Why didn't the Buddha write a book outlining his teaching?

Because oral transmission was the only available transmission media at that time. Any form of documented paper printing didn't exist until a couple centuries later.

Why didn't the Buddha write a book?

Well, he needed writing material and he needed a written script.

Now, a written script has to be one that many people can read and comprehend, otherwise it's useless.

How do languages and written scripts get standardized, taught widely and made official? Through the ruler or government of a country of course.

The following article proves that during the Buddha's lifetime (c. 563 BCE to 483 BCE), there was no official or ubiquitous written script in the region where the Buddha lived, till Emperor Asoka over two centuries later, who had inscriptions carved on stone pillars in the Brahmi script.

Earlier versions of the Brahmi script may have existed in Tamil Nadu (south India) and Sri Lanka at the time of the Buddha, but certainly not in the region where the Buddha lived.

From "The Story of India’s Many Scripts" by Akhilesh Pillalamarri in The Diplomat:

India has a long history of writing. While India has been a literate culture for millennia, it has also greatly valued oral knowledge. The ancient Hindu scriptures, the Vedas, the oldest of which dated to around 1500 BCE were memorized verbatim for at least a thousand years, if not more, before being committed to writing. The oldest writing found in the subcontinent is the as yet undeciphered script of the Indus Valley Civilization (IVC), which seems to have been somewhat logo-syllabic in nature. The script fell out of use by 1500 BCE.

The linguistic landscape of the subcontinent changed dramatically during the 2nd millennium BCE, so that is is impossible to determine if there is a connection between the IVC script and the next clearly attested script in India, the Brahmi script found in the inscriptions of the Mauryan Emperor Ashoka (ruled 268-232 BCE), especially since they probably represented vastly different, unrelated languages.

The sudden appearance of the Brahmi writing system is one of the great mysteries of writing in India, as there is no evidence of inscriptions beforehand. Another script, the (extinct, childless) Kharosthi of northwest Pakistan and Afghanistan seems to be clearly derived from the imperial Aramaic script used by the Persians who ruled over parts of the Indus Valley for two centuries until the arrival of Alexander the Great. It is unclear if the fully developed Brahmi script was invented by the Mauryan Empire as a result of exposure to Aramaic, but this seems unlikely, particularly since there were advanced states in the Ganges valley and a corpus of Vedic literature dating from before the Mauryan period.

It is more likely that pre-Mauryan inscriptions may still be discovered, and in fact, some Brahmi inscriptions have been found in Tamil Nadu and Sri Lanka dating to the 6th century BCE. Is it possible then, that writing spread from the south to the north, countervening the traditional notion that the Indic scripts originate in the Ganges valley? This may quite possibly be the case, especially since the coasts of southern India were more exposed to foreign trade from the Middle East than northern India, and scripts from traders could have been brought to India this way (the same way the Phoenicians brought their script to Greece). This long gestation period and overland route from southern to northern India may explain why the Brahmi script, even if it is vaguely derived from Middle Eastern alphabets, is so different and nativized, especially relative to the more obviously Middle Eastern-inspired Kharosthi.

Once the Brahmi script was spread throughout India by the subcontinent-wide Mauryan Empire, it was used by the subcontinent’s elites. However, unlike imperial China with its unified central government and bureaucratic exam system, and Christian and Muslim societies that were united by a written scripture, oral culture and regional differences in India led to the Brahmi script differentiating and evolving into different scripts in various regions of India, a phenomenon that was already occurring by the end of the Mauryan period in the 2nd century BCE. This phenomenon—each literary language having a particular and unique script — is not actually that unique to India, as the various languages of the ancient Near East and Mediterranean also evolved their own scripts from a common source.

The article above continues with a commentary on writing material.

The following paper excerpt shows that Emperor Asoka's stone pillar inscriptions referred to suttas by name, from the oral tradition.

From Wynne, A. (2004), "The Oral Transmission of Early Buddhist Literature", Journal of the International Association of Buddhist Studies, Volume 27, Number 1, p. 97:

In his Calcutta-Bairat edict, Asoka names some texts, the study of which he considers advantageous. The general consensus seems to be that what Asoka calls Munigatha correspond to the Munisutta (Sn 207-21), Moneyasute is probably the second half of the Nalakasutta (Sn 699-723), and Upatisapasine may correspond to the Sariputtasutta (Sn 955-975). The identification of most of the other titles is less certain, but Schmithausen, following Oldenberg before him, identifies what Asoka calls the Laghulovada with part of a prose text in the Majjhima Nikaya, the Ambalatthika-Rahulovada Sutta (M no.61). It is hardly likely that Asoka knew these texts in exactly the same form as they are found in the extant Suttapitaka, but this does not matter. What matters is that Asoka was able to name texts, and this surely means that their content was more or less fixed.

As a side topic, while a common written script was not available, spoken dialects were ubiquitous and subject to the phenomena of dialect continuums. This is the reason why there was no one single language or dialect called Pali, but it was formalized later, based on the dialects of the Buddha's time and region, plus later partial Sanskritization. So, the other thing to note is that there was no formal Pali language at the time. Please also see this answer.

Some kind of formal Sanskrit (pre-Panini) may have been present at the time, but not available to the common folk, so the Buddha had forbidden its use for transmission of his teachings - please see this question.

it's hard to believe that the cause for not writing a book to be lack of written script or writing material

There seems to be circumstantial evidence that the Brahmi script existed:

• Brahmi script (Wikipedia)
• Pāṇini (Wikipedia)
• Lipi (Wikipedia

It's hard to date exactly or with certainty without physical (e.g. archaeological) evidence, therefore perhaps it's arguable. The earliest surviving evidence of writing are the Pillars of Ashoka from the 3rd century BCE -- and also surviving, possibly as much writing as would fit on some coins (i.e. not very much) -- as you say though I'm not certain that the Mauryan Empire was the first time writing was used in India, even if that's the oldest surviving archaeological evidence.

Inks and Palm leaves and so on aren't especially durable!

I read though that Pali was a spoken language and not a literary language -- I've no reason to believe that isn't true, if the literary or written language at the time was Sanskrit.

Being of noble birth I think it's even possible that Gautama might have been one of the (perhaps few) people who might have learned to read and write, as a youth.

That's not to imply that he wanted to, though -- i.e. I suppose he didn't think that writing was the best way to communicate and to share what he'd discovered.

because the method (i.e the oral transmission) is not about dialectic rather repeating the exact word and phrase

I think the Buddha's doctrine was absolutely about dialectic, dialogue.

If you read (or hear recited) any of the suttas, they are someone's asking the Buddha a question to which he replies, or the Buddha's correcting someone's wrong view (or more rarely the Buddha's giving a talk to an assembly).

The Buddha's dialectic often even sounds like the 'Socratic' method, asking "what do you think" as a kind of rhetorical question.

That's true not only in the suttas but in the origin stories for the vinaya rules, i.e. they result from the Buddha's interacting with things people say or do.

And the Buddha's first question after becoming enlightened was, "Who can I explain this to -- who can understand?"

So it seems to me very much in-person, person-to-person -- not writing and publishing in the modern sense, broadcasting via some medium in the hope than some audience might pick it up.

"Repeating the exact word and phrase" comes afterwards, i.e. that (memorising word for word) is how the monks chose to preserve the doctrine. Or rather, I should say, it's one of the ways: because they also tried to memorialise it in practice; translate it into other languages; write summaries and commentary; and give "dhamma talks" in their own words.

I guess Buddha's teaching was pretty smart and simple and your question looks like "Why didn't P.Fermat write a book outlining his teaching about Fermat's Last Theorem". Seems in both cases Buddha and P.Fermat gave a lot of hints about Buddhism and Fermat's Last Theorem and those hints can be written in several pages (maybe few lines), therefore they didn't write books.

The four nikayas do not mention "writing". However they do mention "reciting" many times. One mention, in particular is quite interesting:

AN8.20:6.4: It’s impossible, mendicants, it can’t happen that a Realized One could recite the monastic code in an impure assembly.

In another sutta, the Buddha instructs:

DN29:17.1: So, Cunda, you should all come together and recite in concert, without disputing, those things I have taught you from my direct knowledge, comparing meaning with meaning and phrasing with phrasing, so that this spiritual path may last for a long time. That would be for the welfare and happiness of the people, for the benefit, welfare, and happiness of gods and humans.

What is notable about these and many other quotes is the amount of care mandated in transmitting the Dhamma. The transmission of the Dhamma goes beyond mere written duplication via Gutenberg printing press or mindless scribe. The transmission of the Dhamma requires transmission of pure understanding, with care taken to ensure the meaning and phrasing of what is spoken and heard. Specifically, what is required is:

AN2.126:1.3: The words of another and proper attention.

In my own experience, I have found that hearing, reciting and discussing the Dhamma to be far more powerful and effective than mere reading. I have found the eye to be fickle in its grasping as it flits from here to there seeing what it wants to see as it skims the written word. In contrast, hearing and reciting and discussing demand far more attentiveness. And although the Dhamma is available for reading, I myself study the Dhamma by listening, reciting and discussing. The difference in understanding is remarkable. I cannot say what the Buddha intended. All I can say is that reading is too superficial for me. Instead, I rely on hearing, memorizing and speaking the Dhamma in concert. And that oral tradition does indeed work for me. The oral tradition facilitates proper attention.

We can get a glimpse from Wikipedia article on Vedas:

The Vedas were written down only after 500 BCE,[101][66][20] but only the orally transmitted texts are regarded as authoritative, given the emphasis on the exact pronunciation of the sounds.[68] Witzel suggests that attempts to write down the Vedic texts towards the end of 1st millennium BCE were unsuccesfull, resulting in smriti rules explicitly forbidding the writing down of the Vedas.[66] Due to the ephemeral nature of the manuscript material (birch bark or palm leaves), surviving manuscripts rarely surpass an age of a few hundred years.

so sounds like writing was considered unreliable back then.

We can get another hint from Buddha's prohibition to use Sanskrit as the official language of transmission. He wanted the Dharma to be spoken in a language of the audience, to make it more accessible¹.

If you write it, you have to write it in one language but which one? Not everyone understood Sanskrit. Not all local languages had a written form back then, and I suspect not many people could read.

¹ (BTW this also supports Mahayana rephrasing the teaching in the way most accessible to each audience, instead of passing down the same exact words. This is how I answer questions here, explaining in new words every time as much as I can.)

The Buddha responded to the questions of others. Writing a book to present a position is a ‘modern’ undertaking. The Buddha’s oral teachings were memorized (thus all the enumerations that exist in Buddhist doctrines because it made memorization easier) because his was predominately an oral culture, but also because the teachings were not a static doctrine, but were transmitted/crafted by the Buddha and later teachers to meet the context of the questioner and audience.