August 10, 2008
Tangency
Having just suggested that Descartes was not important as a mathematician, I thought I'd try to back that up via a consideration of the influence of his work on the invention of the calculus.
When calculus is taught, differentiation is commonly the first aspect of it to be covered, and this is in terms of it being a method for finding tangents to algebraic curves. Prior to the invention of the calculus there were other ways—less general and more difficult—for finding tangents. One of these was given by Descartes in La Géométrie. This depended on an insight that was partially geometric, and partially algebraic.
Suppose you have a curve, and a point on the curve at which you want to find the tangent. Now suppose there is a circle that passes through this point. The circle will probably cross from one side of the curve to the other, and then cross back again somewhere else, so intersecting with the curve twice. However, if the circle just touches the curve once then the tangent of the circle at this point will be the same as that of the curve, and hence (perpendicular to this) the normal of the circle will be the same as that of the curve. The centre of the circle will lie on this normal.
In algebraic terms, the intersections between the curve and the circle will be given by roots of some equation. If there are two intersections then there will be two roots. However, if the circle just touches the curve, and so there is only one intersection, then this corresponds to the equation having a repeated root. [This is like, for example, the equation x2-2x+1=0, which can be factorised as (x-1)(x-1)=0, having a repeated root at x=1.]
Putting this together: given a curve and a point, if you can contrive a way to find a circle that passes through the point, and which gives a repeated root in the equation describing how it intersects with the curve, then you can find the tangent to the curve as being perpendicular to the line passing through the point and the centre of the circle.
Applying this method wasn't terribly straightforward. However, Jan Hudde devised a refinement to this method that made it easier to find a suitable circle. As far as I can tell (i.e. going by MA290 TV5) it went something like this:—
Suppose, for example, that you want to find the tangent to the curve y=x3 for some particular value of x. Let there be a circle with its centre on x-axis at (c,0), and which cuts the curve at the point you want the tangent, and also somewhere else. Call the difference in ordinates of the two intersections e:
The first step is to find an equation relating e to c. The intersections are at (x,x3) and (x+e,(x+e)3). And as they are both on the circle, their respective distances from (c,0) must be the same. This gives
(x+e–c)2 + ((x+e)3)2 = x2 + (x3)2Expanding this out and then simplifying gives
2e(x–c) + e2 + 6x5e + 15x4e2 + 20x3e3 + 15x2e4 + 6xe5 + e6 = 0i.e.
2e(x–c) + 6x5e + e2f = 0where f=1+15x4+20x3e+15x2e2+6xe3+e4.
Then dividing through by e and re-arranging a bit gives
x–c = –3x5 - ef/2The point of all of this is so that we can now apply one of Fermat's tricks. If the second intersection ‘adequates’ to the first then the circle be just touching the curve:
Also e, the difference between them, disappears. This gives
x–c = –3x5So the slope of the normal is x3/(x–c)=–1/(3x2), and hence the slope of the tangent is 3x2 (as you might have expected).
As I recall, Isaac Newton's initial work on what came to be called differentiation was done in 1665 when he was working through this stuff from his copy of La Géométrie. Instead of going from two adequal points on a curve to a circle passing through them, and then (via its normal) to the tangent, Newton realised that you could get the tangent from the chord joining the two points.
In the example above, the slope of the chord is
((x+e)3 – x3) / eso as e disappears this becomes 3x2.
= (x3 + 3x2e + 3xe2 + e3 – x3) / e
= 3x2 + 3xe + e2
Whilst it might appear that Newton was building on Descartes' work, closer inspection of the developments shows a different picture:
- Fermat had developed ways of reasoning about 'adequal' quantities that he had applied to tangency problems, although he did not develop his ideas very far. More importantly, he had not published them. However, they were presumably known amongst Mersenne's circle of correspondents.
- Descartes devised a different way of dealing with tangency problems which he did publish, but which was very cumbersome to apply.
- Hudde used Fermat's ideas to simplify Descartes' method. This was included in the later editions of La Géométrie.
- In his reading of La Géométrie, Newton extracted Fermat's ideas from Hudde's method. He generalised them, whilst excising Descartes' one contribution to the solution of tangency problems.
Posted by robin2 at August 10, 2008 07:32 PM