Note 409 — Chapter IX · 1504-1524.

“This distance,” he remarks, “calculated geometrically upon the ratio that three and one seventh times the diameter of a circle is equal to its circumference, gives 92⁵⁴¹⁶⁴⁄₄₇₂₇₃₃ degrees. For if we take 114⁶⁄₁₁ degrees as the chord of an arc of a great circle, we have by the same ratio 95²⁴⁸³⁄₄₉₅₀ degrees as the chord of an arc on the parallel of 34°, being that on which we first made land, and 300²³³⁄₁₅₇₅ degrees as the circumference of the whole circle, passing through this plane. Allowing then, as actual observations show, that 62½ terrestrial miles correspond to a celestial degree, we find the whole circumference of 300²³³⁄₁₅₇₅ degrees, as just given, to be 18,759³¹⁄₁₂₆ miles, which, divided by 360, makes the length of a degree of longitude on the parallel of 34° to be 52⁹⁸⁹⁄₉₀₇₂ miles, and that is the true measure. Therefore, by a right line to the said rock which stands in 32°, we have to calculate the distance, the said 1,200 leagues which we have found, from the thirty-fourth parallel, from west to east, hence I should have run 92⁵⁴¹⁶⁴⁄₄₇₂₇₃₃ degrees, and this many therefore we have sailed to the West, which was not known to the ancients.”

114⁶⁄₁₁ × 3¹⁄₇ = 360. 300²³³⁄₁₅₇₅ × 7 ÷ 22 = 95²⁴⁸³⁄₄₉₅₀. 300²³³⁄₁₅₇₅ × 62½ = 18,759³¹⁄₁₂₆. 18,759³¹⁄₁₂₆ ÷ 560 = 52⁹⁸⁹⁄₉₀₇₂. 4,800 by 52⁹⁸⁹⁄₉₀₇₂ = 92⁵⁴¹⁶⁴⁄₄₇₂₇₃₃.