Reading old or ancient texts in mathematics

Question

I am recently wondering whether it is a good idea to read old (or very old) books in mathematics. In high school level, some people suggest students who are interested in geometry to read Euclid's Elements. Euclid does have some beautiful theorems and proofs, but the way they are originally stated in Elements are much more awkward than modern versions of the same theorems/proofs. For example, when proving that a triangle with two angles (say, $ABC,ACB$) equal must be isoceles, Euclid constructed $AC'=AB$ along the side $AC$ and showed that if $C'$ and $B$ does not coincide, then there is a contradiction. This is clearly more awkward than the modern version of (essentially) the same proof, which does NOT involve any construction of new lines or points, but just use the fact that triangles $ABC$ and $ACB$ are congruent (note that the order of $A,B$ and $C$ matters, so $AB=AC, BC=CB$, and so on). People back then seem to have a hard time accepting that a triangle is congruent to itself. This is sort of an "automorphism". (Well, the stabilizer subgroup of the set $\{A,B,C\}$ in the group of all isometries of $\mathbb R^2$ is of order 2.)

The same thing is also true for university-level texts. An old text (by "old", I actually mean something like before 1980s) could be quite good if the subject discussed in the book is well-established. However, sometimes, a branch of mathematics might be slightly "reworded" over time, with some possibly significant shifts in its emphasis. To give an example, Allen Hatcher remarks in the preface of his Algebraic Topology book the following:

In a sense, the book could have been written thirty years ago since virtually all its content is at least that old. However, the passage of the intervening years has helped clarify what the most important results and techniques are.

Now, here is the problem: how to tell if a book is too old to be used today?

Answer 1

This has been a question for many, many years. Dodgson (i.e. Lewis Carroll) goes to typically wacky lengths to defend using a two-thousand-year-old text (see a helpful review here).

Honestly, it really depends a lot upon the subject matter. Some people are still using Granville's Calculus (apparently), but try to read an algebraic geometry text from 80 years ago with a recent education and you will have lots of trouble.

So I think that while, as a beginning instructor, recent books may be a good idea, as one gains experience using older texts may be suitable if the older text lends insight in some useful way, or is simply no worse and no better than modern texts (again, particularly in calculus this is likely to be true). But it is a judgment call, and really dependent upon content. Obviously disciplines like game theory and operations research didn't even exist a hundred years ago ...

In fact, properly annotated, older texts can enhance understanding, because one might see how something came to be, not just handed a formula. In that regard, I recommend the TRIUMPHS project, which has dozens of primary source materials put into a form that "modern" students can really benefit from, going all the way from the Babylonians, Greeks, and the Nine Chapters to near today.

Finally, one should note it also depends on the goals of a course. For me, teaching a linear algebra or number theory course without talking about quite modern computational applications would be dereliction of duty, so if I were to choose a text (new or old) which didn't have some resources on this, I would need to create my own or use web resources appropriately. But not everyone would agree with me on this, and for them using in some cases quite old texts could be perfectly acceptable. A similar comment applies for things like exercises, which tend not to be a big factor in the pre-industrial textbook industry!

To sum up, as long as one as instructor is aware of the modern approaches, notations, and results, and can effectively connect them to the text and translate them for the students, there shouldn't be an arbitrary time limit, though for some topics the further back you go, the less realistic this will be.

Answer 2

For me, generally, using any book that is easy to use, for a new topic is the ideal strategy. Worrying too much about missing some particular insight is counter-productive versus just moving forward. There will always be a tension of completeness versus ease of first entry. I just think you are better off going forward and then filling in, versus worrying too much about learning it all perfectly (or at least as the Internet answerers define things) to start.

In terms of practical implications, I would draw the line where the language or notation is too difficult. I think that is earlier than the 80s. You see huge amounts of Dover paperbacks or Schaum's Outlines from the 50s that still get a lot of high reviews on Amazon from students (often dissatisfied with their prescribed modern, longer, texts). I find them very accessible, sometimes more so.

If anything, I would probably draw the line in the 30s (approximately). Although some earlier books (e.g. Granville calculus) were very readable and thus used into the 60s. And I find Granville still accessible now, kind of the first modern book. But some older books I find hard to use because of language (Edwards Treatise on Integral Calculus, Taylor DiffyQs, etc.) I would draw the rough age cut later for a text in "British" than a text in "American" (but maybe the reverse for subjects of the Queen). This is not to say not lots of fun nuggets inside old books. But just that starting with such a text for first exposure may be non-optimal.

Also on your specific topic, I think learning high school geometry from a normal text, not Euclid, is more time-efficient and more likely to be tolerable (meaning you won't get frustrated and give up). I would leave the direct Euclid study for later, if you ever really care enough. Or you can go to St. Johns and do it in Ancient Greek. Then again, that school remains rather idiosyncratic...so not like it's caught on.

Of course if you learn geometry deeply or become a teacher of it, you will eventually see different texts and have some perspective on which is good at what. Which is missing what (and being complete is not always a feature, can be a bug). That is a wonderful thing about an experienced teacher that they have this perspective. But not what a neophyte should concentrate on, now.

But really you should just see how things work for you. You can tell if you are turned off/on by a text. It's a free country, still (sorta, barely). So pick something you like. For instance, it is charming to see how many people from India still use books from British math writers from 1860-1930 and get a lot out of them and presumably don't struggle with the older language.

But at the end of the day, if you use a non-optimal book and learn it perfectly (work every problem of the homework, etc.) than you will do WAY better than someone who dabbles (or even just works at a non-mastery level) in a book that is optimal. It's like sports. Yeah, having a great coach is nice and helpful. But still, if you are a good athlete and train diligently, that is way more important than facilities or who your coach is (versus other athletes).

Answer 3

I can only answer from my own experience but for me reading Euclid in college was life-changing. I had always been "good" at math -- went as far as I could go in high school (Calculus BC) and did very well in it. But I also never entirely saw the point, besides challenging myself, since I didn't have any intention of going into a math-intensive profession. (I am in fact a software developer but the business code that I write doesn't involve much math)

Reading Euclid opened up a very different perspective -- for the first time I saw math as worth knowing in its own right.

I like to reflect on the way that Euclid approaches the Pythagorean theorem vs the way one typically encounters it in high-school geometry. The conventional approach might spend a little time on a proof or two, but seems mainly aimed at providing students with a useful tool for calculating.

Euclid, on the other hand, presents the theorem as the culmination of Book I of the Elements. The first 46 propositions lead directly to it, and then once he has proven the theorem (prop. 47) and its converse (prop. 48), he's done, and moves on to another branch of geometry in Book II. There are no calculation exercises that follow because, it seems, Euclid is mainly interested in helping us to see the theorem. In other words, the Elements presents a fundamentally speculative approach to math rather than a practical one. And this is beautiful and delightful.

So to answer your bottom-line question: Any text that can open a student's eyes to the beautiful and delightful world of speculative math is worth studying, no matter how old it is.

Answer 4

I think this depends a lot on the book and the author. Some books from the 80s already read as old-fashioned and out-of-date, while some older books were really ahead of their times. It also depends on the field, since fields go through times of change which can make earlier books obsolete. The earliest book that I would happily use as a text in a normal mathematics class is Dirichlet's Lectures on Number Theory from 1863. It's truly a joy to read, and other than using quadratic forms instead of ideals, it's modern enough for advanced undergraduates.