The Mathematical Psychology of Gratry and Boole: Translated from the Language of the Higher Calculus Into That of Elementary Geometry

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This is an OCR edition with typos.
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CHAPTER III THE DOCTRINE OF LIMITS of the mathematical artifices which is most important psychologically is what is called the doctrine of " limits." It may be illustrated by trying to sum up the series i + + £ + £ + etc. It will be noticed, first, that the series itself has no natural termination ; however long we write, we shall never write its last term. Next, that, however many terms we write, we shall never make up a sum quite equal to 2; the sum is always 2 minus a fraction equal to the last term we have written. Thirdly, that the more terms we write, the nearer their sum will approach to 2 ; so that by writing more terms we can make the sum as near to 2 as we please. The sum of the series approaches the limit 2, as the last term approaches the limit zero. These facts are expressed by the equations:— Sumofi + |+J . . . adinfn. = 2. Last term of series— i + +1 . . . ad infn. = o; or by the statement that 2 is the limit of the series i + £ + £ as the last term approaches the limit o. These are fictitious statements; no series is ever written out ad infn. In the same way it is stated that the parabola touches a certain imaginary straight line, the asymptote, " at infinity," or that the asymptote is the "limit" of breadth of the parabola, as it approaches oo in length. The parabola is a re-al line, the curve traced by a projectile. In any given case it comes to an abrupt termination, because the projectile is stopped by the earth ; but it does not naturally join ends like an ellipse; it is, potentially, of indefinite length. Man, in investigating the parabola, finds it useful to invent the imaginary asymptote, and then to show that, supposing the asymptote existed, the parabola would touch it " at infinity" (i.e., never), but be always approaching nearer and ...

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ISBN-13: 9781145399280 Creative Media Partners, LLC 02/17/2015 134 7.44(w) x 9.69(h) x 0.29(d)

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