--- title: "Meaning & Key Takeaway of: On what may be the last page he wrote in…" description: "\"“On what may be the last page he wrote in his notebooks, Leonardo drew four right triangles with bases of differing lengths (fig. 143). Inside of each ..." image: "https://sms4smile.com/quotes/i/85f/walter-isaacson-384_geometry_on-what-may-be-the-last-page-he-wrote-in_1380321.png" --- # On what may be the last page he wrote in his notebooks… ![“On what may be the last page he wrote in his notebooks, Leonardo drew four right triangles with bases of differing lengths (fig. 143). Inside of each he fit a rectangle, and then he shaded the remaining areas of the triangle. In the center of the page he made a chart with boxes labeled with the…” quote by Walter Isaacson](https://sms4smile.com/quotes/i/85f/walter-isaacson-384_geometry_on-what-may-be-the-last-page-he-wrote-in_1380321.png) [Download](https://sms4smile.com/quotes/i/85f/walter-isaacson-384_geometry_on-what-may-be-the-last-page-he-wrote-in_1380321.png) [Open image](https://sms4smile.com/quotes/i/85f/walter-isaacson-384_geometry_on-what-may-be-the-last-page-he-wrote-in_1380321.png) > ““On what may be the last page he wrote in his notebooks, Leonardo drew four right triangles with bases of differing lengths (fig. 143). Inside of each he fit a rectangle, and then he shaded the remaining areas of the triangle. In the center of the page he made a chart with boxes labeled with the letter of each rectangle, and below it he described what he was trying to accomplish. As he had done obsessively over the years, he was using the visualization of geometry to help him understand the transformation of shapes. Specifically, he was trying to understand the formula for keeping the area of a right triangle the same while varying the lengths of its two legs. He had fussed with this problem, explored by Euclid, repeatedly over the years. It was a puzzle that, by this point in his life, as he turned sixty-seven and his health faded, might seem unnecessary to solve. To anyone other than Leonardo, it may have been.”” [Walter Isaacson](https://sms4smile.com/quotes/author/walter-isaacson/) ## About This Quote This interpretation was drafted with AI assistance. It is one reading of the quote, not the author's own explanation. Leonardo used geometric drawings to explore how to keep a right‑triangle’s area constant while changing its leg lengths, reflecting his lifelong habit of visual problem solving. **In simple terms:** He tried to keep triangle area same while legs change. Key Takeaway Use visual models to explore complex problems. Themes creativity problem solving geometry visualization Mood curious reflective Type historical analytical When to use this quote - mathematical research - engineering design - educational illustration - personal reflection Key Concepts mathematics proportional reasoning invariant properties Questions to Reflect On - How can visual thinking aid abstract reasoning? - When is a visual approach unnecessary? 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