upon the water within the cylinder, tending to produce a vacuum there, this water is pushed up by the pressure of the air upon the water outside the cylinder, and follows the rising piston, until the column of water inside the cylinder exerts a pressure equal to that of the atmosphere upon an equal area.  So much for the computation; does it correspond with the fact?  It is found that at the sea level water can be pumped to the height of 33 ft; and that such a column of water has a pressure of 15 lb. to the square inch.  We may show further that, at the sea level, spirits of wine may be pumped higher according to its less specific gravity; and that if we attempt to pump water at successive altitudes above the sea level, we can only raise it to less and less heights, corresponding with the lessened atmospheric pressure at those altitudes, where the column of air producing the pressure is shorter.  Finally, if we try to work a pump, having first produced a vacuum over the water outside the cylinder, we shall find that the water inside will not rise at all; the piston can be raised, but the water does not follow it.  The verification thus shows that the computed effect corresponds with the phenomenon to be explained; that the result does not depend upon the nature of water only, but is true (allowing for differences of specific gravity) of other liquids; that if the pressure of the outside air is diminished, the height of pumping is so too (canon of Variations); and that if that pressure is entirely removed, pumping becomes impossible (canon of Difference).

Any text-book of Astronomy or Physics furnishes numerous illustrations of the deductive method.  Take, for example, the first chapter of Deschanel’s Optics, where are given three methods of determining the velocity of Light.  This was first deduced from observation of Jupiter’s satellites.  The one nearest the planet passes behind it, or into its shadow, and is eclipsed, at intervals of about 42-1/2 hours.  But it can be shown that, when Jupiter and the Earth are nearest together on the same side of the Sun, an eclipse of this satellite is visible from the earth 16 min. 26.6 sec. earlier than when Jupiter and the earth are furthest apart on opposite sides of the Sun:  16 min. 26.6 sec, then, is the time in which light traverses the diameter of the Earth’s orbit.  Therefore, supposing the Earth’s distance from the Sun to be 92 millions of miles, light travels about 186,000 miles a second.  Another deduction, agreeing with this, starts from the fact of aberration, or the displacement of the apparent from the actual position of the stars in the direction of the earth’s motion.  Aberration depends partly on the velocity of light, partly on the velocity of the Earth; and the latter being known, the former can be computed.  Now, these two deductive arguments, verifying each other, have also been verified experimentally.  Foucault’s experiment to measure the velocity of light is too elaborate to be described here:  a full account of it will be found in the treatise above cited, Sec. 687.