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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)2
Expanding this out and then simplifying gives
2e(x–c) + e2 + 6x5e + 15x4e2 + 20x3e3 + 15x2e4 + 6xe5 + e6 = 0
i.e.
2e(x–c) + 6x5e + e2f = 0
where f=1+15x4+20x3e+15x2e2+6xe3+e4.

Then dividing through by e and re-arranging a bit gives

x–c = –3x5 - ef/2
The 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 = –3x5
So 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) / e
= (x3 + 3x2e + 3xe2 + e3 – x3) / e
= 3x2 + 3xe + e2
so as e disappears this becomes 3x2.

Whilst it might appear that Newton was building on Descartes' work, closer inspection of the developments shows a different picture:

This is why the ‘Descartes effect’ in early modern mathematics cannot really be understood simply in terms of Descartes' own work as a mathematician.

Posted by robin2 at 07:32 PM | Comments (0)

August 07, 2008

TMA 2.2

Question:

To what extent is it fair to claim that mathematics developed in the sixteenth and seventeenth centuries primarily as a means for responding to technological and scientific needs?

The influence of science and technology on mathematical development in the 16th and 17th centuries

The 16th and 17th centuries were times of great change in Europe: the Reformation; the impact of printing; exploration of the New World; loosening of scholastic dogmas; the rising urban middle class; the development of technology and the invention of modern science. Specific to mathematical history there was the ongoing recovery of ancient Greek works; an increase in vernacular texts; innovations in symbolism; the emergence of an pan-European mathematical community; developments in algebra and its new use in geometry. These things were bound up with each other in complex ways; when it comes to mathematical development over the course of these two centuries there is no reason to think that the needs of technology and science were its sole cause.

A clear-cut case of new mathematics being developed due to such needs is that of Napier’s logarithms, which were invented and refined with the explicit aim of easing the tiresome calculations needed in many applications of mathematics. However, it is also possible to point to important mathematics developed without application. The researches in algebra in 16th century Italy are such: no problem in commercial arithmetic would require cubics for their solution. Later algebraic work, such as Fermat’s on number theory, can similarly be regarded as pure research.

The requirements of technology and science are only likely to result in new mathematical developments if the requirements themselves are new. In some cases this was true: Kepler’s computationally intensive work was made possible by increasing accuracy in astronomical observation; and the investigation of spherical geometry in relation to navigation became important to the various European nations that were developing their interests in the New World (and so had to negotiate the Atlantic). However, some important practical requirements were not new: in surveying there was a tradition of “sub-Euclidean” geometry going back through the Middle Ages to Roman agrimensores; and so, after the development of trigonometry in the 15th century, it was not the creation of new knowledge that was important so much as the changing distribution of existing knowledge.

Knowledge was propagated in the form of printed manuals for literate artisans or embodied in the use of instruments. In itself this is not a properly mathematical development, but is important nonetheless. If nothing else, the increasing importance of mathematical knowledge for diverse practical applications resulted in improvements in its teaching: a tide on which all boats rise.

Mathematical developments certainly made some scientific advances possible. Newton’s mathematical physics would have been inconceivable without the algebraic approach to geometry instigated by Descartes. However, this does not make the science the cause of the mathematics: Descartes was not motivated by a desire to produce such a physics, and his own physics was not mathematical in this sense. For Descartes mathematics was a source of example problems which could be used to demonstrate his method and to sharpen his logic. In his philosophical schema this logic provided the basis for his metaphysics which in turn founded his physics.

In this example it is plausible that the emergence of algebraic geometry was in part due to styles of study arising from the invention of printing. Learning became more book-based, and—as Peter Ramus noted—the synthetic proofs of classical geometry did not produce much insight in themselves. Moreover, the distribution of books enabled scholars to work outside of traditional institutions; but, again, the use of synthesis in Greek works did not furnish them with a method of invention. Using algebra resolves this by allowing wholly analytic solutions: a problem could be assumed solved—as normal—but the unknowns of this solution could be represented symbolically, manipulated, and solved in the course of analysis. This made it ideal for independent scholars like Descartes. Newton, although belonging to a Cambridge college, seems to have largely taught himself from books: hence his early enthusiasm for Descartes’ Geometry over Euclid’s Elements.

Pure maths research and practical applications could be closely related. Two contrasting examples of this are provided by calculus and projective geometry. Calculus originated in the theoretical treatment of tangents and areas of curves, in the work of mathematicians such as Fermat and Cavelieri; and Newton brought this work to fruition in order to provide a basis for his work in physics. Whereas the origins of projective geometry were practical, in the theory of perspective drawing created by Leon Battista Alberti in the 15th century; but Desargues, who wrote a manual of perspective, used this as a starting point for his theoretical endeavours.

It is conspicuous that much mathematical thought had a mechanistic flavour: Napier conceived of his logarithms as being generated by concurrent motions, and Newton thought of tangents to curves in terms of instantaneous direction of motion. Although this was not universal—Kepler’s account of logarithms and Descartes earlier treatment of tangency were both statically geometric—such a conceptualisation was not considered disreputable. Similarly, Kepler’s and Galileo’s view of the world as inherently mathematical meant that—for all the avowed Platonism—physics was not a separate field of investigation.

Overall, it is too simplistic to say that practical needs drove all the mathematical developments in this period. It is true that it drove some, but it is also true that the influence could be much less direct than this thesis suggests, and also that technology, science and mathematics were all influenced by other aspects of these changing times.

Thoughts

I'm reasonably happy with that, although there are certain things in it that I now think are wrong. I don't even think I could have had much of a problem with the word count (otherwise I can't see why I would have given Alberti's name in full).

I think I over-estimated the important of Descartes (easily done) in a way that tends to undermine my comments about institutional changes in the period. I also think I missed a big bit of the picture about the invention of the calculus, and consequently the influence of applied maths was more pervasive that I realised.

The comment about “the algebraic approach to geometry instigated by Descartes” is misleading if not outright nonsense. The phrasing of it comes about because I knew that François Viète's work predated Descartes' (by about half a century), but that Descartes' approach was significantly different, chiefly in its essentially dimensionless treatment of magnitudes. (For example, for Descartes two lengths could be multiplied to give a third length; whereas for Viète they would give an area that could not be compared directly with lengths, only with other areas.) I am now not convinced by this, and am more inclined to believe John Wallis' famous assessment that there was little or nothing in Descartes that wasn't to be found in Viète, William Oughtred or Thomas Harriot.

There were lively algebraic traditions in England and the Netherlands, and—as far as I can see—the main reasons for Descartes' standing as a mathematician was that he was treated as a figurehead by certain Dutch mathematicians (van Schooten, Hudde, van Heurat) and that the ‘commentaries’ included in the later editions of Descartes' Geometry were used as a vehicle for propagating their research results. (With the third edition of 1683, only a quarter was actually by Descartes.) But for that it is conceivable that Descartes' views on geometry would have had of no greater impact that Bishop Berkeley's views on the calculus.

The reason that this undermines my point about universities vs independent scholars is that the Dutch mathematicians were working in the context of a university system. I still think I had a point—apparently John Wallis taught himself mathematics from Oughtred's Clavis Mathematicae—but I didn't really substantiate it.

I said that my picture of build-up to the calculus was decificient. I won't go into detail here, but it seems likely that development of numerical methods, particularly of interpolation techniques—used in the production of trigonometric tables required in navigation and astronomy—formed an important part of the calculus' conceptual basis.

Posted by robin2 at 11:35 PM | Comments (0)