Worksheet 3/23/2018–3/26/2018 - History of Math (Spring 2018)

The purpose of this assignment is to understand early methods of differentiation, which predate the modern notion of a limit by at least 150 years, and to compare similarities and differences with current methods. As a word of caution, the reader should be aware that the methods of Barrow, Newton, Leibniz and their contemporaries are considered non-rigorous by modern standards... you may be asked to reflect, at some point, on why.

From several different perspectives, we will consider the curve defined by

y − x3 + 5x2 − 6 = 0.

Our goal, in each of the following five problems, is to find the slope of the tangent line at any given point (x, y) on the curve.

1. Pretend you are a modern Calculus I student. Set y = f(x) = x3 − 5x2 + 6, and differen- tiate using the power rule to find an expression for y0.

2. Pretend you are a modern Calculus I instructor, trying to illustrate to your students how the slope of a tangent line may be found by taking a limit of slopes of secant lines. (a) Set y = f(x) = x3 − 5x2 + 6, and compute a simplified expression for the difference quotient

f(x + h) − f(x) . h

Sketch a picture of what this different quotient is supposed to represent.

(b) Find y0 by taking the limit as h → 0 of your expression for the difference quotient. 1 2

3. Pretend you are Isaac Barrow (see Burton pp. 389–391 for additional guidance on this problem). Consider the curve defined by y − x3 + 5x2 − 6 = 0, and let P = (x, y) denote a point on this curve.

(a) Construct the “differential triangle”: let Q = (x − e, y − a) be another point on the curve, and let R be a point so that 4P QR becomes a right triangle, with right angle ∠R, and the legs have length PR = a and QR = e. Sketch a picture of the triangle with two vertices on the curve. (The picture need not be to scale.)

(b) Now supposing the distances e and a to be infinitely small (whatever that means), we ←→ agree it is reasonable to regard the line PQ as being equal to the tangent line to the a curve at P . So we want to find its slope e . To find it, observe that since Q lies on the curve, we must have

(y − a) − (x − e)3 + 5(x − e)2 − 6 = 0.

Expand out the equation above.

(c) Use the fact that P lies on the curve to write y − x3 + 5x2 − 6 = 0, and remove as many terms as possible from your expression in part (b).

(d) Looking at the reduced expression from part (c), now “reject all terms in which a and e are above the first power, or are multiplied together (for they are no value with the rest, as being infinitely small.” (Barrow 1670) In other words, drop all terms where the degree of a plus the degree of e is more than 1.

a (e) Lastly, solve for e to find an expression for the slope of the tangent line. 4. Pretend you are Sir Isaac Newton (see Burton p. 419 for additional guidance on this prob- lem). Consider the curve defined by y − x3 + 5x2 − 6 = 0. From your viewpoint, each of the variables x and y is fluent, i.e. in continuous motion along the path of the curve. The flux- ion, or instantaneous rate of change, of y is denoted byy ˙ and the fluxion of x is denoted byx ˙.

y˙ To find the slope of the tangent line at a given point (x, y), it is our task to find x˙ , the ratio of the fluxion of y to the fluxion of x.

(a) Suppose now the fluents x and y are altered by an infinitely small change o (whatever that means). After the change is effected, the new values of x and y will be x +xo ˙ and y +yo ˙ , respectively. Since the new values still lie on the curve, we must have

(y +yo ˙ ) − (x +xo ˙ )3 + 5(x +xo ˙ )2 − 6 = 0.

Expand out the equation above.

(b) Use the fact that y − x3 + 5x2 − 6 = 0, and remove as many terms as possible from your expression in part (b).

(c) Divide o out from all the remaining terms in your reduced expression from part (c). 3

(d) Looking at the reduced expression from part (d), note that “whereas o is supposed to be infinitely little, that it may represent the moments of quantities, the terms that are multiplied by it will be nothing in respect to the rest; I therefore reject them.” (Newton 1671). In other words, drop all terms where o appears.

y˙ (e) Lastly, solve for x˙ to find an expression for the slope of the tangent line. 5. Pretend you are Gottfried Wilhelm Leibniz (see Burton pp. 415–416 for additional guid- ance on this problem).

Consider the curve y − x3 + 5x2 − 6 = 0. Your method for finding the slope of the tangent line at (x, y) is almost identical to Barrow’s, but instead of considering a “differential trian- gle” with two vertices (x, y) and (x − e, y − a) an infinitely small distance apart, you prefer considering a “characteristic triangle” (which is the same idea with a different name) with two vertices (x, y) and (x + dx, y + dy). Here again dx and dy are considered infinitely small distances. Mimic the methods of the previous problems, to find the slope of the tangent line, dy which is given by the ratio dx .

6. The use of infinitesimals, or “infinitely small quantities,” like Newton’s o and Leibniz’s dx were criticized by many authors. One of the major criticisms is that the inventors of the calculus treated infinitesimals as if they were zero, while simultaneously treating them as if they were non-zero.

For example, in George Berkeley’s famous critique of calculus, The Analyst (1734), the au- thor writes:

“...the fallacious way of proceeding to a certain Point on the Supposition of an Increment, and then at once shifting your Supposition to that of no Increment . . . Since if this second Supposition had been made before the common Division by o, all had vanished at once, and you must have got nothing by your Supposition. Whereas by this Artifice of first dividing, and then changing your Supposition, you retain 1 and nxn−1. But, notwithstanding all this address to cover it, the fallacy is still the same. ...

And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?”

Explain in concrete terms exactly how Barrow, Newton, and Leibniz treat infinitesimals both as zero, and as non-zero.