(1802-1829) Norwegian mathematician. He demonstrated that the general quintic equation
ax5 + bx4 + cx3 + dx2 + ex + f = 0
could not be solved algebraically. Subsequent work covered elliptic functions, integral equations, infinite series, and the binomial theorem.
Abel was born at Finnöy, a small island near Stavanger, and studied at Oslo. In 1823 he provided the first solution in the history of mathematics of an integral equation and published a paper demonstrating that a radical expression to represent a solution to fifth- or higher-degree equations was impossible.
In 1825, Abel went to Berlin, where he met Leopold Crelle (1780–1855), a privy councillor and engineer much taken with problems in mathematics. Together they brought out the first issue of Crelle’s Journal, which was to become the leading 19th-century German organ of mathematics. A year later Abel moved on to Paris, where he wrote Mémoire sur une propriété générale d’une classe très-étendue de fonctions transcendantes. It dealt with the sum of the integrals of a given algebraic function and presented the theorem that any such sum can be expressed as a fixed number of these integrals with integration arguments that are algebraic functions of the original arguments.
Abel transformed the theory of elliptic integrals by introducing elliptic functions, and this generalization of trigonometric functions led eventually to the theory of complex multiplication, with its important implications for algebraic number theory. He also provided the first stringent proof of the binomial theorem. A number of useful concepts in modern mathematics, notably the Abelian group and the Abelian function, bear his name.

Množina: Abels

Cain and Abel were the first children of Adam and Eve born after the Fall of Man; Abel was killed by Cain.
In the Old Testament, the second son of Adam and Eve; as a shepherd, he made burned offerings of meat to God which were more acceptable than the fruits offered by his brother Cain; he was killed by the jealous Cain.

(English form of Pierre Abélard) (1079-1142)
French scholastic philosopher who worked on logic and theology. His romantic liaison with his pupil Héloïse caused a medieval scandal. Details of his life are contained in the autobiographical Historia Calamitatum Mearum/The History of My Misfortunes.
Abelard, born near Nantes, became canon of Notre Dame in Paris and master of the cathedral school 1115. When his seduction of, and secret marriage to, Héloïse became known, she entered a convent. He was castrated at the instigation of her uncle Canon Fulbert, and became a monk. Resuming teaching a year later, he was cited for heresy and became a hermit at Nogent, where he built the oratory of the Paraclete, and later abbot of a monastery in Brittany.
Abelard died at Châlon-sur-Saône on his way to defend himself against a new charge of heresy. Héloïse was buried beside him at the Paraclete 1164; their remains were taken to Père Lachaise cemetery, Paris, 1817.
He opposed realism in the debate over universals, and propounded conceptualism, whereby universal terms have only a mental existence. He asserted the importance of reason in religious belief, and his skillful application of logic and dialectic to such doctrines as the Trinity and atonement, though controversial, gave theology a new breadth.

Množina: abeles

A poplar that is widely cultivated in the United States; has white bark and leaves with whitish undersurfaces.

Množina: abelias

Any of various deciduous or evergreen ornamental shrubs of the genus Abelia having opposite simple leaves and cymes of small white or pink or purplish flowers; Asia and Mexico.