Chapter 9

Multiply 8023 by 4638 using the method of al-Uqlidis?

Solve the following problems due to al-Khw?rizmi: a. \(x^{2}+(10-x)^{2}=58\) b. I have divided 10 into two parts, and have divided the first by the second, and the second by the first and the, sum of the quotients is \(21 / 6\). Find the parts.

Solve \(\frac{1}{2} x^{2}+5 x=28\) by multiplying first by 2 and then using al- Khw?rizmi's procedure. Similarly, solve \(2 x^{2}+\) \(10 x=48\) by first dividing by 2

Solve the following problems of Ab? K?mil: a. Suppose 10 is divided into two parts and the product of one part by itself equals the product of the other part by the square root of 10 . Find the parts. b. Suppose 10 is divided into two parts, each one of which is divided by the other, and the sum of the quotients equals the square root of 5 . Find the parts. (Ab? K?mil solves this in two ways, once directly for \(x\), and a second time by first setting \(y=\frac{10-x}{x}\).)

Solve the following problems of Ab? K?mil: a. \([x-(2 \sqrt{x}+10)]^{2}=8 x\) (First substitute \(x=y^{2}\).) b. \(\left(x+\sqrt{\frac{1}{2} x}\right)^{2}=4 x\) (Ab? K?mil does this three different ways; he first solves directly for \(x\), next substitutes \(x=\) \(y^{2}\), and finally substitutes \(x=2 y^{2}\).)

Complete the solution of Ab? K?mil's problem in three variables given in the text by now beginning with the assumption that \(z=1\).

Solve the following problem in three variables due to Ab? K?mil: \(x

Complete al-Samaw'al's procedure of dividing \(20 x^{2}+30 x\) by \(6 x^{2}+12\) to get the result stated in the text. Prove that the coefficients of the quotient satisfy the rule \(a_{n+2}=-2 a_{n}\) where \(a_{n}\) is the coefficient of \(\frac{1}{n}\)

Give a complete inductive proof of the result $$ \sum_{i=1}^{n} i^{3}=\left(\sum_{i=1}^{n} i\right)^{2} $$ and compare with al-Karaji's proof.

Use ibn al-Haytham's procedure to derive the formula for the sum of the fifth powers of the integers: $$ 1^{5}+2^{5}+\cdots+n^{5}=\frac{1}{6} n^{6}+\frac{1}{2} n^{5}+\frac{5}{12} n^{4}-\frac{1}{12} n^{2} $$

Show, using the formulas for sums of fourth powers and squares, that $$ \begin{aligned} \sum_{i=1}^{n-1}\left(n^{4}-2 n^{2} i^{2}+i^{4}\right) &=\frac{8}{15}(n-1) n^{4}+\frac{1}{30} n^{4}-\frac{1}{30} n \\ &=\frac{8}{15} n \cdot n^{4}-\frac{1}{2} n^{4}-\frac{1}{30} n \end{aligned} $$

Show that one can solve \(x^{3}+d=c x\) by intersecting the hyperbola \(y^{2}-x^{2}+\frac{d}{c} x=0\) with the parabola \(x^{2}=\sqrt{c} y\). Sketch the two conics. Find sets of values for \(c\) and \(d\) for which these conics do not intersect, intersect once, and intersect twice.

Show that \(x^{3}+c x=b x^{2}+d\) is the only one of al-Khayy?mi's cubics that could have three positive solutions. Under what conditions do these three positive solutions exist? How many positive solutions does the equation \(x^{3}+\) \(200 x=20 x^{2}+2000\) have? (The solution of this equation enabled al-Khayy?mi to solve his quadrant problem.)

Show that one can solve \(x^{3}+d=b x^{2}\) by intersecting the hyperbola \(x y=d\) and the parabola \(y^{2}+d x-d b=0 .\) Assuming that \(\sqrt[3]{d}

Show using calculus that \(x_{0}=\frac{2 b}{3}\) does maximize the function \(x^{2}(b-x)\). Then use calculus to analyze the graph of \(y=x^{3}-b x^{2}+d\) and confirm Sharaf al-Din's conclusion on the number of positive solutions to \(x^{3}+d=b x^{2}\).

Show, as did Sharaf al-Din al-T?si, that if \(x_{2}\) is the larger positive root to the cubic equation \(x^{3}+d=b x^{2}\), and if \(Y\) is the positive solution to the equation \(x^{2}+\left(b-x_{2}\right) x=\) \(x_{2}\left(b-x_{2}\right)\), then \(x_{1}=Y+b-x_{2}\) is the smaller positive root of the original cubic.

Analyze the possibilities of positive solutions to \(x^{3}+d=\) \(c x\) by first showing that the maximum of the function \(x(c-\) \(x^{2}\) ) occurs at \(x_{0}=\sqrt{\frac{c}{3}}\). Use calculus to consider the graph of \(y=x^{3}-c x+d\) and determine the conditions on the coefficients giving it zero, one, or two positive solutions.

Show that 17,296 and 18,416 are amicable by using ibn Qurra's theorem.

Show that 1184 and 1210 are amicable numbers that are not a consequence of the theorem of Th?bit ibn Qurra.

Find a pair of amicable numbers different from those in the text.

Ab? Sahl al-K?hì knew from his own work on centers of gravity and the work of his predecessors that the center of gravity divides the axis of certain plane and solid figures in the following ratios: Tetrahedron: \(\frac{1}{4}\) Segment of a parabola: \(\frac{2}{3} \quad\) Paraboloid of revolution: \(\frac{2}{6}\) Hemisphere: \(\frac{3}{8}\) Noting the pattern, he guessed that the corresponding value for a semicircle was \(3 / 7\). Show that al-K?hi's first five results are correct, but that his guess for the semicircle\\} implies that \(\pi=31 / 9\). (Al-K?h? realized that this value contradicted Archimedes' bounds of \(310 / 71\) and \(31 / 7\), but concluded that there was an error in the transmission of Archimedes' work.)

Use al-B?r?ni's procedure to determine the qibla for Rome (latitude \(41^{\circ} 53^{\prime} \mathrm{N}\), longitude \(12^{\circ} 30^{\prime} \mathrm{E}\) ).

Show that the radius \(r_{\alpha}\) of a latitude circle on the earth at \(\alpha^{\circ}\) is given by \(r_{\alpha}=R \cos \alpha\), where \(R\) is the radius of the -earth.

The latitudes of Philadelphia and Ankara, Turkey, are the same \(\left(40^{\circ}\right)\), with the first at longitude \(75^{\circ} \mathrm{W}\) and the second at longitude \(33^{\circ} \mathrm{E}\). Calculate the distance between Philadelphia and Ankara along the latitude circle, by first calculating the radius of that circle, using 25,000 miles for the circumference of the earth. Then calculate the distance along a great circle, by noting that the chord connecting the two cities can be thought of as a chord of that circle as well as a chord of the latitude circle. (Hint: You will have to convert the chords to the appropriate sines to make this calculation.)

Show directly, without the use of Ptolemy's theorem, that in an isosceles trapezoid, the square on a diagonal is equal to the sum of the product of the two parallel sides plus the square on one of the other sides.

Use al-B?r?ni's nontrigonometric procedure for calculating distances on the earth to find the great circle distance. between New York (latitude \(41^{\circ} \mathrm{N}\), longitude \(74^{\circ} \mathrm{W}\) ) and London (latitude \(52^{\circ} \mathrm{N}\), longitude \(0^{\circ}\) ). Assume that the circumference of the earth is 25,000 miles.

Al-B?r?ni devised a method for determining the radius \(r\) of the earth by sighting the horizon from the top of a mountain of known height \(h\). That is, al-B?r?ni assumed that one could measure \(\alpha\), the angle of depression from the horizontal at which one sights the apparent horizon (Fig. 9.38). Show that \(r\) is determined by the formula $$ r=\frac{h \cos \alpha}{1-\cos \alpha} $$ Al-B?r?n? performed this measurement in a particular case, determining that \(\alpha=0^{\circ} 34^{\prime}\) as measured from the summit of a mountain of height \(652 ; 3,18\) cubits. Calculate the radius of the earth in cubits. Assuming that a cubit equals \(18^{\prime \prime}\), convert your answer to miles and compare to a modern value. Comment on the efficacy of al-Bir?ni's procedure.

Use al-T?si's method to solve the spherical triangle with known sides of \(40^{\circ}\) and \(50^{\circ}\) and with the angle between those sides equal to \(25^{\circ}\).

Al-T?si demonstrated a method to solve a spherical triangle if all three angles are known. Suppose the three angles of triangle \(A B C\) are given (Fig. 9.39), where we assume that all three sides of the triangle are less than a quadrant. We extend each side of the triangle two different ways to form a quadrant. That is, we extend \(A B\) to \(A D\) and \(B H\); \(A C\). to \(A E\) and \(C G\); and \(B C\) to \(B K\) and \(F C\), where all of the six new arcs are quadrants. We then draw great circle arcs through \(D\) and \(E, F\) and \(G\), and \(H\) and \(K\) to form the new spherical triangle \(L M N\). Now the vertices of the original triangle are the poles of the three sides of the new triangle. Then, for example, \(M D=E N=90^{\circ}-D E=90^{\circ}-A\), or \(M N=180^{\circ}-A\). Thus, the three sides of triangle \(L M N\) are known, and therefore the triangle can be solved by the procedure sketched in the text. But we also know that the vertices of triangle \(L M N\) are the poles of the original triangle. So, for example, \(B F=C K=90^{\circ}-B C\), and \(L=F K=180^{\circ}-B C\). We therefore can determine the sides of the original triangle. Use this procedure to solve the triangle \(A B C\), where \(A=75^{\circ}, B=80^{\circ}\), and \(C=85^{\circ}\).

Why did it take many centuries after its introduction for the decimal place value system to become the system of numeration universally used in the Islamic world?