A Treatise on Fluxions

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J. Smith, 1818 - 400 páginas
 

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Página 17 - I have no controversy about your conclusions, but only about your logic and method. How you demonstrate? What objects you are conversant with, and whether you conceive them clearly? What principles you proceed upon; how sound they may be; and how you apply them? It must be remembered that I am not concerned about the truth of your theorems, but only about the way of coming at them; whether it be legitimate or illegitimate, clear or obscure, scientific or tentative.
Página 1 - 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.
Página 1 - Therefore considering that Quantities, which increase in equal Times, and by increasing are generated, become greater or less according to the greater or less Velocity with which they increase and are generated; I sought a Method of determining Quantities from the Velocities of the Motions or Increments, with which they are generated; and calling these Velocities of the Motions or Increments Fluxions, and the generated Quantities Fluents, I fell by degrees upon the Method of Fluxions, which I have...
Página 110 - The surface of a sphere is equal to four times the area of a circle...
Página 3 - one quantity increase uniformly., and another of the ' same kind increase -with an accelerated or retarded velocity, and two increments be assumed which are generated in the same time; if those increments be diminished till they vanish, 'that ratio to which they approach as their limit is the ratio of the fluxions of those quantities.
Página 120 - Multiply the weight of the several particles by the squares of their distances from the centre of motion, and divide the sum of the products by the weight of the whole mass ; the square root of the quotient will be the distance of the centre of gyration...
Página 8 - Hence it appears, that whether the root be a simple or a compound quantity, the fluxion of any power of it is found by the following general Rule : Multiply by the index, diminish the index by unity, and multiply by the fluxion of the root. Thus the fluxion...
Página 16 - ... to demonstrate any proposition, a certain point is supposed, by virtue of which certain other points are attained; and such supposed point be it self afterwards destroyed or rejected by a contrary supposition; in that case, all the other points attained thereby, and consequent thereupon, must also be destroyed and rejected, so as from thence forward to be no more supposed or applied in the demonstration.
Página 303 - Prove, that the velocity of the earth continually increases from the aphelion to the perihelion distance. 17. Compare the velocity in an ellipse, at the greatest distance, with the velocity of a body revolving in a circle at the same distance.
Página 30 - Prove that the problem, to describe a circle with its centre on the circumference of a given circle so that the length of the arc intercepted within the given circle shall he a maximum, is reducible to the solution of the equation 0 = cot 0.

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