MIRA IS DEAD! (Long Live PCV)
In Book VII of Elements, Euclid defined units and numbers, yet not addition. So, in accordance with the doctrine of Aristotle, (you can't define something with an undefined term), Euclid could not have said 'Multiplication Is Repeated Addition', (MIRA).
Also, Euclid's proofs for his Book VII propositions involving multiplication, never mention addition! To arrive at this surprising discovery, I checked every printed English edition of Euclid's Elements, as listed below.
1570 | Henry Billingsley
1660 | Isaac Barrow
1661 | John Leeke and George Serle
1731 | Edmund Stone
1788 | James Williamson
1908 | Thomas Little Heath
2008 | Richard Fitzpatrick*
The following Book VII propositions mentioning multiplication, in either proposition, or proof, have no mention of addition in any English translation.
Euclid's Book VII Propositions Involving Multiplication:
(Text as per Heath, T. L., (1908). The thirteen books of Euclid's elements: translated from the text of Heiberg. Vol. 2, Cambridge: At the University Press.pp. 296-344.)
#16 If two numbers by multiplying one another make certain numbers, the numbers so produced will be equal to one another.
#17 If a number by multiplying two numbers make certain numbers, the numbers so produced will have the same ratio as the numbers multiplied.
#18 If two numbers by multiplying any number make certain numbers, the numbers so produced will have the same ratio as the multipliers.
#19 If four numbers be proportional, the number produced from the first and fourth will be equal to the number produced from the second and third; and, if the number produced from the first and fourth be equal to that produced from the second and third, the four numbers will be proportional.
#22 The least numbers of those which have the same ratio with them are prime to one another.
#24 If two numbers be prime to any number, their product also will be prime to the same.
#25 If two numbers be prime to one another, the product of one of them into itself will be prime to the remaining one.
#26 If two numbers be prime to two numbers, both to each, their products also will be prime to one another.
#27 If two numbers be prime to one another, and each by multiplying itself make a certain number, the products will be prime to one another; and, if the original numbers by multiplying the products make certain numbers, the latter will also be prime to one another.
#30 If two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers.
#33 Given as many numbers as we please, to find the least of those which have the same ratio with them, and
#34 Given two numbers, to find the least number which they measure.
To understand Euclidean multiplication, and how it should have evolved into PCV upon the arrival of zero and negative numbers, read
* A PDF of Richard Fitzpatrick’s self-published edition is at