Chapters on the Modern Geometry of the Point, Line, and Circle (Volume 1); Being the Substance of Lectures Delivered in the University of Dublin to the Candidates for Honors of the First Year in Arts - Softcover
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This historic book may have numerous typos, missing text or index. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. 1863. Not illustrated. Excerpt: ... negative increase is of course a positive decrease, and conversely, and a negative maximum consequently a positive minimum, and conversely; and the two values = 0 or co are not real minima or maxima values at all (45), but merely the particular values through which the variable magnitude in continuous decrease or increase passes at the moment of changing sign. Of course if absolute values of magnitudes only were taken into account, as in arithmetic and in the geometry of ancient times, the particular values 0 and oo would be the least of all minima and the greatest of all maxima for magnitudes of every kind; but in the geometry of the present day, in which magnitudes of certain kinds are regarded as having not only absolute value but also sign, they are looked on as in no way differing from any other particular values through which variable magnitudes in continuous decrease or increase may happen to pass. In the case of magnitudes incapable of change of sign, the values 0 and oo are of course the extreme minima and maxima values in modern as in ancient geometry, and it might at first sight appear questionable whether it would not be better to regard them as such for magnitudes of all kinds as well. The advantages, however, resulting from the convention of signs in modern geometry are so numerous and considerable, that in the present state of the science it could scarcely be regarded as optional to forego them or not. 59. The extreme maxima and minima values of variable magnitudes, in whichever light regarded, give evidently in all cases the extreme limits of possibility and impossibility in the solutions of all problems involving the magnitudes; it being of course impossible to construct a magnitude of any kind greater than the extreme maximum or less than the e...
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