The 8th Book of Tan by Sam Loyd
The unsolved problems
The Second Book of Tan, like the lost Euclid, was given to elementary problems, the correctness or fallacy in the construction of which was to be discovered by the student, who was required to decide between the correctness of two conflicting propositions. The squaring of the circle appears to have been treated according to the following kindergarten method, which will furnish food for reflection for the young folks as well as the brainy mathematicians.
To make it clear to all, it may be explained that the squaring of the circle is merely the problem of figuring out how many square feet are contained in a circle of a given diameter--viz., how many square feet of sod are there in a round grass-plot 100 feet in diameter?
"If the Pons Asinorum, as shown, is correct," says Tan, "let us describe these semicircles upon the sides of the triangle A. Knowing that B is equal to C and D combined in Fig. 1, we will divide B by a straight cut from X to Z. Then the remaining half of B, as shown in Fig. 2, will be equal in area to C.
"Let us now, as shown in Fig. 3, continue the circle from X to Z and draw the vertical line from X to Y so as to cut the semicircular pieces D and D from both C and B. Those pieces which are removed being exactly similar, taken from forms of the same areas, prove that the remainders C and B must be of the same areas, although the one is a triangle and the other a crescent.
"The space A being of similar size and construction to B we proceed to place the pieces as shown in Fig. 4 and halve them by a straight cut from C to X, so as to have the remaining pieces as shown in Fig. 5, which have been proven by construction to be of the same areas.
"We will then cut the piece C by a diagonal line, and, as shown by the dotted lines, proceed to describe the circle enclosed within a square.
"Now let us show by analysis what has been proven: CC have been shown by Figs. 3, 4, and 5 to be equal to A.
"As BB added to CC would form a square of the same size as AD, then BB must be equal to D, as CC are equal to A. It shows that B must be equal to B and C equal to C, because BB equals A or BC equals A.
"Then if the segments B, B, C, and C are all equal, each piece represents the sixteenth part of the large square, and we find that twelve of these sixteen pieces are contained within the circle.
"Therefore, if the square is 100 in diameter, 100 x 100 = 10,000 square feet, of which our grass-plot, which was said to be 100 feet in diameter, would contain twelve-sixteenths, or just 7,500 square feet."
And, as Tan says, "There you are! the truth or fallacy of Pons Asinorum is submitted to the judgment of the students who will kindly pass in their examination papers to the professor."