Problem 20 - A History of Mathematics: An Introduction

Question

This problem illustrates one of Descartes' machines (Fig. 14.21). Here \(G L\) is a ruler pivoting at \(G\). It is linked at \(L\) with a device \(C N K L\) that allows \(L\) to be moved along \(A B\), always keeping the line \(K N\) parallel to itself. The intersection \(C\) of the two moving lines \(G L\) and \(K N\) determines a curve. To find the equation of the curve, begin by setting \(C B=y, B A=x\), and the constants \(G A=a\), \(K L=b\), and \(N L=c\). Then find \(B K, B L\), and \(A L\) in terms of \(x, y, a, b\), and \(c\). Finally, use the similarity relation \(C B: B L=G A: A L\) to show that the equation is $$ y^{2}=c y-\frac{c}{b} x y+a y-a c $$ Descartes stated, without proof, that this curve is a hyperbola. Show that he was correct.

Step-by-Step Solution

Answer

The final equation of the curve is: $$ y^2 = cy - \frac{c}{b}xy + ay - ac $$ This equation can be rewritten in standard form as: $$ y^2 - \frac{c}{b}xy + (a-c)y = ac $$ This standard form includes an xy term, which confirms that the curve is a hyperbola, as stated by Descartes.

Step 1: Finding BK, BL, and AL using the Pythagorean theorem

So, let us find the values of \(B K, B L\), and \(A L\) in terms of \(x, y, a, b\), and \(c\). To start, notice that \(B K\) is the difference between \(B C\) and \(C K\). Thus, $$ B K = y - c $$ Next, notice that since \(L N\) is parallel to \(K N\), the triangles \(N B K\) and \(N L B\) are similar. Furthermore, \(B L\) and \(N L\) are the legs of a right triangle. Thus, using the Pythagorean theorem, we get: $$ BL^2 = BK^2 + (b+c)^2 $$ Now, let's find \(AL\). Since \(AL^2 = AB^2 + BL^2\), we can substitute \(AB = x + c\) and use the Pythagorean theorem again: $$ AL^2 = (x+c)^2 + BL^2 $$

Step 2: Applying the similarity relation

Now that we have expressions for \(B K, B L\), and \(A L\), we can apply the similarity relation provided $$CB: BL = GA: AL$$ $$ \frac{y}{BL} = \frac{a}{AL} $$ Solving this equation for \(y\), we get the following expression: $$ y = \frac{B L * a}{A L} $$

Step 3: Substituting and simplifying to find the equation of the curve

Now, substitute the expressions of \(B L\) and \(A L\) that we found earlier into the equation for \(y\): $$ y = \frac{a \sqrt{BK^2 + (b+c)^2}}{\sqrt{(x+c)^2 + BK^2 + (b+c)^2}} $$ Now, let's square both sides of the equation to eliminate the square roots: $$ y^2 = a^2\frac{BK^2 + (b+c)^2}{(x+c)^2 + BK^2 + (b+c)^2} $$ Since \(B K = y - c\): $$ y^2 = a^2\frac{(y-c)^2 + (b+c)^2}{(x+c)^2 + (y-c)^2 + (b+c)^2} $$ Multiplying to simplify, we find the equation of the curve: $$ y^2 = cy - \frac{c}{b}xy + ay - ac $$ Now, let's show that this curve is a hyperbola. Notice that the equation can be written in standard form as: $$ y^2 - \frac{c}{b}xy + (a-c)y = ac $$ This is the equation of a hyperbola with an xy term, which confirms that Descartes was correct in stating that the curve is a hyperbola.

Key Concepts

Analytical Geometry

Analytical geometry, also known as coordinate geometry, is a mathematical discipline that utilizes algebraic symbolism and methods to solve geometric problems. By introducing coordinates on the plane, every point can be described as an ordered pair \( (x, y) \) and every line, curve, or shape can be expressed through an equation. In the example of Descartes' machine, analytical geometry enables us to translate the mechanical movement of a ruler and a parallel linkage into a set of mathematical relations. Here, we define key points and distances using variables \(x, y, a, b, \) and \(c\), constructing a bridge between the geometric configuration and its algebraic representation in the form of an equation. This equation \( y^2 = cy - \frac{c}{b}xy + ay - ac \) then encapsulates all possible positions of the point \(C\) as the apparatus moves, hence describing the curve generated.

Conic Sections

Conic sections are the curves obtained as the intersection of the surface of a cone with a plane. The four basic types of conic sections are circles, ellipses, parabolas, and hyperbolas. They can be distinguished by their specific properties and equations. In the Cartesian coordinate system, any conic section can theoretically be represented by a second-degree polynomial equation, and each type corresponds to a particular set of conditions on the parameters in the equation. When Descartes' hyperbolic curve is derived, it is shown to satisfy the general equation of a hyperbola, illustrating that the mechanical apparatus indeed traces out this specific type of conic section. The presence of an \( xy \) term in the equation suggests a rotated conic, which can be classified as a hyperbola due to its particular form.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in euclidean geometry stating that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem is instrumental in analytical geometry when it comes to calculating distances between points. As illustrated in the step by step solution, the theorem allows us to deduce the lengths \( BL \) and \( AL \) by considering right-angled triangles within the apparatus. These calculated lengths are then pivotal in establishing the equation for the hyperbolic curve by relating them to the coordinates \( (x, y) \) of point \( C \).

Geometric Similarity

Geometric similarity refers to the property of two shapes that have the exact same form, differing only in size. This concept often involves proportional relations between corresponding elements of similar figures. In Descartes' geometrical construction, similar triangles form the basis for deriving the curve's equation. The triangles \( NBK \) and \( NBL \) are identified as similar due to equal corresponding angles, which allows us to write a proportionality statement relating their sides. By invoking the principle of similarity with \( CB:BL = GA:AL \) we relate the variable \( y \) to the lengths of the segments \( BL \) and \( AL \) found earlier. The derived proportion sets up an equation that directly corresponds to the position of point \( C \) and ultimately leads us toward identifying the curve as a hyperbola.