Author(s):

Daniel J. Curtin (Northern Kentucky University)

### Translation

Download the author's translation of Hudde's Second Letter.

### Classroom Use

Several student exercises and projects can make use of Hudde’s second letter. It also can be used as an aid to those working on learning mathematical Latin. Some of these issues are addressed in [Curtin].

- First, students can verify the mathematical preliminaries in modern terms, then see how Hudde did the calculations. In fact at each stage it is useful for the student to solve the problem using modern methods, both to reinforce former learning and to help see where Hudde’s methods are similar and where they differ.
- Hudde’s proof of the first theorem was a proof by example, but it is not hard to see that it generalizes readily to any polynomial with a double root. Then he began working out some examples of his method. Students should study these examples carefully to see what Hudde was doing, since his notations and concepts are a bit different from ours.
- Most important is to understand Hudde’s other examples in the section Example 2 , where the denominators have several terms. At first it appears Hudde has committed an algebraic error, namely writing \[\frac{a}{b+c}\quad{\rm{as}}\quad\frac{a}{b}+\frac{a}{c},\] but a more careful analysis will illuminate his thinking and show that it is correct.
- After working through the examples, students can discuss exactly what Hudde has done, both in his own terms and in modern terms. For example, in what sense has he developed the quotient rule?
- The final example, finding the maximal width of Descartes’ folium, is a nice extension of Hudde’s method and a good exercise in itself.
- Finally, note that Hudde most often used an arithmetic progression that multiplies \(x_n\) by \(n\) (thus operating like the modern derivative). He could thus ignore all constant terms, including the actual maximal value. He often did eliminate the constant terms separately, since they do not affect the \(x\)-value at which the maximum occurs. Hudde discussed the possibility of using other progressions, thus choosing positive powers of \(x\) to be eliminated, and students could explore whether this approach could be used to any advantage in the types of max/min problems typical in Calculus courses.

Daniel J. Curtin (Northern Kentucky University), "Jan Hudde’s Second Letter: On Maxima and Minima - Classroom Use - Translation," *Convergence* (June 2015)