Search Loci: Convergence:
Now that practical skills have developed enough to provide adequately for material needs, one of these sciences which are not devoted to utilitarian ends [mathematics] has been able to arise in Egypt, the priestly caste there having the leisure necessary for disinterested research.
Jordanus de Nemore's Arithmetica
14. If a number is divided into two parts and the whole number is multiplied by one of the parts, the result is that number multiplied by itself added to the product of the two parts.
15. If a number is divided into two parts and the whole is multiplied by itself, the result is the sum of each part multiplied by itself and twice one part multiplied by the other.
16. If a number is divided into two parts, then the sum of the whole multiplied by itself and one part multiplied by itself is twice the whole multiplied by that part and the second part multiplied by itself.
17. If a number is divided into two parts, then the whole multiplied by itself is the same as four times the product of one part multiplied by the other and the difference between the two parts multiplied by itself.
18. If a number is divided into two parts, then the smaller multiplied by itself and the whole number multiplied by the difference of the two parts is the same as the larger multiplied by itself.
19. If a number is divided into two equal parts and also into two unequal parts, then one of the equal parts multiplied by itself is the same as the product of the unequal parts and the product of the differences between the unequal parts and the equal parts.
Note that illustrations of each theorem are given by diagrams with numbers in the left margin. Thus the first diagram illustrates that 2(2+4) = 4 + 8 = 12.