Ramble, Part 5: Exhausting!
Nov. 17th, 2006 02:25 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowI mentioned earlier that Eudoxus made two major contributions to mathematics. The first was the theory of proportionality I discussed earlier; the second was a simple observation which led to a procedure, the method of exhaustion, for computing the areas or volumes of regions with curved boundaries. Archimedes used this method extensively, solving such problems as the volume of a sphere and the area of the region lying between a parabola and a chord of the parabola. I discuss these ideas in detail under the cut.
Eudoxus' key insight in this respect appears in Euclid as Proposition 1 of Book X:
A typical application of this idea involved approximating a curvilinear area - say, a circle - with the areas of polygons. Let me walk through Archimedes' use of this method to compute the area of a circle. First, we need a definition and a theorem. Consider a regular polygon, say with n sides. Draw the line segment from its center to the midpoint of one of its sides; this is the apothem of the polygon. Then the area of the polygon is equal to one-half the product of its apothem and its perimeter - or, to put it another way, to the area of a right triangle whose legs are the apothem and the perimeter. To see this, connect the center of the polygon to each of its vertices. This divides the polygon into n congruent triangles, each with height the apothem and base one of the sides of the polygon. Each triangle has area one-half the apothem times one side of the polygon; the area of the polygon is n times this, which is as described.
Next, let A and B be two points on a circle, and consider the shorter of the two arcs between them. (You may need to draw a picture here.) Let C be the midpoint of the arc. Then the area of the triangle ABC is more than half the area between the chord AB and the arc. To see this, draw the tangent to the circle at C, and drop perpendiculars to the tangent from A and B, forming a rectangle ABDE. The rectangle contains the region between the chord and the arc, and the area of the triangle is precisely half the area of the rectangle.
Now, take a circle, and inscribe a square in it. We apply Eudoxus' idea to the area lying between the square and the circle, as follows. The vertices of the square divide the circle into four arcs. Take the midpoints of those arcs, and connect them to the adjacent vertices of the square, forming a regular octagon. The area between the octagon and the circle is less than half the area between the square and the circle. Repeat the process, to obtain a regular 16-sided figure, a regular 32-sided figure, and so on. Doing it enough times, we may make the area between the polygon and the circle as small as desired.
This brings us to Archimedes' result: The area of a circle is equal to the area of a right triangle whose legs are the radius and the circumference of the circle. Suppose the area of the triangle is less than the area of the circle, falling short by some quantity c. Then we may construct a regular polygon, as described above, whose area differs from the area of the circle by less than c; that is, the area of the polygon is greater than the area of the triangle, but less than the area of the circle. But the apothem of the polygon is shorter than the radius of the circle - i.e., one leg of the triangle - and the perimeter of the polygon is shorter than the circumference of the circle - i.e., the other leg of the triangle. That means that the area of the polygon is less than the area of the triangle, a contradiction. So the area of the triangle can't be less than the area of the circle. We can use a similar argument, starting with a square circumscribed around the circle and cutting off corners, to show that the area of the triangle can't be greater than the area of the circle either.
Archimedes was able to exploit this technique to compute the volumes of cylinders (approximate them with prisms), of cones (use pyramids), and ultimately of spheres. (For the last, revolve a regular polygon around a diagonal; the resulting figure can be dissected into numerous segments of cones, and it can be made to approximate the sphere as closely as desired.) Each such computation required greater and greater ingenuity, however. Furthermore, the method could really only be used when the expected answer was already known; making the appropriate guess required some other technique, which will be the subject of the next Ramble post.
In developing the method of exhaustion, Eudoxus came close to the notion of limit. He did not, however, come up with the idea in full generality; that had to wait until the time of Newton or later. I believe that there were good reasons why it took so long; there were specific developments in the intervening time that made it easier to develop the limit concept. I'll get to that later too.
Previous Next
Ramble Contents
Eudoxus' key insight in this respect appears in Euclid as Proposition 1 of Book X:
Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process be repeated continuously, there will be left some magnitude which will be less than the lesser magnitude set out.In more modern language, if you start with a positive number and repeatedly reduce it by a factor of at least 1/2, then it will eventually be made arbitrarily small. This is in essence a statement about limits: if a sequence of positive numbers a1,a2,... has the property that each is less than or equal to half its predecessor, then the limit of the sequence is 0.
A typical application of this idea involved approximating a curvilinear area - say, a circle - with the areas of polygons. Let me walk through Archimedes' use of this method to compute the area of a circle. First, we need a definition and a theorem. Consider a regular polygon, say with n sides. Draw the line segment from its center to the midpoint of one of its sides; this is the apothem of the polygon. Then the area of the polygon is equal to one-half the product of its apothem and its perimeter - or, to put it another way, to the area of a right triangle whose legs are the apothem and the perimeter. To see this, connect the center of the polygon to each of its vertices. This divides the polygon into n congruent triangles, each with height the apothem and base one of the sides of the polygon. Each triangle has area one-half the apothem times one side of the polygon; the area of the polygon is n times this, which is as described.
Next, let A and B be two points on a circle, and consider the shorter of the two arcs between them. (You may need to draw a picture here.) Let C be the midpoint of the arc. Then the area of the triangle ABC is more than half the area between the chord AB and the arc. To see this, draw the tangent to the circle at C, and drop perpendiculars to the tangent from A and B, forming a rectangle ABDE. The rectangle contains the region between the chord and the arc, and the area of the triangle is precisely half the area of the rectangle.
Now, take a circle, and inscribe a square in it. We apply Eudoxus' idea to the area lying between the square and the circle, as follows. The vertices of the square divide the circle into four arcs. Take the midpoints of those arcs, and connect them to the adjacent vertices of the square, forming a regular octagon. The area between the octagon and the circle is less than half the area between the square and the circle. Repeat the process, to obtain a regular 16-sided figure, a regular 32-sided figure, and so on. Doing it enough times, we may make the area between the polygon and the circle as small as desired.
This brings us to Archimedes' result: The area of a circle is equal to the area of a right triangle whose legs are the radius and the circumference of the circle. Suppose the area of the triangle is less than the area of the circle, falling short by some quantity c. Then we may construct a regular polygon, as described above, whose area differs from the area of the circle by less than c; that is, the area of the polygon is greater than the area of the triangle, but less than the area of the circle. But the apothem of the polygon is shorter than the radius of the circle - i.e., one leg of the triangle - and the perimeter of the polygon is shorter than the circumference of the circle - i.e., the other leg of the triangle. That means that the area of the polygon is less than the area of the triangle, a contradiction. So the area of the triangle can't be less than the area of the circle. We can use a similar argument, starting with a square circumscribed around the circle and cutting off corners, to show that the area of the triangle can't be greater than the area of the circle either.
Archimedes was able to exploit this technique to compute the volumes of cylinders (approximate them with prisms), of cones (use pyramids), and ultimately of spheres. (For the last, revolve a regular polygon around a diagonal; the resulting figure can be dissected into numerous segments of cones, and it can be made to approximate the sphere as closely as desired.) Each such computation required greater and greater ingenuity, however. Furthermore, the method could really only be used when the expected answer was already known; making the appropriate guess required some other technique, which will be the subject of the next Ramble post.
In developing the method of exhaustion, Eudoxus came close to the notion of limit. He did not, however, come up with the idea in full generality; that had to wait until the time of Newton or later. I believe that there were good reasons why it took so long; there were specific developments in the intervening time that made it easier to develop the limit concept. I'll get to that later too.
Previous Next
Ramble Contents
