The Apex: Third Century Hellenistic Mathematics - THE ANCIENT ROOTS OF MATHEMATICS - MATHEMATICS IN HISTORY - Mathematics for the liberal arts

Mathematics for the liberal arts (2013)

Part I. MATHEMATICS IN HISTORY

Chapter 1. THE ANCIENT ROOTS OF MATHEMATICS

1.4 The Apex: Third Century Hellenistic Mathematics

There is no permanent place in the world for ugly mathematics.

G. H. HARDY (1877–1947)

The Third Century BCE.

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Greek mathematics, especially geometry, achieved its highest expression in the 3rd century BCE. The century started with the writing of the most famous math book ever, continued with the work of the greatest of the ancient mathematicians, and finished with the definitive Greek text on conic sections.

Alexander the Great

Macedonia was a small kingdom on the northern boundary of Greece. Under King Philip II, who ruled from 356 to 336 BCE, Macedonia gained effective control of most of Greece, and began a war against the Persian empire to the east. Shortly after the start of this war, Philip was assassinated and his army elected his son Alexander as his successor. Alexander, already a veteran general, was twenty years old.

The war against Persia continued. Alexander won a great battle at Issus, in modern-day Turkey, in 333. He continued his conquests, as far as modern-day Uzbekistan and Pakistan in Asia, and Egypt in Africa, creating the largest empire up to that time. He caught a fever and died in 323 BCE. He was 33 years old.

Among the many stories told of Alexander, perhaps the most famous is that of the Gordian knot. The legend has it that when Alexander entered the city of Gordium, he was shown a sacred knot tied around a pole. Supposedly, the man who could untie the knot was destined to become the king of Asia. Alexander’s response was to take out his sword and slice the knot. The phrase “cutting the Gordian knot” is now used to indicate an audacious solution of a complicated problem. Perhaps it can also be considered a metaphor for Alexander’s ruling philosophy. Another legend has it that as Alexander lay dying he was asked to whom would he leave his empire. His answer: “to the strongest.”

In fact, the empire fell apart after Alexander’s death. Major pieces were ruled by several of his generals, including Antigonus in Macedonia, Seleucus in the east, and Ptolemy in Egypt. What followed was a period of empire, much different than the time of the great Greek city-states. The successors of Seleucus, the Seleucids, ruled much of west Asia, gradually declining in power until their last holdings in Syria were conquered by the Romans in 64 BCE. The Ptolemys in Egypt ruled until the death of Cleopatra in 31 BCE, again falling to the Romans.

The civilization of this time, from Alexander until the rise of the Romans, is known as Hellenistic, distinguishing it from the earlier Hellenic period of Greek culture. Hellenistic culture spread across all of Alexander’s empire; a form of Greek became the common language of trade and government. The rulers were educated in classical Greek culture.

Alexandria and Its Museum and Library

The greatest city of this time, both commercially and culturally, was Alexandria in Egypt. The city was founded by Alexander in 331, the first of seventeen cities of that name. It was a major port, located where the Nile empties into the Mediterranean. As such, it was situated to profit from Egypt’s large surplus of grain, which was shipped to many Mediterranean ports.

Egypt was ruled by the Macedonian general Ptolemy I Soter from the time of Alexander’s death in 323 until 283 BCE. His capital was Alexandria, and it was there that he built the Museum (“temple of the Muses”) and Library. The Museum recruited the leading scholars of the Greek world, paying them a salary, providing free board and freedom from taxes. Originally, it was not a school; it has been compared to the modem Institute for Advanced Study in New Jersey, a place for scholars to discuss, and invent, ideas. Over time, students were attracted to the Museum, to learn from the experts.

The Library at Alexandria was the largest in the ancient world, eventually housing over 500,000 manuscripts. Ships sailing from Alexandria were instructed to gather any manuscripts they could, to add to the Library. One story has it that the Library borrowed manuscripts, copied them, then returned the copies, retaining the originals.

Alexandria quickly eclipsed Athens as the chief center of GreekJeaming. There were other centers, including Syracuse where Archimedes worked, but Alexandria was the greatest. The city and its Museum remained influential even after the rise of Rome.

Euclid’s Elements

The Elements by Euclid is the most successful textbook in history. Written around 300 BCE, it is second only to the Bible in number of editions, certainly over one thousand. It was the mathematical textbook into at least the 19th century. It inspired many students from Abraham Lincoln to Albert Einstein. Edna St. Vincent Millay wrote a sonnet entitled Euclid Alone Has Looked on Beauty Bare.

The Elements is the culmination of the early period of Greek mathematics. In it, Euclid summarized much of the mathematics developed in the preceding three centuries. But he did more than that; he put this mathematics into a rigorous and consistent logical framework, starting with unproved assumptions and carefully, step-by-step, deducing more advanced theorems from them. It is this structure that gives the book its special character, and is no doubt the reason why high school geometry courses to this day often are students’ first introduction to formal mathematical reasoning. It is also far from the modem multicolored, image-laden, mathematical textbook. In fact, the modem reader will find it dry as dust. It remains, however, one of the great intellectual milestones in history.

Euclid (c. 330–270 BCE)

Even less is known of Euclid than of many of his predecessors. He worked at the Museum in Alexandria, probably arriving some time around 300 BCE. It is reasonable to think that he had studied in Athens, perhaps at the Academy, because he was certainly familiar with the works of Eudoxus and other Athenians.

Euclid’s date and place of birth are unknown, as is the date of his death. He wrote on many mathematical subjects, including astronomy and optics, perhaps a dozen books in all, but only the Elements has survived. Of course, tales are told of him, but they are all from many hundreds of years later. One of the most often repeated is that the king Ptolemy asked him if there were a shorter way to learn geometry than through the Elements.He was said to reply that there “is no royal road to geometry.” He probably did not add that kings at least can afford tutors like Euclid.

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The Elements is divided into thirteen “books.” Book I starts with five postulates and five “common notions,” both of which we would call axioms today. They are given below.

Postulates

1. To draw a straight line from any point to any point.

2. To produce a finite straight line continuously in a straight line.

3. To describe a circle with any center and radius.

4. That all right angles equal one another.

5. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Common Notions

1. Things which equal the same thing also equal one another.

2. If equals are added to equals, then the wholes are equal.

3. If equals are subtracted from equals, then the remainders are equal.

4. Things which coincide with one another equal one another.

5. The whole is greater than the part.

Some of this language may need translation. For example, Postulate 1 means that, given any two points, we can draw a straight line joining them. Common Notion 1 can be stated algebraically: If a = b and c = b then a = c.

Postulate 5 sticks out like a sore thumb. In modem language, it might be stated thus: if a line intersects lines 1 and 2, as in Figure 1.17, and the angles α and β sum to less than 180 degrees, then lines 1 and 2 must eventually intersect, on the same side of the third line as α and β. Postulate 5 is also called the parallel postulate, because Euclid used this postulate to prove theorems about parallel lines.

If you think about this postulate a bit, you may be able to convince yourself of its truth. Postulates and common notions, however, are supposed to be self-evident. This one seems a bit too involved. Over the centuries many people have attempted to show that it could be derived from the other postulates and axioms. In the 19th century, mathematicians developed Non-Euclidean geometries, in which the other axioms hold but the parallel postulate is false. Euclid was vindicated.

The rest of Book I is a careful, step-by-step, argument, culminating in the Pythagorean Theorem.

Book II contains a number of results which we would think of as algebraic, but in geometric form. In general, the Greeks preferred geometry to algebra, and would often cast problems into geometric form that we now would handle algebraically. As an example, consider the following.

Figure 1.17 Parallel postulate.

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Proposition II-4 If a straight line is cut at random, then the square on the whole equals the sum of the squares on the segments plus twice the rectangle contained by the segments.

We can picture this as in Figure 1.18, where the line at the left is cut into pieces of lengths a and b. The proposition states that the area of the big square equals the sum of the areas of the two smaller squares and the two rectangles. Of course, drawn as it is in the diagram, this is rather evident, but let’s translate this into equations. The area of the larger square is (a + b)2. The smaller squares have areas a2 and b2, and each of the rectangles has area ab. So the proposition states that

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This is our familiar way of squaring a binomial.

Figure 1.18 Proposition II-4.

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Book II contains other results of this nature, including a geometric form of the quadratic formula, which we now write as

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giving solutions of the equation ax2 + bx + c = 0.

Book III contains thirty-seven propositions on circles, starting from basic definitions. Here is one that you may have seen before.

Proposition 111–20 In a circle the angle at the center is double the angle at the circumference when the angles have the same circumference as base.

Figure 1.19 Proposition 111–20.

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Using the notation of Figure 1.19, angle ABC is twice angle ADC.

Book IV has results about inscribing polygons in circles and circles in polygons, and gives constructions of some regular polygons. The regular polygons of three and four sides, i.e., equilateral triangles and squares, are relatively easy to construct. The book ends with the construction of a regular pentagon (5-gon) and 15-gon.

Book V presents some basic results on magnitudes, which we would think of as lengths or areas. Here is an example.

Proposition V-l If any number of magnitudes are each the same multiple of the same number of other magnitudes, then the sum is that multiple of the sum.

An algebraic version of this is, for a positive integer n and any magnitudes a1, a2, …, ak, we have na1 + na2 + … + nak = n(a1 + a2 + ak). For us this follows from the associative law of numbers. Euclid didn’t consider magnitudes to be the same as numbers, however. Magnitudes and numbers were different things for him.

Book V also presents Eudoxus’ theory of proportions, and uses it to prove other results on magnitudes. Many of these results are used in Book VI, which is about similarity, defined as follows (see Figure 1.20).

Definition VI-1 Similar rectilinear figures are such as have their angles severally equal and the sides about the equal angles proportional.

Figure 1.20 An example of similar triangles.

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In Proposition VI-19 it is proved that the ratio of the areas of two similar triangles is the square of the ratio of their corresponding sides. As an illustration, the larger triangle in Figure 1.20 has four (= 22) times the area of the smaller.

Books VII, VIII, and IX cover number theory. These books do not rely on the results of the first six books, because they are about numbers, not magnitudes, which Euclid viewed as entirely different entities. Thus, Euclid proved again theorems such as the distributive law, even though he had a similar theorem about magnitudes. The number theory chapters are based on Pythagorean work, although this work had been later logically reorganized, perhaps by Theaetetus.

We mention four important results from these three books. The first has to do with greatest common divisors. Recall that the greatest common divisor of two integers is the largest integer dividing them both. Book VII contains the Euclidean Algorithm, a method of determining greatest common divisors that is still important. Details on this algorithm can be found in Section 5.4.

Two fundamental results on prime numbers are in these books. The first is a proof that the number of primes is infinite. We give this proof in Section 5.5. The second is the Fundamental Theorem of Arithmetic, that every positive integer can be expressed as the product of primes, and in only one way. This theorem is the subject of Section 5.8.

The culmination of Euclid’s number theory chapters is a study of perfect numbers. Recall that a positive integer n is a perfect number if it is the sum of all its divisors, excluding n itself. Euclid was able to connect perfect numbers to certain types of primes. This topic is explored further in Section 5.7.

Book X undertakes a study of incommensurable magnitudes. This book starts with the following definition.

Definition X-1 Those magnitudes are said to be commensurable which are measured by the same measure, and those incommensurable which cannot have any common measure.

Here is the idea. Suppose that we have two lines, of lengths a and b. Then a and b (thought of as magnitudes, not numbers) are commensurable if there is another line of length c that fits into both a and b a whole number of times. For example, 6 and 10 are commensurable because 2 fits into both, i.e., 6 = 3 · 2 and 10 = 5 · 2. In general, if a and b are commensurable, say by c, then a = mc and b = nc, for some integers m and n. But then a/b = m/n. In other words, the ratio of a to b is a rational number. Studying incommensurable magnitudes is tantamount to studying irrational numbers. It should therefore come as no surprise that Euclid relies on Eudoxus’ theory of proportions in this chapter.

Books XI–XIII concern solid (3-dimensional) geometry. Books XI and XII prove some extensions of earlier theorems of plane geometry, and obtain results on volumes of a number of solids. For example, proofs are included for the theorems that the volume of a cone is one-third that of the related cylinder, and the volume of a pyramid is one-third that of its related prism. The following important result is also proven.

Proposition XII-18 Spheres are to one another in triplicate ratio of their respective diameters.

“Triplicate ratio” means cube, so the proposition is that the volume of a sphere is proportional to the cube of its diameter. This is equivalent to writing V = kr3, where V is the volume, r the radius, and k some constant of proportionality. We now know that k = 4π/3, but this was not known to Euclid.

Many of the results in Books XI and XII are proved by the method of exhaustion, that is, by exhausting the area or volume of shape being studied by figures of known properties.

Finally, Book XIII is a study of convex regular polyhedra, the Platonic solids. These are 3-dimensional equivalents of regular polygons, solids whose faces are all congruent regular polygons, arranged the same way around each vertex. See Figure 1.21. These polyhedra make good dice and are used in various role-playing games.

Figure 1.21 The Platonic solids.

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Book XIII shows how to construct the Platonic solids, and gives some of their properties. It culminates in the proof that, unlike the regular polygons, there are only a finite number of convex regular polyhedra. In fact, the five in Figure 1.21 are the only ones.

The Elements contains some mistakes, and Euclid made some implicit assumptions without stating them. Still, Euclid skillfully surveyed the whole of Greek mathematics of his time. Most of the content of his book was originally due to others, but he put it together so well that previous summaries have not even been preserved.

Archimedes and Higher Geometry

Euclid was not the last word in Greek geometry. The 3rd century BCE was the high water mark for all of ancient geometry. The two most notable figures were Archimedes and Apollonius.

Archimedes (c. 287–212 BCE)

Widely considered to be the greatest mathematician of antiquity, Archimedes was bom and raised in the Greek settlement of Syracuse in Sicily. His father was an astronomer, and no doubt his first mathematics teacher. Later, Archimedes traveled to Egypt, probably studying at the Alexandria Museum with students of Euclid. He then returned to Syracuse to work.

In addition to mathematics, Archimedes made important discoveries in physics, especially hydrostatics (the physics of water) and was an engineer of note. The most famous, probably apocryphal, story told of Archimedes concerns his solution of a problem presented to him by King Hiero. The king had recently acquired a golden crown, and wanted to verify that it was indeed made entirely of gold, without in any way modifying the shape. Archimedes was pondering the problem while in his bath, and it occurred to him that he could use the displacement of water when the crown was lowered into a bath to determine its density, hence its composition. He was so excited that he jumped out of the bath and ran naked through the streets yelling “Eureka! Eureka!” Eureka means “I have found it.”

Among his many mechanical inventions was the Archimedean screw, a mechanical device for lifting water. Although it has now been mostly replaced for that purpose by other sorts of pumps, it is still used in a variety of applications from sewage treatment plants to fish hatcheries.

Archimedes was also noted for his prowess in designing and building war machines. They were a factor in allowing Syracuse to hold off a Roman siege for three years during the second Punic War. In the end, however, Syracuse was overrun, and Archimedes was slain by a Roman soldier.

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Archimedes wrote many works on a wide variety of topics. Unlike Euclid, he did not write textbooks, but rather original research monographs. As with the other ancients, none of the originals have survived, and many of his works have been lost.

He was a pioneer of mathematical physics, the creation of mathematical models for physical situations. He proved the law of the lever, a result that had been previously known but not rigorously studied. This law states that the two sides of the lever are in balance when the product of the weight and the distance from the fulcrum on one side of the lever equals the similar product from the other side. Archimedes used this law in many of his mechanical inventions. He is said to have claimed: “Give me a lever long enough and a fulcrum on which to place it, and I shall move the world.”

Archimedes also studied centers of gravity of various shapes, a topic that combined his interest in physics and geometry. The center of gravity of a body is the point from which it can (at least conceptually) be suspended and be at rest.

Hydrostatics is yet another area of science where Archimedes excelled. He discovered the Archimedes principle, that a body immersed in water will displace a volume of fluid that weighs as much as the body would weigh in air. (This was the theoretical principle behind the solution of the golden crown problem.)

Among his many mathematical works, we will visit three. The first is a short treatise, called Measurement of the Circle. In the tradition of studying the squaring of the circle, Archimedes proves that the area of the circle is equal to one-half the radius times the circumference. If we define π as the ratio of the circumference of circle to its diameter, π = C/2r, this gives

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It was previously known that the area was proportional to r2, i.e., A = kr2. Archimedes showed that constant of proportionality is π.

He then goes on to numerically approximate π. He bounds the area of the circle above and below by inscribing polygons in the circle, and inscribing the circle in other polygons, as in the method of exhaustion. By considering polygons with an increasing number of sides (ultimately 96 sides), he arrives at the bounds images images. For many years after this, the estimate of images was commonly used for π; it is called the Archimedean value of π.

Archimedes’ masterpiece was On the Sphere and Cylinder. In it he studied a sphere and its circumscribed cylinder (Figure 1.22). He proved that the surface area of the sphere is 2/3 that of the cylinder (including the two caps), and the volume of the sphere is 2/3 that of the cylinder.

Let us see how we can use Archimedes’ results to find formulas for the surface area and volume of a sphere. First, we figure the surface area of the cylinder. Here is the approach:

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If the radius of the sphere is r, each of the caps (top and bottom) is a circle of radius r, so has area πr2. To figure the area of the sides of the cylinder, mentally unroll it. You will get a rectangle whose height is the height of the cylinder, 2r, and width is the circumference of the circle, 2πr. So the area of the side is the product of the width and height, or (2πr)(2r). Putting this together,

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Thus the surface area of the sphere is 2/3(6πr2) = 4πr2.

The volume of the cylinder is the area of the base times the height, or πr2(2r) = 2πr3. Hence, Archimedes’ theorem tells us that the volume of the sphere is images. As with the circle, before this theorem, the Greeks knew that the volume of the sphere was proportional to r3 but did not know the constant of proportionality.

Figure 1.22 A sphere and its circumscribed cylinder.

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Archimedes was so proud of these results that he requested the diagram of the sphere and cylinder be carved on his gravestone.

Archimedes proved the above results in the Greek fashion, in a way that did not reveal how he discovered them. In particular, he perfected the method of exhaustion. This method is useful, however, only after you know what the answer is. His method of discovery was mysterious until about 100 years ago. In 1899 a treatise containing a copy of his work called The Method was discovered in Constantinople, in the library of a Greek monastery. It had been written on a parchment in the 10th century, but then overwritten by a religious work in the 13th century. The practice of reusing parchments was not uncommon, since parchment was quite valuable. Such a reused parchment even has a name, palimpsest.

In The Method Archimedes gives an ingenious technique for computing areas and volumes, one that combines his work in geometry and physics. He mentally slices the shape he is studying into thin cross sections, then balances these against cross sections of a known shape, using the law of the lever. Some modem mathematicians have seen hints of the integral calculus (developed in the 17th century) in this process. It was not, however, rigorous, so after he used this method to discover results, he proved them in the usual deductive way.

Marcellus, the general who commanded the army that overran Syracuse, was upset to find that Archimedes had been killed by one of his soldiers. He erected a small column to mark Archimedes’ grave, and had the figure of the sphere inscribed in the cylinder carved on it, as Archimedes had wanted. Cicero, the Roman philosopher and statesman, upon being appointed governor of Sicily in 75 BCE, 137 years later, found the grave neglected and overgrown. He cleaned it up and restored the marker. But it was again neglected, to be found only in 1965 during excavation for the construction of a hotel.

Conic Sections

Recall that conic sections were originally studied by Menaechmus to solve the Delian problem, doubling the cube. From the algebraic point of view, they are a natural topic of study, after straight lines. Specifically, a straight line is the set of points solving an equation of the form

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If we allow second powers of the variables, we get conic sections, which are solutions of equations of the form

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The Greeks did not have this algebra, so they studied conic sections geometrically.

Apollonius (c. 240–174 BCE)

Apollonius was reportedly bom in the Greek city of Perga, in modern-day Turkey, and lived as a young man in Pergamum. In the late 3rd century, Pergamum, also in Turkey, became a major intellectual center, modeled after Alexandria. It housed the second largest library in the Greek world.

After Pergamum, Apollonius moved to Alexandria, where he spent most of his career. He published a number of works on astronomy, geometry, and arithmetic, most of which have been lost.

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Apollonius’ masterwork was the Conics. It was the first known mathematical text to systematically and exhaustively treat a single topic–conic sections. It contains eight books and 487 propositions. The first part covers what was known to previous mathematicians, and the second part consists of original contributions.

Figure 1.23 An ellipse and its foci.

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Apollonius gave us the terms for the three conics: ellipse, hyperbola, and parabola. Among the many results in the Conics, the ellipse is shown to be the set of points the sum of whose distances from two fixed points is constant. The two points are called the foci (singular focus) of the ellipse. An ellipse and its foci are shown in Figure 1.23; the sum a + b is the same for any point on the ellipse. As the two foci get closer together, the ellipse becomes less elongated, until, when the foci are the same point, the ellipse is a circle.

You can use the idea of the foci to construct an ellipse physically. Take a pad of paper, and stick two thumb tacks in where the foci are to be, as in Figure 1.24. Then take a length of string, and tie the two ends to the tacks. If you stretch the string taut, the two parts of the string will give the distances to the two tacks. The sum of these distances is the length of the string. So, if you take a pencil and trace out the curve you get by moving the pencil, always keeping the string taut, the resulting figure will be an ellipse.

Figure 1.24 Drawing an ellipse.

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Apollonius gave a similar result for the hyperbola. In that case, the difference of the distances to the foci is a constant. (See Figure 1.25.)

Apollonius did not have a similar result for a parabola, but his contemporary Diodes showed that the parabola is the set of points equidistant from a point (its focus) and a line (its directrix). See Figure 1.26.

At the heart of every large, modem telescope is a mirror in the shape of a paraboloid, a solid figure obtained by rotating a parabola. This uses an important reflection property, illustrated in Figure 1.27. In the figure, we have a cross-section of a paraboloid, i.e., a parabola, with the focus marked by a dot. All light that comes from a direction perpendicular to the directrix, after bouncing off the mirror, passes through the focus. This property is used to focus the light to form an image of distant objects. The word “focus” is from the Latin, meaning fireplace or hearth.

Figure 1.25 A hyperbola and its foci.

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Figure 1.26 A parabola with focus and directrix.

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The Conics was very influential for many years. In the 17th century, Kepler drew on it when he discovered that planetary orbits are ellipses, with the Sun located at one focus.

Figure 1.27 Reflection property of a parabola.

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EXERCISES

1.39 In the text, we restated Common Notion 1 in algebraic terms. Do the same for Common Notion 2.

1.40 In the text, we restated Common Notion 1 in algebraic terms. Do the same for Common Notion 3.

1.41 Given that ABC is a right angle, find θ.

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1.42 Given that the two triangles below are similar, find x and y.

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1.43 The two right triangles below are similar. Find x and y. (Hint: the Pythagorean Theorem may help.)

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1.44 Euclid’s Proposition II-4 is a geometric version of the binomial identity

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In this problem, we consider the analogous trinomial identity

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a) Draw a version of Figure 1.18 to represent the trinomial identity.

b) By considering areas, show that your diagram illustrates the trinomial identity.

1.45 The first proposition of the Elements is the construction of an equilateral triangle using straightedge and compass. Show how to carry out such a construction.

1.46 Given three points in the plane, not all on a line, show how to construct, using straightedge and compass, the circle that passes through them.

1.47 The law of the lever, proved by Archimedes, applies to playground teeter totters. Suppose that a 50 pound child sits 9 feet from the fulcrum (center) of the teeter totter. How far from the fulcrum would a 75 pound child sit in order to balance the other child?

1.48 The Earth weighs about 1.3 × 1025 pounds (13 followed by 24 zeroes). Suppose that Archimedes had a fulcrum placed 8000 miles from the Earth (about one diameter), and he can manage 200 pounds of weight on his end of the lever. How long should he be from the fulcrum to “move the world?”

1.49 Suppose that a sphere is inscribed in a cylinder, and that we measure the surface area of the cylinder to be 27.1434 cm2 and the volume to be 10.8573 cm3. Using Archimedes’ theorem, find the surface area and volume of the sphere.

1.50 Find the surface area and volume of a sphere of radius 2 feet.

1.51 An ellipse has foci at (2,0) and (–2,0), and contains the point (3,0), as in the figure below.

a) What is the sum of the distances from (3,0) to the two foci?

b) If the point (0, y) is on the ellipse, what is y?

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1.52 Below is a parabola with the x-axis as its directrix. Its equation is y = imagesx2 + 2. Find the coordinates of the focus. (Hint: the point on the parabola with x = 0 is equidistant from the focus and the directrix.)

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1.53 Conic sections can be created by folding papers. Try typing “folding conic sections” in your favorite Internet search engine to find out how to do this.