Johann Faulhaber's Academia Algebrae

This is the title page of the Academia Algebrae by Johann Faulhaber (1580 - 1635).  Faulhaber was a German cossist, who evidently had influence on both Johann Kepler and René Descartes.  In this work, he exhibited formulas (in the German cossist notation) for sums from 1 to n of the kth powers of the positive integers for k = 13, 14, 15, 16, and 17.  He had earlier exhibited the formulas for smaller values of k.  Unfortunately, he left little indication as to how he had developed these formulas.

On this double page, signature B, f. i (verso) and ii (recto) of the Academia Algebrae, Faulhaber gave the formula for the sum of the 13th powers of the integers, with some steps leading to this formula.  The final formula appears in the third from the last line of the paragraph with all the algebraic notation.  In translation to modern notation, it reads that the sum of the 13th powers from 1 to n is equal to (30n¹⁴ + 210n¹³ + 455n¹² – 1001n¹⁰ + 2145n⁸ – 3003n⁶ + 2275n⁴ –  691n²)/420.  For more information on Faulhaber and sums of powers in general, consult the article in this magazine by Janet Beery, "Sums of Powers of Positive Integers."