Chinese money, with the square, triangle, and rhomboid holes through the centre, was originally coined with certain values based upon the ratio of these Tangram shapes to the square.
That Tangrams were used in the teaching of mathematics is clearly shown by the presence of Euclid's famous 47th problem, known as Pons Asinorum, as shown in the above seventh figure, accompanied by simple kindergarten progressive steps.
"Tan" had two squares, the size of Fig.3, which he desired to unite into one large square, so he cut one of the squares in two on the bias, and arranged the pieces in the form of a pyramid, as shown, and discovers that it forms half of a larger square. He then makes another pyramid from the other square and builds the square, which he knows to be exactly twice as large as the first, because it has twice as many pieces. Then he marked off a right-angled "corner" on a piece of paper and found that by placing the small squares on the sides and the large squares connecting them, so as to form the right-angled triangle B in the center, he proved the proposition that when three squares form a right-angled triangle the largest square will be equal to the other two combined. Then he formulated the following invaluable rule of mechanics : when, having two squares of different sizes, it is required to find the dimensions of one square equal in area to the other two, draw a diagram of a right-angled corner, and measure off the diameter of one of the squares on one arm of the angle and the diameter of the other square on the other arm; then a straight line drawn from one of those points to the other, forming the right-angled triangle A, will show the side of a square equal in area to the other two.
Then he found, as shown in Figs. 8 and 9, that this rule applied to polygons, triangles, circles, or any irregular shapes, so long as they were of the same form; viz., draw the right-angled corner, and upon one arm measure off a point which will indicate the shortest diameter of any irregular design. Then on the other arm measure off the shortest diameter of a larger representation of the same design, and the distance from one of those indicated points to that of the other arm will show the shortest diameter of an enlargement of the design equal to the other two. All of which was known many, many thousands of years ago.
This brief reference to the mathematics of Tangrams shows how Li Hung Chang would have the book interpreted according to the age and mental calibre of the students.
The working out of Tangram puzzles and the originating of new designs fills the long-felt want for a refined and instructive relaxation from overwork, as well as a good antidote for ennui.
Tangram parties become very enjoyable when sets of pieces with full instructions are sent out with the invitations a couple of weeks in advance, requesting the guests to furnish a number of original designs, unaccompanied by descriptions or names of authors; the intention being for an impromptu art committee to award some humorous prize for the best work.
By this plan, it may be mentioned, some choice and interesting contributions to the Eighth Book of Tan have been secured from some very distinguished sources.
Now, scholars, run away and amuse yourselves, for I am invited to say a few words to the children of a larger growth.