Transmuting squares and rectangles into equal figures
Reshaping a square or rectangle to a given height, a method useful for metal plates
This recto develops the transmutation of rectilinear figures: turning a square or rectangle into an equal figure of a different proportion fixed by a given point. Leonardo's method draws indefinite lines beyond the base, transfers the height c e onto c n, and finds a circle-centre h from which the new width follows, invoking the definition of the circle. A related construction extends a line b o equal to the remainder a o of another line, and a marginal note remarks that the method is good for plates of any metal reduced to various widths and lengths. Numerous lettered semicircle and square diagrams accompany the two columns of text.
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Transmuting a square to a given height to find its new width
Given a square a b c d and a point e set at a chosen height, Leonardo finds the width into which the square is transformed while keeping its quantity. He extends an indefinite line n g beyond the base, lays the height c e onto c n, and from n and the corner a draws two equal lines meeting at h, the centre of the governing circle.
Extending a line equal to the remainder of another
From the end of the line c b, Leonardo draws b o equal to the remainder a o of the line a c upon which b c rests. The equality of b o and a o is secured by the definition of the circle.
Problem: the width of a square carried to a given height
A heading poses the task of a square with equal angles and sides drawn up to the height of a given point above it, asking the width into which it is transmuted.
Problem: the width of a rectangle produced to a given length
A companion heading asks, of a 'long square' or rectangle produced to the length of a point outside its surface, what its resulting width will be.
The construction applied to metal plates
A short phrase beside the upper figure notes that the method is good for plates of any metal reduced to various widths and lengths, linking the area-preserving geometry to practical metalworking.