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In the publishing industry, is it usual for authors to have a complete prescriptive non-fiction manuscript before seeking an agent, or after seeking an agent?

I am a professor interested in writing a popular (i.e. non-technical) math book. I have a fully fleshed-out idea for a book. I have some writing experience, having won cash prizes in international writing competitions for works of 40000 to 60000 words. I do not have a social media platform.

I wouldn't mind just writing the manuscript and hawking it at publishers, but I've heard that getting an agent should be your first step if you want to get published. So which should come first, a manuscript or an agent search?

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The normal process for a non-fiction book is to sell it first, then write it. The reason is that non-fiction books sell based largely on the author's qualifications, the soundness of the idea, and the size of the author's platform (their existing audience). It also allows the publisher to shape the book to their requirements before it is written. All of my non-fiction books have been sold before they were written.

I'm not sure if you need or could get an agent for a book of this sort. Agents do fiction and some of the more popular non-fiction categories like biographies and cookbooks. My impression is that more technical and specialist works are sold directly to publishers. Reasons for this include that most agents don't understand this market well enough to sell into it effectively, and that publishers have less need of agents to act as a filter are they are not (AFAIK) inundated with thousands of unpublishable submissions every day in these fields.

Non-fictions sells largely on credentials and platform. These can be described to a publisher in a one page email. The next step after that is to work up a book proposal outlining the scope and market for the work. If you get initial interest from a publisher, the will probably tell you what they want to see in a book proposal.

If you don't have a platform already, you always have the option of using your book ideas to build your platform by starting a blog and publishing your material there. If it gains a substantial following, you can then use your site stats and comment chains to demonstrate to a publisher that there is an audience for you work.

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