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Newton’s work on Tangents

Newton’s work on Tangents. The Brachistochrone problem. Newton’s Answer.

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Newton’s work on Tangents

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  1. Newton’s work on Tangents

  2. The Brachistochrone problem

  3. Newton’s Answer • From the given point A let there be drawn an unlimited straight line APCZ parallel to the horizontal, and on it let there be described an arbitrary cycloid AQP meeting the straight line AB (assumed drawn and produced if necessary) in the point Q, and further a second cycloid ADC whose base and height are to the base and height of the former as AB is to AQ respectively. This last cycloid will pass through the point B, and it will be that curve along which a weight, by the force of its gravity, shall descend most swiftly from the point A to the point B.

  4. Newton’s Answer • In English : The cycloid. • What is a cycloid. • What Newton found. C:\Users\vincenzopie\Documents\PGCE\Newton Project\the inverted cycloid.gif http://archives.math.utk.edu/visual.calculus/0/parametric.5/cycloid.html

  5. Newton and the motion of particles • Newton wanted to find his own method to find a tangent of a curve at a point. He did this by looking at a curve as the path a moving particle take. • http://www.fearofphysics.com/Proj/projectile.html In what directions does this ball travel?

  6. Breaking down a particles movement

  7. Breaking down a particles movement

  8. Finding the Gradient using velocities. • We can call these velocities as “the change” in x and y over time. • The velocity vector of a particle lies on the tangent line at that point. • The common notation for this is. • How can we find the gradient using these quantities.

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