Descartes: A Biography

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 Descartes: A Biography

geometry, he assumes that all three kinds of problems are soluble, although

only the first two – those that are constructible using a ruler and compass,

or using one of the new compasses – belong properly to geometry. The neat

division into three parallel types, for discrete and continuous magnitude

respectively, might suggest that the problems in each group were mathe-

matically equivalent, or that problems in geometry could be resolved by

using equations of the corresponding level in algebra. However, Descartes

seems not to have made enough progress in applying algebraic methods

to geometry to realize that this was not the case.

What was new in this programmatic statement, apart from the scope

and ambitiousness of the proposal, was the acceptance of ‘new compasses’

as valid instruments of construction, in addition to the traditional compass

used in geometry for drawing a circle. These compasses made it possible

to solve two kinds of problem that had been known in the literature for

centuries: (a) constructing mean proportionals to given numbers, and (b)

dividing a given angle into more than two equal parts. Apart from such

technical issues that are specific to mathematics, however, the letter to

Beeckman also shows that Descartes was thinking of mathematics as a

model for resolving all kinds of theoretical problems, and that his pri-

mary interests inwere mathematical or involved the application of

mathematical techniques to problems in physics.

Beeckman appreciated the efforts of his younger collaborator enough to

copy this letter into his journal. In the subsequent months Beeckman had

other worries on his mind, both professional and personal. The Synod

of Dordrecht concluded onMayand, in November of that year,

Beeckman was appointed to teach at Utrecht. In April,hemarried

Cateline in Middelburg.

Beeckman as a ‘Special Friend’

Following the relatively short boat journey in March,when Descartes

returned from Middelburg to Breda, he reported that he ‘crossed the

waves without getting sea-sick’ and that, as a result, he felt ‘more coura-

geous about taking a longer journey.’His new plan was to travel by

sea to Denmark and onward by land to Germany. He was not deterred

bystormy seas or troop movements associated with the early stages of

the Thirty Years’ War; however, he was forced to wait a little longer

before embarking. He planned to sail northward and thereby bypass the