I used to believe that geometry wasn’t possible without algebra, that geometry was essentially a representation of algebra. I didn’t have an experience of geometry that focused on the creativity and ingenuity that I’ve come to now know. Chapter 6 of Journey Through Genius reiterates how impressive the discipline of geometry truly is. Cardano’s solution to the general solution was astonishing, to me, because of the literal use of geometry to find the appropriate algebraic identities and the verbal recipe for the solution.
I believe that the verbal recipe was surprising because he created a rule to an algebraic equation without using an algebraic equation. He describes, in words, step for step what we would do algebraically. He also does it in …show more content…
When we think of the mathematical disciplines learned in high school, algebra typically reigns supreme. We use algebra to explain geometric concepts, usually the geometry element in the process. This was never more evident than our discussion in class about completing the square, and how only one of us had been taught the geometry behind the concept. However, Cardano does the exact opposite of what we would think to do today. He provides us with his verbal recipe, and rather than derive and generate a formula, he attacks the problem geometrically. In his proof, Cardano describes sectioning off a large cube into six different sections, consisting of “cubes, slabs, and blocks”. He uses these figures and their volumes to derive a formula for the area of the entire cube. I found this really intriguing because he sliced the cube in such a way that the solution would come out of it. We get the completed work, but the amount of struggle that he likely went through to know to cut the cube in this fashion isn’t shown. I’ve come to appreciate this aspect of mathematics more and more as I’ve gone through upper level courses. I also love how the algebra just follows from the geometric work. I believe that the visuals serve as a tangible solution, so that someone can see that it works. It also takes away the thought that all of this was somehow pulled out of thin
I believe that the verbal recipe was surprising because he created a rule to an algebraic equation without using an algebraic equation. He describes, in words, step for step what we would do algebraically. He also does it in …show more content…
When we think of the mathematical disciplines learned in high school, algebra typically reigns supreme. We use algebra to explain geometric concepts, usually the geometry element in the process. This was never more evident than our discussion in class about completing the square, and how only one of us had been taught the geometry behind the concept. However, Cardano does the exact opposite of what we would think to do today. He provides us with his verbal recipe, and rather than derive and generate a formula, he attacks the problem geometrically. In his proof, Cardano describes sectioning off a large cube into six different sections, consisting of “cubes, slabs, and blocks”. He uses these figures and their volumes to derive a formula for the area of the entire cube. I found this really intriguing because he sliced the cube in such a way that the solution would come out of it. We get the completed work, but the amount of struggle that he likely went through to know to cut the cube in this fashion isn’t shown. I’ve come to appreciate this aspect of mathematics more and more as I’ve gone through upper level courses. I also love how the algebra just follows from the geometric work. I believe that the visuals serve as a tangible solution, so that someone can see that it works. It also takes away the thought that all of this was somehow pulled out of thin