
Bill Thayer


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p806 Norma

Article by James Yates, M.A., F.R.S.,
on p806 of

William Smith, D.C.L., LL.D.:

[image ALT: An engraving of a stone monument, broken at the top, on which are carved an assortment of tools: an adze, a ruler, a compass, a curve, a T‑square, a template, a punch, an awl.]

The caption below the sculptured panel, which I suspect to be by the authors of the Dictionary rather than on the original monument, reads

("Carpenters' Instruments")

NORMA (γνώμων), a square, used by carpenters, masons, and other artificers, to make their work rectangular (Philo de 7 Orb. Spect. 2; Vitruv. VII.3; Plin. H. N. XXXVI.22 s51; Prudent. Psychom. 828). It was made by taking three flat wooden rulers [Regula] of equal thickness, one of them being two feet ten inches long, the others each two feet long, and joining them together by their extremities so as to assume the form of a right-angled triangle (Isid. Orig. XIX.19).a This method, though only a close approximation, must have been quite sufficient for all common purposes. For the sake of convenience, the longest side, i.e. the hypotenuse of the triangle, was discarded, and the instrument then assumed the form, in which it is exhibited among other tools in woodcut at p283. A square of a still more simple fashion, made by merely cutting a rectangular piece out of a board, is shown on another sepulchral monument, found at Rome and published by Gruter (l.c. p229), and copied in the woodcut which is here introduced. The square was used in making the semicircular striae of Ionic columns [Columna], a method founded on the proposition in Euclid, that the angle contained in a semicircle is a right angle (Vitruv. III.5 §14).

From the use of this instrument a right angle was also called a normal angle (Quintil. XI.3 p446, ed. Spalding). Any thing mis-shapen was called abnormis (Hor. Sat. II.2.3).

Thayer's Note:

a 2′10′′ long . . . (Isid. Orig. XIX.19): It's not often that my curmudgeonly nature gets treated to catching two authors out in the same sentence, but here goes. Smith's reference is wrong, he means Isid. Orig. XIX.18 (q.v.); that's quite minor. More importantly, Isidore himself has given us something very strange, nor can it be explained by manuscript corruption, and our dictionary author has barreled blithely by it.

A perfect right triangle the legs of which are 24 inches each will have a hypotenuse of

(242 + 242)-2 = 1152-2 = 33.941 inches.

    Now that's pretty good. That third piece of wood, at 34 inches, is only six hundredths of an inch too long, and some of you must be wondering: why quibble? Well . . . every schoolchild past a certain age knows that you can build an absolutely perfect right angle with three pieces of wood in the ratio of 3:4:5; after all, although we want to obtain a right angle, nothing requires us to do it in any particular way, and that includes by the use of a square instrument when a rectangular one will do. Why on earth then does Isidore suggest these peculiar measurements, to get an inferior result? and why does Yates not catch him out, leaving the work to me 127 years later?

Now that I've taken you down this garden path — rectilinear though it may be — let's get Isidore off the hook (in honor of his being proposed as the patron saint of the Internet) and in so doing discover something else.

First of all, given normal tolerances on a construction site, even today let alone in Antiquity, the angle produced by three boards in the ratio 24:24:34 is so close to 90° that, as our article says, it just wouldn't matter. Isidore is giving us a useful carpenter's trick: how to make a right angle with only two measurements, not the three required for the perfect 3‑4‑5 figure; although even then we are still left wondering why not just 12:12:17.

Far more interesting, though, is what I think underlies the oddly measured norma in the Origines. It was already known well before Isidore's time that the diagonal of a square is incommensurable with its sides; to put it another way, that the square root of 2 is an "irrational" number. This irrefutable fact, however, flew in the face of a certain kind of order, much loved by the Pythagoreans, where numbers represented patterns: surely if one worked hard enough at it, one could find a square in which the diagonal would come out "even", to a nice round number. Now Isidore — whose Book III, on Mathematics, Music, and Astronomy is a Pythagorean's playground — is too much of a scholar ever to say he's found one; but gee he comes close! The curious measurements of his T‑square, ever so slightly off, are a testimony to a wistful Christian scientist, wishing the world were perfect and knowing full well it's not.

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Page updated: 16 Nov 04