Euclid's common notions: the axioms of equality
Nine mind-notions on equal and unequal magnitudes, with line diagrams
The recto (folio 139r) sets out the nine "common notions of the mind" that Leonardo drew from Euclid: things equal to a same thing are equal to one another, equals added to or taken from equals give equals, and every whole is greater than its part. Down the right margin runs a column of short line segments marked with crosses and the numeral 2, illustrating each axiom about equal and unequal lengths. Faint geometric figures visible at the left are offsets showing through from the facing leaf.
On this page
Equals added to or taken from equals
The first notions state that magnitudes equal to a same magnitude are equal to one another. Adding equals to equals yields equal wholes; subtracting equals from equals leaves equal remainders. Taking equals from unequals, or adding equals to unequals, leaves the results unequal.
Superposition and the whole greater than its part
Two things each equal to a same thing are equal to each other, and two halves of the same thing are equal. If one figure laid upon another coincides with it exactly, neither exceeding the other, the two are equal. Finally, every whole is greater than its part.
Line-segment diagrams of the axioms
A vertical column of short horizontal strokes accompanies the notions, each pair marked with crosses or the figure 2 to show equal and unequal lengths. The diagrams turn the verbal axioms into a visual demonstration of comparison of magnitudes.