Counting Rectangles:
To count the number of rectangles in an 8 by 8 chessboard, we can consider the dimensions of the rectangle. We can choose any two vertical lines and any two horizontal lines on the chessboard to form the sides of the rectangle.

Counting Horizontal Lines:
There are a total of 9 horizontal lines on the chessboard, labeled from 1 to 9.

Counting Vertical Lines:
Similarly, there are 9 vertical lines on the chessboard, also labeled from 1 to 9.

Calculating Combinations:
To find the number of rectangles, we need to select 2 horizontal lines and 2 vertical lines out of the available 9 lines each.

The number of ways to choose 2 horizontal lines out of 9 is given by the combination formula:
C(9, 2) = 9! / (2! * (9-2)!) = 36

The same formula applies to choosing 2 vertical lines out of 9, resulting in 36 combinations.

Total Number of Rectangles:
To calculate the total number of rectangles, we need to multiply the number of horizontal line combinations by the number of vertical line combinations:
Total number of rectangles = 36 * 36 = 1296

Therefore, there are 1296 rectangles in an 8 by 8 chessboard.

Counting Squares:
To count the number of squares in an 8 by 8 chessboard, we can consider the dimensions of the square. The sides of the square will be the same length.

Counting Possible Side Lengths:
Since the chessboard is 8 by 8, the maximum possible side length of a square can be 8. Therefore, we need to count the number of possible side lengths from 1 to 8.

Calculating the Number of Squares with Each Side Length:
For a side length of 1, there will be 8x8 = 64 squares.
For a side length of 2, there will be 7x7 = 49 squares.
For a side length of 3, there will be 6x6 = 36 squares.
And so on, until a side length of 8, where there will be 1 square.

Total Number of Squares:
To find the total number of squares, we need to sum up the squares for each side length:
Total number of squares = 64 + 49 + 36 + ... + 1

This is an arithmetic series with a first term of 64, a common difference of -15, and a last term of 1.

Using the formula for the sum of an arithmetic series:
n = (last term - first term) / common difference + 1
n = (1 - 64) / -15 + 1
n = 63 / 15 + 1
n = 4.2 + 1
n = 5.2

Therefore, the total number of squares is approximately 5.2 or 5 squares.

Final Answer:
The number of rectangles and squares in an 8 by 8 chessboard respectively are 1296 and 5.