The correct option is
C impaired
'...and that it has vitiated the reasoning of geologists on some points of great interest in the ancient history of the world.' This sentence means that the assumption has damaged or spoiled the reasoning of the geologists.
Here, 'vitiated' means spoiled or damaged.
Option A - 'Infiltrated' means to enter or gain access to something in order to acquire secret information. Thus option A is incorrect.
Option B - 'Occupied' means being used by someone or with someone in it. It also means 'full in use or busy'. Thus option B is incorrect.
Option D -'Invigorated' means to make someone feel fresher, healthier and more energetic. Thus option D is incorrect.
Option C - 'Impaired' means damaged. For example, 'The device is impaired.' Thus option C is the correct answer.
'Vitiated' most nearly means 'impaired'.