What scientific tools and methods does the astronomer use in "When I Heard the Learn'd Astronomer" by Walt Whitman?

Walt Whitman’s poem “When I Heard the Learn’d Astronomer” contrasts the technical expertise of an astronomer with the sense of wonder experienced by the poem’s speaker. The speaker of the poem attends a lecture by a “learn’d astronomer,” who is an expert in the study of the planets and stars that dot the night sky. The astronomer is rewarded with “much applause” as he outlines his expansive knowledge and shares it with his audience. The clearest answer to your question can be found in lines 2-3 of the poem.

When the proofs, the figures, were ranged in columns before me, When I was shown the charts and diagrams, to add, divide, and measure them,

The lecturing astronomer is well-prepared for his presentation. He provides his audience with “proofs” and “figures” to mathematically support his theories. The data is “ranged in columns” and well-organized. The astronomer displays “charts and diagrams” that help his audience visualize his ideas. But the unnamed speaker of the poem is not enthralled with the lecture. Instead, he finds himself “tired and sick”. Presumably, he finds the data-driven lecture sterile and devoid of wonder. The speaker leaves the lecture in favor of looking up “in perfect silence at the stars.”
I hope this helps!