How was perpendicularity expressed in classical Latin?

Question

In today's mathematics, two lines are said to be normal to each other if they are at a right angle (perpendicular) to each other. I want to know how this can be expressed in classical Latin.

Closely related questions have been asked on mathematical SE sites:

• How did “normal” come to mean “perpendicular”?
• Etymology of the word “normal” (perpendicular)

What expressions are there for normality in Latin, preferably classical? I would like to see examples of mathematical mentions of normality, as old ones as possible. Looking at dictionaries gives a list of possible translations to "right angle", "perpendicular" and "normal", but it is not clear which ones could be used in mathematical context. (Some possible expressions: derectus, directus, praeceps, perpendicularis, perpendicularius, perpendiculatus, cathetus, angulus rectus, angulus normalis.)

To be able to write mathematics reasonably fluently, it is good to know several (possibly grammatically different) ways to express the same thing. The accepted MathOverflow answer gives a classical example of angulus normalis (straight angle), but I have not been able to find more classical expressions for perpendicularity.

Since my question about finding ancient mathematical texts in Latin has produced no example texts so far, I will have to accept post-classical examples. Some amount of geometry does appear in non-mathematical texts, so I hope to find classical examples as well. (Maybe I should finally start reading my Vitruvius…)

Answer 1

You're right to look at Vitruvius for this. The best expression in Latin for "perpendicular" is actually the Greek πρὸς ὀρθᾶς, as Vitruvius uses in 9.7.

Itaque in quibuscumque locis horologia erunt describenda, eo loci sumenda est aequinoctialis umbra, et si erunt quemadmodum Romae gnomonis partes novem, umbrae octo, linea describatur in planitia et e media πρὸς ὀρθᾶς erigatur ut sit ad normam quae dicitur gnomon.

Hence, wherever a sundial is to be constructed, we must take the equinoctial shadow of the place. If it is found to be, as in Rome, equal to eight ninths of the gnomon, let a line be drawn on a plane surface, and in the middle thereof erect a perpendicular, plumb to the line, which perpendicular is called the gnomon.

[...]

haec erit linea πρὸς ὀρθᾶς radio aequinoctiali.

This will be a line perpendicular to the equinoctial ray, and it is called in mathematical figures the axis.

_9.7

The concept is related to plumb (i.e. vertical in normal English), as you can see above as well.

If it must be strictly Latin, he chiefly uses perpendiculum:

Tum insuper alternis trabibus ex quattuor partibus angulos iugumentantes et ita parietes arboribus statuentes ad perpendiculum imarum educunt ad altitudinem turres, intervallaque, quae relinquuntur propter crassitudinem materiae, schidiis et luto obstruunt.

Then upon these they place sticks of timber, one after the other on the four sides, crossing each other at the angles, and so, proceeding with their walls of trees laid perpendicularly above the lowest, they build up high towers.

_2.1.4

cum autem, uti supra scriptum est, in fronte inclinata fuerint, tunc in aspectu videbuntur esse ad perpendiculum et normam.

But when the members are inclined to the front, as described above, they will seem to the beholder to be plumb and perpendicular.

_3.5.13

Here it seems he uses ad perpendiculum recte for a stronger version of "perpendicular and vertical."

ea habet ancones in capitibus extremis aequali modo perfectos inque regulae capitibus ad normam coagmentatos, et inter regulam et ancones a cardinibus compacta transversaria, quae habent lineas ad perpendiculum recte descriptas pendentiaque ex regula perpendicula in singulis partibus singula, quae, cum regula est conlocata aeque, tangendo aeque ac pariter lineas descriptionis indicant libratam conlocationem.

At the extremities it has legs, made exactly alike and jointed on perpendicularly [i.e. vertically, but perpendicularity is implied - CW] to the extremities of the straightedge, and also crosspieces, fastened by tenons, connecting the straightedge and the legs. These crosspieces have vertical lines drawn upon them, and there are plumblines hanging from the straightedge over each of the lines. When the straightedge is in position, and the plumblines strike both the lines alike and at the same time, they show that the instrument stands level.

_8.5

For what it's worth, Caesar uses directis lateribus for perpendicularity to the ground:

Fossam pedum viginti directis lateribus duxit, ut eius fossae solum tantundem pateret quantum summae fossae labra distarent.

He dug a trench twenty feet deep, with perpendicular sides, in such a manner that the base of this trench should extend so far as the edges were apart at the top.

_{Caesar De Bello Gallico 7.72}