According to Pappus, " Apollonius, having completed Euclid's four books of conics and added four others, handed down eight volumes of conics.

According to the 4th-century report of Pappus of Alexandria, Apollonius ' own book on this problem — entitled (, " Tangencies "; Latin: De tactionibus, De contactibus )— followed a similar progressive approach.

According to Cauchy's functional equation theorem, the logarithm is the only continuous transformation that transforms real multiplication to addition.

According to (), the first historical mention of the statement of this theorem appears in ().

According to Birkhoff's theorem, it is the only vacuum solution that is spherically symmetric.

According to the theorem, it is possible to expand the power ( x + y )< sup > n </ sup > into a sum involving terms of the form ax < sup > b </ sup > y < sup > c </ sup >, where the exponents b and c are nonnegative integers with, and the coefficient a of each term is a specific positive integer depending on n and b. When an exponent is zero, the corresponding power is usually omitted from the term.

According to the theorem, it is possible to expand any power of x + y into a sum of the form

According to Noether's theorem, the conservation of energy is a consequence of the fact that the laws of physics do not change over time.

According to the theorem, within every sufficiently powerful logical system ( such as Principia ), there exists a statement G that essentially reads, " The statement G cannot be proved.

* According to a theorem of Nikolay Nikolov and Dan Segal, in any topologically finitely-generated profinite group ( that is, a profinite group that has a dense finitely-generated subgroup ) the subgroups of finite index are open.

According to the spin-statistics theorem, particles with integer spin occupy symmetric quantum states, and particles with half-integer spin occupy antisymmetric states ; furthermore, only integer or half-integer values of spin are allowed by the principles of quantum mechanics.

According to Rice's theorem, if there is at least one computable function in a particular class C of computable functions and another computable function not in C then the problem of deciding whether a particular program computes a function in C is undecidable.

According to the Rouché – Capelli theorem, the system is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix.

According to the optical Raychaudhuri's theorem, this requires a violation of the averaged null energy condition.

According to the theorem, the reconstructed signal will match the original signal provided that the original signal contains no frequencies at or above this limit.

According to the Nyquist theorem, error-free detection of the line code requires a channel bandwidth of at least the Nyquist rate, which is half the line code pulse rate.

According to Shannon's source coding theorem, the optimal code length for a symbol is − log < sub > b </ sub > P, where b is the number of symbols used to make output codes and P is the probability of the input symbol.

According to the CPT theorem of particle physics, antihydrogen atoms should have many of the characteristics regular hydrogen atoms have ; i. e., they should have the same mass, magnetic moment, and transition frequencies ( see atomic spectroscopy ) between their atomic quantum states.

According to Liouville's theorem, any bounded complex analytic function defined on the whole complex plane is constant.

According to Allen Weiss, in Mirrors of Infinity, this optical effect is a result of the use of the tenth theorem of Euclid ’ s Optics which asserts that “ the most distant parts of planes situated below the eye appear to be the most elevated .” In Fouquet ’ s time, interested parties could cross the canal in a boat, but walking around the canal provides a view of the woods that mark what is no longer the garden and shows the distortion of the grottos previously seen as sculptural.

According to the virial theorem, the gravitational binding energy of a star is about two times its internal thermal energy.

According to the correspondence principle and Ehrenfest's theorem as a system becomes larger or more massive ( action >> Planck's constant ) the classical dynamics tends to emerge, with some exceptions, such as superfluidity.

According to the Kutta – Joukowski theorem, lift is the product of circulation, airspeed, and air density.

According to the Shannon – Hartley theorem, the channel capacity of a properly-encoded signal is proportional to the bandwidth of the channel and the logarithm of the signal-to-noise ratio ( SNR ) ( assuming the noise is additive white Gaussian noise ).

According to Ptolemy's second theorem,

According to the inverse function theorem, the matrix inverse of the Jacobian matrix of an invertible function is the Jacobian matrix of the inverse function.

According to one leading historian of statistics, he may have been the earliest discoverer of Bayes theorem.

According to Vietnamese Ministry of Foreign Affairs, 4. 8 million Vietnamese people were exposed to Agent Orange, resulting in 400, 000 people being killed or maimed, and 500, 000 children born with birth defects.

According to Article 27 of the Universal Declaration of Human Rights, " everyone has the right to the protection of the moral and material interests resulting from any scientific, literary or artistic production of which he is the author ".

According to some, results of keno games in great cities were sent to outlying villages and hamlets by carrier pigeons, resulting in its Chinese name 白鸽票 báigē piào, literally “ white dove ticket ”, pronounced baak-gap-piu in Cantonese ( which the Western spelling ' pak-ah-pu ' / ' pakapoo ' was based on ).

According the Heitler-London theory, so-called two-body molecular-orbitals are formed, namely the resulting orbital is:

According to the Source-Filter Theory, the resulting sound excites the resonance chamber that is the vocal tract to produce the individual speech sounds.

According to a Symbolics employee, the reason for the change in policy was Richard Stallman's making changes with which they disagreed, such as removing Symbolics ' copyright notices on Symbolics ' produced enhancements and transferring the resulting enhancements to the other commercial licensees, and at one point leaving the software in a state where it would not compile.

According to utilitarianism the moral worth of an action is determined only by its resulting outcome although there is debate over how much consideration should be given to actual consequences, foreseen consequences and intended consequences.

According to this series, an area of Earth stretching from Florida to Venezuela will be devastated by an alien attack in 2153, resulting in the death of 7 million people.

According to scholar Ervand Abrahamian, although there were several decades of prohibition of torture that spread from Europe to most parts of the world, by the 1980s, the taboo against torture was broken and torture " returned with a vengeance ," propelled in part by television and an opportunity to break political prisoners and broadcast the resulting public recantations of their political beliefs for " ideological warfare, political mobilization, and the need to win ' hearts and minds.

According to the Internet Movie Database, Fisher had said that her costumes in the previous two films had been so long that viewers could not tell " she was a woman ", resulting in the skimpy outfit for Return of the Jedi.

According to Israel, about 9, 000 attacks were launched from 1949 to 1956, resulting in hundreds of Israeli civilian casualties.

According to Vietnamese Ministry of Foreign Affairs, 4. 8 million Vietnamese people were exposed to Agent Orange, resulting in 400, 000 people being killed or maimed, and 500, 000 children born with birth defects.

According to one report, Louis ordered Pitau to " make him a piece to remember ", and Pitau took two years on the piece, resulting in a " triangular-shaped 69-carat gem the size of a pigeon's egg that took the breath away as it snared the light, reflecting it back in bluish-grey rays.

According to the National School Safety Center, since the 1992-1993 U. S. school year there has been a significant decline in school-associated violent deaths ( deaths on private or public school property for kindergarten through grade 12 and resulting from schools functions or activities ):

According to records of the Führer Conferences on Naval Affairs, 4, 980 of these were executed, resulting in the destruction of the organised resistance movement in Germany for the remainder of World War II.

According to Kendall Bennett, developer of the VBE / AF standard, the VESA Software Standards Committee was closed down due to a lack of interest resulting from charging high prices for specifications.

According to various sources, the resulting death toll was likely between 20 and 40 million.

According to Banks ' appendices to Consider Phlebas, the war began in 1327 AD, and continued for 48 years and one month, resulting in an eventual but total victory for the Culture.

According to these scriptures, Amitābha possesses infinite merits resulting from good deeds over countless past lives as a bodhisattva named Dharmakāra.

According to former City Clerk Beverly Masters, an explosion and the resulting house fire in the early 1980s prompted expansion of the community's public safety offerings.

According to published reports, bookkeeping by the government was spotty, resulting in losses from unpaid property taxes the town failed to assess.

According to articles, reports and a biography, Turpin couldn't deal with the obscurity resulting from the loss of his crown.

According to the Sutta Pitaka, the " ten wholesome courses of conduct " will disappear and people will follow the ten unwholesome concepts of theft, violence, murder, lying, false speech, sexual misconduct, abusive and idle talk, covetousness and ill will, wanton greed, and uncontrolled lust resulting in poverty and the breakdown of law and cooperation or regard for the dharma.

According to documents filed with the Securities and Exchange Commission, in 1999 the company had revenue of $ 3. 5 million, with a resulting net loss of $ 26. 3 million.

According to local mythology, the Rock of Cashel originated in the Devil's Bit, a mountain 20 miles ( 30 km ) north of Cashel when St. Patrick banished Satan from a cave, resulting in the Rock's landing in Cashel.