Let A, B, C, _fig. 74._, be three small pieces of metal each attached by threads to two points of suspension, and let them be placed in the same vertical line under the point O; suppose them so adjusted that the distances O A, O B, and O C shall be in the proportion of the numbers 1, 4, and 9. Let them be removed from the vertical in a direction at right angles to the plane of the paper, so that the threads shall be in the same plane, and therefore the three pendulums will have the same angle of vibration. Being now liberated, the pendulum A will immediately gain upon B, and B upon C, so that A will have completed one vibration before B or C. At the end of the second vibration of A, the pendulum B will have arrived at the end of its first vibration, so that the suspending threads of A and B will then be separated by the whole angle of vibration; at the end of the fourth vibration of A the suspending threads of A and B will return to their first position, B having completed two vibrations; thus the proportion of the times of vibration of B and A will be 2 to 1, the proportion of their lengths being 4 to 1. At the end of the third vibration of A, C will have completed one vibration, and the suspending strings will coincide in the position distant by the whole angle of vibration from their first position. So that three vibrations of A are performed in the same time as one of C: the proportion of the time of vibration of C and A are, therefore, 3 to 1, the proportion of their lengths being 9 to 1, conformably to the law already explained. (212.) In all the preceding observations we have assumed that the material of the pendulous body is of inconsiderable magnitude, its whole weight being conceived to be collected in a physical point. This is generally called a simple pendulum; but since the conditions of a suspending thread without weight, and a heavy molecule without magnitude, cannot have practical existence, the simple pendulum must be considered as imaginary, and merely used to establish hypothetical theorems, which, though inapplicable in practice, are nevertheless the means of investigating the laws which govern the real phenomena of pendulous bodies.