Chapter 14

Consider the cubic equation \(a a a-3 r a a=2 x x x .\) Show that if one sets \(a=e+r\), the resulting cubic equation in \(e\) does not have a square term. For example, show that the equation \(a a a-6 a a=400\) can be reduced to the equation eee \(-12 e=416\). Find a solution of the last equation for \(e\) and therefore find a solution for the equation in \(a\).

Solve \(x^{3}=300 x+432\) using Girard's technique, given that \(x=18\) is one solution.

Solve \(x^{3}=6 x^{2}-9 x+4\) using Girard's technique. First, determine one solution by inspection.

Assuming that \(x y=c\) represents a hyperbola with the \(x\) and \(y\) axes as asymptotes, show that \(x y+c=r x+x y\) also represents a hyperbola. Find its asymptotes.

Determine the locus of the equation \(b^{2}-2 x^{2}=2 x y+y^{2}\) (Hint \(:\) Add \(x^{2}\) to both sides.)

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.

To solve the fourth-degree equation \(x^{4}-p x^{2}-q x-\) \(r=0\), Descartes considered the cubic equation in \(y^{2}\). \(y^{6}-2 p y^{4}+\left(p^{2}+4 r\right) y^{2}-q^{2}=0 .\) If \(y\) is a solution, show that the original polynomial factors into two quadratics: \(r_{1}(x)=x^{2}-y x+\frac{1}{2} y^{2}-\frac{1}{2} p-\frac{q}{2 y}, r_{2}(x)=x^{2}+y x+\) \(\frac{1}{2} y^{2}-\frac{1}{2} p+\frac{q}{2 y}\), each of which can be solved. Apply this method to solve the equation \(x^{4}-17 x^{2}-20 x-6=0\) Note that the corresponding equation in \(y, y^{6}-34 y^{2}+\) \(313 y^{2}-400=0\), has the solution \(y^{2}=16\)

Solve the equation \(x^{3}-\sqrt{3} x^{2}+\frac{26}{27} x-\frac{8}{27 \sqrt{3}}=0\) by first substituting \(y=\sqrt{3} x\) and then \(z=3 y\) to get an equation in \(z\) with integral coefficients.

In de Witt's substitution \(z=y+\frac{b}{a} x+c\), which simplifies the equation $$ y^{2}+\frac{2 b x y}{a}+2 c y=b x-\frac{b^{2} x^{2}}{a^{2}}-c^{2} $$ he has rotated one of the axes through an angle \(\alpha\). Find the sine and cosine of \(\alpha\)

Show that de Witt's equation $$ y^{2}+\frac{2 b x y}{a}+2 c y=\frac{f x^{2}}{a}+e x+d $$ represents a hyperbola. (Use the substitution \(z=y+\frac{b}{a} x+\) \(c\) and show that this substitution, when combined with a substitution of the form \(x^{\prime}=\beta x\), converts the original oblique \(x-y\) coordinate system into a new \(x^{\prime}-z\) coordinate system based on perpendicular axes.) Sketch the curve.

Prove by induction on \(n\) that $$ \left(\begin{array}{l} n \\ k \end{array}\right)=\sum_{j=k-1}^{n-1}\left(\begin{array}{c} j \\ k-1 \end{array}\right) $$ for all \(k\) less than \(n\).

Prove that $$ \left(\begin{array}{l} n \\ k \end{array}\right):\left(\begin{array}{c} n \\ k+1 \end{array}\right)=(k+1):(n-k) $$

Prove that $$ \left(\begin{array}{l} n \\ k \end{array}\right):\left(\begin{array}{c} n-1 \\ k \end{array}\right)=n:(n-k) $$

Pascal stated that the odds in favor of throwing a six in four throws of a single die are 671 to 625 . Show why this is true.

Show that the odds against at least one 1 appearing in a throw of three dice is \(125: 91\). (This answer was stated by Cardano.)

Suppose three players play a fair series of games under the condition that the first player to win three games wins the stakes. If they stop play when the first player needs one game while the second and third players each need two games, find the fair division of the stakes. (This problem was discussed in the correspondence of Pascal with Fermat.)

For a roll of three dice, show that both a 9 and a 10 can be achieved in six different ways. Nevertheless, show that the probability of rolling a 10 is higher than that of rolling a \(9 .\) (A discussion of this idea is found in a fragment of a work of Galileo.)

If two players play a game with two dice with the condition that the first player wins if the sum thrown is 7 , the second wins if the sum is 6 , and the stakes are split if there is any other sum, find the expectation (value of the chance) of each player.

There are 12 balls in an urn, 4 of which are white and 8 black. Three blindfolded players, \(A, B, C\) draw a ball in. turn, first \(A\), then \(B\), then \(C\). The winner is the one who first draws a white ball. Assuming that each (black) ball is replaced after being drawn, find the ratio of the chances of the three players.

There are 40 cards, 10 from each suit. \(A\) wagers \(B\) that he will draw four cards and get one of each suit. What are the fair amounts of the wagers of each?

Prove that if \(p\) is prime, then \(2^{p} \equiv 2(\bmod p)\) by writing \(2^{p}=(1+1)^{p}\), expanding by the binomial theorem, and noting that all of the binomial coefficients \(\left(\begin{array}{l}P \\ k\end{array}\right)\) for \(1 \leq k \leq\) \(p-1\) are divisible by \(p\). Prove \(a^{p} \equiv a(\bmod p)\) by induction

For a proof of the Fermat Little Theorem in the case where \(a\) and \(p\) are relatively prime, consider the remainders of the numbers \(1, a, a^{2}, \ldots\) on division by \(p .\) These remainders must ultimately repeat (why?), and so \(a^{n+r} \equiv a^{r}(\bmod p)\) or \(a^{r}\left(a^{n}-1\right) \equiv 0(\bmod p)\) or \(a^{n} \equiv 1(\bmod p) .\) (Justify each of these alternatives.) Take \(n\) as the smallest positive integer satisfying the last congruence. By applying the division algorithm, show that \(n\) divides \(p-1\).

Construct a tangent to a point \(P\) on a conic section by using Pascal's hexagon theorem. Consider the tangent line as passing through two neighboring points at \(P\). Then pick four other points on the conic and apply the theorem.

The best-known quotation from Descartes is "I think, therefore I am," from the Discourse on Method. The context is Descartes' resolve only to accept those ideas that are selfevidently true. There is a well-known joke based on this quote: Descartes goes into a restaurant. The waiter asks him, "Would you like tonight's special?" He replies, "I think not" and disappears. Comment on the logical validity of this joke. \(^{44}\)

Outline a lesson on the principle of mathematical induction using material from Pascal's Treatise on the Arithmetical Triangle.

Compare Pascal's use of mathematical induction to the use of it by ibn al- Haytham, al-Samaw'al, and Levi ben Gerson.