Chapter 13

Kepler gave the following construction for a hyperbola with foci at \(A\) and \(B\) and with one vertex at \(C\) : Let pins be placed at \(A\) and \(B\). To \(A\) let a thread with length \(A C\) be tied and to \(B\) a thread with length \(B C\). Let each thread be lengthened by an amount equal to itself. Then grasp the two threads together with one hand (starting at \(C\) ) and little by little move away from \(C\), paying out the two threads. With the other hand, draw the path of the join of the two threads at the fingers. Show that the path is a hyperbola.

This problem provides details on constructing a Mercator chart to represent the region between the equator and \(30^{\circ} \mathrm{N}\)latitude and between \(75^{\circ}\) and \(85^{\circ} \mathrm{W}\) longitude. Draw a line \(10 \mathrm{~cm}\) long to represent the equator between those meridians. Divide it into intervals of \(1 \mathrm{~cm}\) and draw the meridians perpendicular to the line of the equator. Then \(1^{\circ}\) of longitude is taken as \(1 \mathrm{~cm}\). To find the distance on the map to the \(10^{\circ}\) parallel, note first that since \(1 \mathrm{~cm}\) corresponds to \(1^{\circ}\) on a great circle, the radius of the corresponding sphere must be \(\frac{180}{\pi}\). One must therefore multiply this value by \(D\left(10^{\circ}\right)\), computed by Equation 13.1. Similarly, to calculate the distance to the parallel at \(20^{\circ}\) from that at \(10^{\circ}\), find \(D\left(20^{\circ}\right)-D\left(10^{\circ}\right)\) and multiply by the radius. Also, calculate the distance of the \(30^{\circ}\) parallel from the equator. To make the chart somewhat more precise, determine the distances of the parallels at \(5^{\circ}, 15^{\circ}\), and \(25^{\circ}\).

In triangle \(A B C\), suppose the ratio \(\angle A: \angle B=10: 7\) and the ratio \(\angle B: \angle C=7: 3\). Find the three angles and the ratio of the sides. (This problem and the next two are also from \(O n\) Triangles.)

In triangle \(A B C\) with \(A D\) perpendicular to \(B C\), suppose \(A B-A C=3, B D-D C=12\), and \(A D=30\). Find the three sides.

Show that if the sum of two arcs is known and the ratio of their sines is known, then each arc may be found. In particular, suppose the sum of the two arcs is \(40^{\circ}\) and ratio of the sine of the larger part to that of the smaller is \(7: 4\). Determine the two arcs. (Although Regiomontanus only used sines, it is probably easier to do this using cosines and tangents as well.)

Show that Regiomontanus's versine formula is equivalent to the spherical law of cosines: \(\cos a=\cos b \cos c+\) \(\sin b \sin c \cos A\)

The following problem is from Pitiscus's Trigonometry: Find the area of the field \(A B C D E\) given the following measurements: \(A B=7, B C=9, A C=13, C D=10, C E=\) 11, \(D E=4\), and \(A E=17\). Begin by drawing \(B F \perp A C\), \(C G \perp A E\), and \(D H \perp C E\) (Fig. 13.35).

This problem is from Copernicus's De revolutionibus. Given the three sides of an isosceles triangle, to find] the angles. Circumscribe a circle around the triangle and draw another circle with center \(A\) and radius \(A D=\frac{1}{2} A B\) (Fig. 13.36). Then show that each of the equal sides is to the base as the radius is to the chord subtending the vertex angle. All three angles are then determined. Perform the calculations with \(A B=A C=10\) and \(B C=6\).

Prove that $$ \sin \alpha \sin \beta=\frac{1}{2}[\cos (\alpha-\beta)-\cos (\alpha+\beta)] $$

Given that the period of the earth is 1 year, and given that Mars's mean distance from the sun is \(1.524\) times that of the earth's mean distance, use Kepler's third law to determine the period of Mars.

According to Kepler's second law, at what point in the planet's orbit will the planet be moving the fastest?

Use the law of tangents to solve a triangle with sides 10 and 13 and included angle \(35^{\circ}\).

Prove: The times of motion of a moveable starting from rest over equal planes unequally inclined are to each other inversely as the square root of the ratio of the heights of the. planes.

Show that a projectile fired at an angle \(\alpha\) from the horizontal follows a parabolic path.

Galileo states that if a projectile fired at an angle \(\alpha\) from the horizontal at a given initial speed reaches a distance of 20,000 if \(\alpha=45^{\circ}\), then with the same initial speed it will reach a distance of 17,318 if \(\alpha=60^{\circ}\) or \(\alpha=30^{\circ}\). Check this statement.

Galileo states that if a projectile fired at a given initial speed at an angle \(\alpha\) to the horizontal reaches a maximum height of 5000 if \(\alpha=45^{\circ}\), then with the same initial speed it will reach a height of 2499 when \(\alpha=30^{\circ}\) and a height of 7502 , when \(\alpha=60^{\circ}\). Check this statement.

Given that the distances traveled in any times by a body falling from rest are as the squares of the times, show that the distances traveled in successive equal intervals are as the consecutive odd numbers \(1,3,5, \ldots\)

Find out how the use of logarithms was mechanized in the seventeenth century by the invention of the slide rule. Give examples of the various types of slide rules used. When did the slide rule itself become obsolete and why?

Compare Galileo's and Kepler's attitudes toward the interaction of experiment (or observation) and theory in developing a new body of knowledge.

Look up a treatment of geometrical perspective in a modern text on techniques of painting. How does it compare to the discussion of Alberti?