# The Rule of False Position and Geometric Problems

## Introduction

Historical examples in college mathematics courses can help students understand the process of creation in mathematics and how mathematicians have grappled with problems over the ages. This can enliven mathematics and humanize it [1].  Swetz proposes to have students solve some of the problems that were of interest for early mathematicians as a direct approach to enrich mathematics teaching and learning through history [7]. Historical remarks can help students understand the material better and help them see how it fits into the wider domain of mathematics [5]. Historical examples of problems can also provide an interesting background to tell the history of mathematical ideas [4]. Future teachers can benefit especially from the historical perspective. They will learn solution methods alternative to the ones usually taught in schools. Let us look at a problem Simon Stevin published in 1583 [6].

Problem:  Construct a square knowing the difference PQ between its diagonal and its side.

Solution: Let BCDE be any square. On the diagonal EC take the point F so that EF = ED.

If FC = PQ, then the square BCDE is the solution to the problem. If not, the side of the solution square (say y) will be the fourth proportional with respect to the segments FC, PQ and ED. That is, FC:PQ = ED:y.

The method used by Stevin is called the rule of false position. The solution is especially interesting because he used this method in a geometrical problem; the method was mainly used in problems of algebraic nature. It is also interesting because our students do not use geometrical proportionality when they face problems about constructing figures that have some restrictions. To solve the problem of the square and its diagonal, students in high school (ages 16-18) frequently use methods involving trigonometry and algebra to compute the side of the square. For them the real challenge is to construct the figure.

## Simon Stevin (1548-1620)

Stevin was born in Bruges, Flanders (now Belgium) in 1548 and died at The Hague (Netherlands) in 1620. He introduced the systematic use of decimal numbers into European mathematics, and proposed the unification of the system of weights and measures with a method based on the decimal subdivision of the unit. He also published one of the first tables of interest with many practical examples and with rules for simple and compound interest. In the field of physics he made important contribution in Statics and Hydrostatics. As engineer he made important contributions to civil and military engineering.

## Construction of the Fourth Proportional

Let a, b, and c be three line segments.

On ray Or trace segments OA = a  and  AC = c. On ray Os trace segment OB = b. From point C trace CX parallel to AB. We can verify that x is the fourth proportional with respect to the segments a, b, and c. That is, a:b = c:x.

Indeed, triangles OAB and OCX  are similar. Therefore a:b = (a + c):(b + x).  So a:b = [(a + c) - a]:[(b + x) - b] = c:x, as desired.

## The Rule of False Position

The rule of false position, regula falsi, rule of one false position, or simple rule of false position was very popular in mathematics texts of the sixteenth century, and was still present in some books of elementary mathematics in the first half of the twentieth century [8]. Lumpkin [2] traces the development of the method of false position from its uses in ancient Egypt, the Hellenistic world, medieval Islam, to its transmission to Europe and subsequent appearance in mathematical texts in the eighteenth century, to Benjamin Banneker’s advancement of the rule of false position through his own efforts.

In general, regula falsi was used to solve equations of first degree with one unknown, without using algebraic symbolic notation. The statements of problems that were solved using the rule of false position can be translated to an equation of the type ax = b  or more precisely, a1x  + a2x  + . . . + anx = b.

We describe the simple rule of false position with modern algebraic notation:

Suppose we want to solve the equation ax = b  (1).

If we make x = c, then ac = b (2).

There are two possibilities:

• If bequals b, then x = c is the solution of the equation.
• If b1 does not equal b, dividing equation (1) by equation (2) gives x/c = b/b1, so x = bc/b1

Regula falsi can be justified geometrically using the following figure:

Using similar triangles we have c/b1 = x/b; therefore x = bc/b1.

## Regula Falsi and Geometric Problems

The mathematician, physicist, engineer, and inventor Simon Stevin used the rule of one false position in the solution of additional problems of geometric nature. He included his geometric investigations in the Problematum geometricorum (1583), which he structured in five books.

In the second book (De continuae quantitatis regula falsi) Stevin used the rule of one false position in the solution of four problems, doing some geometrical constructions with the help of similar figures. In the solution of each of the problems there are five phases: (1) Display of the data, (2) Display of the problem, (3) Construction, (4) Demonstration, (5) Conclusion. In the third phase the simple rule of false position is used.  He generally began by constructing a figure similar to the desired one.  If this figure satisfied all the required conditions, then the problem is solved.  Otherwise, Stevin made use of proportionality theory to construct a new figure that did satisfy all the conditions.

We will present problems 1, 3, and 4 (problem 2 was presented at the beginning of this article).  Given the expository nature of this article, we adapted the problems from the original text to modern language. We have tried to adhere to the spirit of the author.

## First Problem

Problem:  Build an equilateral triangle knowing the length of a line segment PQ equal to the side of the triangle less its height plus a third of its height.

Construction: Let BCD be any equilateral triangle. Draw the height BE and segment FG that joins the center of the triangle with the midpoint of the side BC. On the side CD take the point H so that CH = BE. On the extension of CD take the point I so that DI = FG.

If triangle BCD is the solution to the problem, then HI will be equal to the line segment PQ. If not, the side of the equilateral triangle solution (say x) will be the fourth proportional with respect to the segments HI, PQ and BC. In other words,

HI:PQ = BC:x.

## Third Problem

Problem:  Build a regular pentagon knowing one segment PQ whose endpoints are one of the vertices of the pentagon and the midpoint of the opposite side.

Construction: Let BCDEF be any regular pentagon. Draw the line segment BG that joins the vertex B with the midpoint of the side ED.

If  If BG = PQ, then the regular pentagon BCDEF is the solution to the problem. If not, the side of the pentagon solution (say z) will be the fourth proportional of the segments BG, PQ and ED. That is,

BG:PQ = ED:z.

## Fourth Problem

Problem:  Let RSTUV be a given polygon and PQ a given line segment. Construct a polygon MNKIL similar to the previous and equally arranged so that if segment MN, homologous to RS, is taken from LN, homologous to VS, and to the rest you add segment LI, homologous to VU, you obtain a segment equal to PQ.

Construction:  Let BCDEF be any polygon similar to RSTUV and equally arranged. On segment CF take point G so that CG = CB. Extend segment CF to point H so that FH = FE.

If HG = PQ, then the polygon BCDEF is the solution to the problem. If not, we will determine the fourth proportional (say w) with respect to the segments HG, PQ and ED. Then, w = IK will be the segment homologous to ED in the solution polygon. From it the required polygon will be constructed.

## Conclusions

When students solve problems similar to the ones presented, the following contents of conceptual, procedural, or attitudinal nature can be introduced, firmed, or developed.

Conceptual contents

• Proportional magnitudes.
• Congruence and similarity of figures. Similarity ratio.
• Fourth proportional with respect to three line segments.

Procedural contents

• Identify the measure characteristics of similarity of plane figures in concrete situations.
• Apply similarity of figures to the solution of construction problems.
• Construction of the fourth proportional with respect to three line segments.
• Identification of similarity in plane figures.
• Search for relations in geometric figures and configurations.
• Construction of plane figures.

Attitudinal contents

• Value the usefulness of proportionality for the solution of problems.
• Skillful utilization of some drawing instruments (ruler, construction triangle, compass, computer programs…).
• Recognize and value the usefulness of regula falsi for the solution of geometrical problems.
• Flexibility to solve geometric problems with different approaches.

In addition to the above benefits, the use of regula falsi can encourage teachers and students to experiment in mathematics. Experimenting in the context of similarity problems can lead to a better understanding. Discovery in mathematics is frequently preceded by multiple experiments.

## References

Print references:

1. Avital, S. (1995). History of mathematics can help improve instruction and learning. In F. J. Swetz, J. Fauvel, O. Bekken, B. Johansson & V. Katz (Eds.), Learn from the masters! (pp. 3-12). Washington, DC: Mathematical Association of America.

2. Lumpkin, B. (1996). From Egypt to Benjamin Bannaker: African origins of false position solutions. In R. Calinger (Ed.), Vita mathematica (pp. 279-289). Washington, DC: Mathematical Association of America.

3. Meavilla, V. (2000). Historia de las Matemáticas: métodos no algebraicos para la resolución de problemas. SUMA. Revista sobre la enseñanza y el aprendizaje de las Matemáticas, nº 34, pp. 81-85.

4. Meavilla, V (2001). Aspectos históricos de las matemáticas elementales. Zaragoza, Spain: Prensas Universitarias de Zaragoza.

5. Rickey, V. F. (1996). The necessity of history in teaching mathematics. In R. Calinger (Ed.), Vita mathematica (pp. 251-256). Washington, DC: Mathematical Association of America.

6. Stevin, S. (1583). Problematum geometricorum. Antverpiae, Apud Ioannem Bellerum.

7. Swetz, F. J. (1995). Using problems from the history of mathematics in classroom instruction. In F. J. Swetz, J. Fauvel, O. Bekken, B. Johansson & V. Katz (Eds.), Learn from the masters! (pp. 25-38). Washington, DC: Mathematical Association of America.

8. Winicki, G. (2000). The analysis of regula falsi as an instance for professional development of elementary school teachers. In V. Katz (Ed.), Using history to teach mathematics: An international perspective (pp. 129-133). Washington, DC: Mathematical Association of America.

Online references:

1. Biographies of Simon Stevin

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Stevin.html

2. Works of Simon Stevin

http://www.library.tudelft.nl/ws/a/resources_guide/treacutesor/digital_works/principal_works_stevin/index.htm