### 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 new model of multiplication via Euclid

^{* }A PDF of Richard Fitzpatrick’s self-published edition is at

http://farside.ph.utexas.edu/Books/Euclid/Euclid.html