Geometric Dissections and Transpositions
Research articles
Geometry

On Geometric Dissections and Transformations

Extracted from The Messenger of Mathematics, New Series, No. 19, 1872.

«The present series of papers consist chiefly of notes and diagrams extracted from my note-books and sketches made during the last forty years. I have devoted much time and thought to the solution of geometric theorems and problems by dissections and transpositions: viz. in proving the equality of areas in equivalent rectangles, &c., and investigating how the figures could be best dissected, so that their component parts might be fitted together in either form; and I have often contemplated proving all the suitable Theorems and Problems in Euclid by such dissections and transpositions: so as to render them self-evident by ocular demonstration. I worked on paper ruled all over in small squares, which I found useful in facilitating dissections; and I have alway had a fancy that the ancient Egyptians and early Greck geometers adopted some such expedient in their geometrical researches. So that I deem it probable that the property of the right-angled triangle may have been discovered by similar means; and the solution I hit upon forty years ago was perhaps the very one discovered forty centuries ago, and re-discovered by Pythagoras nearly twenty centuries afterwards; and not unlikely, as was also the case with me, in the endeavour to find geometrically a square demonstrably equal in area to a circle by dissection and transposition.»

Geometric Dissections and Transpositions