Leibniz's Formula for Radius of Curvature

The radius of a circle drawn through three infinitely nearby points on a curve in the -plane satisfies

where denotes the arclength.  For example, if , so , then

If  the curve is given parametrically, and , so , then

Changes

Consider three points on a curve with equal distances between the points.  Let and denote the angles between the horizontal and the segments connecting the points as shown.  We have the relation between the changes in and :

(2)

The difference between these angles, , is shown near .  

The angle between the perpendicular bisectors of the connecting segments is also , because they meet the connecting segments at right angles.

These bisectors meet at the center of a circle through the three points on the curve whose radius we denote .  The small triangle with hypotenuse gives

(3)

Small Changes

Now we apply these relations when the distance between the successive points is an infinitesimal . The change

,

(4)

with , by smoothness of sine (see above).  Smoothness of sine also gives,

, with

Combining this with formula (1.1.3) for the infinitesimal case (assuming ), we get

, with

Now substitute this in (1.1.4) to obtain

, with

By trigonometry, , so

, as long as is not infinitely large.

Keisler's Function Extension Axiom allows us to apply formulas (1.1.3) and (1.1.4) when the change is infinitesimal, as we shall see.  We still have a gap to fill in order to know that we may replace infinitesimal differences with differentials (or derivatives), especially because we have a difference of a quotient of differences.  

First differences and derivatives have a fairly simple rigorous version in Robinson's theory, just using the differential approximation (1.1.1).  This can be used to derive many classical differential equations like the tractrix,  catenary, and isochrone, see: Chapter 5 Differenital Equations from Increment Geometry in Projects for Calculus: The Language of Change on my website at http://www.math.uiowa.edu/%7Estroyan/ProjectsCD/estroyan/indexok.htm

Second differences and second derivatives have a complicated history.  See

H. J. M. Bos, Differentials, Higher-Order Differentials and the Derivative in the Leibnizian Calculus, Archive for History of Exact Sciences, vol. 14, nr. 1, 1974.

This is a very interesting paper that begins with a course in calculus as Leibniz might have presented it.

The natural exponential

The natural exponential function satisfies

y[0] = 1

We can use (1.1.1) to find an approximate solution,

Recursively,

:

, for

This is the product expansion , for .

No introduction to calculus is complete without mention of this sort of "infinite algebra" as championed by Euler as in

L. Euler, Introductio in Analysin Infinitorum, Tomus Primus, Lausanne, 1748. Reprinted as L. Euler, Opera Omnia, ser. 1, vol. 8. Translated from the Latin by J. D. Blanton, Introduction to Analysis of the Infinite, Book I, Springer–Verlag, New York, 1988.

A wonderful modern interpretaion of these sorts of computations is in

Mark McKinzie and Curtis Tuckey, Higher Trigonometry, Hyperreal Numbers and Euler's Analysis of Infinities, Math Magazine, vol. 74, nr. 5, Dec. 2001, p. 339-368

W. A. J. Luxemburg's reformulation of the proof of one of Euler's central formulas

appears in our monograph, Introduction to the Theory of Infinitesimals, Academic Press Series on Pure and Applied Math. vol 72, 1976, Academic Press, New York.

Created by Mathematica  (September 22, 2004)