There can be very little doubt that the most significant maths textbook ever written is *The Elements of Euclid*. First composed in the fifth century BCE it was still in use throughout the world in the nineteenth century CE. One maths historian thinks that there have been more printed editions of the Elements than any other book except the Bible. Whether this is true or not, it is certainly a serious candidate for one of the most often printed and widely distributed books of all time. It was put together by Euclid, about whom we know almost nothing, in Alexandria as a compendium of most of the known mathematics of the time expressed in geometrical form. Thus it contains such oddities as a geometrical presentation of incommensurables lengths that parallels our modern presentation of real numbers or geometrical solutions to algebraic quadratic equations. However what most distinguishes this book is the fact that its entire contents are deduced logically from a handful of primitive terms thus becoming the first and by far and away the most famous presentation of the axiomatic deductive method. Mathematicians such as Moritz Pasch and David Hilbert demonstrated in the nineteenth century that the whole edifice was not quite as logical as had always been assumed but that is a topic for another post.

Although Greek science already went into general decline in the second century CE there was a new edition of *The Elements* from Theon of Alexandria in the fourth century CE eight hundred years after the original. In fact Theon’s edition was the only one know throughout the centuries until another earlier edition was discovered by François Peyrard at the beginning of the nineteenth century. *The Elements* were not lost with the collapse of the Roman Empire and the general collapse of science and it was one of the first Greek mathematics books translated into Arabic. It returned to the European sphere in the twelfth century mathematical renaissance translated by Adelard of Bath around 1120 CE. The first printed edition in Latin was published by Erhard Ratdolt in 1482 and was the first scientific book that contained illustrations and also the first book to be printed in more than one colour. The first printed edition in Greek appeared in 1533 edited by Simon Grynäus. Some of the vernacular translations are famous in the own right. The first Italian edition 1543 was produced by the great algebraist Niccolò Tartaglia and the first English edition by Henry Billingsley from 1570 contains John Dee’s brilliant preface. In the late nineteenth century as *The Elements* was being finally replaced in English schools and university Charles Dodgson, better known as Lewis Carroll, wrote a spirited and highly sarcastic defence of Euclid in the form of a play, *Euclid and his Modern Rivals*.

When the great Jesuit educational reformer Christoph Clavius was introducing mathematicus into the Catholic schools and universities at the end of the sixteenth century he wrote a series of textbook to be used by his teachers. One of the earliest was his Euclid edition, which was not a simple translation but an improved, modernised and streamlined version of the original. Clavius’ *Elements* was one of the most popular textbooks throughout the seventeenth century used by both Catholics and Protestants. It is even said that both Leiniz and Newton learnt their geometry from the Clavius *Elements*. One of Clavius’ students Matteo Ricci produced a Chinese translation of the first part of the Clavius *Elements* with Xu Guangqi in 1607. Another product of Clavius’ Jesuit mathematics programme was Giovanno Girolamo Saccheri who was born on 5 September 1667 who in his confrontation with *The Elements* made an amazing mathematical discovery but failed to realise what he had done.

Born in Sanremo, Saccheri entered the Jesuit order in 1685 and was ordained in 1694. He taught philosophy at the University of Turin from 1694 till 1697 and philosophy, theology and mathematics at the University of Pavia from 1697 until his death in 1733. He published some significant works on logic but it was his final publication shortly before his death that established his fame or possibly infamy, *Euclides ab omni naevo vindicatus* (*Euclid Freed of Every Flaw*).

I said above that the whole of *The Elements* are deduce from a few primitive terms the point being that these terms should be obviously true and in need of no further proof. Book I of *The Elements*, there are thirteen books in all, starts with twenty-three definitions, five postulates and five common notions. The definitions and the common notions need not bother us here as they are fairly obvious and have always been accepted by the readers, the same applies to the first four of the postulates but the fifth postulate poses a problem; this is the famous parallel postulate. In the Euclidian original version in the English translation of Thomas Heath it reads as follows:

*That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles*.

This postulate can be more simply expressed with the adage that two parallel lines never meet.

From the very beginning there were people who did not like this postulate. They thought it was too complex to be obvious and it would be preferable if it could be proved, that is deduced from the other postulates, as a theorem. Over the centuries many attempts were made either to prove the fifth postulate or to replace it with something intuitively more obvious; none of them succeeded. Enter Saccheri!

Saccheri decided that he would prove the fifth postulate with a reductio ad absurdum proof. This is the type of proof where you assume the opposite of the theorem you wish to prove deduce a contradiction from this assumption and then by the logical law of the excluded middle it follows that the theorem must be true. This assumes that all correctly formed theorems are either true or false, that is if the opposite of a theorem is false then the theorem must be true. There are several geometrical theorems that can be shown to be equivalent to the fifth postulate one of which is that the sum of the internal angles of a triangle is 180° and this is the theorem that Saccheri chose for his reductio ad absurdum.

Saccheri first assumed that the angle sum is greater than 180° and fairly quickly deduced that this led to the statement that all straight lines are finite. This contradicts Euclid’s second postulate that straight lines can be extended indefinitely and so Saccheri thought he had his desired proof. Unfortunately for him he missed his first chance to become famous as this consistent geometry, in which the second and fifth Euclidian postulates are not valid, is the so-called elliptic geometry and is the geometry of the surface of a sphere where great circles are taken to be the ‘straight lines’. This is one of the non-Euclidian geometries the discovery of which are credited to János Bolyai, Nicolai Ivanovich Lobachevsky, Carl Friedrich Gauss and Georg Friedrich Bernhard Riemann in the nineteenth century.

Saccheri now followed the other path and assumed that the angle sum was smaller than 180°. His investigations led to some totally bizarre results but no obvious contradiction. Confronted with this Saccheri declared his results as repugnant and thus contradictory and conclude that Euclid had been vindicated. The fifth postulate had been proved. This was a bad decision on Saccheri’s part as what he had actually done was to discover another consistent non-Euclidian geometry, the hyperbolic geometry. If Saccheri had realised what he had in reality achieved he would have gone down in history as one of the greatest mathematicians instead of which he is only an unfortunate footnote.

Although Saccheri published his results nobody took any notice of them and it was only long after the (re-)discovery of the non-Euclidian geometries that somebody realised that Bolyai and Lebachevsky had been anticipated by a Jesuit priest.

Nice post. I knew the basic story, but this was chockful of interesting details.

When I was into all things Lewis Carroll, I ran across mentions of

Euclid and his Modern Rivals, but I didn’t realize it was a play, and I could never find a copy. Now I see it’s online.A couple of minor nits:

Bolyai and Lobachevsky discovered hyperbolic geometry but not elliptic geometry. Riemann was the first to point out how space could be “finite but unbounded”. (Also, elliptic geometry isn’t

quitethat of great circles on a sphere, since great circles interesect in two points, violating the euclidean result that two line can intersect in only one point. You need to identify pairs of antipodal points, leading to what’s known as the real projective plane.)Rephrasing Euclid’s fifth as “parallel lines never meet” strikes me as being a bit too rough-and-ready; after all, hyperbolic geometry also has parallel lines in this sense. Maybe the so-called Playfair version of euclid’s fifth is more understandable: Given a line and a point not on it, at most one parallel to the given line can be drawn through the point. (So-called because, of course, Playfair wasn’t the first to come up with it; it appears already in Proclus, 5th C.)

Anyway, nice post.

Spherical geometry is a valid model for elliptic geometry. Antipodal point pairs are treated as single points.

Parallel lines never meet is the absolute ‘street version’ of Euclid’s fifth postulate, which is why I quoted it but called it an adage and not a postulate.

“Antipodal point pairs are treated as single points.”

Yes, I said that (“You need to identify pairs of antipodal points”).

But this is not at all an obvious thing to do. It’s an interesting historical point, or I wouldn’t pursue it. Traditional spherical geometry goes back to Appolonius, and was well-known during the high middle ages (e.g., Sacrobosco). But no one thought of regarding it as a model for geometry before Riemann, afaik.

If one does identify points, then the topology changes dramatically: cutting the “plane” along a “straight-line” no longer disconnects it.

Digging down into the details of Saccheri’s proof that “angle sum can’t be greater than 360”, it turns out to rely crucially, but implicitly, on just this topological fact.

Euclid never stated his topological assumptions (so-called “between-ness postulates”) explicitly, and as you noted, nobody really did before Pasch and Hilbert around 1900.

So Saccheri really did quite well up until the very end of “Euclid Vindicated”.

“If Saccheri had realised what he had in reality achieved he would have gone down in history as one of the greatest mathematicians instead of which he is only an unfortunate footnote.”

An interesting what-if. If he had realized what he had, he would have

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