## SEARCH

#### Subject

##### ( see all 10)

- Mathematics [x] 35 (%)
- Geometry 34 (%)
- Astronomy 1 (%)
- Astrophysics 1 (%)
- Chemistry/Food Science, general 1 (%)

## CURRENTLY DISPLAYING:

Most articles

Fewest articles

Showing 1 to 10 of 35 matching Articles
Results per page:

## Geometrical problem solving — the state of the art c. 1635

### Redefining Geometrical Exactness (2001-01-01): 211-221 , January 01, 2001

The present chapter concludes my discussion of the early modern tradition of geometrical problem solving before Descartes. Problem solving constituted the primary context of Descartes’ geometrical studies to which Part II is devoted. His contributions, however, changed the theory and practice of geometrical problem solving in so fundamental a manner that they eclipsed many of the techniques, concepts, and concerns of the earlier tradition. It is therefore appropriate to conclude Part I by a sketch of the state of the art of geometrical problem solving around 1635, that is, just before Descartes published his innovations (and also before Fermat’s new techniques began to circulate among cognoscenti).

## Arithmetic, geometry, algebra, and analysis

### Redefining Geometrical Exactness (2001-01-01): 119-134 , January 01, 2001

The adoption of algebraic methods of analysis induced a gradual fusion of arithmetic, algebra, and geometry, which in turn gave rise to a number of technical, terminological, and conceptual questions. The fusion also spurred issues of legitimation, in particular the question whether it was allowed to use numbers in geometry. The present chapter deals with the technical, terminological, and conceptual questions; the issues of legitimation are the subject of Chapter 7. We find the first successful merging of the fields of algebra, arithmetic, and geometry in the work of Viète; Chapter 8 explains how Viète overcame the obstacles and the legitimation issues involved.

## Clavius

### Redefining Geometrical Exactness (2001-01-01): 159-166 , January 01, 2001

Having reviewed in the previous five chapters the early modern tradition of geometrical problem solving, the emergence of algebraic analysis as the principal dynamics in that field, and the questions of legitimation that were raised in relation to this process, I now turn to the actual debates on the interpretation of geometrical exactness with respect to constructions.

## Back Matter - Redefining Geometrical Exactness

### Redefining Geometrical Exactness (2001-01-01) , January 01, 2001

## Descartes’ solution of Pappus’ problem

### Redefining Geometrical Exactness (2001-01-01): 313-334 , January 01, 2001

Descartes first studied Pappus’ problem during late 1631 and early 1632, on the instigation of Golius. In Chapter 19 I argued that the confrontation with the problem was decisive for the final stage of the development of his programmatic ideas on geometry. I now turn to his treatment of the problem in the *Geometry*, where he used it as the central example for illustrating his techniques and showing their power.

## Front Matter - Redefining Geometrical Exactness

### Redefining Geometrical Exactness (2001-01-01) , January 01, 2001

## Curves and the demarcation of geometry in the Geometry

### Redefining Geometrical Exactness (2001-01-01): 335-354 , January 01, 2001

I now come to a crucial issue in Descartes’ geometrical doctrine: the demarcation of geometry.^{1} His program required a reinterpretation of geometrical exactness concerning constructions. Constructions were to be performed by means of curves; they had to be geometrically acceptable and as simple as possible. Consequently, his new doctrine had to provide clear answers to the following two questions:
A.

Which curves are acceptable as means of exact construction in geometry?

B.When is one curve simpler than another?

I will discuss the second question in Chapter 25. The first question concerned the demarcation between exact, geometrical procedures, on the one hand, and non-exact, non-geometrical ones on the other. Descartes gave such a demarcation in terms of the curves used in the procedures. He distinguished between “geometrical” and “mechanical” curves; the former were acceptable in geometry, in particular for use in constructions, the latter were not. In effect, his distinction was straightforward: “geometrical” curves were those that, with respect to rectilinear coordinates, had an algebraic equation; all ethers (in particular the spiral and the quadratrix) were “mechanical.” This Cartesian demarcation of geometry had a great influence in the second half of the seventeenth century, especially as the background for discussions on the exactness of various methods for tracing curves.^{2}

## The legitimation of geometrical procedures before 1590

### Redefining Geometrical Exactness (2001-01-01): 23-36 , January 01, 2001

Sixteenth- and seventeenth-century mathematicians were not the first to struggle with the interpretation of exactness of geometrical constructions; also in the classical Greek period there existed a rich variety of constructional procedures and many opinions on their acceptability. It appears that these opinions never converged to a clear *communis opinio*. This lack of uniformity and the relative scarcity of sources meant that sixteenth-century mathematicians had no clear classical examples in developing their ideas about the exactness of geometrical constructions. Therefore, it is not necessary to survey the classical ideas on the legitimacy of constructions here; rather I can restrict myself to dealing with those few classical sources that actually influenced the early modern debates. Besides, and fortunately, I can refer to an excellent recent study on the classical tradition of geometrical problem solving, Wilbur Knorr’s *The ancient tradition of geometric problems*._{1}

## Editorial statement

### Archive for History of Exact Sciences (1997-03-01) 51: 1 , March 01, 1997

## Early modern methods of analysis

### Redefining Geometrical Exactness (2001-01-01): 95-117 , January 01, 2001

Two kinds of analysis were distinguished in early modern geometry: the classical and the algebraic.^{1} The former method was known from examples in classical mathematical texts^{2} in which the constructions of problems were preceded by an argument referred to as “analysis;” in those cases the constructions were analysis called “synthesis.” Moreover, a few classical sources^{3} spoke in general about this arrangement. The most important of these texts was the opening of the seventh book of Pappus’ *Collection*; I quote this passage here in full:

Now, analysis is the path from what one is seeking, as if it were established, by way of its consequences, to something that is established by synthesis. That is to say, in analysis we assume what is sought as if it has been achieved, and look for the thing from which it follows, and again what comes before that, until by regressing in this way we come upon some one of the things that are already known, or that occupy the rank of a first principle. We call this kind of method “analysis,” as if to say *anapalin lysis* (reduction backward). In synthesis, by reversal, we assume what was obtained last in the analysis to have been achieved already, and, setting now in natural order, as precedents, what before were following, and fitting them to each other, we attain the end of the construction of what was sought. This is what we call “synthesis.”