Chapter 15: The Beginnings of Calculus

1. With the advent of analytic geometry in the 17th century, what new mathematical possibilities arose?

2. What does it mean to adequate a function? What does it mean to adequate a curve? What did the process of adequation enable Fermat to more easily calculate?

3. How did the ancient Greek and Islamic mathematicians determine areas of regions bounded by curves? How did Kepler do it?

4. What was Torricelli's “Infinitely Long Solid”, and what was so surprising about it?

5. For what types of regions was Fermat able to compute area? What about Wallis, Roberval, Pascal, and Gregory of St. Vincent?

6. How are areas related to logarithms?

7. In Descartes' Geometry, what does he state that no human mind could discover rigorously and exactly? When and by whom was this statement shown to be incorrect?

8. What fundamental mathematical relationship was discovered by Gregory, and what investigation led him to it? How did Barrow discover this relationship?

Chapter 16: Newton and Leibniz

1. When did Newton work out his basic ideas on Calculus? To what extent were they “published”?

2. By what analogy was Newton particularly “struck”?

3. Newton's discovery of power series came out of reading what source?

4. What were some of the power series Newton found?

5. What is a “fluxion”, and what are some of its applications? What is a “fluent”?

6. In Newton's Treatise on Methods, what is one missing idea that appears later in his Principia?

7. What material is covered in Newton's Principia?

8. What truth is there to the statement that Newton developed the Calculus in order to work out his system in the Principia? 9. Who brought Leibniz to the frontiers of mathematical research? What texts did Leibniz read?

10. Leibniz's calculus grew out of what idea?

11. What did Leibniz use the symbols d and ∫ to represent? What is the geometric significance of the expressions ∫ d y = y and d ∫ y = y ?

12. When did Leibniz discover all the basic ideas of his calculus? The product and quotient rule? The power rule? The power series for the sine function?

13. Briefly describe the priority controversy between Newton and Leibniz.

14. Whose system was easier to work with, and what was the result?

15. Describe the significance and the content of L'Hospitals's Analyse des Infiniment Petits.

(No Exercises this time.)