Descartes Method of Solving the Pappus Locus Problem

Descartes Method of Solving the Pappus Locus Problem

Avery Cooper

Rene Descartes a French mathematician who lived from 1596-1650. Not only was he a mathematician, but also a philosopher and scientist. He said his three approaches to mathematics were philosophy, being a student of nature, and being a man concerned with the uses of science [5].He is famous for many reasons, but, I’ll be discussing his work in the development of analytic geometry, and his solution to the Pappus Locus Theorem problem, which was discussed in one of his greatest works La Geometriefrom 1637 [1].

In the late 1500s and early 1600s, many mathematicians found the need to develop more rigorous proofs for the many geometrical problems addressed by the ancients. One of most motivating reasons for this was the translation by Commandino in 1588 of Pappus’s Collection from the 4th century CEinto Latin [2]. This work contained many of the ancient procedures for constructing curves as well as some geometrical proofs and methods of analysis or synthesis in geometrical problem-solving [2]. One of the more notable problems addressed in Collection was in the 7th book, where Pappus discusses plane loci, or analysis of a set of point in comparison to a set of lines [6]. In particular, Pappus was able to address the four-line problem, which is as follows:

Given there are four fixed lines and four fixed angles. Pick an arbitrary point and let four variables lines be the distance between the arbitrary point and the given lines, such that the fixed angles exist between the given lines and the variable lines. There is a set of points such that difference of the product of two variable lines with the product of the other two variable lines is equal to a constant.Find this set of points, or locus, that satisfies the given constraints.

Pappus credits Apollonius with the creation of this problem, as Apollonius had addressed many problems associated with loci and conics in works such as Plane Loci. However, Apollonius only addressed specific cases where the plane loci resulted in a line or a circle [6].

Fermat and Descartes both were interested in the problem, and found that Apollonius’s findings were incomplete. Both agreed that the work of Pappus was not generalized enough. Not only this, but Fermat also disliked the restricted scope of the theorems, while Descartes found the wordiness and prolixity of the arguments needed some refining [6].In fact, in 1631 a Dutch mathematician named Golius urged Descartes to address the Pappus problem due to its lacking in generality [2].

Descartes knew he was capable of solving the problem, but in order to understand how he did it, one must understand what advances in mathematics had happened recently that allowed himsucceed. Descartes needed to understand many facets of Greek mathematics in order to understand Pappus’s problem. Understanding conic sections, planes, lines, and solids were very important to the problem. This is because the final form of the locus must be of the form

in order for the locus to be a conic section. But algebra was missing from Pappus’s time, and Pappus would not have understood it in these terms. Descartes says in La Geometrie that his use of algebra is what allowed him to solve the problem for a more general case, and goes on to say that anyone else who has come to the solution he has could not have been ignorant to algebra [2]. But, few had forged a connection with algebra and geometry at the time of Descartes. One of the few was Francois Viete (sometimes seen as Vieta).

Viete was a French mathematician who was born before Descartes, and lived from 1540 to 1603. Viete’s work with geometrical problem solving paved the way for mathematicians like Descartes because of his ability to combine algebra with geometric ideas. By the time Descartes was around, Viete had developed a way to show cubic equations had roots using geometric proofs [2]. Maybe even more influential, Viete’s methods used symbols to represent magnitudes as well as for angles and line segments, as shown in his work Isagoge from 1591 [2]. Labeling magnitudes proved to be one of the most important developments in analytic geometry because it allowed mathematicians to give a name to something that was unknown. Problems could then be easily put in their simplest, or conical form [6]. Viete was one of the first to recognize that the basic operations (subtraction, multiplication, division, and square roots) for magnitudes could all be expressed algebraically. This was a big step compared to the Greeks, since they had no concept of expressing these operations in any way other than geometrically, but Viete was able to solve geometric problems algebraically. Viete in fact gave a technique for problems involving loci, but Viete’s techniques only accounted for analysis and synthesis of determinate problems, meaning there was only one unknown [6].In order for Descartes to find a more general solution to the Pappus problem, he must solve the problem with indeterminate equations with two unknowns.

Descartes came to understand Viete’s ideas, but his method of problem solving was also key to his success in solving the Pappus problem. As noted earlier, Descartes was not only a mathematician, but a scientist and philosopher too, and in all areas of his life, he used his method of problem solving. It is a set of steps as follows:

1. Draw a figure of the problem
2. Identify clearly what you are trying to find
3. Give each quantity, whether it is known or unknown, a name
4. List all known relations between these quantities using the names you’ve given them
5. Apply various techniques to these relations until you have the unknowns in a set of equations that can be solved. [3]

Using this method, Descartes attempted to tackle the thousand-year-old problem of Pappus’s locus problem.

Descartes published his solutions in a three volume book called La Geometrie in 1637 [1]. The first volume, contained problems that can be constructed using only circles and straight lines, while the second volume was about the nature of curves, and finally the third volume, which contains solid and super-solid problems [4]. In the book, he starts where Viete left off, and continues on to discuss what he calls “geometrical calculus”, or analytic geometry [2].But most importantly, he addresses the Pappus Locus theorem and how to construct the curves they yield.

To solve the Pappus Locus Theorem, Descartes used his method as he would any other problem. So, first he drew a figure of the problem. Figure 1 (from [7]) is an example of a figure Descartes might have drawn.

Using the notation used in Figure 1, we can see that lines are the given lines, while the dotted lines are the variable lines. We are also given , the angles of the known and unknown lines’ intersection.Next, Descartes went to step two, where he clearly identified that he was trying to find the curve associated to the locus of all such points with these restrictions.The problem is to find the locus of all such points C, where

where J is a given constant. Hence his third step: name everything. He did this by working backwards, starting by naming C, an arbitrary point. He then named two unknown line segments: and .

Descartes noticed that all of the triangle ’s angles are easily found from , and the fact that s are given lines. Hence, he wrote

ie,

Here we see Descartes starting to name known values as ect.

Next, he sees that it is also true to say

He notes that triangle then will be

Using substitution, Descartes found that

He then goes on to say that because are fixed lines, he knows that and . Descartes continues to label everything. He had , , , and . Remember that are all known magnitudes.
Now that Descartes had everything labeled, he moved to step four: list all known relations between magnitudes.

Looking back at the figure, Descartes says that

Using substitution, he found that this means

Next, he saw that

Using substitution again, Descartes saw that

By looking at the diagram again, he saw

Which by substituting it into he got

Substituting yet again, he found that

Finally

So, Descartes had found the equations of the unknowns namely

Since each equation is in terms of x and y, you can take any pairwise product from and create a quadratic equation in terms of x and y. Therefore, the point C must lie on a conic section, since each equation is of that form [7].

This was groundbreaking at the time. Clearly x and y are both unknown at the beginning of the problem, but Descartes still names them, exercising step two of his method. Before Descartes, algebraic equations in two unknowns were considered to be indeterminate, so you could only use substitution with a known value to find any other value.

But Descartes wasn’t done. The Greeks had already figured out how to do the four-line problem, Descartes needed to take it a step further, finding loci for more than four lines. He ran into a problem here. Viete and Greek mathematicians thought of the product of two magnitudes as a square, and the product of three magnitudes was a cube, but what about the product of four or more magnitudes?To solve the problem of writing numbers like , Descartes thought of two line segments of length and write respectively. He then wrote them as

This allowed him to construct any number in this way. For example, could be constructed as the length such that

is true [3]. Descartes could now easily construct expression of degree five or higher, which he need in order to prove the Pappus locus theorem for more than four lines. Before Descartes, this could not be constructed for unknown quantities. Not only this, but the Greeks had used the Archimedean axiom “Quantities cannot be compared unless some multiple of one can exceed the other”, ie, a point cannot be added to a line, or an area to a solid [3]. In todays notation, this means they could not write. He solved this by following his method. First he drew a figure with two rectangles: one with side length he deemed x, the other side=1, while the other rectangle had two side lengths of x. He claimed that would be equal to the area of the sum of these two rectangles.

He went on to solve the problem with five, six, and then an arbitrary number of lines [3]. He went about this in his usual matter, labeled line segments, and worked out the equations using techniques like substitution and reduction. He found for a specific case of 5 lines, the locus curve was no longer a conic section. The construction was with four parallel known lines with another known line perpendicular. The curve was of degree three, and in fact is now called the cubical parabola of Descartes [3]. Descartes was concerned with the process of constructing a curve like this, because for him, a geometric problem is not solved until the curve is constructed. This cubic curve he deemed to be mechanical, rather than geometrical, because he needed to use many instruments (and theorems) in order to construct it. Here’s how he did it.

He drew two straight lines: GA and AK, then drew a line KN, which will be the fixed distance KL from a ruler, which starts at the line GL in the figure. The ruler can rotate from a fixed point G. The point L can move along the ruler, hence the segment KL moves up the line AB. As KL moves up, the ruler rotates about G. Note that the angle between KL and KN is fixed, as well as the segment lengths themselves. Then, at the point C, which is the point at which the ruler GL intersects KN, will be a point on the generated curve.

Descartes continued by labeling everything (letting y=CB, x=BA as unknowns, as well as other lengths labels as known magnitudes) and show that since CNK is a straight line, then the curve generated by the set of points at C will be a hyperbola [3]. He continued by saying if CNK was instead a parabola whose axis was the straight line KB, then the curve constructed was the curve he had found for the Pappus five-line problem [3]. He went on to discover that five or six lines yielded a cubic equation, while seven or eight lines yielded a quartic equation, nine or ten lines a quantic equation, and so forth [4]. Descartes had taken Pappus’s work with four linesfound a way to construct curves from the loci of many more lines. Although he was unable to generalize it for an n-line Pappus problem, (his proof was wrong), he did come up with a construction method that allowed him to construct any curve that can be expressed algebraically [3].In addition, he had made a geometric construction using algebra concerning indeterminate equations with two unknowns, which had had no connection to geometry until now [6].

Descartes’ work on the Pappus locus theorem is still one of the most influential in mathematics. He helped pave the way for other two variable indeterminate problems to be solved, as well as moving the notation and problem solving method forward. Fermat later looked at the Pappus problem after reading about it in La Geometrie. In his work Introductionhe said “Whenever two unknown quantities are found in final equality, there results a locus in place, and the endpoint of one of these describes a straight line or a curve. [6]” As Boyer says, this statement is “’one of the most significant statements in the history of mathematics’” [6]. This is because like Descartes, Fermat had used the technique of applying algebra to locus problems, and translated geometric conditions into the general language of algebra. Today, our understanding of geometric curves can be completely understood as sets of equations with two unknowns, and this is primarily thanks to the work of Descartes and his work on problems like the Pappus locus theorem.

WORKS CITED

[1] Boyer, Carl B.The History of the Calculus and Its Conceptual Development. New York: Dover, 1959. Print.

[2] Domski, Mary, "Descartes' Mathematics",The Stanford Encyclopedia of Philosophy(Winter 2015 Edition), Edward N. Zalta(ed.), URL = <

[3] Grabiner, Judith. “Descartes and Problem-solving”.Mathematics Magazine68.2 (1995): 83–97. Web.

[4]John W. Cooley, Descartes' Analytic Method and the Art of Geometric Imagineering in Negotiation and Mediation, 28 Val. U. L. Rev. 83 (1993). Available at:

[5] Kline, Morris.Mathematical Thought from Ancient to Modern times. New York: Oxford UP, 1972. Print.

[6] Mahoney, Michael S. "Fermat's Analytic Geometry: The Introduction To Plane and Solid Loci."The Mathematical Career of Pierre De Fermat (1601-1665). Princeton, NJ: Princeton UP, 1973. N. pag. Print.

[7] Sidoli, Nathan Camillo. "Descartes’ Treatment of the Pappus Locus Theorem." (n.d.): n. pag. Waseda University, 2007. Web.