Sums of Powers of Positive Integers - Johann Faulhaber (1580-1635), Germany

Johann Faulhaber was an arithmetic teacher (rechenmeister), who spent his entire life in Ulm, Germany. He also was an enthusiastic adherent of the new Protestantism who highlighted the number 666 in all of his books, whether religious or mathematical, and whose numerological predictions put him in constant conflict with religious and civil authorities.

In his 1631 Academia Algebrae, Faulhaber presented formulas for sums of powers of the first n positive integers from the 13^th to the 17^th powers (Faulhaber, folios Bi verso-Cii recto). He had presented formulas for sums of powers up to the seventh power in his 1614 Newer Arithmetischer Wegweyser and up to the 12^th power in his 1617 Continuatio seiner neuen Wunderkunsten (Schneider, p. 138; see also Faulhaber, ff. Aiii verso-Aiv recto). Furthermore, D.E. Knuth has argued that, in Academia Algebrae, Faulhauber actually encoded sums of powers of positive integers up to the 23^rd power (Knuth, pp. 282-283).

Figure 10. Faulhaber’s presentation of his formula for the sum of the 13^th powers in Academia Algebrae, folios Bi verso-Bii recto, is printed across a two-page spread. An electronic copy of Academia Algebrae, including folios Bi verso and Bii recto, is available from the University of Dresden Digital Library. (Reproduced by permission of Columbia University Library. For images of these two pages and the title page with rulers alongside to show size, see "Johann Faulhaber's Academia Algebrae" in "Mathematical Treasures.")

Faulhaber's formulas in his Academia Algebrae are presented both in terms of n and in terms of n(n + 1)/2. Actually, rather than being given in terms of an unknown n or x, they are presented using an early algebraic notation for powers of the unknown called "cossist" notation. Faulhaber's symbols for these powers are close to Recorde's (see Figure 8) and are listed in folio Dii verso of his Academia Algebrae, available electronically from the University of Dresden Digital Library. In his notation, the symbol for the square is a script "z" that stands for zensus (square), and the symbol for the cube a script "c" that stands for cubus (cube). The symbol for the fourth power, actually two script "z"s, stands for zensus de zensus (square of square), and that for the sixth power, "zc," zensicubus (square of cube). The symbol for the fifth power, similar to the German "ss," stands for sursolidum; for the seventh power, this symbol is preceded by a "b", so we get b-sursolidum. Similarly, Faulhaber's symbol for the eleventh power is c-sursolidum, for the 13th power d-sursolidum, and so on for the other prime powers. Continuing in this manner, we see that the 14th power symbol, then, stands for zensus de b-sursolidum (square of seventh power).

We can rewrite Faulhaber's instructions for the sum of the 13th powers (see Figure 10) in modern notation as follows:

Given n, multiply $${{n^2 + n} \over 2}$$

by 2764 to obtain 1382n² + 1382n, then subtract 691 from this expression to obtain 1382n² + 1382n – 691.

Next, subtract $${{n^4 + 2n^3 + n^2 } \over 4} \cdot 4720,$$

or $$1180n^4 + 2360n^3 + 1180n^2;$$

add $${{n^6 + 3n^5 + 3n^4 + n^3 } \over 8} \cdot 4592,$$

or $$574n^6 + 1722n^5 + 1722n^4 + 574n^3;$$

subtract $${{n^8 + 4n^7 + 6n^6 + 4n^5 + n^4 } \over {16}} \cdot 2800,$$

or $$175n^8 + 700n^7 + 1050n^6 + 700n^5 + 175n^4;$$

and add $${{n^{10} + 5n^9 + 10n^8 + 10n^7 + 5n^6 + n^5 } \over {32}} \cdot 960,$$

or $$30n^{10} + 150n^9 + 300n^8 + 300n^7 + 150n^6 + 30n^5,$$

to obtain the expression $$30n^{10} + 150n^9 + 125n^8 - 400n^7 - 326n^6 + 1052n^5 + 367n^4 - 1786n^3 + 202n^2 + 1382n - 691,$$

which we are to divide by 105. Note that $${{n^2 + n} \over 2} = {{n(n + 1)} \over 2}$$

and that the successive quotients whose products by certain numbers are being added and subtracted are $$\left( {{{n(n + 1)} \over 2}} \right)^2, \left( {{{n(n + 1)} \over 2}} \right)^3, \left( {{{n(n + 1)} \over 2}} \right)^4,\ {\rm and} \left( {{{n(n + 1)} \over 2}} \right)^5.$$

The polynomial we have obtained so far cannot be the correct formula for the sum of the 13^th powers because it is of degree 10 rather than degree 14. Faulhaber’s next instruction is to multiply the expression obtained so far by $${n^4 + 2n^3 + n^2 \over 4}\,\,\, {\rm or}\,\,\, \left(n(n + 1)\over 2 \right)^2$$

to obtain the sum of the 13^th powers, $${{30n^{14} + 210n^{13} + 455n^{12} - 1001n^{10} + 2145n^8 - 3003n^6 + 2275n^4 - 691n^2 } \over {420}}.$$

As D.E. Knuth pointed out (Knuth, p. 277), Faulhaber also described in his presentation above the formula $${{960N^7 - 2800N^6 + 4592N^5 - 4720N^4 + 2764N^3 - 691N^2 } \over {105}},$$

where N = n(n + 1)/2, for the sum of the 13^th powers. In Academia Algebrae, Faulhaber presented similar formulas for sums of 14^th through 17^th powers, and gave a formula for the sum of the eighth powers as well (f. Diii).

According to Faulhaber scholar Ivo Schneider, Faulhaber's early training as a weaver helped him see relationships between long columns containing numerical values of powers, sums of powers, and powers and products of sums of powers (see Schneider, pp. 131-139). Indeed, when these columns were arranged side-by-side, they would have resembled somewhat the warp threads – the vertical threads – in a loom, and Faulhaber's alternating pattern of addition and subtraction of terms may have corresponded, for him, to the weaving of weft threads – horizontal threads – over and under the warp threads. Even so, patterns or formulas for writing sums of 18th and higher powers have never been readily apparent to readers of Faulhaber's presentation of his formulas for sums of 13th through 17th powers in Academia Algebrae!