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In the past days I've been wondering about math books of the antique, e.g. the well-known Euclid's Elements. I looked at a few pages of the original Euclid's Elements, but of course was not able to understand anything due to the Greek language. However, some basic questions came in my mind:

How would Euclid's Elements (or antique math books in general) compare to modern high school / university books?

To specify what I mean by compare, let me give you some criteria:

  1. The information density. I have seen that the pages are 'filled with letters', but I can hardly image that Euclid's Elements includes more information than a common high school book. Is that assumption wrong? Have the Greek mathematician used many examples? Or did they 'blather' a lot?

  2. The size. If we would reduce the writing to a modern 10pt font size and use the modern page formats, how many pages would Euclid's Elements have? Would it be more like a pocketbook or like a huge 1000 page university coursebook?

  3. Up-to-dateness. Would a good high school student know most of the information of Euclid's Elements or at least a average university student? Would we call the information given in Euclid's Elements basic knowledge today? And/Or could a university math student derive most of the results by himself (since for university students it is very easy to derive e.g. high school theorems)?

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    $\begingroup$ It partly depends on presentation: Oliver Byrne's colourful 1847 edition uses 256 pages for the first six books of Euclid, while the Heiberg/Fitzpatrick bilingual Greek English version needs 192 pages for the same content (544 pages for 13 books and a lexicon) and so presumably about half that in one language. $\endgroup$
    – Henry
    Apr 13, 2017 at 11:01
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    $\begingroup$ Examples of pages from the manuscript version of the elements from the 800s claymath.org/library/annual_report/ar2004/… $\endgroup$
    – James K
    Apr 13, 2017 at 14:12
  • $\begingroup$ Remember that alot of the Elements isn't really about what we would now call geometry. A certain amount of it is essentially concerned with constructing the real numbers, making it an introductory analysis book as well. $\endgroup$
    – Jack M
    Apr 13, 2017 at 19:57
  • $\begingroup$ Why don't you just read a few of the old books? Euclid, Diophantus, ..., Newton, Euler? $\endgroup$
    – ShreevatsaR
    Apr 14, 2017 at 5:51

6 Answers 6

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When reading papers, even ones that were written only recently, one of the first tasks is to translate what the author did into his own mathematical language, into his own terms. For most concepts, there are many different ways to see it, and different authors might be doing the exact same thing without realizing it at first, as they have different approaches and languages.

If you go farther back and read a paper from, say, 100 years ago, you will notice that it gets even harder to translate it. The results in there might (or might not) be easy to proof given todays knowledge and understanding, but to first see what the author is doing, how that translates into your own understanding and your own pool of knowledge, get increasingly difficult.

Going back even further, one might notice the problem that mathematics changes over time, the terms, the very thinking evolves. To give one example, the topic "group theory" was originally the study of the symmetric group and its subgroups. The axiomatic approach to group theory we know today got introduced much later and one important result (by Cayley?, I'm not entirely sure right now...), showing that every group in the new sense is indeed a subgroup of the symmetric group, allowed this new concept to survive and gain popularity among group theorists of the time, that at first did not care that much about this strange concept.

Another example, way farther back, is the proof by example (regarding your second point). What is considered a rookie mistake nowadays was common in the old days. A theorem got proven by computing it and showing that it is true for "enough" numbers. The whole idea of proving theoretically for all (e.g.) natural numbers got introduced only later.

And last but not least, you should consider that mathematics was part of philosophy for a long period of time. It has still some relations today, even though it got closer to natural science over time.

All in all, the farther you go back, the more different mathematics get. Thus it might be able for a high school student or a university student to derive the results by himself, if he would be able to understand them at first that is. Translating the old works to your own language (and I mean that mathematically, not Greek to English) is in my eyes the most difficult part here - from there, using todays results and methods, many things might seem trivial (and others might not).

Standard disclaimer: This answer is based on what I myself heard and learned about the history of mathematics during my studies, I am in no means an expert on the field.

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    $\begingroup$ Actually, "proof by example" is underestimated. While I'm not arguing that ancient mathematician's were thinking this way by any means, there are several cases where a (universal) statement can be proven by checking examples. The most obvious (and relevant to ancient mathematicians) situation is the equality of polynomials can be checked by simply comparing the values at a finite set points. Much less obviously this is also true for broad classes of identities involving hypergeometric functions, recurrences, and summations. $\endgroup$
    – Derek Elkins left SE
    Apr 13, 2017 at 22:36
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    $\begingroup$ @DerekElkins A=B is such a cool book $\endgroup$
    – qwr
    Apr 14, 2017 at 5:01
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    $\begingroup$ One example of a proof by example is Euclid's proof of the infinitude of primes (today we might ask ourselves: why did he prove that in a geometry textbook at all?). The way we phrase his(!) proof today is that given $n$ primes $p-1,\ldots,p_n$, we can always find yet another prime $p_{n+1}$ as divisor of $p_1\cdots p_n+1$. But whyt is actually shown in the Elements is merely the case $n=3$, IIRC. Of course, this is in part owed to the lack of even the language to express the general case in sufficient precision. $\endgroup$
    – Hagen von Eitzen
    Apr 14, 2017 at 13:10
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Ancient math had almost no symbols.

A simple equation:

X + 3 = 5
X = 2

was written and read like:

If a quantity increased by three becomes five, then that quantity has to be two.

The power of modern math is in its careful use of symbols, which allows for a much easier life.

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    $\begingroup$ +1 I believe I read some of Tartaglia work on cubic equations and it was a mess. At the moment I'm not sure if he used symbols already or not, but this is an example of how people used to express themselves (talking about the solution to cubics): "When the cube and the things together Are equal to some discrete number Find two other numbers differing in this one. Then you will keep this as a habit That their product shall always be equal Exactly to the cube of a third of the things The remainder then as a general rule Of their cube roots subtracted Will be equal to your principal thing $\endgroup$
    – Ant
    Apr 13, 2017 at 14:09
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    $\begingroup$ Related question on lack of symbols in old mathematical literature $\endgroup$
    – user332714
    Apr 14, 2017 at 9:44
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The first two points are quite simple to dispose of: one can simply look at the text. Let's take the Elements as an example, since it's well-known and easily available (although it is not typical of the style of Greek mathematics, its very likely the only Greek mathematics work most people have even vague familiarity with). The Greeks didn't really do examples: you get propositions split into cases at best. You also don't get any of the extraneous stuff like motivation and so on, so the text is always very dense, at least in the Elements. Remember that until relatively recently, paper was extremely expensive, so only the essentials are normally written down!

Your third point rather more contentious.

You can look at a translation of Euclid's Elements online here. Some of it is taught as standard in schools (the similar triangles stuff in Book VI), some is now regarded (not necessarily fairly) as obsolete (the theory of proportion in Book X), and some stuff is taxing enough that it would almost certainly never be taught in the style of Euclid (Book XII's use of exhaustion on circles, pyramids and cones: this will invariably be done using calculus.) X.1 and XII.2 are regarded as some of the high points of Greek mathematics of Euclid's time and before, but the systems we use these days to describe the objects involved are so different from the Greek system that you need the context to understand what the point is. Thinking geometrically with straightedge and compass is very different from the algebraical thinking that is taught these days.

So asking if a work is "up-to-date" is fraught with assumptions about what that means. You know the area of a circle is $\pi r^2$; Euclid knows that the areas of two circles are in the same ratio as the areas of the squares on their diameters. Are these the same result? The answer is normally "no" if you're an historian (they have different contexts, different interpretation, live in different logical systems, and are proved in different ways), and "yes" if you're writing a popular maths book and want to mention some ancient dudes with beards and togas (they're basically the same, right?).

As an even stronger example, Apollonius's Conics is one of the most difficult books on mathematics ever written (some of the reasons for this become apparent on reading Heath's preface to his translation). When it returned to Europe in the Renaissance, it was so difficult to understand that people ended up not trying, and worked on it using "modern" methods using algebra, rather than the "correct way" of using geometry that he uses. But you probably know some of the results from it using calculus (how do you construct the tangent to a parabola? Differentiation, as anyone who has studied calculus knows...). Most university students would struggle with it if you let them use trigonometry (because they haven't done that much geometry with it in years). Make them use geometry? No chance (we just don't do geometry to that level).

(Full disclosure: most of my historical knowledge of mathematics pre-nineteenth century is thanks to Piers Bursill-Hall's excellent lecture series at Cambridge, with some of my own reading)

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Here is a translation of the Elements with Greek and English text shown side by side. As you can see, the English text is more or less the same length as the Greek text - if anything, the English is usually slightly longer. I believe this is because the Greek text uses a lot of abbreviations - which makes sense as it had to be copied out by hand.

This translation is 538 pages less 4 pages of introduction so 534 pages. The English text takes up half of this, but there is a small amount of wasted space in the "gutter" down the middle of each page. So the English text alone would probably be around 260 pages.

A good high school student could understand most of the propositions and demonstrations, but they wouldn't already know all of them because geometric constructions are no longer taught in such detail. For example, Book 13 Proposition 13 "To construct a regular pyramid (i.e., a tetrahedron), and to enclose it in a given sphere, and to show that the square on the diameter of the sphere is one and a half times the square on the side of the pyramid" is not something a high school student will just know, but they should be able to follow Euclid's argument.

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  • $\begingroup$ +1 but I don't see any abbreviations in the Greek text. Can you clarify? $\endgroup$
    – ypercubeᵀᴹ
    Apr 13, 2017 at 16:27
  • $\begingroup$ A nice comparison. But one should also be aware that miniscule letters, spacing between words, and punctuation all came centuries after the time of Euclid. If you were to see the text as originally written, it would look much different from this. $\endgroup$
    – Paul Sinclair
    Apr 13, 2017 at 17:18
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To add onto GameDeveloper's answer, a direct translation of old enough sources can feel uncomfortably wordy. As an example, I came across the following translations on Wikipedia of parts of books by Brahmagupta, an ancient Indian mathematician:

The difference between rupas, when inverted and divided by the difference of the unknowns, is the unknown in the equation. The rupas are [subtracted on the side] below that from which the square and the unknown are to be subtracted.

This corresponds to $bx + c = dx + e$ is equivalent to $x = \frac{e - c}{b - d}$, where rupas refers to the constants $c$ and $e$.

Two equivalent solutions to the general quadratic equation,

18.44. Diminish by the middle [number] the square-root of the rupas multiplied by four times the square and increased by the square of the middle [number]; divide the remainder by twice the square. [The result is] the middle [number].

18.45. Whatever is the square-root of the rupas multiplied by the square [and] increased by the square of half the unknown, diminish that by half the unknown [and] divide [the remainder] by its square. [The result is] the unknown.

are written today as solutions to $ax^2 + bx = c$ given by $x = \frac{\sqrt{4ac+b^2}-b}{2a}$ and $x = \frac{\sqrt{ac+\tfrac{b^2}{4}}-\tfrac{b}{2}}{a}$.

He also gave a recurrence relation for forms of Pell's equation, using the Euclidean algorithm:

The nature of squares:

18.64. [Put down] twice the square-root of a given square by a multiplier and increased or diminished by an arbitrary [number]. The product of the first [pair], multiplied by the multiplier, with the product of the last [pair], is the last computed.

18.65. The sum of the thunderbolt products is the first. The additive is equal to the product of the additives. The two square-roots, divided by the additive or the subtractive, are the additive rupas.

I find his work particularly impressive having proceeded some European works by centuries. The wordings are also like fun little puzzles to try to decipher. I definitely recommend reading through the whole Wikipedia article. Some other math to read about include that of Bhāskara II and the Chinese mathematical book The Nine Chapters on the Mathematical Art. A treatment much broader in scope is given on the page about the history of mathematics.

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This sea was so disturbed that no quantification will be conclusive.

Very interesting -------> http://www.nationalgeographic.com.es/historia/grandes-reportajes/la-biblioteca-de-alejandria_8593

Sum up $3$ hypotheses that shuffle some Arab authors according to wikipedia and those that would say the Romans or the caliphs conquerors of shift in his day plus his henchmen, misrepresentations of scientific conclusions for not agreeing with the conception that suited the powerful, etc etc ... --------> https://en.wikipedia.org/wiki/Euclid

The war and religion misunderstood blurred / muddled everything (I hope the past and the present do not infect the future).

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