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XIV. Researches on the Tides. — Twelfth Series. On the Laws of the Rise and Fall 
of the Sea's Surface during each Tide. By the Rev. W. Whewell, B.D. F.R.S., 
Fellow of Trinity College, Cambridge. 
Received June 13, — Read June 18, 1840. 
The subject of the present communication is different in its nature from those of 
previous memoirs on the tides presented by me, and printed by the Society ; since it 
refers, not to comparison of the times and heights of different tides, but to the rate 
of the rise and fall of the surface of the water in successive stages of the same tide. 
This inquiry has often been prosecuted at particular places by naval observers, and is 
of very material importance to navigation. For even supposing the time and height 
of high water to be known, it is still often requisite, for nautical purposes, to know 
the height of the water at a given interval before or after the moment of high water. 
And this inquiry may be the more useful, inasmuch as the laws of rise and fall of the 
surface are nearly the same at all places ; the differences being, for the most part, of 
such a kind as can be ascertained and allowed for without much difficulty. Hence 
these laws, once stated, will be applicable on every coast ; and the knowledge of them 
may supersede those laborious trains of observation which have often been instituted 
in order to ascertain the laws at particular places. 
The materials of the present investigation (which is principally founded upon ob- 
servation) are the following : — Five months’ tide observations made at Plymouth, in 
which, besides the time and height of high and low water, the time of the surface 
passing two lines above the level of mean water was carefully observed ; these latter 
observations being made, at my request, by direction of the Lords Commissioners of 
the Admiralty : — Three months’ observations (taken out of a larger series) made at 
Liverpool, under the direction of Capt. Denham, R.N., in which the height of the 
surface was noted every half hour : — and twelve months’ observations made at Bristol 
by Mr. Bunt, by means of his tide-gauge. The latter observations were reduced by 
Mr. Bunt himself ; the others were discussed under my direction by Mr. Dessiou 
and Mr. Ross, of the Hydrographer’s Office, with their usual care and skill. 
Whatever be the exact law of the rise and fall of the tide, if the rise and fall be 
nearly symmetrical (which it usually is), the height of the surface of the water at a 
given place will increase nearly as the sine, while the time increases as the arc. Hence, 
if we were to make the time the abscissa and the height the ordinate, we should obtain 
(for one tide) the J figure of sines for our curve. Or, to put the matter otherwise : if 
we suppose a point to move uniformly through the circumference of a circle in a tidal 
