Rational Explanation of the Language of Leibnitz

Excerpts

• Rational Explanation of the Language of Leibnitz

  We shall proceed to a strict proof of this; but in the meanwhile, as a familiar illustration, imagine a small arc to be cut off from a curve, and its extremities joined by a chord, thus forming an arch, of which the chord is the base.

• Rational Explanation of the Language of Leibnitz

  However the original arc may be diminished, let the magnified base continue of a given length. This is possible, since on any line a figure may be constructed similar to a given figure.

• Rational Explanation of the Language of Leibnitz

  All which must be interpreted to mean that, the chord and arc being diminished, approach more and more nearly to a ratio of equality as to their lengths; and also that the greatest separation between an arc and its chord may be made as small a part as we please of the whole chord or arc, by sufficiently diminishing the chord.

• Rational Explanation of the Language of Leibnitz

  Hence $PQ$ can be taken so small that $VQ$ shall contain $VP'$ as often as we please, or the ratio of $VQ$ to $VP'$ shall be as great as we please.

• Rational Explanation of the Language of Leibnitz

  Let $PP'$ (4) be a part of a curve, whose equation is $y = \phi(x)$, that is, $PM$ may always be found by substituting the numerical value of $OM$ in a given function of $x$.

Equations

• Rational Explanation of the Language of Leibnitz

  y = \phi(x)

  The curve PP' is the graph of y equal to a given function of x, so PM is found by substituting OM into that function.

• Rational Explanation of the Language of Leibnitz

  MM' = dx

  The increment of x, taken as the segment MM', is written dx; it may be supposed as small as we please.

• Rational Explanation of the Language of Leibnitz

  P'Q = \phi' x\, dx + \phi'' x\, \frac{(dx)^{2}}{2} + \phi''' x\, \frac{(dx)^{3}}{2·3} + \etc.

  The increment P'Q of the ordinate equals a series in powers of dx, whose first term is phi'(x) dx, with higher derivatives divided by factorials.

• Rational Explanation of the Language of Leibnitz

  dx \sqrt{1 + \left(\frac{dy}{dx}\right)^{2}}

  The chord PP' equals dx times the square root of one plus the square of dy over dx.

• Rational Explanation of the Language of Leibnitz

  \tan VPQ·PQ = VQ

  The tangent of the angle VPQ times PQ gives VQ, so the first term of the series is the segment VQ along the tangent.

Problems

No exercises in this chapter.