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The following is exactly what we mean by a <small>LIMIT</small>. ...let the several values of x... bea<sub>1</sub> a<sub>2</sub> a<sub>3</sub> a<sub>… - Augustus De Morgan
" "The following is exactly what we mean by a <small>LIMIT</small>. ...let the several values of x... bea<sub>1</sub> a<sub>2</sub> a<sub>3</sub> a<sub>4</sub>. . . . &c.then if by passing from a<sub>1</sub> to a<sub>2</sub>, from a<sub>2</sub> to a<sub>3</sub>, &c., we continually approach to a certain quantity l [lower case L, for "limit"], so that each of the set differs from l by less than its predecessors; and if, in addition to this, the approach to l is of such a kind, that name any quantity we may, however small, namely z, we shall at last come to a series beginning, say with a<sub>n</sub>, and continuing ad infinitum,a<sub>n</sub> a<sub>n+1</sub> a<sub>n+2</sub>. . . . &c.all the terms of which severally differ from l by less than z: then l is called the limit of x with respect to the supposition in question.
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About Augustus De Morgan
Augustus De Morgan (June 27 1806 – March 18 1871) was an Indian-born British mathematician and logician; he was the first professor of mathematics at University College London. He formulated De Morgan's laws and was the first to introduce the term, and make rigorous the idea of mathematical induction. De Morgan crater on the Moon is named after him.
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Additional quotes by Augustus De Morgan
A great many individuals ever since the rise of the mathematical method, have, each for himself, attacked its direct and indirect consequences. ...I shall call each of these persons a paradoxer, and his system a paradox. I use the word in the old sense: ...something which is apart from general opinion, either in subject-matter, method, or conclusion. ...Thus in the sixteenth century many spoke of the earth's motion as the paradox of Copernicus, who held the ingenuity of that theory in very high esteem, and some, I think, who even inclined towards it. In the seventeenth century, the depravation of meaning took place... Phillips says paradox is "a thing which seemeth strange"—here is the old meaning...—"and absurd, and is contrary to common opinion," which is an addition due to his own time.
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I have throughout introduced the Integral Calculus in connexion with the Differential Calculus. ...Is it always proper to learn every branch of a direct subject before anything connected with the inverse relation is considered? If so why are not multiplication and involution in arithmetic made to follow addition and precede subtraction? The portion of the Integral Calculus, which properly belongs to any given portion of the Differential Calculus increases its power a hundred-fold...
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