Here are my suggested translations:
- You ask further, esteemed colleague, if imaginary quantities are neither nothing nor bigger or smaller than zero, what can they be? I reply that they are imaginary for just that reason; for if they were zero, or bigger or smaller than zero, they would be real and therefore not imaginary.
I translated the salutation vir amplissime as "esteemed colleague".
There might be something more idiomatic, but I'm not sure what would fit.
Perhaps "good man" would also work and be less dependent on context.
The turn should be tum; it is easy to mistake m for rn in old printed or handwritten text.
Both humans and computers make this mistake.
Checking the original indeed shows that it's tum.
I translated nihil as "zero".
You can also use "nothing" which is more literal, but it is clear that here the zero of the field of real numbers (and by extension of complex numbers) is meant.
- If then there are those things that ought to be expressed by negative numbers, addition and subtraction carried out in the usual manner is hindered by no difficulties.
A verb seems to be missing.
I supplied sunt (a form of esse is typical to leave out) in the meaning "there are".
It would be useful if you could show the original printed text so we could take a closer look at some details.
It is also possible that some other verb was intended, but for that we should look at the original text, including that surrounding this sentence.
The subjects of premitur are additio and subtractio.
I would read it as additio premitur et subtractio premitur, "addition is pressed and subtraction is pressed".
This is why I chose "is" instead of "are" in the translation.
More literally it would be "pressed" rather than "hindered".
I will leave it for others to judge what would make the most idiomatic English translation.
Mine is pretty faithful to the Latin original.
The point of this passage is clearly that addition and subtraction work in the usual manner with complex numbers — in the presence of things expressed by (square roots of) negative numbers.
Upon seeing it in context, this passage is about arithmetic operations on positive and negative real numbers.
Either way, the point is: Once a number system is extended (from positive reals to all reals or from reals to complex numbers), the operations stay "the same".
He does discuss complex numbers too, but in my reading the quote you picked is not about them per se.
I am a mathematician myself and I would be interested in seeing how these quotes are used.
Please share when you are ready to do so!