PHYSICAL AND MATHEMATICAL SCIENCES. 1 (71

deduced mathematically, that the spaces described mustbe as the squares of the times, and that the space fallenthrough in one portion of time is exactly half thatwhich would be described in the same time with thevelocity last acquired, continued uniformly. He soonsaw that a body descending on an inclined plane mustbe, in like manner, accelerated: he therefore adoptedthis as a simple mode of putting his theory of fallingbodies to the test of experiment.

From his knowledge of the inclined plane, it easilyfollowed that the times of falling down all the chords ofa circle, terminating at its lowest point, must be* thesame. He, however, fell into an error in maintainingthe same' of the circular arcs, and in applying this tothe vibrations of the pendulum.

Knowing the law of falling bodies to give spacesproportional to the squares of the times, he deduced themotion of projectiles, and showed their path to he aparabola. This reasoning involved a principle whichGalileo does not expressly refer to by name, but whichis, in fact, one of the cases of the general principle ofcomposition of forces.

The question as to the motion of projectiles had beenput by Aristotle , but without any solution. Some ofthe early writers on' gunnery noticed that for certaindistances the gun must be pointed upwards: and ThomasDigges , in his “ New Artillerie” (1591), had remarkedthat the ball has, from the beginning, a downward mo-tion, which, though insensible at first, draws it from itsdirect course. Tartalea supposed the ball’s path to bemade up of an ascending and a descending straight line,connected in the middle by a circular arc. To theseimperfect notions succeeded the sound and philosophicaltheory of Galileo .

The mechanical and dynamical researches of thisgreat philosopher are unquestionably those in whichwe recognise the first real union of experimental andmathematical reasoning in investigating the laws offorce and motion.