Euclid's Elements - Book VII: Proposition I.

Proposition 1. Δύο αριθμών ανίσων εκκειμένων, ανθυφαιρουμένου δε αεί του ελάσσονος από του μείζονος, εάν ό λειπόμενος μηδέποτε καταμετρή τόν πρό εαυτού, έως ου λειφθή μονάς, οί εξ αρχής αριθμοί πρώτοι πρός άλληλους έσονται. Heath: Two unequal numbers (being) laid down, and the lesser being continually subtracted, in turn, from the, greater, if the remainder never measures the (number) preceding it, until a unit remains, then the original numbers will be prime to one another. The proof by Euclid is ingenious and needs no improvement whatsoever. Proposition 1 is about the use of obelus (÷) division to demonstrate the prime number algorithm. So let’s focus on an example. Are 8 and 19 prime to each other? 19=8(2)+3 3=2(1)+1 So a unit remains implying that 8 and 19 are prime to each other. The first step is to subtract the greatest multiple (2) of the smaller number (8) from the larger (19). Then repeat the process with the multiple (2) until a unit remains or no unit remains. The multiple of the previous step (2) becomes the divisor of the next. The remainder (3) is what gets to be measured in the subsequent step. Notice that in spite of dealing with numbers, the basic operations of arithmetic are demonstrated on the measure of line segments (numbers!). dividend = divisor (multiple)+remainder The divisor measures the dividend and if there is a remainder, then the remainder becomes the new dividend and the previous multiple becomes the divisor with the process repeating until there is no remainder or a unit remainder (in which case the numbers are prime to each other).