By revealing ( in modern terms ) that numbers could be irrational, this discovery seems to have provoked the first foundational crisis in mathematical history ; its proof or its divulgation are sometimes credited to Hippasus, who was expelled or split from the Pythagorean sect.

By never revealing the decrypted program instructions, the cryptoprocessor prevents tampering of programs by technicians who may have legitimate access to the sub-system data bus.

By revealing the competing bids in real time to every participating supplier, reverse auctions promote “ information transparency ”.

By filing the lawsuit, Alabama band attorneys mistakenly included copies of band contracts as exhibits along with their lawsuit papers, thus allowing fans a chance to look at the inner workings of the band and revealing that Herndon actually had a contractual full band share of the farewell tour.

By revealing such large numbers of very young galaxies, the HDF has become a landmark image in the study of the early universe, with the associated scientific paper having received over 800 citations by the end of 2008.

By the late 1960s, Jones began at least partially openly revealing in Temple sermons his " Apostolic Socialism " concept.

By revealing only a part of their identity, strippers attempt to avoid being characterized by the stigmatizing attributes associated with exotic dancing.

In two of the books in which he appears — The Mysterious Affair at Styles and The ABC Murders — Hastings plays a prominent role in the resolution of the mystery, with a casual observation he makes at one point in the novel leading Poirot to realise the guilty party: By mentioning that Poirot had to straighten some spill holders and ornaments in Styles, he prompts Poirot to realise that someone had moved them, thus allowing Poirot to discover a crucial piece of evidence, and when he suggests that an incorrectly addressed letter revealing the latest crime in ABC Murders was addressed that way on purpose, Poirot realises that the letter had indeed been wrongly addressed deliberately so that it would not be received until after the murderer had committed his crime, revealing that the murderer had attached greater importance to that particular murder, and wanted to be certain that it was committed.

By 1969, the average white family income had risen to $ 10, 953, while average black family income lagged behind at $ 7, 255, revealing a continued racial disparity in income amongst various segments of the American population.

By revealing the information when he did, he avoided a much bloodier war, and he cemented his reputation for prescience and as a man for peace.

By that which we understand, the sixty-six books of the Old and New Testaments are given by divine inspiration, inertly revealing the will of God concerning us in all things necessary to our salvation, so that whatever is not contained therein is not to be enjoined as an article of faith.

By this capacity of hiding or revealing the main working area the spinet desk could be said to be a smaller, less obtrusive cousin of the rolltop desk and the cylinder desk.

By simultaneously impersonating a duty-bound and an evil twin sister, Rökk was able to act out her entire theatrical talent, accompanied by several dance interludes which were quite revealing at that time.

By revealing she was using magic and not the powers of a god ( by casting a spell of his own ), Raistlin proved that she was a charlatan, destroying her cult and at the same time invoking the wrath of the townspeople against himself, coming dangerously close to being burned at the stake.

" The review concluded, " By revealing the scope and opening up the operations of the N. S. A.

By revealing only a part of themselves, strippers may avoid being characterized by the stigmatizing attributes associated with exotic dancing.

By Season 2, he becomes principal and tricks Jake into revealing himself as a dragon to him.

By revealing his attitudes to the subject he reveals much of himself.

The largest species are red alder ( A. rubra ) on the west coast of North America, and black alder ( A. glutinosa ), native to most of Europe and widely introduced elsewhere, both reaching over 30 m. By contrast, the widespread Alnus viridis ( green alder ) is rarely more than a 5 m tall shrub.

By the late 1980s, the cellular industry's subscriber base had grown into the millions across America and it became necessary to add channels for additional capacity.

By the mid 20th century, with the growing market for rare coins, the Sheldon system was adopted by the American Numismatic Association and most coin professionals in the North America.

By 2005 Colombia had the highest mobile phone density ( 90 percent ) in Latin America, as compared with the region ’ s average density of 70 percent.

By far, the best known representation is the animated Wile E. Coyote, Super Genius, whose popularity has spread the three-syllable Spanish pronunciation of the word coyote throughout English-speaking North America.

By the Oligocene, all three subfamilies of canids ( Hesperocyoninae, Borophaginae, and Caninae ) had appeared in the fossil records of North America.

By 0. 3 Mya, Canis lupus ( the gray wolf ) was fully developed and had spread throughout Europe and northern Asia ; and Beringia offered a way to North America.

By 1524 an association of conquest regarding South America was formalized among Almagro, Pizarro and Luque.

By this time, Parsons's own use of drugs had increased to the extent that new songs were rare and much of his time was diverted to partying with the Stones, who briefly relocated to America in the summer of 1969 to finish their forthcoming Let It Bleed album and prepare for an autumn cross-country tour, their first series of regular live engagements since 1967.

According to the Harvard University Library, " By the late 19th century, 70 to 90 % of the urban populations of Europe and North America were infected with the TB bacillus, and about 80 % of those individuals who developed active tuberculosis died of it.

By the time Monroe became president, several European powers, in particular Spain, were attempting to re-establish control over South America.

*' U. S. By the Numbers: What's Left, Right & Wrong with America, ( ISBN 9781892123145, Capital Books, Incorporated, 2000 ) with Raymond J. Keating, and Thomas N. Edmonds

By the 1970s and 1980s the sport had expanded beyond Japan and had reached North America and Europe.

By the end of the 20th century, almost the only steam power still in regular use in North America and Western European countries was on heritage railways largely aimed at tourists and / or railroad hobbyists, known as ' railfans ' or ' railway enthusiasts ', although some narrow gauge lines in Germany which form part of the public transport system, running to all-year-round timetables retain steam for all or part of their motive power.

By the late 1960s, few major railroads in North America, Europe and Oceania continued to operate steam locomotives, although significant numbers still existed outside these areas.

By the end of the last ice age ( 10, 000 – 12, 000 years ago ), camelids were extinct in North America.

By the end of the last ice age about 11, 000 years ago, much of northern Europe and North America was covered by ice sheets up to 3 km thick.

By the end of the 1970s, electronic musical devices were becoming increasingly common and affordable in North America, Europe and Japan.

By the time Elijah Muhammad died in 1975, there were 75 NOI centers across America.

*" America ’ s Opera Boom " By Jonathan Leaf, The American, July / August 2007 Issue

By the early 1900s, the movement included England, Canada, Croatia, France, Denmark, Italy, America, the Philippines, China, and Hungary.

By 1981, hardcore was the dominant punk rock style not only in California, but much of the rest of North America as well.

By mid-Paleozoic, the collision of North America and Europe produced the Acadian-Caledonian uplifts, and a subduction plate uplifted eastern Australia.

By the late 19th century, 70 to 90 % of the urban populations of Europe and North America were infected with M. tuberculosis, and about 40 % of working-class deaths in cities were from TB.