346
FROM PLATO TO EUCLID
Again, says Aristotle,
A will move B over the distance \G in the time
and^M „ \B a distance G ,, „ D; 1
and so on.
Lastly, we have in the Mechanica the parallelogram of
velocities:
£ When a body is moved in a certain ratio (i, e. has two linear
movements in a constant ratio to one another), the body must
move in a straight line, and this straight line is the diameter
of the figure (parallelogram) formed from the straight lines
which have the given ratio.’ 2
The author goes on to say 3 that, if the ratio of the two
movements does not remain the same from one instant to the
next, the motion will not be in a straight line but in a curve.
He instances a circle in a vertical plane with a point moving
along it downwards from the topmost point; the point has
two simultaneous movements; one is in a vertical line, the
other displaces this vertical line parallel to itself away from
the position in which it passes through the centre till it
reaches the position of a tangent to the circle; if during this
time the ratio of the two movements were constant, say one of
equality, the point would not move along the circumference
at all but along the diagonal of a rectangle.
The parallelogram of forces is easily deduced from the
parallelogram of velocities combined with Aristotle’s axiom
that the force which moves a given weight is directed along
the line of the weight’s motion and is proportional to the
distance described by the weight in a given time.
Nor should we omit to mention the Aristotelian tract On
indivisible lines. We have seen (p. 293) that, according to
Aristotle, Plato objected to the genus ‘ point ’ as a geometrical
fiction, calling a point the beginning of a line, and often
positing ‘indivisible lines’ in the same sense. 4 The idea of
indivisible lines appears to have been only vaguely conceived
by Plato, but it took shape in his school, and with Xenocrates
1 Phijs. vii. 5. 250 a 4-7. 2 Mechanica, 2. 848 b 10,
3 Jb. 848 b 26 sq. 4 Metaph. A. 9. 992 a 20.
