History of Math Midterm 1 Prep
When did Ptolemy live?
100-170 AD
(Egypt) Because division is the inverse of multiplication, a problem such as 156 ÷ 12 would be stated as
"multiply 12 so as to get 156.
Euclid: Finish this statement. In particular, on the basis of the equilateral triangle construction, we can construct the regular hexagon. This is an example of an important technique of drawing a regular polygon by.....
"walking" around the circumference of a circle with a fixed length (stored by the compass)
Euclid: Please define the concept exhibited below (imagine the ()'s are pebbles/objects): () () () () () vs. () () () () () ()
() () () () () - prime (cannot split in half) vs. () () () () () () - not prime (can split in half.
Which number system was used for each operation (Egpyt) (1) addition (2) subtraction (3)multiplication
(1) base 10 (convenient) (2) base 10 (3) base 2... because they used the continual doubling process
What did Apollonius use conic sections for?
(1). Doubling a cube (2). (Also can't do with straight edge and compass) used conic sections and their intersections to tri-sect an angle. Given an angle give three equal parts.
Using only a straight edge and a compass, one can construct (A) a line with length square root 2 (B) a line with length square root 2 index 3 (C) a square with area pi
(A) a line with length square root
According to your textbook by V. Katz (see p.39), the Pythagoreans, members of the Pythagorean school founded by Pythagoras (~ 572 - 497 BC) in what is now southern Italy (then a Greek colony, Magna Greaca) suspected that the square root of 2 was a rational quantity (a measurable "magnitude;" they didn't consider it to be a number - or maybe they did?). A proof that square root of 2 is irrational (or "incommensurate," as the Greeks would have called it) was known: (A) already at the time of Aristotle (~ 384 - 322 BC) B) already to the architects of Ramses Il, but they left no record about it; C) only after a proof was found by Hypatia in 400 AD.
(A) already at the time of Aristotle (~ 384 - 322 BC)
A sphere of radius r contains 2/3 of the volume of the smallest cylinder into which the sphere can be placed. This theorem was proved by (A) the Greek mathematician Archimedes of Syracuse (B) an unnamed court mathematician of King Hamurabi (C) the Greek mathematican Euclid of Alexandria.
(A) the Greek mathematician Archimedes of Syracuse
Write a formula for a... Parabola Hyperbola Elipse Circle
(Apollonius section) Parabola: y^2=4ax Hyperbola: x^2/a^2 - y^2/b^2=1 Elipse: x^2/a^2 + y^2/b^2=1 Circle: x^2+y^2=a^2
Q: How to construct y^2=2ax (parabola)
(Apollonius) Draw semi circle on x axis, with diameter 2a. Draw new point on x axis. Where it hits y axis record point, again.
mapping eccentricity: Epsilon Conic Section 0 0<ε<1 1 >1
(Apollonius) mapping eccentricity: Epsilon Conic Section 0 circle 0<ε<1 elipse 1 parabola >1 hyperbola
Who can be attributed to the following? Fill in the blanks Modus ponens a.)If p, then q p. Therefore, what? b.) What if it was not p, what does this tell us about q? Modus tollens c.) If p, then q Not q. Therefore, ....what?
(Aristotle) a.) If p, then q p. Therefore, q. (means p is true hence q is true too). b.) If p, then q not p. Tells us nothing about q. c.) Therefore, not p
Archimedes of Syracuse is (A) an honorary professor title at Syracuse University in New York State; (B) one of the greatest Greek mathematicians of ancient times (C) the founder of the Greek city of Syracuse on the island of Sicily.
(B) one of the greatest Greek mathematicians of ancient times
The "Almagest," which was the standard work for astronomers in Europe until the revolutionary ideas of Copernicus, was authored by (A) Aristotle; (B) Thales of Miletus; (C) Ptolemy of Alexandria.
(C) Ptolemy of Alexandria.
Thales of Miletus (born before 620 BC, died after 550 BC) is generally credited with having introduced A) the base 60 system of numbers; B) the transcendental numbers; (C) the concepts of rigorous proof and scientific inquiry.
(C) the concepts of rigorous proof and scientific inquiry.
Prior to the introduction of analytic geometry (i.e. using coordinates) the study of Euclidean geometry was conducted by describing constructions (in principle) of the objects under study using straight edge and compass. The mathematician who perfected the study of conic sections using this method was (A) the Greek mathematician Pythagoras; (B) the Greek mathematician Thales of Miletus; (C)the Greek mathematician Appolonius of Perga.
(C)the Greek mathematician Appolonius of Perga.
Q: Can you construct square root of 2 with a straight edge and compass?
(Euclid) Yes! Using circle with radius 1. Triangle inside with and triangle with adjacent and opposite sides equal to length 1 and hypotenuse equal to square root of 2. CIRCLE TRICK
Can we double the area of a given square with a straight edge and compass? What about a cube?
(Euclid): Square yes cube no.
Who discussed the 5 regular polyhedra?
(Greeks) Euclid devoted the last book of the Elements to the regular polyhedra, which thus serve as so many capstones to his geometry. In particular, his is the first known proof that exactly five regular polyhedra exist
The five Platonic solids (regular polyhedra) are the.....
(Pythagoras - written in Euclid's elements) tetrahedron, cube, octahedron, icosahedron, and dodecahedron. The regular polyhedra are three dimensional shapes that maintain a certain level of equality; that is, congruent faces, equal length edges, and equal measure angles. Sides are pentagons, squares and equilateral triangles.
Explain the significance of the city Alexandria? -who worked here? -why was it so awesome?
-Alexander the Great ordered Alexandria (the city) to be great! -Euclid worked in Alexandria -When Euclid was in his 20's, Alexandria was booming. Throwback: Aristotle taught Alexander the great which is why Alexander valued discovery and school so much! This is why access to this in Alexandria (the city) was so amazing. :)
Euclid completed Book IV with: Construction of n-gons with compass and straight edge. - the construction of a ..... and a ..... in a circle but did not mention the construction of other regular polygons. - -If m, he presumably he was aware of the construction of a polygon of 2... sides -If m and l, it was straightforward to construct a polygon of.... sides -What are the missing polygons?
-Euclid completed Book IV with the construction of a regular hexagon and a regular 15-gon in a circle but did not mention the construction of other regular polygons. -Presumably, he was aware that the construction of a polygon of 2^k*m sides (m = 3, 4, 5) was easy, beginning with the constructions already made. hexagon. No pentagon or 9-gon -It was straightforward to construct a polygon of m*l sides (m, l relatively prime) if one can construct one of k sides as well as one of m sides. 9-gon, pentagon -What are the missing polygons? prime #'s: 7,11,13 (cannot draw with straight edge and compass) He was able to construct shapes with composite number of sides (now prime).
Euclid's Postulates: Fill in the blanks 1. A straight line segment can be drawn joining..... 2. Any straight line segment can be ..... 3. Given any straight line segment, a .... can be drawn having the segment as ...... and one endpoint as ...... 4. All right angles are ..... 5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must .... if extended far enough. This postulate is equivalent to what is known as the .....
1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 4. All right angles are congruent. 5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.
Euclid's Book 1: Finish the Definition 16. ) And the point is called the ..... of the circle. 17.) A ..... of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle. 18.) A ...... is the figure contained by the diameter and the circumference cut off by it. And the center of the semicircle is the same as that of the circle. 19.) Rectilinear figures are those contained by straight lines, ...... figures being those contained by 3, ...... those contained by 4, and ...... those contained by more than 4 straight lines 20.) Of trilateral figures, an ...... triangle is that which has its three sides equal; an ..... triangle that which has two of its sides equal, and a ...... triangle which has its three sides unequal.
16. ) And the point is called the center of the circle. 17.) A diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle. 18.) A semicircle is the figure contained by the diameter and the circumference cut off by it. And the center of the semicircle is the same as that of the circle. 19.) Rectilinear figures are those contained by straight lines, trilateral figures being those contained by 3, quadrilateral those contained by 4, and multilateral those contained by more than 4 straight lines 20.) Of trilateral figures, an equilateral triangle is that which has its three sides equal; an isosceles triangle that which has two of its sides equal, and a scalar triangle which has its three sides unequal.
What Egyptian fractions had special symbols?
2/3, 1/2 and 1/4
Timeline (part 1): 5000 BC: ...... & ..... civilization takes hold 1650 BC:..... copied from a 200-year-old original 624-546 BC: Person - traveled to Egypt and traveled back to Greece, teaching others what they learned. Credited for developing more sophisticated explanations, than just "it works, it must be true". Possibly introduced the notion of proof into mathematics 570-495 BC: Person - (school named after them). In Italy (but Greek), since the Greeks colonized the Mediterranean. Believed nature could be understood in whole numbers.
5000 BC: Egyptians and Mesopotamia civilization takes hold 1650 BC: Rhine papyrus copied from a 200 year original 624 - 546 BC: Thales of Miletus.... 570-495 BC: Pythagoras...
(Timeline part 2) 530-450 BC: Person Person 490-430 BC: Person Person 470-410 BC: Person 465-398 BC: Person 460-370 BC: Person 428-348 BC: Person - student of Socrates 408-355 BC: Person 384-322 BC: Person - student of Plato 320 - 280 BC: Person 287-212 BC: Person 276-194 BC: Person 262-190 BC: Person 190-120 BC: Person
530-450 BC: Hippasus Parmenidis 490-430 BC: Zeno Socrates 470-410 BC: Hippocrates 465-398 BC: Theodorus 460-370 BC: Democritus 428-348 BC: Plato (student of Socrates) 408-355 BC: Endoxus 384-322 BC: Aristotle( student of Plato) 320 - 280 BC: Euclid 287-212 BC: Archimedes 276-194 BC: Eratosthenes 262-190 BC: Apollonius 190-120 BC: Hipparchus
(Roman times) 60 - 120 AC 200 - 284 AC 350-415 AC 476
60 - 120 AC Nicomadus 200 - 284 AC Diophantus 350-415 AC Hypatia (female) 476 The end to the West Roman Empire (beginning of the Dark ages)
Timeline (part 3) 60-120 AD: Person 100-170 AD: Person 200-284 AD: Person 350-415 AD: Person - killed for being educated 412-485 AD: Person (historian)
60-120 AD: Nicomachus 100-170 AD: Ptolemy 200-284 AD: Diaphanous 350-415 AD: Hypatia (female) - killed for being educated 412-485 AD: Proclus (historian)
When was Rome founded?
753 BC
Euclid's Book 1: Finish the Definition 8.) A ...... is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line. 9.) And when the lines containing the angle are straight, the angle is called ........ 10.) When a straight line meeting another straight line makes the adjacent angles equal to one another, each of the equal angles is ......, and the first straight line standing on the other is called a ........ to that on which it stands 11.) An ...... angle is an angle greater than a right angle 12.) An ...... angle is an angle less than a right angle 13.) A ...... is that which is an extremity of anything 14.) A ...... is that which is contained by any boundary or boundaries 15.) A ....... is a plane figure contained by one line such that all the straight lines meeting it from one point among those lying within the figure are equal to one another.
8.) A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line. 9.) And when the lines containing the angle are straight, the angle is called rectilinear. 10.) When a straight line meeting another straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the first straight line standing on the other is called a perpendicular to that on which it stands 11.) An obtuse angle is an angle greater than a right angle 12.) An acute angle is an angle less than a right angle 13.) A boundary is that which is an extremity of anything 14.) A figure is that which is contained by any boundary or boundaries 15.) A circle is a plane figure contained by one line such that all the straight lines meeting it from one point among those lying within the figure are equal to one another.
(Greeks) Given a square of pebbles (N wide and N long), that had N pebbles in the diagonal, the number of pebbles in the triangle below the triangle are= The number of pebbles in the triangle that include the diagonal and below are=
=((N^2-N)/2) =((N^2-N)/2) +N
Parabola Complete the formula (involving directrix)- distance from M(P) to P =
=distance from P to F. (Epsilon equals 1)
A flashlight has a mirror that is the shape of a.... and we expect the light to emerge in the shape of... But the light emerges in the shape of an..... Why?
A light has a mirror that is the shape of a parabola and we expect the light to emerge in the shape of an elipse. It actually emerges in the shape of a parabola Why: side of flashlight dont go on forever
Already the earliest ancient Greek mathematicians established combinatorial identities, such as 1+2+3+...+ N =N(N + 1)/2 and 1+3+5+...+ (2N -1) = N^2 which are also listed in Euclid's Elements. Their method of proof was A) using geometrical arguments to effectively count (say) pebbles arranged in rectangular lattice; B) the method of complete induction; C) by building the first mechanical calculator, the famous astrolab.
A) using geometrical arguments to effectively count (say) pebbles arranged in rectangular lattice;
Euclid's number theory What are #'s for Euclid and his mathematical forbear? What did Euclid not consider as a natural #?
A: 2,3,4..... Did not consider 1 as a natural number, 2 was the first number for Euclid. 1 was the unit.
Construct formula for distance between Points and Foci for Hyperbola
Absolute value (distance between chosen P and F1 - chosen P and F2) = C
Who ruled Egypt after Alexander the Great died?
After Alexander died in 323 B.C., his generals (known as the Diadochoi) divided his conquered lands amongst themselves. Soon, those fragments of the Alexandrian empire had become three powerful dynasties: the Seleucids of Syria and Persia, the Ptolemies (Ptolemy) of Egypt and the Antigonids of Greece and Macedonia
Egypt: Civilization - .... years ago 1st unified kingdom - .... BC
Agriculture emerged in the Nile Valley in Egypt close to 7000 years ago, but the first dynasty to rule both Upper Egypt (the river valley) and Lower Egypt (the delta) dates from about 3100 BC
Why is the year 323 BC significant?
Alexander the Great dies (in battle)
What book did Ptolemy write?
Almagest
What did we study; WHO's conic sections?
Apollonius
Who created a cone that was no longer "regular"
Apollonius
Who gave a new definition that generalized the conic section definition used by Euclid?
Apollonius
Who generalized the notion of a cone as follows: If from a point a straight line is joined to the circumference of a circle which is not in the same plane as the point, and the line extended in both directions, and if, with the point remaining fixed, the straight line is rotated about the circumference of the circle..., then the generated surface composed of the two surfaces lying vertically opposite one another ... [is] a conic surface. The fixed point [is] the vertex and the straight line drawn from the vertex to the center of the circle [is] the axis....The circle [is] the base of the cone.1
Apollonius
Who is responsible for Epicycles?
Apollonius
Who was a contemporary of Archimedes?
Apollonius
Who, in his Conics, decided to define the conic sections slightly differently. He decided that it was not necessary to restrict oneself to a cutting plane perpendicular to a generator, nor even to a right circular cone, to determine the curves.
Apollonius
Who says the eccentricity of a circle is 0? What variable is used for eccentricity
Apollonius. Epsilon
Apollonius; how many focus points? (Conic sections) 1 focus: 2 foci:
Apollonius; how many focus points? (Conic sections) 1 focus: circle, parabola 2 foci: elipse, hyperbola(two branches of parabola)
Who created a machine for raising water used for irrigation. What was it called?
Archimedean Screw (Archimedes)
Suppose a sphere with radius r is placed inside a cylinder whose height and radius both equal the diameter of the sphere. Also, suppose that a cone with the same radius and height also fits inside the cylinder Suppose we look at a cross-section of the three solids obtained by slicing the three solids with a plane containing point S and parallel to the base of the cylinder. SOMEONE discovered that if the cross-sections of the cone and sphere are moved to H (where |HA| = |AC|), then they will exactly balance the cross-section of the cylinder, where HC is the line of balance and the fulcrum is placed at A.
Archimedes
The methods of discovery of several of WHO'S results are collected in a treatise called The Method, which was unexpectedly discovered in 1899 in a Greek monastery library in Constantinople
Archimedes
Whose first postulate is an example of what is usually called the Principle of Insufficient Reason?
Archimedes
Fill in the blank and answer the question: Who discovered the volume of a solid sphere is ..... the volume of the smallest cylinder that surrounds it. And how?
Archimedes the volume of a solid sphere is two- thirds the volume of the smallest cylinder that surrounds it Cone is 1/3 of a cylinder. 2/3 of volume is left...AKA the sphere. Archimedes figured this out using a balance experiment.
Who said the following? The area of a circle is equal to the area of the right triangle whose legs (that meet at the right angle) have lengths equal to the radius and to the circumference of the circle respectively
Archimedes (Proposition)
Whose responsible for the following Postulate Postulate #1: That is, one assumes that equal weights at equal distances balance because there is no reason to make any other assumption
Archimedes - this relates to the lever.
How was the Law of the lever used in battle? Who created it?
Archimedes - threw heavy stones.
Who wrote the formula for the circumference of a circle, approximating pi? (Using the ancient contributions from other mathematicians throughout time)
Archimedes envisioned a hexagon inscribed within a circle with radius ½. The formula for circumference is 2πr. Hence, with ½ as a radius, the circumference of his circle would be π. He then conjectured that the hexagon's perimeter would approach the circumference of the circle (π).
Who proposed that a segment of a parabola is 4/3 of the triangle inscribed in it?
Archimedes in the Method
What great mathematician came after Euclid? Where did he live?
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. 280 - 220 BC
The king asked a gold smith to make a gold crown. He wanted who to check if it was 100% real?
Archimedes realizes that a crown made with silver will have a different density than the gold, and would sink faster --> Buoyancy
Archimedes was the first mathematician to derive quantitative results from the creation of....
Archimedes was the first mathematician to derive quantitative results from the creation of mathematical models of physical problems on earth.
What was the occupation of Archimedes father?
Archimedes' father was Phidias, an astronomer
Was Archimedes or Euclid's work more widespread? Why?
Archimedes' work: not widespread until hundreds of years later (better discoveries though) Euclid's work: Much more widespread because he was in Alexandria (which became the center of learning)
The relationship between the parameter of a circle and area was proven through what method and by whom?
Archimedes, method of exhaustion
Who used the tools of Euclidean geometry to show that the area inside a circle is equal to that of a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius. How was this proven?
Archimedes. Proven through the method of exhaustion
Who built war devices that helped the Greeks hold off Roman soliders?
Archimedes: His legendary war machines struck fear into the Roman soldiers and sailors and ensured that Syracuse held out for three years against an extended Roman siege. - he was then killed...many different stories how though. (Roman army under Marcellus) Marcellus gave explicit orders that Archi-medes be spared, but Plutarch relates that, "as fate would have it, he was intent on working out some problem with a diagram and, having fixed his mind and his eyes alike on his investi-gation, he never noticed the incursion of the Romans nor the capture ofthe city. Andwhen a soldier came up to him suddenly and bade him follow to Marcellus, he refused to do so until he had worked out his problem to a demonstration; whereat the soldier was so enraged that he drew his sword and slew him"
The volume of a cylinder = What is the relationship between the volume of the cone and the volume of the cylinder? Who discovered this relationship?
Archimedes: VOLUME of a cone volume of a cone = 1/3piR^2 volume of a cone = piR^3 Euclid says: (Answer: The cone takes up one-third of the volume of the cylinder.)
Area of the cross section of a cylinder = Area of the cross section of a sphere = Area of the cross section of a cone = according to the ......., in order for the above balancing relationship to hold we need to following equation to be true:
Area of the cross section of a cylinder = Area of the cross section of a sphere = Area of the cross section of a cone = according to the ......., in order for the above balancing relationship to hold we need to following equation to be true:
Who can be attributed to the following? Fill in the blanks Hypothetical syllogism a.) If p, then q. If q, then r Therefore, if p, then....what? what property does this exhibit? Hypothetical syllogism b.) p or q Not p. Therefore, ...what? Above, what is property does this exhibit?
Aristotle a.) therefore, if p, then r. sufficiency (If true, implies something. If not, nothing) b.) therefore, q exclusivity (This OR That) if this was not q... therefore, p.
Who was the teacher of Alexander the Great (D:323)C?
Aristotle (D:322)
...... was the student of Platos. Platos was the student of.......
Aristotle was the student of Platos. Platos was the student of Socrates
Who designed procedure for deducing logic?
Aristotle's logic For Aristotle, logical argument according to his methods is the only certainway ofattaining scientific knowledge. There may be other ways of gaining knowledge, but demonstration via a series of syllogisms is the one way by which one can be sure of the results
Who called out on falling stones, that if someone was on the mast of a ship sailing by and drops a stone, it does not fall straight down (compared to me watching on shore). Is this correct?
Aristotle. The stone would actually fall straight down, and not on a curved path. Aristotle's teachings became accepted without criticism, for a long time. Hence this is a problem!
Euclid's common notions are also known as...
Axioms
Using successive approcimations of a circle by inscribed regular polygons, the number 7 was computed to a precision of I significant digits by the Chinese mathematician Zu Chongzhi. The polygon he used had A) 2645 sides; B) 24516 sides; C) 192 sides.
B) 24516 sides;
There are cases where an irrational square root is needed, particularly square root of 2. Q: Tablet YBC 7289 approximates the square root of 2, but how was this determined?
Background: There is, however, an interesting tablet, YBC 7289, on which is drawn a square with side indicated as 30 and two numbers, 1:24,51,10 and 42;25,35, written on the diagonal (Fig. 1.15). The product of 30 by 1;24,51,10 is precisely 42;25,35. It is then a reasonable assumption that the last number represents the length of the diagonal and that the other number represents A: Begin with the algebraic identity (x+y)^2=x^2+2xy+y^2 whose validity was discovered by the Babylonians. Now given the sqare of area N, for which one wants the side square root of N... 1.) chose regular value a < but close to the desired result. b=N-a^2 2.) find c such that 2ac+c^2 is as close to b 3.) if a^2 is close enough to N then c^2 will be small in relation to 2ac so c can = (1/2)b(1/a), that is square root of N = square root of (a^2+b) = a + (1/2)b(1/a) The picture shows the geometric version of square root of N = square root of (a^2+b) = a + (1/2)b(1/a). In notes (a+b)^2 = a^2 +2ab +b^2 is the same thing as this.
How did Archimedes approximate pi?
By doubling the number of sides of the hexagon to a 12-sided polygon, then a 24-sided polygon, and finally 48- and 96-sided polygons, Archimedes was able to bring the two perimeters ever closer in length to the circumference of the circle and thereby come up with his approximation
Zeno's paradox of Archilles and the tortoise was really paradoxical to him because .A) he couldn't imagine that tortoises could really run that fast: B) he couldn't imagine why Achilles would agree to become so humiliated; (C) he knew the conclusion was wrong, but he could not see what was wrong with his reasoning that produced the wrong conclusion.
C) he knew the conclusion was wrong, but he could not see what was wrong with his reasoning that produced the wrong conclusion.
Babylonian mathematicians expressed numbers A) in base 2: B) in base 10: C) in base 60.
C) in base 60.
The Platonic solids have their faces bounded by regular A) triangles, squares, or hexagons; B) triangles, squares, or heptagons; C) triangles, squares, or pentagons.
C) triangles, squares, or pentagons.
According to some authors (see the Wikipedia entry on square root of 2), a Pythagorean, a member of the Pythagorean school founded by Pythagoras (~ 572 - 497 BC) in what is now southern Italy (then a Greek colony), pos- sibly Pythagoras himself, had discovered that square root of 2 was an irrational number an incommensurate "magnitude"), but the Pythagoreans decided to keep it a secret known only to members of the Pythagorean school. According to legend, the proof that square root of 2 is irrational (or "incommensurate") was revealed by Hippasus, who subsequently A) was given the title "Court Mathematician" by Alexander the Great; B) became admired as a genius; C) was murdered for divulging the secret.
C) was murdered for divulging the secret.
Although we seem to have only indirect evidence based on teats written centuries after the putative events, the theorem that the sum of consecutive odd natural numbers, starting with 1, is always a square number, was first shown by A) Archimedes B) Euclid: C)Pythagoras.
C)Pythagoras.
Can the following be constructed with a straight edge and compass? Given a cube, a^3; what is b such that b^3=2a^3?
Cannot construct with straight edge and compass (doubling of a cube)
Apollonius is known for his contributions to
Conic sections and motion of planets
476 BC: End of the West Roman Empire. Who was the last Emperor of the whole Roman empire and what began after this?
Constantine: last emperor to the whole roman empire (also brought Christianity to Roman Empire) Beginning of the dark ages (intellectual dark ages).
What was the capital of the East Roman Empire?
Constantinople
Who brought back the idea that the sun is the center of the universe?
Copernicus (sometimes spelt with K)
What was Copernicus famous for?
Copernicus was a Polish scientist who first promoted the theory that planets revolve around the sun.
What did Copernicus do?
Created heliocentric theory (sun is center)
What is Archimedes' most proud accomplishment?
Cylinder + Sphere... On his tombstone. a sphere has two-thirds the volume of its circumscribing cylinder.
Diocles showed how to construct a parabola with given focal length. His construction in effect uses the focus-..... property of a parabola, that the points of the parabola are equally distant from the focus and a given straight line called the .....
Diocles showed how to construct a parabola with given focal length. His construction in effect uses the focus-directrix property of a parabola, that the points of the parabola are equally distant from the focus and a given straight line called the directrix (mentioned in class during Appollonius section)
What was the Ptolemaic model?
Each planets move on epicycles whose center moved around Earth on larger deferents (geocentric model)
How did Apollonius used conic sections to describe the motion of plants?
Epicycles of Apollonius: Basic circle, earth is in the center. On that basic circle we imagine there is another basic circle (it's center moved along the big circle). Meanwhile plant moved around smaller circle....weird shape. Stars fixed.
Elipse Complete the formula (involving directrix)- distance from point to P times M(P) =
Epsilon times the distance between the point and F2
Who that the cone with the same base and height as a cylinder was one third of the cylinder, but could not find the ratio of a sphere to the circumscribed cylinder.
Euclid
Who said the following? 1.) Any composite number is measured by some prime number. 2.) Any number either is prime or is measured by some prime number.
Euclid PROPOSITION VII-31 Any composite number is measured by some prime number. PROPOSITION VII-32 Any number either is prime or is measured by some prime number.
How did Euclid construct a regular triangle?
Euclid I.1. To construct a regular triangle on a given line segment AB:- With radius equal to d(A, B) draw circles centred at A and B. They intersect at two points, C and C', one on each side of the line, each of which with A and B complete a regular triangle. Euclid I.10. To bisect a given line segment AB:- Construct as above and join CC'. This line bisects AB at M. If d(A,B) is 1 then by applying the theorem of Pythagoras to the triangle ACM we find that the altitude CM of the regular triangle with unit side is √3/2. Euclid I.11. To draw a perpendicular to a given line at a given point M:- Draw a circle centre M, to cut the line at A and B. Then construct the bisector, as above. Euclid I.12. To draw a perpendicular to a given line from a point P outside it:- Draw a circle, centre P, to cut the line at A, B. Then construct the bisector as above.
How did Euclid construct a regular tetragon? (square)
Euclid IV.6 To draw squares in a given circle:- Draw perpendicular diameters and join their ends. Euclid IV.7 To draw squares round a given circle:- Draw perpendicular diameters and perpendiculars at the ends of the diameters. Euclid IV.8 To draw the circle in a given square:- Draw the diagonals, these meet at the centre; perpendiculars from the centre to the sides are radii of the required circle. Euclid IV.9 To draw the circle round a given square:- Draw diagonals, these are diameters of the required circle and meet at the centre.
How did Euclid construct an elipse vs. Apollonius?
Euclid used compass and straight edge only. Here using role, free from constraint. Euclidean geometry in general terms.
Euclid's Book 1: Finish the Definition 1.) A ..... is that which has no part. 2.) A ...... is breadthless length. 3.) The extremities of a line are ........ 4.) A ...... is a line which lies evenly with the points on itself. 5.) A ...... is that which has length and breadth only. 6.) The extremities of a surface are ...... 7.) A ...... is a surface which lies evenly with the straight lines on itself.
Euclid's Book 1: Finish the Definition 1.) A point is that which has no part. 2.) A line is breadthless length. 3.) The extremities of a line are points. 4.) A straight line is a line which lies evenly with the points on itself. 5.) A surface is that which has length and breadth only. 6.) The extremities of a surface are lines. 7.) A plane surface is a surface which lies evenly with the straight lines on itself.
Euclid's Book 1: Finish the Definition 21.) Further, of trilateral figures, a ...... is which has a right angle; an ........ one which an obtuse angle and an ..... has three acute sides 22.) Of quadrilateral figures, a ...... is that which both equilateral and right angles; an ...... is right-angled but not equilateral; a .....; "the rest" Let other quadrilaterals be called ........ 23.) ....... straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.
Euclid's Book 1: Finish the Definition 21.) Further, of trilateral figures, a right-angled triangle is which has a right angle; an obtuse-angles triangle one which an obtuse angle, and an acute-angled triangle has three acute sides 22.) Of quadrilateral figures, a square is that which both equilateral and right angles; an oblong is right angled but not equilateral; a rhomboid..."the rest"... Let other quadrilaterals be called trapezia. 23.) Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.
Euclid's Common Notions: Fill in the blanks 1. Things which equal the same thing also ..... 2. If equals are added to equals, then the wholes are .... 3. If equals are subtracted from equals, then the ..... are equal. 4. Things which coincide with one another ...... 5. The whole is greater than .....
Euclid's Common Notions: 1. Things which equal the same thing also equal one another. --> The first Common Notion could be applied to plane figures to say, for instance, that if a triangle equals a rectangle, and the rectangle equals a square, then the triangle also equals the square. Magnitudes of the same kind can be compared and added, but magnitudes of different kinds cannot. For instance, a line cannot be added to a rectangle, nor can an angle be compared to a pentagon. 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. --> Typically interpreted as superposition; if one thing can be moved to coincide with another, then they are equal. 5. The whole is greater than the part. --> could be interpreted as a definition of "greater than"
Who determined the sum of the sum of a geometric progression? What is this formula?
Euclid's result states: (ar^n − a) : Sn = (ar − a) : a. Modern formula: Sn=(a(r^n-1))/r-1
Who invented the method of exhaustion? (proof)
Eudoxus
Fill in the blanks below Euclid Prop: There are (more/less) prime #'s than finite multiple (of the basic unit). Proof - Suppose p1=2, p2=3,....,pN are all the primes there are. Then z=p1p2p3....pN+1 is either ..... or there is p not in set (p1,p2,...pN) that divides z. i.e will never find the largest prime.
Fill in the blanks below Euclid Prop: There are more prime #'s than finite multiple (of the basic unit). Proof - Suppose p1=2, p2=3,....,pN are all the primes there are. Then z=p1p2p3....pN+1 is either a prime number (only divisible by itself and 1) or there is p not in set (p1,p2,...pN) that divides z. i.e will never find the largest prime.
Fill in the following blanks Book 5: Proposition 5: To circumscribe a circle about a given triangle. Let ABC be the given triangle. It is required to circumscribe a circle about the given triangle ABC. Bisect (half) the straight lines AB and AC at the points D and E. Draw DF and EF from the points D and E at right angles to AB and AC. They will then meet within the triangle ABC, or on the straight line BC, or outside BC. First let them meet within at F (center point). Join FB, FC, and FA. Then, since BLANK equals BLANK, and DF is common and at right angles, therefore the base BLANK equals the base BLANK. Similarly we can prove that BLANK also equals AF, so that FB also equals BLANK, therefore the three straight lines BLANK and BLANK and BLANK equal one another. Therefore the circle described with center F and radius one of the straight lines FA, FB, or FC also passes through the remaining points, and the circle is circumscribed about the triangle ABC. CONCLUSION:
Fill in the following blanks Book 5: Proposition 5: To circumscribe a circle about a given triangle. Let ABC be the given triangle. It is required to circumscribe a circle about the given triangle ABC. Bisect (half) the straight lines AB and AC at the points D and E. Draw DF and EF from the points D and E at right angles to AB and AC. They will then meet within the triangle ABC, or on the straight line BC, or outside BC. First let them meet within at F (center point). Join FB, FC, and FA. Then, since AD equals DB, and DF is common and at right angles, therefore the base AF equals the base FB. Similarly we can prove that CF also equals AF, so that FB also equals FC, therefore the three straight lines FA and FB and FC equal one another. Therefore the circle described with center F and radius one of the straight lines FA, FB, or FC also passes through the remaining points, and the circle is circumscribed about the triangle ABC. Let it be circumscribed as ABC. Where AF = CF = BF
What are the two types Egyptian writing styles?
From the beginning of Egyptian writing, there were two styles, the hieroglyphic writing for monumental inscriptions and the hieratic, or cursive, writing, done with a brush and ink on papyrus.
Fill in the following blanks Book 1: Proposition 2: Given a point and a finite straight line, construct a straight line of equal length that ends at the given point. Let A be the given point, and BC the given straight line. It is required to place a straight line equal to the given straight line BC with one end at the point A. [Post. 1, I.1.] Join the straight line AB from the point A to the point B, and construct the equilateral triangle DAB on it. [Post.2] [Post.3] Produce the straight lines AE and BF in a straight line with DA and DB. Describe the circle CGH with center B and radius BC, and again, describe the circle GKL with center D and radius DG. [I.Def.15] Since the point B is the center of the circle CGH, therefore BLANK equals BLANK. Again, since the point D is the center of the circle GKL, therefore BLANK equals BLANK. [C.N. BLANK] And in these DA equals DB, therefore the remainder AL equals the BLANK. [C.N. BLANK] But BC was also proved equal to BG, therefore each of the straight lines BLANK and BLANK equals BLANK. And things which equal the same thing also equal one another, therefore AL also equals BLANK. Therefore the straight line AL equal to the given straight line BC has been placed with one end at the given point A.
Given a point and a finite straight line, construct a straight line of equal length that ends at the given point. It is required to place a straight line equal to the given straight line BC with one end at the point A. Post. 1, I.1. Join the straight line AB from the point A to the point B, and construct the equilateral triangle DAB on it. Post.2 Post.3 Produce the straight lines AE and BF in a straight line with DA and DB. Describe the circle CGH with center B and radius BC, and again, describe the circle GKL with center D and radius DG. I.Def.15 Since the point B is the center of the circle CGH, therefore BC equals BG. Again, since the point D is the center of the circle GKL, therefore DL equals DG. C.N.3 And in these DA equals DB, therefore the remainder AL equals the remainder BG. C.N.1 But BC was also proved equal to BG, therefore each of the straight lines AL and BC equals BG. And things which equal the same thing also equal one another, therefore AL also equals BC. Therefore the straight line AL equal to the given straight line BC has been placed with one end at the given point A.
How did Euclid construct a pentagon?
Given the isosceles triangle with base angles double the vertex angle, the inscribing of the regular pentagon in a circle is now straightforward. Euclid first inscribed the isosceles triangle ACE in the circle. Next, he bisected the angles at A and E. The intersection of these bisectors with the circle are points D and B, respectively. Then A, B, C, D, E are the vertices of a regular pentagon.
What led to the disappearance of both the native Egyptian writing forms? How did they eventually become readible?
Greek domination of Egypt in the centuries surrounding the beginning of our era was responsible for the disappearance of both of these native Egyptian writing forms. Jean Champollion (1790-1832) was able to begin the process of understanding Egyptian writing through the help of the Rosetta stone—in hieroglyphics and Greek as well as the later demotic writing, a form of the hieratic writing of the papyri
What happened to Socrates?
He attracted lots of attention from influencing youth (had a lot of authority & worshipped), which made citizens angry. He was forced to drink poison and die.
What was Galileo known for?
He provides further evidence to Copernicus's theory by the invention of the telescope.
How did Archimedes die?
He was killed by a Roman soldier in Syracuse.
The astronomical model in which the Earth and planets revolve around the Sun at the center of the universe.
Heliocentric model
Who was among the first to attack the the doubling of the cube and squaring of the circle problems?
Hippocrates As to the first of these, Hippocrates perhaps realized that the problem was analogous to the simpler problem of doubling a square of side a. That problem could be solved by constructing a mean proportional b between a and 2a, a length b such that a : b = b :2a, for then b2 = 2a2. In any case, ancient accounts record that Hippocrates was the first to come up with the idea of reducing the problem of doubling the cube of side a to the problem of finding two mean proportionals b, c, between a and 2a.
How did Hippocrates explain the doubling of the cube problem?
Hippocrates perhaps realized that the problem was analogous to the simpler problem of doubling a square of side a. That problem could be solved by constructing a mean proportional b between a and 2a, a length b such that a : b = b :2a, for then b2 = 2a2. From the fragmentary records of Hippocrates' work, it is evident that he was familiar with performing such constructions. In any case, ancient accounts record that Hippocrates was the first to come up with the idea of reducing the problem of doubling the cube of side a to the problem of finding two mean proportionals b, c, between a and 2a. Find b and c such that a : b = b : c = c :2a, then.. =a:2a = 1:2
What guided Archimedes in his discoveries?
His mechanical experiments
What is Archimedes known for?
Invented the pulley, the lever, water lifts for irrigation, and pi
In particular, Archimedes is responsible for the first proof of the law of the ....... (Fig. 4.1) and its application to finding ......
Law of the lever, finding centers of gravity
Who solved the doubling of the cube problem by cutting planes?
Malechen Can't be constructed at time of Euclid with compass and straight edge. Connecting to Hippocrates' solving of the doubling of the cube using proportions, Find b and c such that a : b = b : c = c :2a, =a:2a = 1:2 He got these two proportions b and c by finding the intersection of curves from the following x^2=ay and y^2=2ax and x*y=2a^2?
Who is credited with the discovery of conic sections
Menaechmus is credited with the discovery of conic sections. Around the years 360-350 B.C.; it is reported that he used them in his two solutions to the problem of "doubling the cube"
In Egypt, most of the early writing concerned....
Much of the earliest writing concerned accounting, primarily of various types of goods This involved arithmetic and to some extent elementary algebra
What are two important cities founded by Greeks (before Alexander the Great) around the Mediterranean?
Napes (Italy) Marseille (France) - conquered by Roman forces under Julius Caesar
Did Apollonius connect conic sections to the plants?
Nope
What are Nicomadus and Diophantus known for today?
Number theiry
What is Archimedes principle of buoyancy?
Objects more dense than water will sink. Objects less dense than water will float. Objects with the same density as water will suspend, neither floating or sinking.
Fill in the following blanks Book 1: Proposition 1: On a given finite straight line to construct an equilateral triangle. Let AB be the given finite straight line. Thus it is required to construction equilateral triangle on the straight line AB. With centre A and distance AB let the circle BCD be described [Post. BLANK ] again, with centre B and distance BA let the circle ACE be described; [Post. BLANK] and from the point C, in which the circles cut one another, to the points A, B let the straight lines CA, CB be joined. [Post. BLANK] Now, since the point A is the centre of the circle CDB, BLANK is equal to BLANK. [Def. BLANK] Again, since the point B is the centre of the circle CAE, BLANK is equal to BLANK. [Def. BLANK] .......therefore each of the straight lines BLANK, BLANK is equal to BLANK. And things which are equal to the same thing are also equal to one another; [COMMON NOTION BLANK] therefore CA is also equal to CB. Therefore the three straight lines BLANK, BLANK, BLANK are equal to one another. Therefore the triangle ABC is equilateral; and it has been constructed on the given finite straight line AB.
On a given finite straight line to construct an equilateral triangle. Let AB be the given finite straight line. Thus it is required to construction equilateral triangle on the straight line AB. With centre A and distance AB let the circle BCD be described [Postulate 3] again, with centre B and distance BA let the circle ACE be described; [Post. 3] and from the point C, in which the circles cut one another, to the points A, B let the straight lines CA, CB be joined. [Post. 1] Now, since the point A is the centre of the circle CDB, AC is equal to AB. [Def. 15] Again, since the point B is the centre of the circle CAE, BC is equal to BA. [Def. 15] But CA was also proved equal to AB; therefore each of the straight lines CA, CB is equal to AB. And things which are equal to the same thing are also equal to one another; [C.N. 1] therefore CA is also equal to CB. Therefore the three straight lines CA, AB, BC are equal to one another. Therefore the triangle ABC is equilateral; and it has been constructed on the given finite straight line AB.
Why did Hippasus die?
One day, a man called Hippasus discovered that the square root of 2 was irrational. This went so much against the Pythagorean world view that they took him out to sea and threw him overboard.
Fill in the conic section: (or point) ... results from cutting top cone with a plane on a steep diagonal. ....results from cutting one cone with a plane perpendicular ...results from plane perpendicular between the two cones ...results from cutting lower cone with a plane on a non-steep diagonal ...results from cutting cones on a diagonal along one of the lines that intersect them.
PARABOLA results from cutting top cone with a plane on a steep diagonal. CIRCLE results from cutting one cone with a plane perpendicular POINT results from plane perpendicular between the two cones ELIPSE results from cutting lower cone with a plane on a non-steep diagonal STRAIGHT LINE results from cutting cones on a diagonal along one of the lines that intersect them.
What was the Persian capital Alexander the Great conquered? What
Persepolis (the capital of the Persian empire) was sacked and burned by Alexander the Great in 330 BC.
Who founded the Academy in Athens?
Plato
Who played into the idea that, "In addition to physical reality, there is math reality that we discover." ?
Plato
Who made the Greeks obsessed with circles and spheres?
Plato - polygons.
Who introduces polyhedrons?
Plato's work on polyhedra and their relationship to the elements had a significant impact on later philosophers and mathematicians, including Euclid
Who said: a body immersed in a fluid experiences an upthrust equal to the weight of the fluid displaced, and this is fundamental to the equilibrium of a body floating in still water. A body floating freely in still water experiences a downward force acting on it due to gravity.
Principle of Archimedes (buoyancy)
What is the problem if 1 is considered prime? Did Euclid consider 2 as a prime number?
Problem: If 1 is a prime #, it disproves many theorems (the way they were written) 2 is prime because it's not divisible by anything but itself
Who concluded that the square root of 2 is irrational (not a natural #)? What method of proof was used?
Proof through contradiction Hippasus discovered that square root of 2 is an irrational number, that is, he proved that square root of 2 cannot be expressed as a ratio of two whole numbers. Pythagoras Theorem applied to a right-angled triangle whose sides are 1 unit in length, yields a hypothenuse whose length is equal to square root of 2
Who wrote the Almagest?
Ptolemy (Greek: Klaudios Ptolemaios; Latin: Claudius Ptolemaeus; c. AD 100- c. AD 160)
Who created the geocentric model?
Ptolemy (earth is the center of the universe)
Who believed that a number was a quantity that could be expressed as a ratio of two integers (a rational number)
Pythagoras
Who had gathered around him a group of disciples, in what was considered both a religious order and a philosophical school
Pythagoras
Who represented numbers by dots, or more concretely, by pebbles?
Pythagoras (Greeks)
Who can be attributed to the notion that a "number was the substance of all things," that numbers, that is, positive integers, formed the basic organizing principle of the universe.
Pythagoras - the Pythagoreans (mathematical doctrines of his school) What the Pythagoreans meant by this was not only that all known objects have a number, or can be ordered and counted, but also that numbers are at the basis of all physical phenomena. For example, a constellation in the heavens could be characterized by both the number of stars that compose it and its geometrical form, which itself could be thought of as represented by a number.
How was the following represented? And by who? (a) The sum of even numbers is even. (b) An even sum of odd numbers is even. (c) An odd sum of odd numbers is odd.
Pythagoras, using pebbles
How did the Mesopotamians record their math?
Records of their math in clay tablets
EUCLID In terms of numbers (not geometry) S (subscript k) := 1 + x + x^2 + .... x^k = ? (simplified) S (subscript k) = S (subscript k-1) + ? = 1 + ? = S(subscript k-1) =
S (subscript k) := 1 + x + x^2 + .... x^k = 1 + x(1 + x +... x^k-1) S (subscript k) = S (subscript k-1) + x^k = 1 + xS(subscript k-1) = S(subscript k-1)(1-x)= 1-x^k or S(subscript k-1)= (1-x^(k+1))/(1-x)
Q: 1+3+5+.....+(2N-1) = N^2 The theorem that the square of an even number is even, while the square of an odd number is odd results from what? Who simplified this?
Squares themselves could also be represented using pebbles. For example, the square of 4, it is easy to see that the next higher square can be formed by adding a row of dots around two sides of the original figure. There are 2 * 4 + 1= 9 of these additional dots. The Pythagoreans generalized this observation to show that one can form squares by adding the successive odd numbers to 1. 1+ 3 + 5 = 3^2 (9)
Q: Vol (cylinder)/Vol(sphere)=? Q: Area(cylinder)/Area(sphere)=? What is surprising about these results?
Surprising... numerical values! (Archimedes)
Although Aristotle emphasized the use of ........ as the building blocks of logical arguments, Greek mathematicians apparently never used them. They used other forms, as have most mathematicians down to the present. What is this form of logic based on?
Syllogisms This form of logic is based on propositions, statements that can be either true or false, rather than on the Aristotelian syllogisms
(Egpyt) Problem 50 of the Rhind Papyrus reads, "Example of a round field of diameter 9. What is the area?
Take away 1/9 ofthe diameter; the remainder is 8. Multiply 8 times 8; it makes 64. Therefore, the area is 64." In other words, the Egyptian scribe was using a procedure described by the formula A= (d − d/9)^2 = [(8/9)d]^2. A comparison with the formula A= (π/4)d^2 shows that the Egyptian value for the constant pi was 3.1605 which is pretty close! We don't know how they figured this out. A=(9/9d)^2 = 3.1605(d/2)^2
(Beginning of Mathematics in Greece) Who impressed Egyptian officials by determining the height of a pyramid by comparing the length of its shadow to that of the length of the shadow of a stick of known height?
Thales of Miletus
Who can be credited for his application of the angle-side-angle criterion of triangle congruence to the problem of measuring the distance to a ship at sea?
Thales of Miletus
Who discovered that the moon is illuminated by the reflection of the sun? This same person also predicted a solar eclipse in 585 BC.
Thales of Miletus
Who is credited with discovering the theorems that the base angles of an isosceles triangle are equal and that vertical angles are equal and with proving that the diameter of a circle divides the circle into two equal parts.
Thales of Miletus
What is the principle behind the Archimedean spiral?
The Archimedean spiral has the property that any ray from the origin intersects successive turnings of the spiral in points with a constant separation distance (equal to 2πb if θ is measured in radians), hence the name "arithmetic spiral". One turn, one unit, another turn, another intersection.... Example: Gramophone (record player) -Archimedean Spiral
What types of fractions did the Egyptians deal with? Were there any exceptions?
The Egyptians only dealt with unit fractions or "parts" (fractions with numerator 1) and a natural number in the denominator, with the single exception of 2/3,
What were Euclid's books called? How many books are there? How are the books broken up?
The Elements "The basics/principles of mathematics" - the most important mathematical text of Greek times and probably of all time. Euclid's Elements is a work in thirteen books. The first six books form a relatively complete treatment of two-dimensional geometric magnitudes while Books VII-IX deal with the theory of numbers, in keeping with Aristotle's instructions to separate the study of magnitude and number
Mesopotamia: The Mesopotamian civilization is perhaps a bit (older/younger) Civilization - ... years ago Around 2000 BC what happened? 1700 BC: who decided to expand their kingdom and create an empire?
The Mesopotamian civilization is perhaps a bit older than the Egyptian. The Ur Dynasty collapsed around 2000 BC, the small city-states that succeeded it still demanded numerate scribes. By 1700 BC, Hammurabi, the ruler of Babylon, one of these city-states, had expanded his rule to much of Mesopotamia and instituted a legal system to help regulate his empire
There was no mention of their calculations for a pyramid but what was the Egyptian formula for the volume of a truncated pyramid? Where was it found?
The Moscow Papyrus, however, does have a fascinating formula related to pyramids, namely, the formula for the volume of a truncated pyramid (problem 14): V=1/3h(a^2+ab+b^2). If we set b=0 this corresponds to the modern formula V=1/2a^2h
The active political life of the city-states encouraged the development of argumentation and techniques of persuasion. And there are many examples from philosophical works, especially those of ......... (late sixth century BC) and his disciple ...... of Elea (fifth century BC), that demonstrate various detailed techniques of argument.
The active political life of the city-states encouraged the development of argumentation and techniques of persuasion. And there are many examples from philosophical works, especially those of Parmenides (late sixth century BC) and his disciple Zeno of Elea (fifth century BC), that demonstrate various detailed techniques of argument.
What major event began the Dark Ages
The fall of the West Roman Empire - various Germanic tribes conquered. Came from North, West and East. Listed in history as different groups but all came from Scandinavia. Other groups contributed too.
The first two in Archimedes' sequence of propositions leading to the law of the .... are very easy: PROPOSITION 1 Weights which balance at equal distances are ...... PROPOSITION 2 Unequal weights at equal distances will not balance but will incline toward the ........ . . . .
The first two in Archimedes' sequence of propositions leading to the law of the lever are very easy: PROPOSITION 1 Weights which balance at equal distances are equal. PROPOSITION 2 Unequal weights at equal distances will not balance but will incline toward the greater weight.
Egyptian Pharaohs were viewed as.....
The legacy of the first pharaohs included an elite of officials and priests, a luxurious court, and for the kings themselves, a role as intermediary between mortals and gods
The most famous and prominent mathematician of antiquity and the Father of Geometry, Euclid lived and flourished in what city during the reign of who? He worked in the what?, which at the time was the world's most illustrious scientific and cultural center next to the what?.
The most famous and prominent mathematician of antiquity and the Father of Geometry, Euclid lived and flourished in Alexandria during the reign of Ptolemy I. He worked in the Library of Alexandria, which at the time was the world's most illustrious scientific and cultural center next to the Academy.
What are the three ancient impossible construction problems of Euclidean geometry?
The three classical construction problems of antiquity are known as "squaring the circle'', "trisecting an angle'', and "doubling a cube''.
The three types of conic sections are....
The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though it was sometimes called as a fourth type. (Apollonius)
According to Pythagoras, what did 10 represent?
The universe.
Who filled in these blanks with their discoveries? Thus, one can double the square by planar means (as in Elements, Book II, proposition 14), but one cannot ......., although a solid construction is possible (as given above). Similarly, the bisection of any angle is a planar construction (as shown in Elements, Book I, proposition 9), but the general ......... of the angle is of the solid type.
Thus, one can double the square by planar means (as in Elements, Book II, proposition 14), but one cannot double the cube in such a way, although a solid construction is possible (as given above). Similarly, the bisection of any angle is a planar construction (as shown in Elements, Book I, proposition 9), but the general trisection of the angle is of the solid type. Apollonius and the use of conic sections!!!!!
Euclid had many hidden assumptions! Concerning Book 1: Proposition 1 what hidden assumption did we use?
Two nonparallel lines intersect (postulates for lines)... *two circles with equal radius interest (must also postulate for circle)
Our knowledge of Egyptian math is based largely on what?
Two papyrus (scrolls/books). -Rhind Papyrus (18 feet long and 13 inches high). named for the Scotsman A. H. Rhind (1833-1863) who purchased it at Luxor in 1858 -Moscow Papyrus (15 feet long and 3 inches high). Purchased in 1893 by V. S. Golenishchev (d. 1947) who later sold it to the Moscow Museum of Fine Arts.
Volume of a Sphere Formula
V=4/3πr³
In a boat in a pool and throw a stone in the water. What happened to the water level?
Water level will go down.
Egyptian division example: Q: How to distribute fairly 6 loaves of bread to 10 workers? A: how would they write this?
Whenever we would use a nonunit fraction, they simply wrote a sum of unit fractions A: 1/2 + 1/10 cut each loaf in half and one loaf in tenths
What convinced Galileo that Copernicus was correct?
With his telescope, he would observe then record what he saw. He noticed that things such as the moon change position, and so do planets relative to stars. He saw Jupiter and little dots behind. He then noticed that the 4 moons of Jupiter rotated around the planet, just like our moon (what Copernicus said) which confirmed!!!
What convinced Galileo that Copernicus was correct?
With his telescope, he would observe then record what he saw. He noticed that things such as the moon's phase changes and the plants move too relative to stars. He saw Jupiter and little dots behind. He then noticed that the 4 moons of jupiter rotated around the planet, just like our moon (what Coponicus said) which confirmed!
When did writing begin in Mesopotamia relative to Egypt? Why did writing begin there?
Writing began in Mesopotamia, quite possibly in the southern city of Uruk, at about the same time as in Egypt, namely, at the end of the fourth millennium BC In fact, writing began there also with the needs of accountancy, of the necessity of recording and managing labor and the flow of goods.
One of the reasons Aristotle had such an extended discussion of the notions of infinity, indivisibles, continuity, and discreteness was that he wanted to refute the famous paradoxes of WHO?
Zeno
Who produced "lots of paradoxes"?
Zeno
The Greek philosopher and logician Aristotle was (a) a student of Plato and the teacher of Archimedes (b) a student of Plato and the teacher of Alexander the Great (c) a student of Pythagoras and the teacher of Ptolemy Alexandria
a student of Plato and the teacher of Alexander the Great
If I have an elipse with point P connecting the two rays F1 to F2 between two directrix, if I move point P along the outside of the elipse....
always add to the same. The sum of the two distances between the directrix stays constant
Egyptian mathematicians expressed numbers in base...
base 10
(Egypt) What was the number system used for the hieroglyphic system?
base 10 system
Mesopotamia -used base..... -They knew the law of...
base 60 they knew the law of Pythagoras/had a list with the Pythagorean triples
(Egypt) what was the number system used for the hieroglyphic system?
ciphered system
Given point, directrix and Epsilon we can construct a
conic section
For conic sections, the ratio of the distances of a point on the section to the directrix closest to the focal point over that distance from the point to that focus is....
constant
In fact, there are several problems in the Rhind Papyrus where the scribe used the rule V = Bh to calculate the volume of a ....... where B, the area of the base, is calculated by this circle rule. The scribes also knew how to calculate the volume of a ........ given its length, width, and height.
cylinder rectangular box
a straight line the distance to which from any point of a conic section is in fixed ratio to the distance from the same point to a focus.
directrix (time of Apollonius)
For Apollonius, a conic surface was what is today called a ......
double oblique cone.
What did Galileo do?
he provides further evidence to Copernicus's theory by the invention of the telescope
Ptolemy formulated a geocentric model of the solar system which remained the generally accepted model in the Western and Arab worlds until it was superseded by the
heliocentric solar system of Copernicus.
What ancient Egyptian writing style were the papyrus written in?
hieratic system.
Babylonia algorithm for square root of m = x
m=x^2 or x=m/x
Fill in the blanks under the n-gon column below Construction of regular n-gons with compass and straight edge n n-gon 3 4 5 6 7 ....17? Which of these n-gons did Euclid not construct in his books? Why? Also, who eventually did?
n n-gon 3. equilateral triangle 4. square 5. pentagon 6. hexagon 7. heptagon 17. He can construct this! There are certain numbers we can construct. Whether Euclid was aware of a construction for the heptagon, however, is not known. In any case, that construction, the first record of which is in the work of Archimedes, would for Euclid be part of advanced mathematics, rather than part of the "elements," because it requires tools other than a straightedge and compass.
Euclid and Number Theory Book VII of the Elements is the first of three dealing with the elementary theory of numbers. The new start that Euclid made in Book VII is evidence of his desire to stick with Aristotle's clear separation between magnitude and number. What did "number" mean to Euclid? What is its unit? Please find the obstacle below: u * n u * n^2 u * n^3 u * n^4
natural number, with unit 1. obstacle: the # of dimensions they understood. Beyond the third dimension, what is the point! To the 4th, what does that mean to them?! (Think in terms of pebbles). This is the same reason why negatives made no sense. ALL GOES BACK TO GEOMETRY
Did Egyptians use negative numbers? Did Egyptians use zero?
no yes, dealing with supply=demand
Please describe the Egpytians procedure for determining the surface area of a hemisphere
problem 10 of the Moscow Papyrus: "A basket with a mouth opening of 4 1/2 in good condition, oh let me know its surface area. First, calculate 1/9 of 9, since the basket is 1/2 of an egg-shell. The result is 1. Calculate the remainder as 8. Calculate 1/9 of 8. The result is 2/31/61/18 [that is, 8/9]. Calculate the remainder from these 8 after taking away those [8/9]. The result is 7 1/9. Reckon with 71/9 four and one-half times. The result is 32. Behold, this is its area. Evidently, the scribe calculated the surface area S of this basket of diameter d = 41/2 by first taking 8/9 of 2d, then taking 8/9 of the result, and finally multiplying by d. As a modern formula this result would be S=2(8/9d)^2
As to geometry, the Egyptian scribes certainly knew how to calculate the areas of ......, ......., and ..... by our normal methods. It is their calculation of the area of a ....., however, that is particularly interesting.
rectangles, triangles and trapezoids circle
Ptolemy formulated a geocentric model of the solar system which remained the generally accepted model in the western and Arab worlds until it was superseded by
the heliocentric solar system of Copernicus
What did the building of Egyptian monuments/architecture lead to?
the need for geometry
The volume of a cone = Who calculated the volume of a cone? And how was this proven?
volume of a cone = 1/3piR^2 Archimedes, method of exhaustion
