HM/MA 004, Assignment 7, due Friday, 25 October 2002, 10:00 AM

(Note on collaboration: If you do the homework in pairs or groups, please list the name(s) of your collaborator(s) on what you each turn in.)

1. Identifying properties of tangents to curves. (10 points) In class, we deduced that the slope of the tangent line to Fermat's parabola (section 11.C1, pp. 358--359) at a given point was twice the length of the ordinate at that point. (Note that I'm not talking about Fermat's result a=2d on the location of the tangent line.) Consider now the tangent line to the ellipse, whose location you found in Problem 2 of Assignment 5. What is its slope in terms of the ordinate at the point where it touches the ellipse?

2. Finding tangents by the "new method". (10 points) Show how you would get the result for the slope of the tangent from Problem 1 by applying Leibniz's calculus to your equation for the ellipse.

3. Reasoning behind the "new method". (15 points) We discussed in class Leibniz's algebraic "proof" of the product rule for his differential calculus, that is, d(xv) = (x + dx)(v + dv) - xv = x dv + v dx. Give a similar "proof" of his quotient rule for the differential d(v/y). (You will need to come up with a plausible excuse for ignoring a (y dy) term.)

   Extra credit. (5 points) Give a similar "proof" for the power rule for the differential d(x^a).

4. Philosophical context of infinitesimals. (15 points) Write a short essay (about a page) explaining how you think the philosophical ideas in the excerpt below from Blaise Pascal's Thoughts might tie in with his (and other seventeenth-century mathematicians') reasoning about the mathematics of infinitesimals.

I, 1. Difference between the geometric spirit and the subtle spirit. In the first, the principles are tangible, but far removed from common usage; so that it is difficult to direct one's attention to them, on account of lack of familiarity. But if one does turn to them, one sees the principles clearly; and one must have an altogether false mind to reason wrongly about principles so coarse that it is almost impossible that they should elude one.

But in the subtle spirit, the principles are in common usage and constantly before everybody's eyes; one does not have to redirect one's attention or make an extraordinary effort, it is only a question of having a proper perception. But one must have that, because the principles are so fine and in such great number that it is almost impossible that they should not elude one. And the omission of a principle leads to error: so one must have very clear perception to see all the principles, and a judicious mind in order not to reason falsely about the known principles.

Thus all geometers would be subtle if they had clear perception, since they do not reason falsely about principles that they know; and subtle minds would be geometers if they could direct their gaze to the unaccustomed principles of geometry.

III, 231. Do you believe that it is impossible that God should be infinite, without parts?---Yes.---Then I will show you something infinite and indivisible. It is a point moving throughout an infinite space; since it is one and the same in all regions and is completely whole in every place. [...]

III, 233. Infinity---nothing. Our soul is thrown into the body, where it finds number, time, dimensions; it reasons thereupon, and calls this nature and necessity, and cannot believe otherwise.

Unity joined to infinity adds nothing to it, any more than one foot to an infinite measure; the finite is annihilated in the presence of the infinite, and becomes pure nothingness. There is not as great a disproportion between our justice and that of God as between unity and infinity. [...]

We know that there is an infinite, and do not know its nature. Since we know that it is false that numbers are finite, then it is true that there is an infinite in number, but we do not know what it is. It is false that it is even, it is false that it is odd; because in adding unity, its nature is not changed. However, it is a number, and any number is even or odd (it is true that that is understood about all finite numbers). Thus one can easily recognize that there is a God without knowing what he is. [...]