Squares on a divided line, with worked numerical examples
Segments a c b / a c d b and squares partitioned by parallels, checked by arithmetic
Two columns treat the division of a line. On the right a segment is marked a c b, then a c d b with the note that 'd b is the excess'; a square is partitioned by two pairs of parallel lines carrying the figures 2, 2, 8. The left column repeats the divided lines and works a numerical 'example' at length, with squarings and additions whose columns sum to 104 and to 256, verifying the areas of the parts. The letters a, b, c, d label the points of division.
On this page
Line divided a c b, then a c d b with the excess d b
A segment is first divided at c as a c b, then further at d as a c d b, the note explaining that d b is the excess. The two divisions set up the algebra of the square worked below.
Square partitioned by two pairs of parallels
A square is divided by two pairs of parallel lines into regions carrying the figures 2, 2, 8. The partition mirrors the divided line, exhibiting the parts of the square geometrically.
Worked numerical example of the divided square
The left column labels a lengthy 'example' of the same partition: squarings and additions such as columns totalling 104 and 256, checking the areas of the parts numerically against 4, 8 and the products 2, 2, 8.