362 THE MATHEMATICAL PRINCIPLES (BOOK 1L and in the time that the pendulum will perform one single oscillation the waves will advance forward nearly a space equal to their breadth. That which I call the breadth of the waves is the transverse measure lying between the deepest — part of the hollows, or the tops of the ridges. Let ABCDEF represent the surface of stagnant water ascending and descend ing in successive waves ; and let A, C, E, &c., be the tops of the waves ; and let B, D, F, &c., be the intermediate hollows. Because the motion of the waves is carried on by the successive ascent and descent of the water, so that the parts thereof, as A, C, E, &c., which are highest at one time become lowest immediately after ; and because the motive force, by which the highest parts descend and the lowest ascend, is the weight of the eleva ted water, that alternate ascent and descent will be analogous to the recip rocal motion of the water in the canal, and observe the same laws as to the times of its ascent and descent; and therefore (by Prop. XLIV) if the distances between the highest places of the waves A, C, E, and the lowest B, D, F, be equal to twice the length of any pendulum, the highest parts A, C, E, will become the lowest in the time of one oscillation, and in the time of another oscillation will ascend again. Therefore between the pas sage of each wave, the time of two oscillations will intervene ; that is, the wave will describe its breadth in the time that pendulum will oscillate twice; but a pendulum of four times that length, and which therefore is equal to the breadth of the waves, will just oscillate once in that time. Q.E.L COR. 1. Therefore waves, whose breadth is equal to 37\ French feet, will advance through a space equal to their breadth in one second of time; and therefore in one minute will go over a space of 1S3J feet ; and in an hour a space of 11000 feet, nearly. COR. 2. And the velocity of greater or less waves will be augmented or diminished in the subduplicatc ratio of their breadth. These things are true upon the supposition that the parts of water as cend or descend in a right line; but, in truth, that ascent and descent is rather performed in a circle ; and therefore I propose the time denned by this Proposition as only near the truth. PROPOSITION XLVIL THEOREM XXX VII. If pulses are propagated through a fluid, the .ve eral particles of the Jluid, goittff and returning with the shortest reciprocal motion, are al ways accelerated or retarded according to the law of the oscillating pendulum. Let AB, BC, CD, &c., represent equal distances of successive pulses, ABC the line of direction of the motion of the successive pulses propagated