**Step 1: Archimedes sets a new, enormous number system.** Archimedes introduces a way to express extremely large numbers far beyond those in common Greek usage. He defines a "myriad" as 10,000, and then constructs numbers as powers of myriads (for example, a myriad-myriad is 10,000 × 10,000 = 100,000,000). He then continues layering powers, eventually reaching numbers as large as 10^(8×10^16). *Explanation:* This step is crucial because Archimedes needs a system large enough to count quantities far beyond those ever considered—such as the grains of sand that could fill the universe. --- **Step 2: He estimates the number of grains of sand that could fit in the universe using his new system.** Archimedes makes scientific estimates: he approximates the size of the universe (then thought to be the sphere containing the fixed stars), the size of the Earth, and the size of a grain of sand. Using these, he calculates an upper bound for the number of grains of sand that could fit in the universe. *Explanation:* This step shows how his new number system is not just theoretical but can handle practical (if extreme) calculations, addressing the challenge that such quantities are "innumerable." --- **Final Answer:** In **The Sand Reckoner**, Archimedes first invents a system to describe enormous numbers, and then uses it to estimate the maximum number of grains of sand that could fill the universe, demonstrating that even the largest conceivable quantities can be counted with proper mathematical tools.