The Geometry of Rene Descartes (English and French Edition) ebook

This is a great read both historically and for the student of math.

More interesting is if you read a modern text analyzing DeCartes to see where he struggled. Such as with complex roots.

Yes there are way better instructional texts.

My one issue with this book, well two actually is the facsimile of the orginal could be better. And each page faces each other, so you can see the translation and the original. One thing they did not do is copy the diagrams from the original. It is minor, but I like working the problems, and I wish they had them in the translation as well. It would help with the alignment and make it a bit more user friendly.

For the price it is a bargain.

Mathematicians have long said that Rene Descartes invention of analytic/coordinate geometry took mathematics forward. Saying that to any man on the street, they'd think Rene Descartes writings would be easy/boring reading. They'd think there's be a few pages showing a two dimensional grid and four quadrants, a few positive and minus signs . . . maybe a second degree equation . .. plotted, and lets get back to reading his philosophy of science book. No way! In fact, his philosophy of science is far more antiquated and obsolete than his mathematics.

His mathematics is kind of obsolete; but, that's disappointing! His ideas of coordinate geometry were obsolete as of the writing of his own book. He realized the two dimensional grid idea only after thinking about going to three dimensions, and he was thinking about solving three dimensional curves, not just plotting third degree equations. And it's the solving of mathematical problems by means of properties of curves that led him to analytic geometry.

Most non-mathematician students get a little geometry in high school and that's it. This is generally plane geometry. Even this plane geometry is watered down compared to what the Greeks did. There's none of the geometric algebra. The greeks solved equations and did basic arithmetic operations by means of this geometric algebra. This is Rene Descartes starting point including finding roots of sqaree roots. The Greeks derived conics from trying to solve the three Delian problems - trisecting an angle, duplicating a cube, and squaring the circle. They came up with various curves and studied them to a certain extent. Rene Descartes solves and goes far beyond what the Greeks did there. His invention of analytic geometry evolved out of solving these problems. Rene Descartes little book here solves the past, and heralds the future.

Descartes uses lines to find intersections of these special curves. These intersections are roots of equations. Descartes basically generalizes the old Greek geometric algebra of finding roots to higher curves. He finds correspondences to properties of the theory of equations and solving these problems - of what's a linear curve, what's a space curve and so on. We're talking about spirals(Archimedes favorite curve), quadratrixes, conchoids, and cissoids.

His treatments of systems of equations are also far ahead of what the average community college degree student will see. And he relates it to his theory of curves as well.

There's new constructions, insights on almost every page.

The Dover book I bought and looks like everyone else has bought in the 20th and 21st centuries uses scholarly commentary from an Eugene Smith. And that's a good thing. Without the footnotes explaining things, one couldn't solve anything, much less see where/when Rene ever actually discovered analytic geometry. He also seems to reference some obscure late 1600 work commentary that gets into all the findings of Descartes more than this little book - elliptic and hyperbolic concoids and other geometric theorems. I need to learn French and buy that gem in a haystack book. That sounds like a great book.

Reading this, and having read Galileo's "Two new Systems" book, there was plenty more that the Greeks could have done geometrically; but, they did not. It's proof that there was a dark ages(there's dark ages denyers, just like moon landing denyers)

A+

Love it!

The main theme of Descartes's geometry is the justification of algebraic methods in terms of the standards of classical, construction-based geometry.

"To treat all the curves I mean to introduce here [i.e., all algebraic curves], only one additional assumption [beyond ruler and compasses] is necessary, namely, [that] two or more lines can be moved, one [by] the other, determining by their intersection other curves. This seems to me in no way more difficult [than the classical constructions]." (p. 43)

This procedure can be described as follows. You have a plane in front of you, say a pinboard. On top of this plane you place a transparency sheet, which you pin to it with a needle. Thus the sheet is free to rotate about this fixed point. On this sheet you draw some curve and then you make a cut along this curve with a knife. Now you get a second transparency sheet a place it on top of the first. On this sheet too you draw a curve. Then you pierce the second sheet in some point and put a peg through the hole. This peg you also make go through the cut in the bottom sheet. Now you move the peg in a linear motion. The peg then moves the top sheet with it in a simple linear translation, while the bottom sheet goes through a rotating motion around its needle as the peg runs along its cut. The point(s) of intersection of the two curves of the transparency sheets now trace a new curve, typically of higher degree than the two original ones. For example, Descartes shows how to generate a hyperbola from a line and a line (p. 52), and a "Cartesian parabola" (a cubic curve) from a line and a parabola (p. 84). And so it continues: once a cubic curve has been generated it can be taken as one of the starting curves, etc. Another way of generating more complex curves is to use intermediate sheets, pegged to each other pairwise so that the original motion propagates through the entire arrangement. Descartes's "mesolabum" is an example of this (p. 46).

The curves generated in these ways, being intersections of algebraic curves under an algebraic initial motion, are themselves algebraic. "I could give here several other ways of tracing and conceiving a series of curved lines, each curve more complex than any preceding one, but I think the best way to group together all such curves and then classify them in order, is by recognizing the fact that all points of those curves which we may call 'geometric', that is, those which admit of precise and exact measurement, must bear a definite relation to all points of a straight line, and that this relation must be expressed by means of a single equation" (p. 48). In other words, the legitimate curves of exact geometry are precisely those representable by algebraic equations in rectilinear coordinates.

The converse implication, that all algebraic curves are traceable in these ways, is apparently seen by Descartes as a consequence of the fact arbitrary points of an algebraic curve can be constructed (by fixing some x-value and constructing the corresponding y; this assumes the general construction procedure for algebraic equations discussed below): "this method of tracing a curve by determining a number of its points taken at random applies only to curves than can be generated by a regular and continuous motion" (p. 91), he asserts without proof.

In contrast to algebraic curves, "the spiral, the quadratrix, and similar curves ... are not among those curves that I think should be included here, since they must be conceived of as described by two separate movements whose relation does not admit of exact determination" (p. 44), "since the ratios between straight and curved lines are not known, and I believe cannot be discovered by human minds, and therefore no conclusion based upon such rations can be accepted as rigorous and exact" (p. 91). In other words, these kinds of curves involve independent linear and circular motions of coordinated speed, and thus essentially depend on pi, which is not an algebraic number and thus unknowable by Cartesian standards. Similarly, one cannot construct arbitrary points on these curves (a given y-coordinate is generally not an algebraic function of the corresponding x-coordinate).

The above establishes algebraic curves as legitimate construction tools on par with ruler and compasses. It remains to carry out the actual constructions themselves, that is, for a given problem to construct the required points as intersections of various algebraic curves. Thus Descartes shows how to find the roots of any third or forth degree equation by intersecting a circle and a parabola (pp. 192-205), and the roots of any fifth or sixth degree equation by intersecting a circle and a Cartesian parabola (pp. 220-237). Constructions for higher degrees are "intentionally omitted so as to leave to others the pleasure of discovery," "for in the case of a mathematical progression, whenever the first two or three terms are given, it is easy to find the rest" (p. 240).

These constructions mean that a problem can be considered solved according to classical construction standards (as enlarged by Descartes) whenever it is reduced to an algebraic equation. Thus, to Descartes, algebraic geometry does not replace classical construction-based geometry, but is rather subsumed by it. A concluding remark shows that this was no mere theoretical point but that the constructions were indeed intended to be carried out in concrete cases: "in many of these problems it may happen that the circle cuts the [Cartesian parabola] so obliquely that it is hard to determine the exact point of intersection. In such cases this construction is not of practical value. The difficulty could easily be overcome by forming other rules analogous to these, which might be done in a thousand different ways." (p. 239)

"While it is true that every curve which can be described by a continuous motion should be recognized in geometry, this does not mean that we should use at random the first one that we meet in the construction of a given problem. We should always choose with care the simplest curve that can be used in the solution of a problem, but it should be noted that the simplest means not merely the one most easily described, not the one that leads to the easiest demonstration or construction of the problem, but rather the one of the simplest class [i.e., degree] that can be used to determine the required quantity." (pp. 152, 155) Remarkably, Descartes's first use of this principle is to effectively rebuke himself: "there is, I believe, no easier method for finding any number of mean proportionals [than the mesolabum mentioned above]" (p. 155), but it can be done by curves of lower degree so "it would therefore be a geometric error to use [the mesolabum]" (p. 156).

Three main applications of the new geometry are discussed by Descartes. The first and by far the most substantial is Pappus's problem, which asks for the locus of all points that have particular distance relations to a set of given lines. This problem is a showcase for algebraic geometry, as it is an extremely general problem with great classical prestige, yet eminently treatable by algebraic means. In fact, Descartes claims (mistakenly) that the set of all possible solutions to Pappus's problem is exactly the set of all algebraic curves (p. 59).

A second application is "a discussion of certain ovals which you will find very useful in the theory of catoptrics and dioptrics" (p. 115), which is interesting as Descartes gives pointwise constructions only of these ovals in place of equations (pp. 114-149).

The third main application is Descartes's double-root method for finding normals (and thereby tangents). As a simple example, suppose we seek the normal to the parabola y=x^2 at the point (1,1). This normal is determined by its intersection with the y-axis, call it (0,Y). Consider the circle centred at this point passing through (1,1). Its equation is x^2+(y-Y)^2=1^2+(Y-1)^2=Y^2-2Y+2, or x^2+y^2-2Yy+2Y-2=0. We take its intersection with the parabola y=x^2 by replacing x^2 by y to obtain y^2+(-2Y+1)y+2Y-2=0. Since there is only one y-value for which the circle intersects the parabola, this equation must have the form (y-r)^2=0, i.e., y^2-2ry+r^2=0. Comparing this with the above we see that r=Y-1/2 and r^2=2Y-2, so 2Y-2=(Y-1/2)^2, so Y=3/2.