Stairs · a rule of 1683 · three building codes

One Inch Up, Two Across

England's stair guidance asks that twice the rise plus the going fall between 550 and 700 millimetres. François Blondel printed that sum in Paris in 1683, with his reason: an easy pace on the level is two feet and a ladder rung is one, so every inch a step climbs must be paid for with two inches of tread. Below, drag a stair across the plane of all possible stairs and nine authorities, from Vitruvius to the US codes, say whether they would let you build it.

A sum on the tape measure

Put a tape on any flight and measure two things: the rise, from the front edge of one tread up to the next, and the going, from one front edge across to the next. England's Approved Document K (2013 edition) gives a range for each, then adds a sentence that is not a range at all:

“The normal relationship between the dimensions of the rise and going is: twice the rise plus the going (2R + G) equals between 550mm and 700mm.”

Approved Document K, 2013 edition, note to Table 1.1

Twice the rise, once the going. The doubling is much older than any code. It is in the fifth part of François Blondel's Cours d'architecture, printed in Paris in 1683, together with the reason for it:

“La longueur du pas aisé d'un Homme qui marche de niveau est de deux pieds c'est à dire de vingt-quatre pouces, & la hauteur de celuy qui monte à une échelle dressée à plomb n'est que d'un pied ou de douze pouces”

The length of the easy pace of a man walking on the level is two feet, that is twenty-four inches, and the height of one who climbs a ladder set upright is only one foot, or twelve inches.

Blondel, Cours d'architecture, part 5, book III, ch. XIV, p. 693 (the translation is ours)

A stair is half walking and half climbing. So, Blondel argues, each unit of height should be counted as two units of level, and the two together should make one natural pace:

2 × rise + going = 24 Paris inches

He works it down a table: one inch of rise with twenty-two of tread, two with twenty, three with eighteen, on to nine with six, and says the result “se trouve faire un parfaitement bon effet par l'experience”, turns out in practice to work perfectly well. He says he had already published the rule in one of his notes to Louis Savot's book on French houses, and in the 1685 edition of that book the note calls it a rule “que j'ay inventée”, that I invented.

A Paris inch (pouce) was 27.07 mm: the law of 10 December 1799 fixed the metre at 443.296 lignes, which makes the king's foot 4500/13853 of a metre. So Blondel's two feet are 649.7 mm, and they sit inside England's 550 to 700.

The plane of all stairs

Every stair is a point: going across, rise up. The gold line is Blondel's rule, running from the level walk (no rise, a 649.7 mm pace) up toward the ladder (no tread, a 324.8 mm rung), with his table's nine rows as dots. Drag the white dot, or use the sliders, and switch the authorities on and off. Dashed shapes are the old books, as Blondel reports them in his inches; solid shapes are modern codes.

Nine authorities, one picture

Vitruvius (probably written 30 to 20 BC) wanted temple steps no higher than a dextans and no lower than a dodrans, ten and nine twelfths of a foot, with treads a foot and a half to two feet. Taken as treads, which is one of the two readings Blondel weighs, that is a box far above the line: twice the rise plus the going comes to 36 to 44 inches, half as much again as Blondel's pace or more. A stair for processions, not for a house.

Alberti (written 1443 to 1452, printed 1485) is the chip that draws nothing. His Latin, in the Strasbourg edition of 1541 and as Blondel quotes it, says steps should be “neque crassiores sextante neque tenuiores dodrante”: no higher than two twelfths of a foot and no lower than nine. No step can be both. Blondel's verdict:

“cet Autheur à esté si religieux copiste du texte de Vitruve en cet endroit, qu'il n'a pas même voulu en oublier les fautes.”

this author was so devout a copyist of Vitruvius's text at this point that he would not even leave out its mistakes.

Blondel, part 5, book III, ch. III, p. 673

The copied mistake, as Blondel tells it, was one letter: Philander read dextante for sextante, ten twelfths for two, and the modern text of Vitruvius on the Latin Library reads “neque crassiores dextante nec tenuiores dodrante”. An empty region on the plane is what a transcription error looks like when you draw it.

Palladio (1570) and Savot (1624) both write rules as boxes, and both put their steepest stair at the same corner: six inches of rise on twelve of tread. Palladio's words, in Blondel's French, are no more than six inches of rise and no less than four, treads between a foot and a foot and a half; Savot's own, as printed in 1685, are a tread “d'un pied pour le moins, & leur hauteur d'un demy pour le plus”. Twice six plus twelve is twenty-four. That corner is exactly on Blondel's line, and the line runs on through Palladio's box to four on sixteen. Blondel does not say he took the rule from either of them; he says he took it from the pace and the ladder, and in the 1685 edition his note hangs from Savot's own sentence about treads.

Scamozzi (1615), again in Blondel's report, rejects Vitruvius's steep stair, the three-on-four one of the Pythagorean triangle (not the temple steps drawn here), and wants four or five inches of rise on a foot or a little more of tread, a gentle stair that sits below the line: twice the rise plus the going is 20 to 22 inches on a one-foot tread.

Then the codes. The 2012 International Residential Code for US houses allows a riser of at most 7¾ inches on a tread of at least 10. Its steepest legal stair has 2R + G = 25.5 inches, which is 647.7 mm: 2.0 mm short of Blondel's two Paris feet. The page does not claim the code descends from Blondel; it has no source for that, and the coincidence may be only a coincidence. The 2012 International Building Code, for other buildings, stops at 7 on 11, and its corner is 635.0 mm, 14.7 mm short.

England's document keeps the sum itself, as a band. Its private-stair box is also capped at a pitch of 42°, and that cap cuts Blondel's line at a rise of 208.9 mm on a going of 232.0 mm; the private box admits the line from a rise of 174.8 mm up to there. Blondel's own table ends in two rows no modern code on this page would pass: eight on eight, a 45° stair, and nine on six, 56°. His row of seven on ten, 189.5 mm on 270.7 mm, passes the IRC and England's private and utility stairs.

Measure your own stair

A tape and a minute. Measure the rise from one tread's front edge to the next one up, and the going from one front edge to the next along the flight, and enter both.

What this page does not show

It does not test whether Blondel was right about legs. His evidence is a pace, a ladder and “l'experience”; whether 2R + G really predicts how comfortable or safe a stair is would need measurements of people on stairs, and none are used here. The old authorities are drawn as Blondel reports them, in Paris inches: Vitruvius's foot, Palladio's and Scamozzi's were not the Paris foot, so their boxes show what Blondel understood them to say, not the millimetres the originals would measure. Palladio's and Scamozzi's rules are taken from Blondel's French summaries, not from their own books. Scamozzi gives no upper limit for the tread (“un pied ou peu plus”); his box stops at fourteen inches, a choice made for drawing. Vitruvius's retractiones are drawn as treads, one of Blondel's two readings. The codes are the editions named (IRC and IBC 2012, Approved Document K 2013); later editions may differ and were not checked, and the codes' other rules (nosings, headroom, winders, handrails) are left out.

Sources

1. François Blondel, Cours d'architecture enseigné dans l'Académie royale d'architecture, part 5 (Paris, completed printing 30 September 1683), book III, chs. III to XIV, pp. 672 to 693. Getty Research Institute copy, Internet Archive.
2. Louis Savot, L'Architecture françoise des bastimens particuliers, with notes by Blondel (Paris, 1685), p. 78. Internet Archive. First published 1624 (catalogue note on the 1642 printing).
3. Leon Battista Alberti, De re aedificatoria libri decem (Strasbourg, 1541), book I. Internet Archive.
4. Vitruvius, De architectura 3.4.4, Latin text at The Latin Library.
5. HM Government, Approved Document K: Protection from falling, collision and impact, 2013 edition, Table 1.1. gov.uk.
6. International Code Council, 2012 International Residential Code, R311.7.5.1 and R311.7.5.2; 2012 International Building Code, 1009.7.2. Copies at the Internet Archive (IRC, IBC).
7. The Paris foot: Wikipedia, “Traditional French units of measurement”, on the law of 19 frimaire an VIII. Dates of Vitruvius, Alberti, Palladio and Scamozzi from the Wikipedia articles on their treatises.