Descartes: A Biography

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The ScientificEssaysand theDiscourse on Method 

strikes a raindrop, it can be refracted twice (on entering and exiting the

drop), and reflected once at the back of the raindrop that faces the observer

(see diagram). Likewise, it may be refracted twice and reflected twice,

and this explains how the secondary bow appears at a different height,

why it is less vivid (since only some of the light rays are reflected in this

way), and why the colours appear in the opposite order in which they are

seen in the primary bow (diagram).

Diagram.

Diagram.

One of the reasons Descartes offered for publishing this first book in

French was to make it accessible to as wide a readership as possible.How-

ever, despite its being in French, the mathematical language in which the

Geometrywas written was bound to inhibit many readers. Descartes ack-

nowledged this, by inserting a special ‘notice’ at the beginning of the text:

Up to this point I have tried to make myself intelligible to everyone. However, for this

treatise, I fear that only those who already know what is in geometry books will be able

to read it. The reason is that they contain many truths that are very well demonstrated,

and I therefore thought it would be superfluous to repeat them and, for that reason, I

have taken the liberty of using them. (vi.)

The caution was appropriate. Only those who were trained in mathe-

matics could understand what the problems were. Besides, it is clear that

Descartes had been working on some of these problems for at least ten

years, that he had shared his interim conclusions with various correspon-

dents, and that he had written theGeometryin its published form ‘only

when theMeteorswas being printed’ and had ‘even invented part of it at

that time’.

One of the most obvious features of theGeometryis that it is not a series

of proofs of theorems, similar to the familiar methods used by Euclid.