md`My “box-packing” layout of the individual marks here is sloppy; hieroglyphs of the same rank ought to be grouped, ordered from large to small, and in clusters of no more than four. Their spacing was better. In stone I guess these would be outlined. It is not at all clear to me when or how these were discovered or deciphered. It seems to predate the [Rosetta stuff](https://en.wikipedia.org/wiki/Rosetta_Stone). Did you know that it was Napoleon’s expeditionary army in Egypt that rediscovered it to Europe in 1799? That they had 167 art experts with them, to help with the, uh, looting? That Joseph Fourier — yes, he of the series and transforms! — pulled together a comprehensive publication of the findings? And that Napoleon inspected the Rosetta stone personally? And that an artist had to figure out how to use the stone as a printing block to get prints to scholars? Until Jean-François Champollion finally cracked it in 1822 by picking out phonetic characters for “Cleopatra”? It’s all very humbling, isn’t it, I know I should know more and be less surprised, but wow! Earlier examples of the hieroglyphic numerals feature very organic repetitions, with flowers sprouting from the same bush, as in this tally of enemies slain by a king circa 2800 BCE: ${quote( "ifrah1998", 0 )}

At first I was afraid that my SVG script was doing grievous harm to the forms by “rationalizing” them to mere mechanical side-by-side repetitions. But it turns out this just recapitulates (even heightens, advances!) the evolution of the hieroglyphics themselves, which became more standardized over the centuries. The numbers “276” and “4,622” show up as plunder tallies on the [Annals of Thutmosis III](https://www.louvre.fr/en/oeuvre-notices/annals-thutmosis-iii), over 1,300 years later; note the identically duplicated flowers: ${quote( "ifrah1998", 1 )}

The hieroglyphics evolved into the [hieratic](https://en.wikipedia.org/wiki/Hieratic) writing, a logographic cursive that mapped each of 1–9, 10–90, 100–900, 1000–9000 to a different symbol, ligating the repeated marks: ${quote( "ifrah1998", 2 )}

Compared to the hieroglyphs, it makes a familiar [space–time tradeoff](https://en.wikipedia.org/wiki/Space%E2%80%93time_tradeoff): more symbols to remember upfront; fewer strokes to draw a number. The [Moscow](https://en.wikipedia.org/wiki/Moscow_Mathematical_Papyrus) (c. 1850 BC) and [Rhind](https://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrus) (c. 1550 BC) mathematical papyruses are written in it, and hieratic seems to have inspired the later [Greek numerals](https://observablehq.com/@tophtucker/greek-numerals), which mapped those same numbers onto the Greek alphabet. The Rhind papyrus includes several algebra problems; this one is labeled #34 (

) in the upper right, and reads right to left:

Cajori has a [detailed transcription](https://archive.org/details/historyofmathema031756mbp/page/n33) of the problem and its solution, translating step by step. ${quote( "cajori", 4 )} I won’t repeat him, and I can’t do it better, so it is worth looking at. Today it reads laboriously, like a notation in travail, not yet cleansed of dross — due mostly to hieratic only being able to represent unit fractions, i.e., reciprocals of integers, marked by a dot over the denominator, like

for ${tex`\frac{1}{7}`}. Also try to spot

, “heap”, working somewhat like our ${tex`x`} but not yet integrated as a full peer to the numbers. Today, for this problem, we might write: ${tex.block`\begin{aligned} \frac{x}{2} + \frac{x}{4} + x &= 10 \\ \left(\frac{1}{2} + \frac{1}{4} + 1\right)x &= 10 \\ \frac{7}{4}x &= 10 \\ x &= \frac{40}{7} \\ \end{aligned}`} `