In 1658, Jan Hudde extended Descartes’ fundamental idea for finding maxima and minima, namely that near the maximum value of a quantity the variable giving that quantity has two different values, but at the maximum these two values become one – algebraically a double root. He introduced more efficient ways of calculating double roots for polynomials and rational functions. His approach was the precursor of ours, equivalent to setting the formal derivative equal to zero, but his procedures were completely algebraic and based on a clever use of arithmetic progressions. Hudde also presented an early version of the Quotient Rule.

Hudde accomplished all of this in a letter written to Frans van Schooten in 1658, the second of two letters from Hudde that would appear in van Schooten's Latin edtion of Descartes' *Geometria* in 1659. Download the author's translation of Hudde's Second Letter.

The title page of van Schooten's 1659 second edition of Descartes' Geometria is shown above. (Source: Convergence's "Mathematical Treasures") |

Jan Hudde (1628–1704) was born in Amsterdam and lived there his whole life, except for a year or two studying law in Leiden as a young man around 1648. There he also learned mathematics from Frans van Schooten, who emphasized the work of René Descartes. They continued to correspond on mathematical matters for another twenty years.

Hudde produced several mathematical works between 1654 and 1663, but largely abandoned the subject once he embarked on a career of political service in Amsterdam. He served several terms as one of the four Burgomasters who ran the city and held a number of other offices. We know that he used his mathematical abilities while in office, reducing Danish shipping charges by showing errors in their calculation, and championing Jan de Witt's new approach to life annuities, which employed careful mathematical analyses.

Hudde wrote two intriguing letters to van Schooten (the second is translated here), both of which van Schooten included in the second edition of his Latin translation of Descartes’ *Géometrie* in 1659. Van Schooten had seen the need for a translation from Descartes’ French into the standard language of scholars, producing his first Latin translation in 1649. In fact, van Schooten related in his preface to his 1659 edition that in order to include Hudde’s two letters, which were written in Dutch (Belgicè), he translated them into Latin as well [Descartes] (see Note 1).

Perhaps more importantly, van Schooten also recognized that a commentary expanding Descartes’ often cryptic explanations was desirable. Florimond de Beaune contributed to van Schooten’s commentary, which included several works extending ideas found in the *Géometrie.* Among these were Hudde’s two letters.

Frans van Schooten inscribed the copy shown above of his *Exercitationes mathematicae libri quinque *(*Five Books of Mathematical Exercises,* 1657) to his student, Jan (or Johann) Hudde. This copy resides in the Columbia University Library; this image is part of *Convergence*'s collection of "Mathematical Treasures." You may use the image in your classroom; for all other purposes, please seek permission from the Columbia University Library.

**Note 1.** I thank a referee for spotting this. See the third page of van Schooten’s *Praefatio* in [Descartes]. This is Screen 11 in the referenced 1695 photocopy, and image 15 in the referenced 1683 photo-image.

Building on the work of Fermat and Descartes [Curtin, pp. 263-264], Hudde considered algebraic expressions set equal to \(z\) and sought the maximum value for \(z.\) Strictly speaking his procedure finds a local maximum, but I will say “maximum” as he does. I will also use functional notation in this section, though Hudde did not.

In our notation Hudde wanted to find the maximum value \(z_{\rm{max}}\) of \(f(x) = z.\) This will occur when \(f(x) = z\) has a double root. For smaller values of \(z,\) \(f(x) = z\) will have two (or more) solutions (see Figure 1).

**Figure 1.** Near its maximum value, \(f(x) = z\) has two (or more) solutions; at the maximum value \(z_{\rm{max}},\) \(f(x) = z\) has a double root.

For a minimum, invert the picture. Hudde recognized that the procedure for finding maxima and minima was the same. His theorem statements refer to both, though his examples use only the word “maximum.”

For algebraic functions the problem of finding a double root is purely algebraic. No limit or infinitesimal quantity is needed.

Hudde’s letter begins with a theorem telling how to do this [Hudde, p. 507]. For a polynomial

\[p(x) = c_0 + c_1x + c_2x^2 + \cdots + c_nx^n\]

and any arithmetic progression \(a, a+b, a+2b, a+3b,\dots,\) combine the two to get

\[{\hat p}(x) = ac_0 + (a+b)c_1x + (a+2b)c_2x^2 + \cdots + (a+nb)c_nx^n.\quad (1)\] In our notation, we can see

\[{\hat p}(x) = ap(x)+bxp^{\prime}(x).\quad\quad\quad\quad\quad\quad (2)\]

Hudde claimed that if \(p(x) = 0\) has a double root at \(x=x_0,\) then \({\hat p}(x) = 0\) has a single root there. The converse, which is what Hudde really needed, is also true, though Hudde left this implicit.

To prove this, we might proceed as follows. Assume \(p(x) = 0\) has a double root \(x=x_0.\) Thus \(p(x) = {{(x-{x_0})}^2}{q(x)},\) where \(q(x)\) is a polynomial that is not zero at \(x_0.\) From (2),

\[{\hat p}(x) = a{{(x-{x_0})}^2}q(x) + bx[2(x-{x_0})q(x) + {{(x-{x_0})}^2}{q^{\prime}(x)}] \]

\[=(x-{x_0})[ a{(x-{x_0})}q(x) + 2bxq(x) + {(x-{x_0})}{q^{\prime}(x)}].\]

Since \(q(x_0)\not=0,\) the quantity in the square brackets will not be zero, unless \(x_0 = 0,\) in which case identifying a double root would be quite easy.

How did Hudde argue? He looked at the simplest example with a double root, namely \(x=y\):

\[p(x)=x^2-2xy+y^2={(x-y)}^2=0.\]

Apply the progression to obtain

\[{\hat p}(x) = (a+2b)x^2-2(a+b)xy+ay^2=0.\]

Hudde considered it obvious that \(y\) is a single root. We might elaborate

\[0=(a+2b)x^2-2(a+b)xy+ay^2=a{(x-y})^2+2b(x^2-xy)\]

\[=(x-y)[a(x-y)+2bx],\]

and this expression has \(y\) as a single root (again excepting the easy case \(y=0\)).

For the general case, Hudde gave an example that easily generalizes. He considered the following equation, which has \(y\) as a double root:

\[{(x-y)}^2(x^3+px^2+qx+r)=0.\]

Expanding the quadratic,

\[(x^2-2xy+y^2)x^3+p(x^2-2xy+y^2)x^2+q(x^2-2xy+y^2)x+r(x^2-2xy+y^2)=0.\]

If the left-hand side is expanded into powers of \(x\) and each term \(x^k\) multiplied by the corresponding term \(a+kb\) of an arithmetic progression, \(k=0,1,2,3,4,5,\) then each of the occurrences of \( x^2-2xy+y^2\) is multiplied by a different progression. For example,

\[(x^2-2xy+y^2)x^3=x^5-2x^4y+x^3y^2\]

becomes

\[[(a+5b)x^2-(a+4b)2xy+(a+3b)y^2]x^3.\]

Thus \(y\) is a single root of this piece, by the argument for \(x^2-2xy+y^2=0\) and any arithmetic progression. Since this works for each piece, it follows that \(y\) is a single root of the entire equation.

Clearly, if the cubic factor is replaced by any polynomial or rational expression (that isn’t \(0\) at \(y\)), the same argument applies. Hudde also observed that if the expression starts with a triple root, then the transformed expression has a double root, etc.

In this section Hudde’s general approach and his first two examples will be explained carefully.

To find the \(x\)-value of the maximum for \(f(x) = z,\) Hudde first removed all constant terms, including the maximum value \(z,\) since these do not affect the \(x\)-value at which the maximum occurs. He then transformed the rest of the expression with an arithmetic progression. The root of the transformed equation is the value at which the maximum occurs.

Hudde gave an example

\[3ax^3-bx^3-\frac{2ab^2}{3c}x+a^2b=\,{\rm{some\,\,maximum}}.\]

In this example he used the progression \(0, 1, 2, 3,\dots.\) The result is

\[9ax^3-3bx^3-\frac{2ab^2}{3c}x=0.\]

Dividing by \(x,\) this yields

\[9ax^2-3bx^2-\frac{2ab^2}{3c}=0,\]

which he gave as the answer. He assumed the reader knows how to solve this. He gave neither the value of \(x\) nor of \(z\) (nor did he verify the value of \(z\) is, in fact, a maximum).

There is no evidence Hudde tried any specific values in his examples, but it is likely he usually assumed \(a, b, c\) and \(x\) are all positive. If we further assume to be positive the coefficient, \(3a-b,\) of \(x^3\) in the original equation, thus avoiding complex numbers, then we would have

\[x=\frac{b\sqrt{2a}}{3\sqrt{c(3a-b)}}.\]

Using our modern techniques we can see this corresponds to a local minimum. In fact, the local maximum occurs at the negative of this value. We know Hudde was aware of negative solutions, calling them “false solutions,” as did Descartes.

Most of the rest of Hudde’s examples have \(x\) in the denominator. For a rational function \(\frac{f(x)}{g(x)},\) we would use the quotient rule to find the derivative and set the result equal to \(0.\) Usually this means the numerator is \(0,\) i.e.,

\[f'(x)g(x)-f(x)g'(x)=0.\]

Hudde’s approach with arithmetic progressions leads to an expression more like

\[[x\,f'(x)]g(x)-f(x)[xg'(x)]=0.\]

However, the factor of \(x\) is again easily removed.

Hudde’s first quotient example is

\[\frac{4a^2b^3+5a^3x+x^5}{x^3}-ax+bx+ab=z.\]

He removed \(ab\) and \(z\) since they do not affect the \(x\)-value for the maximum. He then put the rest over a common denominator:

\[\frac{4a^2b^3+5a^3x+x^5-ax^4+bx^4}{x^3}.\]

Although Hudde didn’t use the notation of negative exponents, we could think of this as

\[4a^2b^3x^{-3}+5a^3x^{-2}+x^2-ax+bx,\]

which would help us understand Hudde’s next step, applying the progression \(-3, -2, -1, 0, 1, 2,\dots\) to the numerator to get

\[-12a^2b^3-10a^3x+2x^5-ax^4+bx^4.\]

Set this equal to \(0\) and solve to find the solution. Hudde left this to the reader!

Hudde’s next examples expanded on this. I refer the interested reader to [Curtin].

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

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.

Information about Hudde’s life and a detailed explanation of the contents of the second letter are in [Curtin], which uses the examples that appear in the original. Jeff Suzuki’s nice article [Suzuki] also covers these ideas and puts them in a broader context. I know of only one translation of the second letter, which is into Dutch [Grootendorst]; the notes are useful even to those of us whose Dutch is minimal indeed.

[Curtin] D. J. Curtin, “Jan Hudde and the Quotient Rule before Newton and Leibniz,” *College Mathematics Journal,* 36 (4) (2005), 262-272.

[Descartes] René Descartes, *Geometria,* with notes by Florimond de Beaune and Frans van Schooten, Fridericus Knoch, Frankfurt am Main, 1695. A complete photocopy is available at *Gallica* (Bibliothèque Nationale de France): http://gallica.bnf.fr/ark:/12148/bpt6k57484n. For Hudde’s Second Letter, see pages 507–516 (Screens 523–532). Photo images of a 1683 edition of the text, published in Amsterdam, are available at* e-rara* (ETH-Bibliothek, Zürich): http://dx.doi.org/10.3931/e-rara-24189. Images (pages) 507–516 contain Hudde’s Second Letter.

[Grootendorst] A. W. Grootendorst, “Johan Hudde’s ‘Epistola secunda de maximis et minimis’.” Text, translation, commentary (Dutch), *Nieuw Archief voor Wiskunde,* vol. 5 (1987), series 4, 303-334.

[Schooten] Frans van Schooten, *Exercitationum Matematicorum,* Elsevier, Batavia, 1657.

[Suzuki] Jeff Suzuki, “The Lost Calculus (1637-1670): Tangency and Optimization Without Limits,” *Mathematics Magazine,* vol. 78, (2005), 339-353.

The author wishes to express his gratitude to the reviewer who greatly improved the translation, correcting several errors and suggesting many improvements. Any remaining errors are, however, the author’s own!