The Same Problem Solved by the Principles of Leibnitz

Excerpts

• The Same Problem Solved by the Principles of Leibnitz

  The inaccuracy of this supposition has been already pointed out; yet it must be confessed that this once got over, the results are deduced with a degree of simplicity and consequent clearness, not to be found in any other method.

• The Same Problem Solved by the Principles of Leibnitz

  The following cannot be regarded as a demonstration, except by a mind so accustomed to the subject that it can readily convert the various inaccuracies into their corresponding truths, and see, at one glance, how far any proposition will affect the final result.

• The Same Problem Solved by the Principles of Leibnitz

  The beginner will be struck with the extraordinary assertions which follow, given in their most naked form, without any attempt at a less startling mode of expression.

• The Same Problem Solved by the Principles of Leibnitz

  But at the same time we observe that every one of these assumptions approaches the truth, as we diminish the angle $A'PA$, so that there is no magnitude, line or angle, so small that the linear or angular errors, arising from the above-mentioned suppositions, may not be made smaller.

• The Same Problem Solved by the Principles of Leibnitz

  Hence the ratio $A\alpha$ to $\beta B'$ or $dp + \mu$ to $dq + \nu$ will continually approximate to that of $dp$ to $dq$, or a ratio of equality.

• The Same Problem Solved by the Principles of Leibnitz

  An infinitely small arc of a circle is a straight line perpendicular to its radius; hence $A'aA$ and $BbB'$ are right-angled triangles, the first similar to $BOA$, the two having the angle $A$ in common, and the second similar to $B'OA'$.

Equations

• The Same Problem Solved by the Principles of Leibnitz

  BP = q

  The length BP is denoted q.

• The Same Problem Solved by the Principles of Leibnitz

  Aa = dp

  The infinitely small increment Aa is denoted dp.

• The Same Problem Solved by the Principles of Leibnitz

  Aa : A'a :: OA : OB :: a : b

  Corresponding sides of the small triangles at A are in the same ratio as the radii OA and OB.

• The Same Problem Solved by the Principles of Leibnitz

  Bb : B'b :: OA : OB :: a : b

  The small arcs at B are in the same proportion as the radii, so this ratio matches the one at A.

• The Same Problem Solved by the Principles of Leibnitz

  \PadTo[r]{BP + Pa}{Bb} : A'a :: \PadTo[l]{a^{2} + b^{2}}{a^{2}} : b^{2}

  The composed proportion obtained from the two proportions above, with Aa equal to B'b, relates the arc Bb to A'a.

• The Same Problem Solved by the Principles of Leibnitz

  BP + Pa : Pa :: a^{2} + b^{2} : b^{2}

  After cancelling the common factor, the segments BP and Pa stand in the same proportion as a² + b² and b².

• This equation is in A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines (A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines)

• The Same Problem Solved by the Principles of Leibnitz

  BP + PA = l

  The segments BP and PA together make up the whole line AB, of length l.

• The Same Problem Solved by the Principles of Leibnitz

  \PadTo[r]{BP + Pa}{BP} : Pa :: \PadTo[l]{a^{2} + b^{2}}{a^{2}} : b^{2}

  Rearranging the composed proportion gives a proportion between the segments BP and Pa and the squares a² and b².

• The Same Problem Solved by the Principles of Leibnitz

  PA = \frac{b^{2}}{l}

  The distance PA from A to the sought point P equals b² divided by l, the result already obtained.

• The Same Problem Solved by the Principles of Leibnitz

  \angle A'PA = d\theta

  The infinitely small angle A'PA is denoted dθ.

• The Same Problem Solved by the Principles of Leibnitz

  A'a = (p - dp)\, d\theta

  The small arc A'a equals the radius PA' times the small angle dθ.

• The Same Problem Solved by the Principles of Leibnitz

  Bb = q\, d\theta

  The small arc Bb equals the radius PB times the small angle dθ.

• The Same Problem Solved by the Principles of Leibnitz

  PA = p

  The length PA is denoted p.

• The Same Problem Solved by the Principles of Leibnitz

  B'b = dq

  The infinitely small increment B'b is denoted dq.

• The Same Problem Solved by the Principles of Leibnitz

  OA = a

  The length OA is denoted a.

• The Same Problem Solved by the Principles of Leibnitz

  OB = b

  The length OB is denoted b.

• The Same Problem Solved by the Principles of Leibnitz

  AB = l

  The length AB is denoted l.

• The Same Problem Solved by the Principles of Leibnitz

  a\alpha = \mu

  The small perpendicular offset a α is denoted μ.

• The Same Problem Solved by the Principles of Leibnitz

  b\beta = \nu

  The small perpendicular offset b β is denoted ν.

• The Same Problem Solved by the Principles of Leibnitz

  dp = dq

  The infinitely small increments dp and dq are equal, which the book states is rigorously true.

• The Same Problem Solved by the Principles of Leibnitz

  dp + \mu &: (p - dp) \sin d\theta &&:: a &&: b

  The exact proportion at A: the small segment A α plus μ, compared with the perpendicular, is in the ratio a : b.

• The Same Problem Solved by the Principles of Leibnitz

  q \sin d\theta &: \PadTo{(p - dp) \sin d\theta}{dq + \nu} &&:: a - da &&: b + db

  The exact companion proportion at B, with the increments of a and b included.

• The Same Problem Solved by the Principles of Leibnitz

  q\left(1 + \frac{\mu}{dp}\right) : (p - dp)\left(1 + \frac{\nu}{dp}\right) :: a(a - da) : b(b + db)

  After composition and division by dp, the two sides of the proportion are written with the ratios μ/dp and ν/dp.

• The Same Problem Solved by the Principles of Leibnitz

  q &: p &&:: a^{2} &&: b^{2}

  In the limit as dθ vanishes, the segments BP and PA stand in the ratio of a² to b².

• The Same Problem Solved by the Principles of Leibnitz

  q + p = l &: p &&:: a^{2} + b^{2} = l^{2} &&: b^{2}

  Since BP + PA = l and a² + b² = l², the limiting proportion gives PA in terms of l and b², the same result as before.

Problems

No exercises in this chapter.