We'll start with the ancient Greeks. The history of mathematics is a vast subject so we have to be selective. The ancient Greeks are the first modern mathematicians. In any event, that's how we trace our lineage.
Euclid lived in Alexandria around 300 B.C. Not the one in Virginia, the one in Egypt. What's a Greek city doing in Egypt? Alexander the Great (356-323 B.C.) conquered a lot of territory. Alexandria is named after him.
Euclid wrote The Elements. Mostly we think of him in connection with geometry, but there is a lot of number theory in The Elements also: the Euclidean algorithm, the infinitude of the primes, perfect numbers, and so on.
There are two things that stand out to me in the geometry of Euclid: the notion of proof and the notion of construction (or algorithm). A proof is an argument that something is true. Euclid required that proofs start from things that were accepted as true and proceed step by step to the thing being proved. The accepted things are called axioms or postulates; the steps are called deductions. Of course the logic underlying the deductions has to be accepted also.
Two of Euclid's postulates for geometry were
- Postulate 4. All right angles are equal.
- Postulate 5. If a straight line falling on two straight lines make 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 the angles are less than two right angles.
- Postulate 1. To draw a straight line from any point to any point.
- Postulate 2. To produce a finite straight line continuously in a straight line.
- Postulate 3. To describe a circle with any center and distance.
A Euclidean construction is one that can be done using only Postulates 1, 2 and 3. Euclid's Proposition 1 says that you can construct an equilateral triangle on a given finite straight line. This restrictive notion of what a construction is lies behind the three famous mathematical problems of antiquity given on page 40 of our text. Euclid's Proposition 9 says that you can bisect an angle (divide it into two equal parts). One of the famous problems is to trisect an angle, that is, to divide it into three equal parts. But you are only allowed to use the constructions of the first three postulates.
Angle trisection was shown to be impossible in 1837 by Pierre Wantzel. He showed that you can't construct a 20 degree angle (but you can construct a 60 degree angle). The reason is that the cosine of 20 degrees is a root of the polynomial 8X3 - 6 X2 - 1 while all the numbers you can construct by Euclidean constructions are roots of polynomials of degree a power of 2, because the new numbers can always be expressed in terms of square roots of the old numbers (when you find the intersection of a line and a circle, or two circles, you solve a quadratic equation).
The same reasoning shows you can't duplicate a cube because that involves constructing a number that is a root of the polynomial X3 - 2.
Ferdinand von Lindemann showed that you can't square a circle in 1882. He did that by showing that the number pi is transcendental. A number is algebraic if it is a root a polynomial with integer coefficients. So the square root of 2, the cube root of 2, and the cosine of 20 degrees are all algebraic. A number is transcendental if it is not algebraic---so it is not a root of any polynomial with integer coefficients. Every number you get by Euclidean constructions is algebraic, so you can't get a line of length pi, which you could if you could square a circle of radius 1.
Lindemann had 60 Ph.D. students. The most famous was David Hilbert who, among many other things, wrote a book on geometry in which he made explicit many of the assumptions that were implicit in Euclid.
Another ancient question is that of constructing regular polygons. Carl Friedrich Gauss, possibly the greatest mathematician of all time, showed how to construct a regular 17-gon when he was around 20 years old (he was born in 1777). The Greeks knew how to construct regular n-gons for n = 3, 4, 5, 6, 8, 10, 12, 15, 16 and 20, but not 17. You can't construct a regular n-gon for n = 7, 9, 11, 13, 14, 18, or 19.
Go to the Mathematics Genealogy Project and find out the mathematical ancestry of your instructor.
Archimedes 287-212 B.C. Lived in Syracuse which is on the island of Sicily just off the toe of the boot that is Italy. "Give me a place to stand and I will move the Earth"
Heraclitus and Parmenides were presocratic philosophers who lived around 500 B.C. (Socrates died in 399 B.C.) Heraclitus believed that everything is changing. He said, "You can't step into the same river twice." The idea there is that the second time you step into it, it isn't the same river because it has changed so much in the meantime. Parmenides, on the other hand, believed that change and motion were illusions. Zeno's paradoxes were meant to support Parmenides' ideas by showing that the idea of motion was self-contradictory.
Incommensurable magnitudes. The magnitudes we have in mind here are lengths. We say that a length u can be used to measure a length a if we can cut a up into a bunch of pieces each of which has length u. We say that two lengths a and b are commensurable if we can find a length u that measures both a and b. The Pythagoreans thought that all lengths should be commensurable, but it turned out that the side of an isosceles right triangle is not commensurable with its hypotenuse. The Pythagorean theorem relates this to the irrationality of the square root of 2. Indeed, if a = mu and b = nu, then a² + a² = b², so 2m² = m² + m² = n² which shows that 2 = (m/n)² so the square root of 2 is equal to m/n, a rational number.
[The difficulties in the study of the infinite arise because] we attempt, with our finite minds, to discuss the infinite, assigning to it those properties which we give to the finite and limited; but this ... is wrong, for we cannot speak of infinite quantities as being the one greater or less than or equal to another.
Galileo, Dialogues Concerning Two New Sciences 1638
A discussion of Galileo's paradox and Georg Cantor can be found in Infinite Reflections.
Here is Carl Friedrich Gauss on completed infinity: As to your proof, I must protest most vehemently against your use of the infinite as something consumated, as this is never permitted in mathematics. The infinite is but a figure of speech . . . .
For more discussion about infinity and its history see You can't get there from here.
Here are the other books by Dirk J. Struik that are in the WebLuis catalog of the State University System of Florida.
- The origins of American science (New England). 1957 First published as "Yankee Science in the Making" 1948
- Lectures on classical differential geometry. 1950
- Lectures on analytic and projective geometry. 1953
- A source book in mathematics, 1200-1800. 1969
- Birth of the Communist manifesto, with full text of the Manifesto, all prefaces by Marx and Engels, early drafts by Engels and other supplementary material. 1971
I used his differential geometry book as a student. It's a pretty little book. Differential geometry is a combination of analysis (calculus) and geometry. Struik's Marxist leanings got him into trouble with the infamous House UnAmerican Activities Committee in 1951. He refused to answer their questions, invoking the Fifth Amendment. He was suspended from MIT for several years. I think you can see the influence of Marxism from time to time in our text in some of the offhand remarks.
Struik died last year at the age of 106.
Square root of 2. There are two parts to the argument that the square root of 2 is irrational (cannot be written as a fraction, that is, as a quotient of two integers).
- If the square of a rational number p/q is 2, then both p and q are even. This uses properties of even and odd numbers under multiplication (the square of an even number is even, the square of an odd number is odd).
- Every fraction can be written in lowest terms, that is, with p and q having no common factor.
The theory of proportions is ascribed (by Archimedes) to Eudoxus. It is presented in Book V of Euclid's Elements. Euclid uses this notion in Book VI when he proves things like Proposition 5: If two triangles have their sides proportional, then the angles opposite corresponding sides will be equal.
What does it mean for the sides to be proportional? This is a relationship among four sides, a, a', b', and b'. It is expressed by saying, for example, a is to a' as b is to b'. Or that the ratio of a to a' is the same as the ratio of b to b'. What is a ratio? Euclid says, in Definition 3,
- A ratio is a sort of relation in respect of size between magnitudes of the same kind.
Two lengths are "magnitudes of the same kind", as are two areas. The notion of ratio is pretty muddy here, but it is clarified in Definition 5 where Euclid says what it means for two ratios to be equal. It really makes no difference what a ratio is, what counts is when they are equal. In fact, if you understand when two ratios are equal, you will have a handle on what a ratio is.
A ratio is given by two magnitudes. A simple ratio occurs when the first magnitude is twice as big as the second. That ratio is sometimes written 2:1. A more complicated ratio is when twice the first magnitude is equal to three times the second. That ratio is written 3:2. Why not 2:3? Well, for one thing, the first magnitude is clearly bigger. Notice that if we break the first magnitude up into three equal pieces, and the second into two equal pieces, then all the pieces have the same size. If the first magnitude is 3 inches long, then the second will be 2 inches long.
Here is Euclid's Definition 6:
- Magnitudes which have the same ratio are called proportional.
So to talk of proportional magnitudes, we need two ratios, hence four magnitudes.
The theory of proportions was developed in order to deal with incommensurable magnitudes. These are magnitudes whose ratios cannot be expressed, like 2:1 or 3:2, in terms of the ratios of whole numbers.
Here is a rephrasing of Euclid's Definition 5. What does it mean to say, of four magnitudes a, a', b, and b', that a is to a' as b is to b'? It means, among other things, that if 7a is greater than 5a', then 7b is greater than 5b'. The same has to be true for any other whole numbers we use instead of 7 and 5. The quantities 7a and 7b are equimultiples of a and b, in the language of Definition 5 (they are both gotten by multiplying by 7). We know what it means for a magnitude to be 7 times as large as another magnitude, and we know what it means for one magnitude to be larger than another.
Definition 5 reduces the question of when two ratios are equal to questions about when one magnitude is greater than another.
Archimedes believed his greatest work was on The sphere and the cylinder. He wanted that inscribed on his tombstone. He showed that the volume of a sphere is equal to 2/3 of the volume of the circumscribed cylinder, and that the area of a sphere is also equal to 2/3 the area of the circumscribed cylinder (including the top and bottom of the cylinder).
Heron of Alexandria lived around 50 A.D. The name is sometimes spelled "Hero" (Similarly, Plato is sometimes spelled "Platon"). He is famous for a formula, given in Book I of his Metrica, that shows how to compute the area A of a triangle if you know the lengths a, b, and c, of its sides. Heron's formula is
Problem II-8 in Arithmetica by Diophantus of Alexandria, who lived around 150 A.D., is
If you can do that, then 16 - x2 will be a square and you will have solved the problem. Expanding the right hand side of the desired equation gives
Pierre de Fermat, around 1637, wrote in the margin of his copy of Arithmetica that
- It is impossible to write a cube as a sum of two cubes, a fourth power as a sum of two fourth powers, and, in general, any power beyond the second as a sum of two similar powers. For this, I have discovered a truly wonderful proof, but the margin is too small to contain it.
This is the famous Fermat's last theorem, so called because it was the last theorem to be proved of the many that Fermat had claimed that he could prove but did not furnish the proof. It was not proved until 1993 (by Andrew Wiles).
In The Greek Anthology, around 500 A.D., Metrodorus proposed the following problem about Diophantus:
- His boyhood lasted 1/6 of his life. He married after 1/7 more. His beard grew after 1/12 more. His son was born 5 years later. The son lived to half his father's age. The father died 4 years after the son. How long did Diophantus live?
Diophantine equations are equations for which we seek integer solutions. Diophantus was looking for rational solutions to the equation 16 = x2 + y2. That is the same as looking for integers a, b, and c satisfying the equation 16c2 = a2 + b2. At the international conference of mathematicians in Paris in 1900, David Hilbert gave a list of 23 problems for mathematicians of the 20-th century to work on. Hilbert's tenth problem was
- Given a diophantine equation with any number of unknown
quantities and with rational integral numerical coefficients:
To devise a process according to which it can be
determined by a finite number of operations whether
the equation is solvable in rational integers.
This problem was settled in 1970 when Yuri Matiyasevich, a Russian mathematician, showed that such a program could not exist.
Pythagorean triples. These are solutions to the diophantine equation x2 + y2 = z2. They are called Pythagorean because of the Pythagorean theorem which relates the sides of a right triangle in exactly this way. When Diophantus wrote 16 as the sum of two squares (of rational numbers) he constructed the Pythagorean triple 12, 16, 20. Do you see why? This triple is 4 times the triple 3, 4, 5, the smallest and most famous Pythagorean triple. If the numbers in the triple have no common factor, the triple is said to be primitive. So 3, 4, 5 is primitive but 12, 16, 20 is not. Clearly the primitive triples are the most interesting ones. See Pythagorean triples
Al-Khwarizmi. His full name is: Abu Ja'far Muhammad ibn Musa Al-Khwarizmi. The "ibn Musa" here means "son of Moses". "Abu" means "father of". So Al-Khwarizmi was Muhammad, the father of Ja'far and the son of Moses. "Al" is "the" in arabic, like "el" in Spanish. "Al-Khwarizmi" is analogous to "El Greco", the Spanish Painter who was Greek, hence the name "El Greco" which means "the Greek". Al-Khwarizmi came from a region called Khwarizm. In Hisab al-jabr w'al-muqabala you can see two instances of the word "al". The "w" before "al-muqabala" means "and".
The continuum hypothesis. The first problem that Hilbert posed at the 1900 Paris conference was to prove Cantor's continuum hypothesis: "Every system of infinitely many real number is either equivalent to the assemblage of natural integers, 1, 2, 3, ... or to the assemblage of all real numbers and therefore to the continuum, that is, to the points of a line."
Today we would probably replace Hilbert's words "system" and "assemblage" by the word "set", which means the same thing. We still use the word "equivalent" the way Hilbert, following Cantor, is using it here: two sets are equivalent if there is a one-to-one correspondence between them. Recall that Galileo had observed that the set of positive integers is equivalent, in this sense, to the set of squares of positive integers.
In some sense, the continuum hypothesis was settled in the twentieth century. In another sense, it remains wide open. In the 1930's Kurt Goedel showed that you couldn't disprove the continuum hypothesis using the commonly accepted axioms of set theory. This was quite impressive because all reasoning about sets was done in accordance with those axioms, and pretty much still is. In the 1960's, Paul Cohen showed that you couldn't prove the continuum hypothesis using the axioms of set theory. Thus we have an independence result: the continuum hypothesis is independent of the axioms of set theory---you can neither prove it nor disprove it.
So is the continuum hypothesis true or not? We can't settle that question without introducing more axioms for set theory, but no one has proposed any axioms for set theory that have achieved any degree of acceptance since Goedel's result in the 1930's. An alternative view is that there is no fact of the matter: it just isn't the case that the continuum hypothesis is either true or false. That seems to be the position we are drifting into, although it is contrary to the way people have always viewed mathematical statements.
Preparation for the test on Friday, September 28.
As the syllabus says, "You should be able to distinguish Euclid from Euler, both as to time and place, and as to their contributions to mathematics." Of course that doesn't mean just Euclid and Euler. You are supposed to be able to do that with all the major players that we have discussed. For the test, these players are: Al-Khwarizmi, Archimedes, Aristotle, Georg Cantor, Paul Cohen, Diophantus, Eratosthenes, Euclid, Eudoxus, Pierre de Fermat, Fibonacci, Galileo, Carl Friedrich Gauss, Kurt Gödel, David Hilbert, Ferdinand Lindemann, The Pythagoreans, Andrew Wiles, Zeno.
You should also be able to explain the ideas that we have encountered. For the test, be prepared for: algebraic and transcendental numbers, the continuum hypothesis, Euclidean constructions, the golden ratio, the harmonic series. incommensurable magnitudes, potential and completed infinity, reductio ad absurdum, squaring the circle, the theory of proportions, Zeno's paradoxes.
On Infinity is an interesting site that mentions a lot of the people and topics concerning infinity that have come up in class.
Kurt Gödel, 1906-1978, was born in what is now Brno in the Czech Republic. Brno is not that far from Vienna, where Gödel was a student and a professor. In 1931 he published what was to become one of the landmark mathematics papers of the twentieth century. He proved that for any system of axioms for mathematics, there will always be statements that you can neither prove nor disprove. In fact, there will be true statements that you cannot prove. This is his incompleteness theorem. Much has been made of this theorem. Some people use it to argue that we are smarter than computers because we can see that the unprovable true statement is true, but the computers presumably cannot.
In 1940, Gödel came to the Institute for Advanced Studies in Princeton where he remained until his death. In 1940, in a Princeton monograph, he showed that you couldn't disprove Cantor's continuum hypothesis. For more about Gödel see Kurt Gödel.
From Descartes' Geometry, 1637:
- Geometry should not include lines that are like strings in that they are sometimes straight and sometimes curved, since the ratio between straight and curved lines are not known and I believe cannot be discovered by human minds
- Who is so blind as not to see that if there are two equal straight lines, one of which is then bent into a curve, that curve will be equal to the straight line?
Christopher Wren, the great English architect who designed Saint Paul's Cathedral, showed in 1658 that the length of the cycloid is four times the diameter of the generating circle.
(All of this material is found in Calculus Gems by George Simmons, McGraw-Hill 1992.)
In his Method of Fluxions (1671), Newton gave the fundamental task of calculus as: "The relation of the fluents being given, to find the relation of their fluxions [and conversely]". The fluents are the (flowing) quantities and the fluxions are their rates of change. This task is what is nowadays called a related rates problem.
The Analyst, by George Berkeley, is a criticism of Newton's method of fluxions, in particular of Lemma II of Principia Mathematica (see Sections 9 and 10).
The Genitum that Newton is talking about is an expression like
consisting of a product of variables raised to various powers. The powers can be fractions, either positive or negative. These variables, and the Genitum itself, are "variable and indetermined". We still use the nouns variable and indeterminate in the same way Newton is using them here. He goes on to say that they are "increasing or decreasing as it were by a perpetual motion or flux". That emphasizes their variable nature.
Now he talks about the "Moment" associated with each variable. The easiest way to translate "Moment" is by "differential". So, in Leibnizian notation, the moment of A is dA. Newton himself denotes the moment of A by a, a notation that becomes awkward after a while (all variables have to be denoted by capital letters).
In the statement of the lemma, Newton uses the phrase "drawn into". That can be translated by "multiplied by". The "generating sides" are the basic variables---so the sides of
are A, B, C, and D. "The indices of the powers of those sides" are the exponents 2, 3/5, -1, and -2/3. Newton explains what the coefficient of a generating side is in the last sentence of the first paragraph: "the quantity which arises by applying the Genitum to that side". What does it mean to apply the Genitum to a side? It means to form the quotient "Genitum over side", or G/S. So applying the Genitum above to the side C gives
Note that Newton says "It will be the same thing, if, instead of moments, we use ... the Velocities of the increments and decrements (... fluxions ...)". What that means is that, in the lemma, we could use dA/dt, the velocity of A, instead of dA, the differential of A. Just divide everything by dt.
Look at his examples in the second paragraph and see if you can calculate the moments using the instructions given in the statement of the lemma. Or maybe I should say, see if you can figure out what those instructions are by looking at the examples!
For the Java applet, I would like to have seen the observation that the rectangles are the same in all the (approximate) parallelograms, and there are the same number of them, so the total areas are the obviously same.
The a in Rouse Ball's example is an ordinary real number, not infinitesimal or infinite. It is the ratio of the two sides of the right triangle.
The ordinate at a point of the base is the line extending from that point to the hypotenuse. We are thinking of the (area of the) triangle as being composed of all those lines.
The reason (1/2)na is "inconsiderable compared with" (1/2)n2a is because the quotient is 1/n which is an infinitesimally small number.
The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, .... Those of the form 4n + 1 are 5, 13, 17, 29, .... The number 29 can be written as the sum of the two squares 25 and 4. Part of this problem was to show that there is no other way to write it as a sum of two squares. ("can be expressed once, and only once, as the sum of two squares"). How do you show that? One way is successively to subtract squares from 29 and see that you only get a square when you subtract 4 or 25. So 29 - 1 = 28, not a square; 29 - 9 = 20, not a square; 29 - 16 = 13, not a square. You can stop when the squares you are trying to subtract are bigger than 29. Actually, you can stop when the squares are bigger than 29/2 (why?).
If the number is not prime, then you might be able to write it as the sum of two squares in two different ways. For example, 65 = 64 + 1 = 49 + 16. Here is a sort of explanation of that phenomenon. The number 65 is 5 times 13. We can write
I think maybe 65 is the smallest number that can be written as the sum of two squares in two different ways (if you don't count 25 = 0 + 25 = 9 + 16). There is a famous story about Hardy's visiting Ramanujan in the hospital and the smallest number that can be written as the sum of two cubes in two different ways.
A converse of Fermat's theorem is that an odd number which is the sum of two squares must be of the form 4n + 1. This follows easily from the fact that, modulo 4, the only squares are 0 and 1, so the only sums of two squares are 0, 1, and 2. So an odd number that is the sum of two squares must be equal to 1 modulo 4. Fermat's theorem is much more difficult to prove than this.
The first calculus book was Analyse des infiniment petits (Infinitesimal analysis) by the Marquis de l'Hospital in 1696. It was in use for almost 100 years. It is here that the famous "L'Hôpital's rule" appears. Johann (John) Bernoulli later claimed it as his own. L'Hospital had presumably paid Bernoulli for the right to use Bernoulli's results. In addition, he gave credit to him in the preface to his book:
"I am obliged to the gentlemen Bernoulli for their many bright ideas; particularly to the younger Mr. Bernoulli who is now a professor in Groningen."
Here is another, more complete, translation of that passage from the preface to L'Hôpital's book: "I must own myself very much obliged to the labours of Messieurs Bernoulli, but particularly to those of the present Professor at Groeningen, as having made free with their Discoveries as well as those of Mr Leibniz: So that whatever they please to claim as their own I frankly return to them."
The first four propositions in the book are
- d(a + x + y - z) = dx + dy - dz
- d(xy) = y dx + x dy
- d(x/y) = (y dx - x dy)/yy
- d(xr) = rxr-1 where r can be any positive or negative integer or rational number.
A nice history of the cycloid can be found in The Helen of geometers including mention of the roles of Newton, Leibniz and de l'Hospital.
Preparation for the test on Friday, November 2.
Know the players: Abel, Berkeley, Jakob Bernoulli, Johann Bernoulli, Cardano, Cavalieri, Descartes, Fermat, Ferrari, Galileo, l'Hospital, Huygens, Kepler, Leibniz, Mersenne, Newton, Pascal, de Roberval, Tartaglia, Torricelli.
Know the topics: Analyzing the cycloid, Cavalieri's principle, Indivisibles, Kepler's laws, Newton's laws, Solving cubic, quartic, and quintic equations, Universal gravitation.
Here is a pendulum applet you can play with. Happy Halloween!
The Legendre Symbol. I will write it here as ( a/p), although it is usually seen more upright, like:
|
If ( a/p) = 1, we say that a is a quadratic residue modulo p. If ( a/p) = -1, we say that a is a quadratic nonresidue modulo p.
So ( 2/7) = 1 because 2 = 32 mod 7, while ( 2/5) = -1 because the squares modulo 5 are 12 = 1, 22 = 4, 32 = 4, and 42 = 1. Because every nonzero square is the square of two different numbers, half of the numbers 1, 2, ..., p-1 are quadratic residues, and half are quadratic nonresidues.
The Jacobi symbol is like the Legendre symbol except that p can be any odd number. The Jacobi symbol takes on the values 0, 1, and -1, just like the Legendre symbol, and it is equal to the Legendre symbol when p is a prime. But if p is not a prime, it doesn't tell you whether or not a is a quadratic residue. It is used in the computation of Legendre symbols.
Euler Line has an applet illustrating the circumcenter, centroid, and orthocenter of a triangle. You can grab a vertex and change the shape of the triangle. See also The Euler Line of a Triangle, another interactive applet which draws in more lines (the altitudes, medians, and perpendicular bisectors of the sides) and provides more explanation.
Correspondence concerning the Petersburg Game has excerpts from the letters between Nicolas Bernoulli, Montmort, Cramer, and Daniel Bernoulli concerning this game. Nicolas Bernoulli told Montmort about the game in a letter of 9 September 1713. He asked Daniel Bernoulli his opinion of it in a letter of 27 October 1728. In 1713 Montmort published the game in his Essay d'analyse sur les jeux de hazard (analysis of games of chance).
Montmort and the Grand Jan has a picture of the 1713 edition of Montmort's book, and a description of some of its contents.
The following passage is from Daniel Bernoulli's paper, "Specimen theoriae novae de mensura sortis", which appeared in 1738 in Commentarii Academiae Scientiarum Imperialis Petropolitanae [Papers of the Imperial Academy of Sciences in Petersburg] 5 175-192. The translation, "Exposition of a new theory of the measurement of risk" appeared in Econometrica 22 (1954) 123-136.
My most honorable cousin the celebrated Nicolaus Bernoulli, Professor utriusque iuris at the University of Basle, once submitted five problems to the highly distinguished mathematician Montmort. These problems are reproduced in the work L'analyse sur les jeux de hazard de M. de Montmort, p. 402. The last of these problems runs as follows: Peter tosses a coin and continues to do so until it should land "heads" when it comes to the ground. He agrees to give Paul one ducat if he gets "heads" on the very first throw, two ducats if he gets it on the second, four if on the third, eight if on the fourth, and so on, so that with each additional throw the number of ducats he must pay is doubled. Suppose we seek to determine the value of Paul's expectation. My aforementioned cousin discussed this problem in a letter to me asking for my opinion. Although the standard calculation shows that the value of Paul's expectation is infinitely great, it has, he said, to be admitted that any fairly reasonable man would sell his chance, with great pleasure, for twenty ducats.
Cover of Instituzioni analitiche ad uso della gioventu' italiana by Maria Gaetana Agnesi. Also some of the pages of the text (click on "Table of Contents").
The story of noneuclidean geometry
Ferdinand Karl Schweikart [1780-1859], memorandum to Gauss, 1818.
There are two kinds of geometry--a geometry in the strict sense--the Euclidean; and an astral geometry. Triangles in the latter have the property that the sum of their three angles is not equal to two right angles. This being assumed, we can prove rigorously:Expository Papers, References, Internet Sites. Has some good links to sites about noneuclidean geometry. From a geometry course taught by Richard Delaware at the University of Missouri - Kansas City.If this constant were for us the radius of the earth, so that every line drawn in the universe from one fixed star to another, distant 90 degrees from the first, would be tangent to the surface of the earth), it would be infinitely great in comparison with the spaces which occur in daily life.
- That the sum of the three angles of a triangle is less than two right angles;
- that the sum becomes ever less, the greater the area of the triangle;
- that the altitude of an isosceles right-angled triangle continually grows, as the sides increase, but it can never become greater than a certain length, which I call the constant.
The Euclidean geometry holds only on the assumption that the constant is infinite. Only in this case is it true that the three angles of every triangle are equal to two right angles.
Galois on abstraction, amateur English translation. Original French version: La Préface Scientifique à la théorie de Galois
The French mathematician by Tom Petsinis is a recent novel about Evariste Galois, written in the first person as if by Galois himself. It looks easy to read and mentions lots of our people: Cardano, Tartaglia, Lagrange, Descartes, Pascal, Archimedes, Euclid, Abel, Mersenne, Cauchy, Euler, Diophantus, etc. It is available in paperback.
From a letter in 1799 from Gauss to the father of János Bolyai (who wasn't born yet):
In my own work theron I myself have advanced far (though my other wholly heterogeneous employments leave me little time therefor) but the way, which I have hit upon, leads not so much to the goal, which one wishes, as much more to making doubtful the truth of geometry.Some excerpts from letters of Gauss; in particular, on noneuclidean geometry. The last one is to Sophie Germain, 1776-1831 (Paris), a mathematician who used the name Antoine-August Le Blanc in the tradition of women writers who assumed men's names so that their work would not be rejected out of hand. Gauss goes on to say
Indeed I have come upon much, which with most no doubt would pass for a proof, but which in my eyes proves as good as nothing.
For example, if one could prove, that a rectilineal triangle is possible, whose content may be greater, than any given surface, then I am in condition, to prove with perfect rigor all geometry.
Most would indeed let that pass as an axiom; I not; it might well be possible, that, how far apart soever one took the three vertices of the triangle in space, yet the content was always under a given limit.
I have more such theorems, but in none do I find anything satisfying.
The scientific notes with which your letters are so richly filled have given me a thousand pleasures. I have studied them with attention and I admire the ease with which you penetrate all branches of arithmetic, and the wisdom with which you generalize and perfect.Arithmetic here means number theory. Sophie Germain worked on Fermat's last theorem, which she divided into two cases, called Case 1 and Case 2 to this day. Her ideas and methods influenced research on this problem for almost 200 years. She has a class of prime numbers named after her: a prime p is called a Sophie Germain prime if 2p + 1 is also prime. So 5 is a Sophie Germain prime because 11 is also prime, but 7 is not a Sophie Germain prime because 15 is not a prime. For more on Sophie Germain see Math's Hidden Woman.
Saccheri eliminates Euclid's flaw. For a picture of the cover of the book, see Euclides ab omni naevo vindicatus.
Lecture Notes 6 of Bill Cherowitzo, University of Colorado at Denver, on noneuclidean geometry, from his course "Higher Geometry I".
Our last topic will be Gödel's incompleteness theorem, possibly the most interesting mathematical result of the twentieth century.
Comments on Gödel's incompleteness theorem
Lost innocence, an article by Keith Devlin for the general public about the incompleteness theorem.
Lecture 8 in 21st-century science talks about Hilbert, Turing, the paradox of the liar, and incompleteness. Pictures of Hilbert and Gödel, and a couple of Escher drawings.
Math in the Movies, a guide to major motion pictures with scenes of real mathematics. Be sure to see "A beautiful mind", the movie about the mathematician John Nash who won the Nobel prize in economics, when it comes out this winter.
Student Perception of Teaching (SPOT) assessment forms will be passed out and completed in class on Friday, November 30.
The final examination, Friday, December 7 from 1:15 to 3:45, will be like the other two tests, but longer:
Know the players: Abel, Agnesi, Al-Khwarizmi, Archimedes, Aristotle, Berkeley, Daniel Bernoulli, Jakob Bernoulli, Johann Bernoulli, Bolyai, Cantor, Cardano, Cavalieri, Paul Cohen, Descartes, Diophantus, Eratosthenes, Euclid, Eudoxus, Euler, Fermat, Ferrari, Fibonacci, Galileo, Galois, Gauss, Sophie Germain, Gödel, William Rowan Hamilton, Hilbert, l'Hospital, Huygens, Kepler, Legendre, Leibniz, Lindemann, Lobachevsky, Mersenne, Newton, Emmy Noether, Pascal, de Roberval, Saccheri, Tartaglia, Torricelli, Turing, Andrew Wiles, Zeno.
Know the topics: algebraic and transcendental numbers; Cavalieri's principle; continuum hypothesis; cycloid; Euclidean constructions; golden ratio; harmonic series; incommensurable magnitudes; incompleteness theorem; indivisibles; Kepler's laws; Newton's laws; noneuclidean geometry; potential and completed infinity; reductio ad absurdum; Saint Petersburg paradox; solving cubic, quartic, and quintic equations; squaring the circle; theory of proportions; transfinite numbers; universal gravitation; witch of Agnesi; Zeno's paradoxes.
Some other notable women mathematicians
- Julia Robinson, 1919-1985. Contributed to the solving of Hilbert's tenth problem. Was president of the American Mathematical Society. Her sister, Constance Reid, wrote a biography of her: Julia: A life in mathematics. See Julia Robinson.
- Sofia Kovalevskaya, 1850-1891. Born in Moskow. Got her doctorate in 1874 at Gottingen. The Association of Women Mathematicians is sponsoring Sonia Kovalevsky High School Mathematics Days with grants to colleges for workshops, talks, and problem solving competitions for high school women students and their teachers. See Kovalevskaya.
- Emmy Noether, 1882-1935. One of the founders of modern algebra. Hilbert, when he had difficulty getting her a position at Gottingen because she was a woman, asked the question, "Is this a university or a bathhouse?" She taught courses there which were listed under Hilbert's name. See Emmy Noether: A tribute to her life and work by Brewer and Smith (that's FAU's own Jim Brewer). See also Emmy Noether.