Which preposition to use with infinitum

by Occurrences 3%

But, after all, the Author of these machines has endowed them with a faculty to reproduce or perpetuate themselves ad infinitum by the conjunction of both sexes.

in Occurrences 2%

What would a man think of a watchmaker who should have the art to make watches, which, of themselves, should produce others ad infinitum in such a manner that two original watches should be sufficient to multiply and perpetuate their species over the whole earth?

to Occurrences 1%

Since, nevertheless, the demands of the (Christian) moral law continue in their unrelenting stringency to be the standard, we are justified in the hope of an unlimited continuation of our existence, in order that by constant progress in goodness we may draw nearer in infinitum to the ideal of holiness.

without Occurrences 1%

From hence it follows that all the atoms placed at first on different lines must pursue ad infinitum those parallel lines without ever coming nearer one another; and that those who are in the same line must follow one another ad infinitum without ever coming up together, but keeping still the same distance from one another.

Which preposition to use with  infinitum