Leibniz' Vis Viva
June 28, 2012 • 6:26PM

A Preliminary Specimen

ON THE LAW OF NATURE CONCERNING THE POWER OF BODIES

When I had discovered that neither does the POWER [pontentia] of bodies consist in the QUANTITY OF MOTION or the product of a magnitude of weight times velocity -as is commonly believed – nor that in transferring power from one body to another the same quantity of motion will be conserved (of which Descartes had been so greatly persuaded,) I judged it was worth the effort to summon the force of my reasonings to gradually lay, using the most evident demonstrations, a foundation for the TRUE ELEMENTS OF A NEW SCIENCE OF POWER AND ACTION – which you may call DYNAMICS.

This LAW OF NATURE, rather, is such that the WHOLE EFFECT IS EQUIVALENT TO ITS FULL CAUSE, lest one comes to violate the order of things through an excess of effect over and above its cause as in perpetual motion (which I certainly take as absurd, and which I show can be brought to an absurdity.)

I have gathered certain first principles of this new science for special treatment, from which I'd like to now excerpt a read specimen in order to inspire gifted individuals to search for truth, and in so doing replace imaginary laws of Nature with more genuine ones. From this it will be obvious how unsafe it is to affirm anything in Mathematics using probable arguments, since the forces [vires] of two equal weights endowed with unequal velocities are not proportional to their velocities, but are proportional to the heights from which they could have acquired those velocities by falling. Moreover, it is well established that velocities are not calculated as proportional to heights, but as proportional to the square roots of those heights. From this another paradox immediately follows, that it is easier to imprint a given degree of velocity on a body at rest than it is to give the same degree of velocity to the same body once it has been struck into motion such that its velocity in the same direction is doubled. The opposite of this, however, can undoubtedly be seen as derived from a poor understanding of the composition of motion.

Lest anyone be subject to a dispute over semantics, or suspect that we are quarreling over the various meanings of the word 'power', it should be known that we seek, (for example) how much velocity should a body of, say one pound previously at rest receive if the total power or action of a four-pound body endowed with one degree of velocity were transferred to it in such a way that the body of four pounds is brought to rest leaving only the body of one pound in motion. It is a common opinion, but also one greatly celebrated in the writings of Cartesians, that such a body would receive a velocity of four degrees. My opinion is that it would not be able to receive more than two degrees. They think the former so as to conserve the same quantity of motion, which they confuse with power. I, however, think in such a way as to conserve the same quantity of power, i.e. the equality of cause and effect, so that perpetual motion does not arise by one exceeding the other. But it is time that we proceed to a demonstration.

Proposition

SUPPOSE ALL THE POWER OF A FOUR-POUND BODY MOVING HORIZONTALLY WITH A VELOCITY OF ONE DEGREE IS TRANSFERRED TO A ONE-POUND BODY PREVIOUSLY AT REST, IS SUCH A WAY THAT ONLY THE ON POUND BODY REMAINS IN MOTION WHILE THE FOUR-POUND BODY, IN TURN, IS BROUGHT TO REST. THEN IT IS NOT POSSIBLE FOR THE SAME QUANTITY OF MOTION AS THERE WAS BEFORE TO BE CONSERVED AND FOUR DEGREES OF VELOCITY TO BE ALLOTTED TO THE ONE-POUND BODY; AND IT IS NOT POSSIBLE FOR IT TO RECEIVE A VELOCITY OF MORE THAN TWO DEGREES.

LEMMA – This, already established, is common to the first of the three following demonstrations:

THE PERPENDICULAR HEIGHTS OF HEAVY BODIES ARE PROPORTIONAL TO THE SQUARES OF THE SPEEDS WHICH THEY CAN ACQUIRE BY FALLING FROM THESE HEIGHTS, OR TO THE SQUARES OF THE SPEEDS BY WHICH THEY CAN RAISE THEMSELVES TO THOSE HEIGHTS BY THEIR OWN POWER. This proposition of Galileo is demonstrated by the nature of the motion of a heavy body accelerated uniformly. It is accepted by mathematicians and confirmed by numerous experiments.

First Demonstration

AXIOM: THE SAME POWER IS REQUIRED TO LIFT FOUR POUNDS ONE FOOT AS IS REQUIRED TO LIFT ONE POUND FOUR FEET.

This granted, let us assume a four-pound body, A, can raise itself to a height (perpendicular, of course,) of, say, one foot by the force of its velocity of one degree, if, for example, it is moved in such way on a pendulum or on an inclined plane as to be able to direct its force upwards. Body A, therefore, has the power to lift a four-pound weight – proper to the body itself, of course - to a height of one foot, or the power to lift a one pound body four feet, which by the preceding axiom, amounts to the same thing. On the contrary, if a body A is lifted one foot by the force of one degree of velocity, the force of four degrees of velocity will lift body B to a height of 16 feet (by the previously established lemma from Galileo.) Consequently, body B has a force to lift on pound – proper to the body itself, of course – to a height of 16 feet. Therefore, there is four times more power in B than I A, which we have shown can only raise one pound four feet.

This is contrary to the hypothesis in which we had postulated that the very same power in A would be transferred to B.

Second Demonstration

AXIOM: THERE IS NO PERPETUAL MECHANICAL MOTION.

This granted, let a four-pound body, A, advance on the horizontal A2A3 (fig. 32) with a velocity of one degree. Assume all of its power is transferred to a one-pound body, B, resting at B1, in such a way that only B moves through B1B2, while A rests at A3. I say it will be impossible for the quantity of motion in B to equal the quantity of motion which was in A, or for B to receive a velocity of four degrees, or for it to receive even more than two degrees of velocity. For let B receives four degrees of velocity, if it is possible, and assume body A receives its velocity of one degree by descending from a perpendicular height A1H of one foot on an inclined plane A1A2. Assume then B, receiving a velocity of 4, ascends as high as it can up the slope B2B3; it ascends to a height B3M of 16 feet (by the previously demonstrated lemma from Galileo.) Assume now a certain balance A3LB3 appears, extending from A3 (the place on the horizontal in which A is at rest) to B3 (the place to which B has ascended,) and thus divided by a fulcrum or center L in such a way that the arm lB3 is a little more than 4 times the length of arm LA3; for example, say it is five times the length. Consequently B is raised to B3 by the force of its own 'impetus,' so as to be able to fall upon the balance where it will outweigh A, positioned on the opposite end, A3, because while A is four times the weight of B, the distance of B from the center L is nevertheless more than four times the distance of A. Thus B descends all the way to B4 on the horizontal HM, and A is raised from A3 to A4. Now A4P, LN and B3M are dropped from A4, L and B3 perpendicular to the horizon. And so, because B3L is, say, five times the distance A3L, A3L will be one sixth of A3B3; and thus LN will be one sixth of B3M. On the other hand, A4B4 is to B4L as 6 is to 5. Now the ratio of LN to B3M has already been shown to be one to six. Accordingly, A4P is to B3M as one is to five, or in other words A4P is 16/5 feet. While at the beginning we had a four-pound body, A, raised to just one foot, A1H, we now have the same lifted to 3 1/5 feet, for such is the height of A4P. And thus, a body, by only the force of its own descent and the descent of others accomplished by its power, has raised itself nearly four times the height that it had before.

This is absurd (by the immediately preceding axiom,) for we will readily have perpetual motion. For instance, it would be possible that as a heavy body A rolls back from A4 to A1, descending through a height of more than 2 feet, it performs certain desired mechanical operation (like raising other weights, splitting wood, and similar tasks) and then returns, nevertheless, back to A1, where it had been at the beginning.

For B can also return again to its previous position of B1, if it comes to be at B4, not having directly descended to the horizon HM, but has stopped just a little higher, so as to be able to roll back from B4 to B1. Thus, having reinstated everything to its prior position, we have an effective machine of perpetual motion. And a similar absurdity will be shown by only changing the values, as long as the height B3M – to which B is able to ascend by the force of the allotted velocity – is more than four feet, i.e. (by the Lemma) as long as B receives double the velocity which we had in A. Q.E.D.