases – up to 100 million documents, each with up to 65,000 paragraphs Fine control of indexing and searching – right down to the word, sentence, and paragraph level Fine control over data security. Document access can be controlled at the database, document, and paragraph level International language support for all 7/8 bit characters sets and customizable language tables Flexible and customizable stop word lists ANSI-compatible thesauri Hypertext links within and between documents and databases (R6.x) Support for natural language parsing of queries Automatic document summarization tools Client/Server development Programming interfaces for World-Wide Web (HTTP, HTML) access to databases
Environmental informatics
Environmental informatics is the science of information applied to environmental science. As such, it provides the information processing and communication infrastructure to the interdisciplinary field of environmental sciences aiming at data, information and knowledge integration, the application of computational intelligence to environmental data as well as the identification of environmental impacts of information technology. Environmental informatics thus acts as a bridge, providing an interdisciplinary means of analysing, describing and understanding the complex interactions between humans, nature and technology. Since each field of applied computer science has its own subject matter, terminology and methods, specialised disciplines, such as environmental, bio- and geoinformatics have emerged, each of which combines computer science with a specific field of application such as environmental, bio- or geosciences. Environmental informatics, bioinformatics and geoinformatics all deal with computer-based processing of environmental phenomena. However, environmental informatics is the only field that pursues normative goals (e.g., political goals of environmental protection, environmental planning, and sustainability). This also influences the choice of methods. This also distinguishes it from application areas such as numerical weather prediction, which is considered an early and important example of computer simulation of environmental phenomena. The UK Natural Environment Research Council defines environmental informatics as the "research and system development focusing on the environmental sciences relating to the creation, collection, storage, processing, modelling, interpretation, display and dissemination of data and information." Kostas Karatzas defined environmental informatics as the "creation of a new 'knowledge-paradigm' towards serving environmental management needs." Karatzas argued further that environmental informatics "is an integrator of science, methods and techniques and not just the result of using information and software technology methods and tools for serving environmental engineering needs." Environmental informatics emerged in early 1990 in Central Europe. Current initiatives to effectively manage, share, and reuse environmental and ecological data are indicative of the increasing importance of fields like environmental informatics and ecoinformatics to develop the foundations for effectively managing ecological information. Examples of these initiatives are National Science Foundation Datanet projects, DataONE and Data Conservancy. == Subject matter and objectives == The subject of environmental informatics are environmental information systems (EIS). An EIS 'is a computer-based system that integrates and stores data collected about the natural environment and provides powerful methods for accessing and evaluating it.' This allows environmental data to be processed by computers for environmental protection, planning, research and technology. According to Jaeschke and Bossel, environmental informatics has three interrelated objectives: Environmental informatics serves to procure data and information for describing the state and development of the environment. Of particular importance is information that is needed to prevent or limit undesirable changes and to support desirable changes. Based on the evaluation and analysis of data, environmental informatics improves our understanding of the environment and the interactions between nature, technology and society. It thus supports environmentally relevant decisions. This enables the influence of development (system correction), the assessment of the effects and side effects of potential measures, and the creation of tools for the routine planning, implementation and monitoring of measures. == History == The simulation model World3, which formed the basis of the highly acclaimed study The Limits to Growth, is considered the starting point of environmental informatics. It incorporated environmental information, among other things, to calculate scenarios for global development. In the mid-1980s, interest grew in structuring environmental protection as an area of application for computer science. One of the first publications in German was the book Informatik im Umweltschutz. Anwendungen und Perspektiven (Computer science in environmental protection. Applications and perspectives) from 1986. The term 'environmental informatics' did not appear until around 1993, which is why the development of environmental informatics is usually referred to as having taken place in the 1990s. In 1993, the first university chair for environmental informatics was established in Cottbus. In 1994, the anthology Umweltinformatik. Informatikmethoden für Umweltschutz und Umweltforschung (Environmental Informatics: Informatics Methods for Environmental Protection and Environmental Research) was published. The development of environmental informatics was 'primarily initiated by German computer science.' In the English-speaking world, the volume Environmental Informatics was published in 1995, mainly based on the German anthology of 1994. An article in the conference proceedings of the World Computer Congress of the International Federation for Information Processing (IFIP) in Hamburg in 1994 describes the initial situation of environmental informatics as follows: 'On the one hand, we suffer from the huge amount of available data – people sometimes speak of data graveyards – on the other hand, the really relevant data may still be missing.' This statement indicates the need that led to the emergence of environmental informatics as a specialised discipline of applied computer science. Furthermore, the specific characteristics and processing requirements of environmental data necessitated the emergence of environmental informatics. The special features of environmental data include: The data structures required are highly heterogeneous due to specific processes and differing perspectives on environmental aspects (e.g., water protection, emission control, hazardous substances). In addition to the heterogeneity of the data, heterogeneous databases also play a role, as environmental data is often obtained and presented in an interdisciplinary manner. Obligations change frequently as a result of new legislation, whether regional (e.g. state regulations on water protection), national (e.g. federal emission control regulations) or international (e.g. Registration, Evaluation, Authorisation and Restriction of Chemicals|REACH). The objects represented are often multidimensional and, therefore, require complex geometric representation using curves or polygons. It is often necessary to process uncertain, imprecise or incomplete data, which is, for example, the result of extrapolations or forecasts. A new "knowledge paradigm" has emerged to meet the requirements of environmental management. Environmental informatics produces its own concepts, methods and techniques and is not merely the result of using information and communication technology methods and tools to meet environmental requirements. The development of environmental informatics since the 1990s has been significantly influenced by the newly established conferences EnviroInfo, ISESS and ITEE and is documented in the respective proceedings. Aspects of sustainability and sustainable development were increasingly integrated into environmental informatics after 2000, thereby expanding the field. In 2004, the Working Group on Sustainable Information Society of the Gesellschaft für Informatik e. V. (German Informatics Society, GI) published the Memorandum on a Sustainable Information Society, which formulates recommendations for an information society that is compatible with human, social and natural needs. Since 2007, environmental informatics has often been described in more detail as informatics for environmental protection, sustainable development and risk management. The increased focus on sustainability has also contributed to the formation of the research focus Information and Communications Technology for Sustainability (ICT4S) and to the emergence of the international conference ICT4S in 2013. ICT-ENSURE, the European Commission's funding measure for the establishment of a European research area on "ICT for Environmental Sustainability Research" (2008–2010), has also contributed to the structuring of environmental informatics. == Environmental informatics and sustainable development == Efforts to place environmental informatics within the context of sustainable development have been growing since 2000 and were significantly influenced by the Memorandum on a Sustainable Information Society. According to this Memorandum, the information society offers great but unevenly distributed opportunities for education, participation and intercultural understanding. In addition, the Memorandum highlighted the material and energy consumption of inf
30 Boxes
30 Boxes is a minimalist calendaring IOS application created by 83 Degrees. Originating as a web application in March 2006, 30 Boxes was founded by Webshots cofounder Narendra Rocherolle. The website shut down some time in 2020, but relaunched for the IOS in February 2021. The original website was tailored towards "social media junkies". == Reception == Barry Collins of The Sunday Times appreciated the website's plain-language event adding feature, but did not appreciate that he was unable to see more than one month of events at a time. Collins was also unhappy that the website was not capable of warning him when he had two events scheduled at the same time. In a list of the best web-based calendar software for small businesses, Forbes ranked 30 Boxes second, after Google Calendar. They described 30 Boxes like “buying a new car with manual transmission and lots of extras—you don't just want to drive it, you want to fool around with it to see what it can do”.
Kunstweg
Bürgi's Kunstweg is a set of algorithms developed by Jost Bürgi in the late 16th century. They are used to calculate sines to arbitrary precision.. Bürgi used these algorithms to calculate a Canon Sinuum, a sine table in increments of 2 arc seconds. It is believed that the table featured values accurate to eight sexagesimal places. Some authors have speculated that the table only covered the range from 0° to 45°, although there is no evidence supporting this claim. Such tables were crucial for maritime navigation. Johannes Kepler described the Canon Sinuum as the most precise sine table known at the time. Bürgi explained his algorithms in his work Fundamentum Astronomiae, which he presented to Emperor Rudolf II in 1592. The Kunstweg algorithm calculates sine values iteratively. In each step, the value of a cell is the sum of the two preceding cells in the same column. The final cell's value is halved before beginning the next iteration. Ultimately, the values in the last column are normalized. Accurate sine approximations are achieved after only a few iterations. In 2015, Menso Folkerts and coworkers demonstrated that this iterative process does indeed converge toward the true sine values. According to them this was the first step towards differential calculus.