The text and content (including all diagrams and illustrations) of the 1847 work (and of any other work published in 1847) are in the public domain in the US and everywhere in the world. You may freely use them verbatim or in any modified form that you wish. You are not even legally required to credit your source, although not to do so would be unethical, in my view.
The version by Nicholas Rougeux that is linked to in the question has the licensing statement:
Posters and website design are copyright Nicholas Rougeux. All other content and diagrams are under the Creative Commons Attribution-ShareAlike 4.0 International license (CC BY-SA 4.0).
This will not apply to any content copied from the 1847 version, of course. You may use any of the new content and diagrams so released, or make and use derivative works based on those elements, provided that you comply with the CC BY-SA 4.0 license. This has a number of provisions, but the major ones are that you must release your work under the same license, must acknowledge your source work, and must not impose any additional conditions or restrictions on users of your derived work. These are spelled out in sections 3.a and 3.b of the license (linked above). Please read the full terms if you intend to use this license.
If this procedure will satisfy your purpose, you do not need any further permission from Nicholas Rougeux, nor to pay any fees or royalties to him.
If you do not choose to place your work under a CC BY-SA 4.0 license, then you must not use the diagrams from Rougeux's version, nor modified versions directly based on them, nor an overall design clearly and directly based on the original design of that version, unless you secure permission from Rougeux (or the current copyright holder of Rougeux's version, whoever that may be). Given that Rougeux chose to release under a CC BY-SA 4.0 license, he may well be unwilling to grant permission under a different license, but that is his choice to make.
Exception: in the US, you may use content from the Rougeux version to the limited extent permitted by fair use. This is not likely to cover the use of all or a large number of diagrams, particularly for a competing version of the same base work. Without specific information on how much content you would be using from that version, and how similar it would be to the original, no one can reliably determine if fair use would apply or not, but fair use is most likely to apply when a strictly limited amount of content is reused, and particularly when it is used for a different purpose than the original.
Also, fair use is a strictly US legal content, and a work that might be held to be fair use by a US court might be considered an infringement by the courts of some other counties. Other countries have their own exceptions to copyright which are different in scope and terms from fair use. Many of them are significantly narrower. Note that a work posted to the internet is in effect published in all countries, and a copyright holder might choose to sue in any country s/he pleases. US courts might well enforce such a judgement even if it would not have been the judgement of a US court.
Rogeux (or any other creator of a new edition) can have no copyright in elements already present in Byrne's 1847 work. Any similarity to Rogeux's work that is because of a similarity to Byrne's 1847 work is not copyright infringement. But any new elements introduced by Rogeux (or anyone else), including the manner of adding interactivity to a diagram, may well be protected by copyright (although the idea of having an interactive version of the diagram will not be). Any new or significantly modified text or diagrams introduced in a later version will be protected.
As to any other versions of the 1847 "Byrne's Euclid" that may have been published, the publishers gain no copyright over the original 1847 work or any of its elements, including text, diagrams, or color scheme. Provided you do not use any original content newly introduced in such editions, you do not need to secure any permission from, or pay any fee to, the copyright holders of such editions. However, you may not use any such original content, or modified versions clearly based on such original content, without permission, unless an exception to copyright, such as fair use, applies. All that I said above about fair use would then apply.
A copyright holder may grant or refuse permission to use a protected work or create a derivative work on any terms that s/he chooses, and charge any fee or royalty rate s/he thinks proper. Lack of response to a request for permission must be treated as if the response was "No".
