Senior Projects 1: Regiomontanus
May. 27th, 2006 06:02 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowI've mentioned before that students here are required to complete Senior Projects as the capstones of their undergraduate careers; I've decided to go into a little detail concerning some of the projects I'm currently overseeing, to give some idea of what the students have to do.
Now, different professors tend to assign different sorts of projects, of course. We have a few people who specialize in Mathematics Education; their projects generally have to do with pedagogy. One of our specialists in Operations Research assigns fairly tough, frequently practical projects. Me, I like to go the historical route.
Student #1 wanted to do something with a geometrical flavor; after some discussion, we settled on the work of Johannes Müller of Königsberg, commonly known as Regiomontanus (from the Latinized name of his home city). He was a fifteenth-century mathematician/astronomer, best known for his work on spherical trigonometry. His On Triangles was the first European text on the subject, and became the model for all future work. My student's assignment is to read and master that book; her paper and oral presentation will involve describing his work in modern language, together with some applications.
This is a bit harder than it sounds. Of course, Regiomontanus wrote in Latin; equally of course, she's reading it in translation. However, most translations of old mathematical works are themselves rather old; couple that with an attempt at faithful translation, and you get prose that, even by mathematical standards, is smothering. Add in the fact that modern algebraic symbolism hadn't yet been developed: everything is in words, with perhaps a few abbreviations...
So, what's the book about? I'll discuss that under the cut, with - as before - the technicalities banished to blockquotes.
The first two sections of the book deal with ordinary plane trigonometry, which is the study of the relationships among the side lengths and angle measures of triangles in the plane. The rest of it discusses the analogue of this for triangles on the surface of a sphere - which has obvious uses in astronomy and navigation. Now, what lines are to planes, great circles are to spheres - that is, circles on the sphere whose center is the center of the sphere. (Meridians of longitude are great circles; so is the equator, but other parallels of latitude are not.)
Triangles formed by arcs of great circles behave rather differently from plane triangles. The angles of a plane triangle always sum to 180 degrees; the angles of a spherical triangle always sum to more than that. (What is the angle between two great circles? Draw the tangent lines to both circles at a point where they intersect; the angle between the circles is, by definition, the angle between the tangent lines. All meridians are perpendicular to the equator; the angle between, say, the 20E meridian and the 30E meridian is, naturally enough, 30-20=10 degrees.) A spherical triangle may have more than one right angle; in fact, all three angles may be right.
Another interesting aspect is this. Each side of a spherical triangle is, after all, an arc of a circle, and hence it can be identified with an angle - the angle it subtends at the center of the sphere. (The segment of a meridian lying between the 45th and 30th north parallels can be thought of as a 15-degree angle, for example.) Where plane trigonometry relates the side-lengths to the sines and cosines of the angles, spherical trigonometry deals with the sines and cosines, not merely of the vertex-angles, but of the sides.
One last comment before we go into detail. The surface of a sphere is "locally Euclidean"; that is, on a small enough scale, measurements are approximately equal to those that you'd find on a plane. In this context, that means that the laws of spherical trigonometry should reduce, approximately, to those of plane trigonometry if the triangles -that is, the angles associated with the sides - are small. I'll touch on that later.
There are three fundamental theorems of plane trigonometry, from which most everything else is derivable: the Pythagorean theorem (a2+b2=c2, where a,b are the legs of a right triangle and c is its hypotenuse); the Law of Sines ((sin A)/a = (sin B)/b = (sin C)/c, where a,b,c are the sides of a triangle and A,B,C are the angles opposite each); and the Law of Cosines (an extension of Pythagoras to non-right triangles: a2 + b2 - 2 a b cos C = c2). (Note that if C is a right angle, cos C is 0, so we get the Pythagorean theorem back in that case.) The core theorems of spherical trigonometry are analogues of these, but it's not quite obvious that that's what they are.
The spherical Pythagorean theorem describes spherical triangles with a right angle at, say, C. The theorem is: cos c = (cos a) (cos b). (Remember, lower case letters refer to the sides, which we're regarding as angles.) Why is this analogous to the usual Pythagorean theorem? Because, for small angles a, b, c this turns out to be, approximately, the usual theorem. Details are in the box.
The spherical version of the Law of Sines is simple: (sin A)/(sin a) = (sin B)/(sin b) = (sin C)/(sin c). If x is small, sin x is approximately equal to x, so this (approximately) reduces to the plane Law of Sines.
The spherical Law of Cosines is more complex: cos a = cos b cos c + sin b sin c cos A. (Again, if A is a right angle, we get the spherical Pythagorean theorem back.) The comparison to the plane Law of Cosines requires the same approximations to sine and cosine as before, but be careful - don't make the approximation for cos A; A doesn't have to be small, even if a, b, c are.
So, that, in essence, is what my student has to master: she has to be able to prove each of those theorems in her own words, show how to use them for traditional problems like "given the side a and the angles B and C, find the other sides and angles", and discuss applications of these ideas to astronomy and navigation. She began working on it in January, and expects to present towards the end of summer semester, in early August.
Now, different professors tend to assign different sorts of projects, of course. We have a few people who specialize in Mathematics Education; their projects generally have to do with pedagogy. One of our specialists in Operations Research assigns fairly tough, frequently practical projects. Me, I like to go the historical route.
Student #1 wanted to do something with a geometrical flavor; after some discussion, we settled on the work of Johannes Müller of Königsberg, commonly known as Regiomontanus (from the Latinized name of his home city). He was a fifteenth-century mathematician/astronomer, best known for his work on spherical trigonometry. His On Triangles was the first European text on the subject, and became the model for all future work. My student's assignment is to read and master that book; her paper and oral presentation will involve describing his work in modern language, together with some applications.
This is a bit harder than it sounds. Of course, Regiomontanus wrote in Latin; equally of course, she's reading it in translation. However, most translations of old mathematical works are themselves rather old; couple that with an attempt at faithful translation, and you get prose that, even by mathematical standards, is smothering. Add in the fact that modern algebraic symbolism hadn't yet been developed: everything is in words, with perhaps a few abbreviations...
So, what's the book about? I'll discuss that under the cut, with - as before - the technicalities banished to blockquotes.
The first two sections of the book deal with ordinary plane trigonometry, which is the study of the relationships among the side lengths and angle measures of triangles in the plane. The rest of it discusses the analogue of this for triangles on the surface of a sphere - which has obvious uses in astronomy and navigation. Now, what lines are to planes, great circles are to spheres - that is, circles on the sphere whose center is the center of the sphere. (Meridians of longitude are great circles; so is the equator, but other parallels of latitude are not.)
Triangles formed by arcs of great circles behave rather differently from plane triangles. The angles of a plane triangle always sum to 180 degrees; the angles of a spherical triangle always sum to more than that. (What is the angle between two great circles? Draw the tangent lines to both circles at a point where they intersect; the angle between the circles is, by definition, the angle between the tangent lines. All meridians are perpendicular to the equator; the angle between, say, the 20E meridian and the 30E meridian is, naturally enough, 30-20=10 degrees.) A spherical triangle may have more than one right angle; in fact, all three angles may be right.
Another interesting aspect is this. Each side of a spherical triangle is, after all, an arc of a circle, and hence it can be identified with an angle - the angle it subtends at the center of the sphere. (The segment of a meridian lying between the 45th and 30th north parallels can be thought of as a 15-degree angle, for example.) Where plane trigonometry relates the side-lengths to the sines and cosines of the angles, spherical trigonometry deals with the sines and cosines, not merely of the vertex-angles, but of the sides.
One last comment before we go into detail. The surface of a sphere is "locally Euclidean"; that is, on a small enough scale, measurements are approximately equal to those that you'd find on a plane. In this context, that means that the laws of spherical trigonometry should reduce, approximately, to those of plane trigonometry if the triangles -that is, the angles associated with the sides - are small. I'll touch on that later.
There are three fundamental theorems of plane trigonometry, from which most everything else is derivable: the Pythagorean theorem (a2+b2=c2, where a,b are the legs of a right triangle and c is its hypotenuse); the Law of Sines ((sin A)/a = (sin B)/b = (sin C)/c, where a,b,c are the sides of a triangle and A,B,C are the angles opposite each); and the Law of Cosines (an extension of Pythagoras to non-right triangles: a2 + b2 - 2 a b cos C = c2). (Note that if C is a right angle, cos C is 0, so we get the Pythagorean theorem back in that case.) The core theorems of spherical trigonometry are analogues of these, but it's not quite obvious that that's what they are.
The spherical Pythagorean theorem describes spherical triangles with a right angle at, say, C. The theorem is: cos c = (cos a) (cos b). (Remember, lower case letters refer to the sides, which we're regarding as angles.) Why is this analogous to the usual Pythagorean theorem? Because, for small angles a, b, c this turns out to be, approximately, the usual theorem. Details are in the box.
For convenience, assume the sphere has radius 1; the computation is slightly more complicated if you don't. The second-order Taylor approximation to cos x is 1 - 1/2 x2; if you substitute this into the spherical Pythagorean theorem, you get the approximation 1 - 1/2 c2 = (1 - 1/2 a2)(1 - 1/2 b2); expand the right side, discard higher-order terms, and do a little algebra.
The spherical version of the Law of Sines is simple: (sin A)/(sin a) = (sin B)/(sin b) = (sin C)/(sin c). If x is small, sin x is approximately equal to x, so this (approximately) reduces to the plane Law of Sines.
The spherical Law of Cosines is more complex: cos a = cos b cos c + sin b sin c cos A. (Again, if A is a right angle, we get the spherical Pythagorean theorem back.) The comparison to the plane Law of Cosines requires the same approximations to sine and cosine as before, but be careful - don't make the approximation for cos A; A doesn't have to be small, even if a, b, c are.
So, that, in essence, is what my student has to master: she has to be able to prove each of those theorems in her own words, show how to use them for traditional problems like "given the side a and the angles B and C, find the other sides and angles", and discuss applications of these ideas to astronomy and navigation. She began working on it in January, and expects to present towards the end of summer semester, in early August.
no subject
Date: 2006-06-02 02:26 am (UTC)