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So, I find myself writing a book, almost by accident. I started writing and found it a lot easier than I expected. I've kept at it for 34K words now. It pulls together a lot of my thoughts on the history/philosophy of math/science, some of which I first shared here. As part of the research for this, I just finished reading through Euclid's Elements. This turned out to be both easier and more pleasant than I would have guessed. It was good to get a refresher in basic geometry, if nothing else. There are a lot of identities that, if I even ever learned them in middle school, I had long since forgotten. That will always be useful in the shop, not to mention the old school science texts I've been reading which rely heavily on geometric proofs. One of the major themes of the book is how different modern, algebraic thinking is from classical modes of geometric proofs.
Classical geometry is a game with simple rules. The only tools are a straightedge and a compass. The straightedge has no markings on it, so you can’t measure distances with it. It isn’t a ruler or a scale. All you can use it for is to draw a straight line, either connecting existing points or extending an existing line. The compass can draw a circle, but only by using an existing point for the center and another existing point to give the radius. It is “collapsible”, which means it can’t be used to transfer distances from one line to another. If you lift it off the page it collapses and loses its setting.
Given just those tools, plus a pencil and a sheet of paper, what shapes can you generate? And what can you prove about those shapes? It isn’t enough to draw something looks a lot like a hexagon, you have to be able to prove it is perfect. These are the questions Euclid spent 13 books answering in his Elements. Finding the answers turns out to be a major branch of mathematics.
These restrictions might seem, well, restricting, and they are. As with a lot of formal math, they’re added to help clarify the system. We list them explicitly so that we know exactly which conditions the proofs apply to. This set has been the definition of pure geometry since the time of the ancient Greeks. Plato sometimes gets the blame for this, though there isn’t much evidence to support the accusation. He was definitely interested in the subject, though, and his followers are credited with discovering many important proofs.
It all starts off with book I, proposition 1: To construct an equilateral triangle on a given finite straight line. Draw a circle centered on A, with a radius of AB. Draw a circle centered on B, with a radius of BA. These circle will intersect in two places. Label the upper intersection C. Connect CA and CB. Because CA is a radius of the first circle, you know it has the same length as AB. Because CB is a radius of the second circle, you know it also has the same length as AB. Thus the triangle has to be equilateral. QED.
[Book I proposition 1 diagram]
Euclid takes this as a foundation and slowly builds a full system of geometry in the Elements. One proposition is set out after another. He will demonstrate how a certain construction can be done, and he proves that it does what he says. Using the new capability of this construction, he then will explore some useful conclusions that can be drawn. The work is grouped into 13 books, each focusing on a different aspect. For instance, book III deals with circles, and book XI looks at solid (3D) geometry. Many of the proofs are known to have existed long before Euclid, but he was the first to bring them all together and present them in a well structured format.
Book I, proposition 9: To bisect a given rectilinear angle. Take an arbitrary point D on AB. Use the compass to swing D over to mark E on AC, so that AD and AE are the same length. Draw a line between D and E. Construct an equilateral triangle on DE, using proposition 1. Draw a line from A to F. We know that AD and AE are the same length, because that’s how we made them. We know that DF and EF are equal, because they’re both legs of an equilateral triangle. If we consider the triangles ADF and AEF, we’ll see that they have the side AF in common. This means that all their sides are the same, and so by proposition 8, their angles are also the same. Thus angle DAF is the same as angle FAE, thus the original angle ABC has been bisected by AF. QED.
[Book I proposition 9 diagram]
Euclid builds proof after proof in this fashion. For many of the books, they’re all pretty straight forward geometric tools, like showing that all the interior angles of a triangle add up to 180 degrees. Anyone with high school geometry will recognize most of the propositions. Later on, he deals with solid (three dimensional) geometry, but the tools are the same. This shows better than anything how abstract the straightedge and compass are, as here they are being freely used to draw lines in thin air.
In parts of the Elements, however, things get a lot more opaque, such as book II. Here Euclid starts working on proofs such as proposition 9: If a straight line is cut into equal and unequal segments, then the sum of the squares on the unequal segments of the whole is double the sum of the square on the half and the square on the straight line between the points of section. That is, given a line, first cut it in half, and then subdivide the second half any way you like, leaving three sections. Euclid claims the square of the first two sections combined plus the square of the third section equals twice the square of the first section plus the square of the second section.
[Book 2 proposition 9 diagram]
The proof Euclid provides does show this to be true, but why, you might be asking, did he care? It doesn’t seem to be a very useful tool for working geometric proofs. For modern eyes this is made much clearer by looking at it algebraically. If we call both of the lengths AC and CB (they’re equal to each other) x and the length CD y, we can rewrite the problem. Translated into this form the proposition becomes (x+y)^2 +(x-y)^2=2(x^2+y^2), which is indeed true. Book 2 is a series of proofs for doing geometric algebra. Specifically, it presents a suite of tools for working with forms of the quadratic equation.
Working geometrically instead of algebraically is a completely legitimate option. For most of history it would actually have been seen as far more legitimate, since all the lines really meant something. Converting everything to abstract symbols was seen with suspicion for a long time. This bias was unfortunate, because as we can see in this example, algebra is far more versatile than geometry.
Proposition 9 takes about a page to demonstrate its proof. Proposition 8, which does the same thing for 4xy+(x-y)^2=(x+y)^2, is about as long, and it does so in a completely different fashion. One is proved primarily with triangles, the other with parallelograms. It takes significant study to see how they even relate to each other. Each relation has to be found independently, without any clue how to get from one to the next. These kinds of proofs were so difficult to discover that people mostly relied on memorizing Euclid. In contrast, algebra provides the ability for anyone who can deduce the identity in proposition 8 to immediately do the same for the one in 9. In this case you only need to know how to square an expression, how to move values back and forth across the equals sign, and some basic arithmetic. Once you have those few tools mastered you can deduce all of book 2, along with a literally infinite number of variations. Algebra makes these identities all so trivial and obvious that there is no point in memorizing them, any more than you have to specifically memorize the fact that 100 + 25 = 150 - 25.
To the modern eye, one thing is conspicuously missing from the Elements: he never gives equations for the volume of the solids he is working with. This is such a common feature of geometry education now that it’s absence is quite striking. The volumes are given in proportion to other solids, such as in book XII proposition 10, which shows that a cone is equal to 1/3 of a cylinder with the same height and same size base. How big is the cylinder, though? Euclid does not provide an absolute answer to this. Using the tools of geometry he couldn’t, nor would the answer probably even occur to him. Geometry inherently leads to thinking in terms of proportions, not absolute value. The equations that we use today, such as V=h*pi*r^2 for a cylinder, are a product of calculus, that triumph of algebraical thinking. They were only formulated in the way we know in the middle of the 18th century by Euler. It wasn’t until the 17th century that pi had started to become thought of as an actual number. Before then it was simply a ratio. It was only assigned the symbol in 1706 by William Jones -- who even then still said that “the exact proportion between the diameter and the circumference can never be expressed in numbers”.
Classical geometry is a game with simple rules. The only tools are a straightedge and a compass. The straightedge has no markings on it, so you can’t measure distances with it. It isn’t a ruler or a scale. All you can use it for is to draw a straight line, either connecting existing points or extending an existing line. The compass can draw a circle, but only by using an existing point for the center and another existing point to give the radius. It is “collapsible”, which means it can’t be used to transfer distances from one line to another. If you lift it off the page it collapses and loses its setting.
Given just those tools, plus a pencil and a sheet of paper, what shapes can you generate? And what can you prove about those shapes? It isn’t enough to draw something looks a lot like a hexagon, you have to be able to prove it is perfect. These are the questions Euclid spent 13 books answering in his Elements. Finding the answers turns out to be a major branch of mathematics.
These restrictions might seem, well, restricting, and they are. As with a lot of formal math, they’re added to help clarify the system. We list them explicitly so that we know exactly which conditions the proofs apply to. This set has been the definition of pure geometry since the time of the ancient Greeks. Plato sometimes gets the blame for this, though there isn’t much evidence to support the accusation. He was definitely interested in the subject, though, and his followers are credited with discovering many important proofs.
It all starts off with book I, proposition 1: To construct an equilateral triangle on a given finite straight line. Draw a circle centered on A, with a radius of AB. Draw a circle centered on B, with a radius of BA. These circle will intersect in two places. Label the upper intersection C. Connect CA and CB. Because CA is a radius of the first circle, you know it has the same length as AB. Because CB is a radius of the second circle, you know it also has the same length as AB. Thus the triangle has to be equilateral. QED.
[Book I proposition 1 diagram]
Euclid takes this as a foundation and slowly builds a full system of geometry in the Elements. One proposition is set out after another. He will demonstrate how a certain construction can be done, and he proves that it does what he says. Using the new capability of this construction, he then will explore some useful conclusions that can be drawn. The work is grouped into 13 books, each focusing on a different aspect. For instance, book III deals with circles, and book XI looks at solid (3D) geometry. Many of the proofs are known to have existed long before Euclid, but he was the first to bring them all together and present them in a well structured format.
Book I, proposition 9: To bisect a given rectilinear angle. Take an arbitrary point D on AB. Use the compass to swing D over to mark E on AC, so that AD and AE are the same length. Draw a line between D and E. Construct an equilateral triangle on DE, using proposition 1. Draw a line from A to F. We know that AD and AE are the same length, because that’s how we made them. We know that DF and EF are equal, because they’re both legs of an equilateral triangle. If we consider the triangles ADF and AEF, we’ll see that they have the side AF in common. This means that all their sides are the same, and so by proposition 8, their angles are also the same. Thus angle DAF is the same as angle FAE, thus the original angle ABC has been bisected by AF. QED.
[Book I proposition 9 diagram]
Euclid builds proof after proof in this fashion. For many of the books, they’re all pretty straight forward geometric tools, like showing that all the interior angles of a triangle add up to 180 degrees. Anyone with high school geometry will recognize most of the propositions. Later on, he deals with solid (three dimensional) geometry, but the tools are the same. This shows better than anything how abstract the straightedge and compass are, as here they are being freely used to draw lines in thin air.
In parts of the Elements, however, things get a lot more opaque, such as book II. Here Euclid starts working on proofs such as proposition 9: If a straight line is cut into equal and unequal segments, then the sum of the squares on the unequal segments of the whole is double the sum of the square on the half and the square on the straight line between the points of section. That is, given a line, first cut it in half, and then subdivide the second half any way you like, leaving three sections. Euclid claims the square of the first two sections combined plus the square of the third section equals twice the square of the first section plus the square of the second section.
[Book 2 proposition 9 diagram]
The proof Euclid provides does show this to be true, but why, you might be asking, did he care? It doesn’t seem to be a very useful tool for working geometric proofs. For modern eyes this is made much clearer by looking at it algebraically. If we call both of the lengths AC and CB (they’re equal to each other) x and the length CD y, we can rewrite the problem. Translated into this form the proposition becomes (x+y)^2 +(x-y)^2=2(x^2+y^2), which is indeed true. Book 2 is a series of proofs for doing geometric algebra. Specifically, it presents a suite of tools for working with forms of the quadratic equation.
Working geometrically instead of algebraically is a completely legitimate option. For most of history it would actually have been seen as far more legitimate, since all the lines really meant something. Converting everything to abstract symbols was seen with suspicion for a long time. This bias was unfortunate, because as we can see in this example, algebra is far more versatile than geometry.
Proposition 9 takes about a page to demonstrate its proof. Proposition 8, which does the same thing for 4xy+(x-y)^2=(x+y)^2, is about as long, and it does so in a completely different fashion. One is proved primarily with triangles, the other with parallelograms. It takes significant study to see how they even relate to each other. Each relation has to be found independently, without any clue how to get from one to the next. These kinds of proofs were so difficult to discover that people mostly relied on memorizing Euclid. In contrast, algebra provides the ability for anyone who can deduce the identity in proposition 8 to immediately do the same for the one in 9. In this case you only need to know how to square an expression, how to move values back and forth across the equals sign, and some basic arithmetic. Once you have those few tools mastered you can deduce all of book 2, along with a literally infinite number of variations. Algebra makes these identities all so trivial and obvious that there is no point in memorizing them, any more than you have to specifically memorize the fact that 100 + 25 = 150 - 25.
To the modern eye, one thing is conspicuously missing from the Elements: he never gives equations for the volume of the solids he is working with. This is such a common feature of geometry education now that it’s absence is quite striking. The volumes are given in proportion to other solids, such as in book XII proposition 10, which shows that a cone is equal to 1/3 of a cylinder with the same height and same size base. How big is the cylinder, though? Euclid does not provide an absolute answer to this. Using the tools of geometry he couldn’t, nor would the answer probably even occur to him. Geometry inherently leads to thinking in terms of proportions, not absolute value. The equations that we use today, such as V=h*pi*r^2 for a cylinder, are a product of calculus, that triumph of algebraical thinking. They were only formulated in the way we know in the middle of the 18th century by Euler. It wasn’t until the 17th century that pi had started to become thought of as an actual number. Before then it was simply a ratio. It was only assigned the symbol in 1706 by William Jones -- who even then still said that “the exact proportion between the diameter and the circumference can never be expressed in numbers”.
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