A History of the Conceptions of Limits and Fluxions in Great Britain, from Newton to Woodhouse |
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Common terms and phrases
algebraic Quantity Analyst appeared assignable become Benjamin Robins Berkeley Berkeley's called Colson conceive concept consider continually controversy D'Alembert definition difference differential calculus diminished doctrine of fluxions edition equal equation evanescent quantities explained expressed finding the fluxion finite quantity flowing quantity Fluent fluxion of xy geometrical given increase Increments or Decrements indefinitely indivisibles infinite Number infinitely little quantities infinitely small quantities infinitesimals inscribed Instant irrational numbers James Glenie John John Landen John Turner Jurin Landen last ratio Leibniz Lemma London Maclaurin magnitude mathematical mathematicians maticians method of fluxions Moments Monthly Review motion nascent never Newton's Method notation Philalethes Philosophical prime and ultimate Principia principles proportion Quadrature of Curves quotes rectangle reply Republick of Letters Robert Heath Robins's says Scholium Simpson Sir Isaac Newton Space supposed term theorem tion translation Treatise of Fluxions ultimate ratios uniformly variable quantity velocity Walton William Rowan Hamilton writers
Popular passages
Page 9 - ... velocity with which the body arrives at its last place, and with which the motion ceases. And in like manner, by the ultimate ratio of evanescent quantities is to be understood the ratio of the quantities not before they vanish, nor afterwards, but with which they vanish. In like manner the first ratio of nascent quantities is that 'with which they'' begin to be. And the first or last sum is that with which they begin and cease to be (or to be augmented or diminished).
Page 63 - And what are these fluxions? The velocities of evanescent increments. And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?
Page 59 - If with a View to demonstrate any Proposition, a certain Point is supposed, by virtue of which certain other Points are attained; and such supposed Point be...
Page 9 - Perhaps it may be objected that there is no ultimate proportion of evanescent quantities; because the proportion, before the quantities have vanished, is not the ultimate, and when they are vanished, is none. But by the same argument, it may be alleged that a body arriving at a certain place, and there stopping, has no ultimate velocity: because the velocity » before the body comes to the place, is not its ultimate velocity; when it has arrived, is none.
Page 10 - ... nascent quantities is that with which they begin to be. And the first or last sum is that with which they begin and cease to be (or to be augmented or diminished). There is a limit which the velocity at the end of the motion may attain, but not exceed. This is the ultimate velocity. And there is the like limit in all quantities and proportions that begin and cease to be.
Page 58 - Now, as our Sense is strained and puzzled with the perception of objects extremely minute, even so the Imagination, which faculty derives from Sense, is very much strained and puzzled to frame clear ideas of the least particles of time, or the least increments generated therein; and much more so to comprehend the moments, or those increments of the flowing quantities in statu nascenti, in their very first origin or beginning to exist, before they become finite particles.
Page 4 - Quantities, and the ratios of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer to each other than by any given difference, become ultimately equal.
Page 21 - ... described by a continued motion. Lines are described, and thereby generated, not by the apposition of parts, but by the continued motion of points; superficies by the motion of lines; solids by the motion of superficies; angles by the rotation of the sides; portions of time by continual flux: and so on in other quantities. These geneses really take place in the nature of things, and are daily seen in the motion of bodies.
Page 21 - I consider mathematical quantities in this place not as consisting of very small parts, but as described by a continued motion. Lines are described, and thereby generated, not by the apposition of parts, but by the continued motion of points...
Page 60 - But it should seem that this reasoning is not fair or conclusive. For when it is said, let the increments vanish, ie let the increments be nothing or let there be no increments, the former supposition that the increments were something, or that there were increments, is destroyed, and yet a consequence of that supposition, ie an expression got by virtue thereof, is retained.