Final answer:
After the dura mater, the surgeon will pass through the arachnoid mater before reaching the pia mater, which directly covers the brain.
Step-by-step explanation:
When the surgeon passes an instrument through the dura mater, the next layer encountered is the arachnoid mater. The arachnoid mater is a membrane of thin fibrous tissue that forms a loose sac around the central nervous system (CNS), beneath which is the arachnoid trabeculae. The space between the arachnoid mater and the pia mater, which is the next layer and directly adjacent to the surface of the CNS, is filled with cerebrospinal fluid (CSF). The pia mater is a thin fibrous membrane that closely follows the convolutions of gyri and sulci in the cerebral cortex and other indentations of the brain.