# How to read mathematics out loud?

Reading symbolic mathematical expressions out loud in any language is mainly folklore: everyone in the field knows how to do it but finding explicit written instructions is surprisingly hard. I have yet to find good written instructions for any language, despite asking at Mathematics Educators SE. I would like to know how to pronounce mathematics in Latin.

Frankly, I do not expect to find a complete treatise on the topic. Instead, I am asking for any documents or insights whatsoever on the subject. Any partial answers are warmly welcome. For example, do we know how Newton, Euler or Gauss read formulas out loud? (If there is a flood of partial answers to this questions, I can make it narrower, but that does not look like a real threat.)

If you can provide material for pronouncing mathematics in other languages than Latin, please answer the linked question at Mathematics Educators instead.

## 2 Answers

For basic mathematics, I’ve found some answers in the Institutiones Physicæ by Floriani Dalham, published in 1752:

• 1+2 = 3 would be read unus plus duo sunt tres

Additio est duorum, vel plurium Numerorum in unum collectio. Indicatur per signum + adjectum, id est : plus. (…)
Dicatur : 4+2+2+7 sunt 15
(Caput III, p. 26)

• 1-2 = -1 would be read unus minus duo sunt minus unus

(…) indicaturque per signum -, id est : minus. (…)
(Caput IV, p. 28-29)

• 7*4 is read septies quatuor or quater septem
Other solutions:
tres multiplicata in quattuor (found by brianpck in Liber Mahameleth, or in this dictionary: multiplicare numerum in se)
According to the Valbuena Latin-Spanish dictionary (or Jeanneau Latin-French): multiplicare numerum cum numero

Jeanneau: ter multiplicati quinquaginta fiunt centum quinquaginta, Aug. : cinquante multiplié par trois font cent cinquante. (50*3=350)

p.33 Note: the tables of logarithms are called *tabularum Neperianarum (…)multiplicando 360 cum 5 prodibit numerus 1800 (…). [p.41]

• 24/3 is 24 divisum per 3 [p.40] or 24 ad 3

Nam sicut se habet 1 ad 3, ita se habent 2 ad 6. [p. 54]

• A^3 is A exponens 3 [p. 37]

• Cross-multiplication is regula trium

• 2>3: 2 majus ad 3

<, >, =: Signum æqualitas est =. Signum similitudinis ~. > notat majus ad minus. < notat minus ad majus. [p. 75]

• sqrt(2) would be radix quadrata duorum. [p. 101]

The Latin version of Wikipedia can also certainly be of some help:

• At least one book seems to use "tres multiplicata in quattuor" to mean "4 * 3", which seems a much better approach when you get to bigger numbers like "quintagies semel duodeviginti" = "51 * 18" Nov 5, 2016 at 2:03
• @brianpck, good find! That becomes even more useful with variables. Translating "x times y" or "a times b times c" by something analogous to "ter sena" or "ter sex" seems impossible. However, fluent and flexible reading seems to call for a translation for "times". I don't know any language uses a different construction for modern mathematics. Nov 5, 2016 at 23:04
• @brianpck I've added your solution for the multiplication and others I've found via dictionnaries and via Google Books.
– Luc
Nov 7, 2016 at 13:52
• Why wouldn't tres multiplicata in quattuor sunt duodecim be tria multiplicata in quattuor sunt duodecim or (numerus) tres multiplicatus in quattuor est duodecim? Mar 30 at 4:56

Partial answer!

As far as I can tell, most mathematical discourse would be done in Greek. Latin was used for engineering purposes, but speaking unambiguously about mathematics became rather awkward.

For example, Vitruvius's description of the Pythagorean formula, from De Architectura IX.6:

namque si sumantur regulae tres e quibus una sit pedes III altera pedes IIII tertia pedes V, eaeque regulae inter se compositae tangant alia aliam suis cacuminibus extremis schema habentes trigoni, deformabunt normam emendatam. ad eas autem regularum singularum longitudines si singula quadrata paribus lateribus describantur, quod erit trium latus, areae habebit pedes VIIII, quod IIII, XVI, quod V erit, XXV. ita quantum areae pedum numerum duo quadrata ex tribus pedibus longitudinis laterum et quattuor efficiunt, aeque tantum numerum reddidit unum ex quinque descriptum.

Translation:

For if three straight rods are acquired (and from these let one be three feet, the other four feet, and the third five feet) and these rods are arranged together so that they touch each other at the very tip, making the shape of a triangle, they will form a perfect right angle. Then, if squares are marked out, with sides equal to the lengths of each of the rods, then the [square with the] side [length] of three will have an area of nine feet, the four, sixteen, and the five, twenty-five. Thus how many feet of area two squares with sides of three and four feet in length contain, is equal to how many one [square] marked out from five [feet] contained.

In other words: a² + b² = c², expressed geometrically and in far too many words.

Frontinus' De Aquis shows similarly how unwieldy numbers could be in Roman numerals. A small excerpt starting from I.50:

Fistula quadragenaria: diametri digitos septem 𐆑 𐆒 ℈III, perimetri digitos XXII 𐆐 𐆐 𐆑, capit quinarias XXXII S 𐆑.
Fistula quadragenum quinum: diametri digitos septem S 𐆒 ℈ octo, perimetri digitos XXIII S 𐆐 𐆑 𐆒, capit quinarias XXXVI S 𐆑 𐆒 ℈ octo; in usu non est.
Fistula quinquagenaria: diametri digitos septem S 𐆐 𐆐 𐆑 𐆒 ℈ quinque, perimetri digitos XXV 𐆒 ℈VII, capit quinarias XL S 𐆐 𐆒 ℈V.
Fistula quinquagenum quinum: diametri digitos octo 𐆐 𐆐 ℈ decem, perimetri digitos XXVI 𐆐 𐆑 𐆒, capit quinarias XLIIII S 𐆐 𐆑 𐆒 ℈II; in usu non est.

Translation from LacusCurtius, since I'm not familiar enough with the fractions:

The 40‑pipe: 7 digits plus 1/12 plus 1/24 plus 3/288 in diameter; 22 digits plus 5/12 in circumference; it has a capacity of 32 quinariae plus 1/2 plus 1/12.
The 45‑pipe: 7 digits plus 1/12 plus 1/24 plus 8/288 in diameter; 23 digits plus 1/2 plus 3/12 plus 1/24 in circumference; it has a capacity of 36 quinariae plus 1/2 plus 1/12 plus 1/24 plus 8/288; is not in use.
The 50‑pipe: 7 digits plus 1/2 plus 5/12 plus 1/24 plus 5/288 in diameter; 25 digits plus 1/24 plus 7/288 in circumference; it has a capacity of 40 quinariae plus 1/2 plus 2/12 plus 1/24 plus 5/288.
The 55‑pipe: 8 digits plus 4/12 plus 10/288 in diameter; 26 digits plus 3/12 plus 1/24 in circumference; it has a capacity of 44 quinariae plus 1/2 plus 3/12 plus 1/24 plus 2/288; is not in use.

So it is more likely that educated Romans preferred to speak about mathematics in Greek—especially as the most important mathematical texts at the time had all been written in Greek.

As Vitruvius's description shows, mathematics at that time was more focused on geometry than on formulas. Rather than calculating a number, the goal would be to construct a line segment of that length, or a triangle of that area, etc. This made certain things far more difficult: numerical methods can approximate the cube root of 2 to arbitrary precision, but with only a compass and straightedge it is impossible to construct exactly. Plato famously mocked this failing when the geometers couldn't double the volume of a cube, as related by Vitruvius, Plutarch, and others.

However, this also had the advantage of making it much easier to read out loud. Vitruvius's description of the Pythagorean formula may be harder to visualize than a diagram, but unlike a diagram it can be spoken without losing any information. As another example, from Euclid's Elements:

κγʹ. Παράλληλοί εἰσιν εὐθεῖαι, αἵτινες ἐν τῷ αὐτῷ ἐπιπέδῳ οὖσαι καὶ ἐκβαλλόμεναι εἰς ἄπειρον ἐφ ̓ ἑκάτερα τὰ μέρη ἐπὶ μηδέτερα συμπίπτουσιν ἀλλήλαις.

Translation:

1. Lines which are in the same plane, and are projected infinitely in both directions, but touch each other in neither direction, are "parallel".

That's pretty much the same way I would describe parallel lines to someone if I didn't have pen and paper on hand.

(More to come when I find more examples: later usage, for instance.)

• Thanks! Those are nice passages, although they don't exactly tell how to read formulas of any kind out loud (instead of describing their content in words). Ancient mathematicians didn't have notation like we do. I believe Vitruvius can be useful for writing more verbal mathematics, but perhaps not with reading symbolic mathematics. Nov 4, 2016 at 18:26
• I am so using this for my math students--especially the one explaining the Pythagorean theorem. May 9, 2019 at 1:49
• ªJoonas llmavirta: interesting as this material is, why would Latin mathematical terms be pronounced any differently from other Latin words--letter-by-letter; syllable-by-syllable; sound-by-sound?
– tony
May 11, 2019 at 9:35