Viète and an ‘elegant and very beautiful’ result

15 February 2026

François Viète (= Franciscus Vieta; 1540–1603) Viète described as ‘elegant and very beautiful [elegans & perpulchræ]’ the result shown here in the original Latin and using his original notation:¹

$Excerpt from Viéte's ‘De Emendatione Æqvationvm Tractatus Secundus’, text (in Latin with much mathematics) as follows: Si $A$ quadrato-cubus $\overline{-B-D-G-H-K}$ in $A$ quad.\,quad. $\overline{+B \text{in} D+B \text{in} G+B \text{in} H+B \text{in} K+D \text{in} G+D \text{in} H+D \text{in}K+G \text{in} H+G \text{in} K+H \text{in} K}$ in $A$ cubum $\overline{-B \text{in} D \text{in} G - B \text{in} D \text{in} H - B \text{in} D \text{in} K - B \text{in} G \text{in} H - B \text{in} G \text{in} K - B \text{in} H \text{in} K - D \text{in} G \text{in} H - D \text{in} G \text{in} K - D \text{in} H \text{in} K - G \text{in} H \text{in} K}$ in $A$ quad. $\overline{+B \text{in} D \text{in} G \text{in} H+B \text{in} D \text{in} G \text{in} K+B \text{in} D \text{in} H \text{in} K+B \text{in} G \text{in} H \text{in} K+D \text{in} G \text{in} H \text{in} K}$ in $A$, æquetur $B \text{in} D \text{in} G \text{in} H \text{in} K$: $A$ explicabilis est de qualibet illarum quinque $B$, $D$, $G$, $H$, $K$. \textit{$1 QC-15 QQ+85 C-225 Q+274 N$, æquatur $120$. Fit $1 N$ $1$, $2$, $3$, $4$, vel $5$.} Atque hæc elegans & perpulchrae speculationis sylloge, tractatui alioquin effuso, finem aliquem & Coronia tandem imponito.$

Translated into English and into modern algebraic notation, Viète’s result says:

If, for an unknown $a$ and known $b$, $d$, $g$, $h$, and $k$, and the equality $a^5 - a^4(b+d+g+h+k) + a^3(bd + bg + bh + bk + dg + dh + dk + gh + gk + hk) - a^2(bdg+ bdh + bdk + bgh + bgk + bhk + dgh + dgk+ dhk + ghk) + a(bdgh + bdgk + bdhk + bghk + dghk) = bdghk$ holds , then $a$ is either $b$, $d$, $g$, $h$, or $k$.

A mathematician today, working with this modern notation, would presumably notice that if one sets $a = b$ and multiplies out the brackets on the left hand side, all terms except $bdghk$ cancel; by symmetry, the same reasoning applies for $d$, $g$, $h$, $k$.²

Thus, to modern eyes, the result collapses into near-triviality and seems to deserve little if any aesthetic praise. Perhaps the notation Viète had to work with made the result seem more mysterious and thus more aesthetically pleasing.²

There are scattered other references to aesthetic value throughout Viète’s ‘Opera’.²

At least one result seems to have earned lasting aesthetic appreciation. Going beyond Archimedes’ inscription of a $96$-gon in a circle to approximate $\pi$, Viète considered inscribing $4$, $8$, $16$, and in general $2^n$-gons and derived the infinite product

\[ \frac{2}{\pi} = \frac{\sqrt{2}}{2}\cdot\frac{\sqrt{2 + \sqrt{2}}}{2}\cdot\frac{\sqrt{2 + \sqrt{2 + \sqrt{2}}}}{2}\cdots, \]

which Eli Maor judged to be

‘a remarkable formula, one which even today arouses our admiration for its beauty’.³

Notes

F. Vietæ. ‘De Emendatione Æqvationvm Tractatus Secundus’. In: Opera Mathematica. Edited by F. à Schooten. Leiden: Ex Officinâ Bonaventuræ & Abrahami Elzeviriorum, 1646. p. 158. ↩
A. J. Cain. Form & Number: A History of Mathematical Beauty. Lisbon, 2024. pp. 271–3. ↩ ↩² ↩³
E. Maor. To Infinity and Beyond: A Cultural History of the Infinite. Princeton University Press, 1991. ISBN: 978-0-691-02511-7 p. 10. ↩