Customer Review

August 18, 2013
This book can be of some use as a compendium of block quotes from Descartes and his contemporaries on many aspects of the general intellectual atmosphere of the time. As a monograph on "Descartes's mathematical thought," however, it is woefully incompetent.

Thus for example in discussing the "new compasses" on which Descartes's geometry is founded, Sasaki claims that Descartes's interest in these instruments "might have reflected his general inclination to respect the practical and utilitarian aspects of knowledge" (p. 219), and that he "tried to provide his compasses with theoretical foundations through exact algebraic considerations" (p. 105).

This is exactly backwards. Descartes's curve tracing tools are obviously hopelessly impractical, as many contemporaries pointed out. Nor did Descartes try to provide an algebraic foundation for them, but rather the other way around: his algebra was founded on his compasses, not vice versa, as Descartes explains very clearly and unequivocally in the Geometrie. It is puzzling indeed how someone could write an entire book on Descartes's mathematical thought and yet fail to grasp even the most rudimentary aspects of it.

On the few occasions where Sasaki attempts to dabble with actual mathematics he again betrays his utter incompetence in even the simplest geometrical matters. Consider for example Descartes's geometrical construction of multiplication: Given two line segments BC and BD to be multiplied, where a portion BA of BD is taken as unity, "I have only to join the points A and C, then to draw DE parallel to AC, and BE is the product of this multiplication." (p. 212)

Along with the accompanying figure this explanation of Descartes's is crystal clear. Nevertheless Sasaki immediately follows it up with some incoherent, pseudo-mathematical gloss of his own, which I am quoting here in full:

"Let BC and BD be two given line segments." Ok, it seems that Sasaki is going to offer his own paraphrase of Descartes's construction for some reason, even though there is no need to.

"Construct a triangle, of which one side is BC and the other side is the unit." Well, a triangle doesn't have two sides, but I guess it's clear enough what is meant. Apparently Sasaki is going to give us a recipe for the preliminary setup assumed by Descartes, although it is childishly simple and obvious from the figure.

"Since AC is parallel to DE by construction, AB is to BC as BD is to BE." Huh? Now we are apparently presupposing all of a sudden that Descartes's construction has been carried out already. But then what was the point of the previous two sentences? Nothing more, evidently, than to fill out the pages of a $260 book with worthless drivel.
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