I generally enjoyed the biography of the young Descartes in the first few dozen pages of this book. But soon I started reading things about the history of mathematics which I know to be incorrect. So I have to conclude that the book is at least somewhat untrustworthy.
When discussing the duplication of the cube, Aczel says that Eratosthenes was a contemporary of Eudoxus in Plato's Academy, whereas he actually lived more than a century later. Elsewhere on this webpage, Dr. Amir Bernstein dismisses this as "a mere hundred years" and says that "dates of Greek mathematicians/philosophers are known only approximately." Actually, we can date Eudoxus and Eratosthenes quite accurately because Aristotle wrote about Eudoxus' astronomical theories in his _Metaphysics_ and Archimedes dedicated his book _The Method_ to Eratosthenes. Not only did they live more than 130 years apart, but they belonged to completely different mathematical cultures. In between their two lives, there was a philosophical revolution (Aristotle proposed the axiomatic method), a mathematical revolution (Euclid's Elements standardized the practice of geometry) and a social revolution (Alexander's multicultural "cosmopolis" became a reality in Alexandria).
Even such a simple fact such as the year in which Euler first went to St. Petersburg is wrong: it's given as 1730 instead of 1727. Aczel also claims that Euler visited Hanover on this journey. Thiele (1982) gives a very complete chronicle of the journey, based on Euler's own notebook, in his German-language biography, but he makes no mention of Hanover. Tellingly, Aczel gives no citation for his claim, which he uses to bolster his questionable theory that Euler somehow learned his theorem that F+V=E+2 from Descartes' lost notebook.
The biggest error is this claim on page 164: after telling us that Descartes gave a ruler-and-compass construction for square roots, he says "this was one of his greatest achievements in mathematics ... which would have stunned the ancient Greeks since they could construct only much simpler things." Unfortunately, it was not Descartes' discovery, because the construction was, in fact, known to the Greeks. What Descartes presents is just a special case of Proposition 14 from Book II of Euclid's Elements. You need only look at the diagram in Descartes' Geometry (p.4 of the Dover edition) and the one in Euclid's Elements (vol. 1, p. 409, also in Dover) to see this. Yes, one is the mirror image of the other and the letters are different. But the construction is the same. And here's why it works. In II.14, Euclid tells you how to construct a square with an area equal to a given rectangle. If the rectangle has sides of length a and b then its area is ab. Since the area of the square is also ab, the length of each side of the square is the square root of (ab). The Greeks called this the "geometric mean" of a and b, where "mean" is a word we still use today for average. Now what Descartes does is the case b=1, so what he gets is the square root of a. It may seem paradoxical to some, but by constructing a square, Euclid gets a square root along its side.
I think Aczel also gives Cardano too little credit and Tartaglia too much in his story about the solution of the cubic equation, but I'll just refer readers to Boyer & Merzbach (1991, p. 282-286) and Katz (1998, p. 358-364) to make up their own minds.