Most mathematicians have heard of Gerolamo Saccheri in the context of non-Euclidean geometry. We know that he set out to prove the parallel postulate by starting with its negation and trying to obtain a contradiction. He eventually convinced himself that he had succeeded, but he is remembered mostly for having been the first to prove things that later turned out to be theorems of hyperbolic geometry. The result, as DeRisi points out, is a picture of Saccheri as “unwitting innovator.”

Vincenzo DeRisi offers several corrections to that standard story. For one thing, he presents us with the entire *Euclides Vindicatus*. It turns out that when Saccheri says he is setting out to remove “every blemish” from Euclid, he does *not* mean only the parallel postulate. This has been forgotten, mostly because G. B. Halstead decided to translate only the first part of Saccheri’s book.

Henry Saville was perhaps the source of the metaphor: he refers to “two blemishes, two moles” on Euclid’s otherwise “beautiful body.” These blemishes are in the foundations of Euclid’s theory of parallels in Book I and the definition of proportion in Book V. For the theory of parallels, the problem is the well-known one: the parallel postulate seems unwieldy and uncomfortable next to the clarity and simplicity of the others. For proportions, the argument was that Euclid’s definition of proportionality was simply too complicated, obscuring what should have been a simple and straightforward notion.

Writing books to “fix” Euclid was not uncommon, apparently: DeRisi mentions several examples from Italy in the 17th and 18th centuries. One striking thing, however, is that several of these books described themselves as “amending” or “reforming” Euclid — in other words, improving on the *Elements*. Saccheri, however, sets out to *vindicate* Euclid.

DeRisi spends some time thinking about that choice of word, especially in reference to the theory of parallels. How exactly, would it “vindicate” Euclid to present a *proof* of something he took as a postulate? Would this not rather demonstrate that Euclid had made a mistake, that the postulate should have been a theorem? DeRisi argues that what Saccheri is trying to do is indeed to “prove a postulate,” that is, to give a convincing argument that should lead us to accept the postulate. Saccheri’s idea is to prove that the negation of the postulate is self-contradictory by showing that *if we assume the negation of the parallel postulate we can then prove the parallel postulate*. This kind of proof by contradiction via “not-A implies A” had been discussed by Saccheri in a previous book, *Logica Demonstrativa*.

There is also an extended discussion of the reception of *Euclides Vindicatus*, something that is missing from many accounts of the history of non-Euclidean geometry. After all, if Saccheri is merely a “predecessor” that no one read, his role in the story becomes much less important. Luckily, there does seem to be a line of influence from Saccheri to Lambert and from there, perhaps, to Gauss, Bolyai, and Lobachevsky.

The meat of the book is the actual text, of course. It is presented in Latin with facing English translation. For book one, the translation is Halstead’s with minor emendations. For book two, DeRisi uses a translation by Linda Allegri, originally part of her PhD thesis at Columbia in 1960. In this case the emendations have been more extensive. DeRisi adds extensive commentary and notes.

Saccheri’s book one is very interesting. As is well known, he bases his analysis on a particular construction, now often referred to as the Saccheri quadrilateral: on a line, erect two perpendicular segments of the same length, and connect their other endpoints. If we assume the parallel postulate, the result will be a rectangle, so that the two angles at the top will be right. Saccheri proves, without the parallel axiom, that the angles are always equal. There are then two non-Euclidean possibilities: they are both acute, or they are both obtuse. All that, I already knew.

What I did not know is the next step: Saccheri proves that if there is *one* Saccheri quadrilateral with acute angles, then *all* Saccheri quadrilaterals have acute angles as well, and similarly for right angles and obtuse angles.

This scratched an old itch of mine: what is the negation of the parallel postulate? Let’s use Playfair’s formulation. The postulate says “given a line and a point not on the line, there exists a unique line through the given point parallel to the given line.” Many books on non-Euclidean geometry say that the negation is then “there exists more than one parallel” or “there exist no parallels.” But that ignores the implied universal quantifier in “given a line and a point”! The true negation would be “there exists one line-point pair for which there is either no parallel line or more than one”

What Saccheri proves, from this point of view, is that indeed what happens in one case must always happen. Of course, he makes implicit use of the fact that space is homogeneous and isotropic, as Euclid also did.

Saccheri goes on to assume the hypothesis of the obtuse angle (and implicitly that lines can be arbitrarily long) and prove that then the two angles must be right angles. In other words he achieves exactly the “not-A implies A” structure he wanted.

As is well known, he does not succeed in doing this under the hypothesis of the acute angle, though he tries twice. In both cases, he is tripped up by notions of infinity: his first contradiction involves what happens infinitely far away, and his second involves an infinitesimal argument.

The second book is also interesting. At issue is Euclid’s theory of ratios, which Saccheri seeks to defend, with some success. One interesting move is the explicit introduction of a law of trichotomy: two ratios are either equal, or the first is greater than the second, or the first is smaller than the second. This is the kind of thing that is easily assumed, and it is remarkable that Saccheri noticed that it needed to be stated and proved. He uses trichotomy to deal with the well-known issue of the existence of a fourth proportional, which Euclid seems simply to assume.

Overall, then, this is a book well worth reading. DeRisi’s annotations and commentary do a great job of putting Saccheri in historical context and give us a chance to really understand him. Well done!

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME. With William P. Berlinghoff, he is the author of *Math through the Ages.*