Step 1: About Nyayasutram.
Nyayasutram is the foundational text of Nyaya Philosophy, which focuses on logic, epistemology, and the systematic method of reasoning.
Step 2: The author.
The Nyayasutram was composed by Sage Gautama (also called Akshapada Gautama). He is regarded as the founder of the Nyaya school of Indian philosophy.
Step 3: Elimination of wrong options.
- Option (1) Gangeshopadhyayah: Known for later commentaries, not the founder.
- Option (2) Kanadah: Founder of Vaisheshika Philosophy, not Nyaya.
- Option (3) Gautamah: Correct, the propounder of Nyayasutram.
- Option (4) Kapilah: Founder of Sankhya Philosophy, not Nyaya.
Step 4: Conclusion.
Thus, Sage Gautama is the propounder of Nyayasutram.