Abstract
As we already saw, in Physics VII.5 Aristotle explains that if effective mover A moves mobile B over distance C in time D, then half of A (call it E) moves half of B (call it Z) over distance C in time D. The same E (½A) will move two Z loads (each being ½B), over half the distance C in time D.
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Notes
- 1.
In a footnote, Cornford adds “[Literally, ‘otherwise one man could move the ship, since both the force of the haulers and the distance of which all of them together make it move are divisible into the (same) number (of parts as there are men).’]”.
- 2.
This does not take into account the time taken for the original mover to go back to the starting point after transporting each of the partial loads. However, since the relation here pertains to the effective moving force in actual operation on the loads and not to any particular original mover, this omission is justified.
- 3.
For example, in On the Heavens II.vii, Aristotle explains that the celestial bodies (stars, planets, sun and moon) are made of ether, and hence do not burn and do not shine. They generate heat and light by chafing diffusions of air in the ethereal region through which they move (for a well-argued claim that such diffusions do not contradict any element in Aristotle’s cosmology, see J. Thorp 1982, pp. 104–123). Starlight and sunlight, then, are byproducts of friction. In On the Heavens II.ix, however, Aristotle rejects the Pythagorean suggestion that the stars produce harmonious music as they move. This cannot be so, Aristotle says, because the stars are carried in their ethereal spheres, and hence do not move through a stationary medium. Had they been moving through air, he says, the noise would indeed be deafening, given the enormous speeds involved. Guthrie, for one, cannot see a way out of the overt contradiction between Book II.vii on the friction-generated light of the stars, and book II.ix that seems to reject any such friction (On the Heavens, Guthrie, 1939, note a, p. 196). However, the difficulty dissolves under the assumption that the threshold for light and heat generation is lower than the threshold for sound generation: the motion may generate sufficient friction to ignite the terrestrial diffusions encountered by the planet, but not sufficient to produce sound, possibly because the diffusions are so rare. That sound-generation will not occur below a certain threshold is explicitly stated by Aristotle in Physics VII.v: “And in this lies the fallacy of Zeno’s contention that every grain of millet must make a sound as it falls (if the whole measure is to do so). For it may well be that in no period of time could the one grain move that air that the whole bushel moves.” (Physics, VII.5. 250a20-23, Wicksteed and Cornford 1929).
- 4.
Time (also space) for Aristotle, does not have an existence independent of matter. Indeed, it is material motion and its variability that renders necessary the continuity of time: “And since every motion is in time, and in every time it is possible for a thing to be moved, and every moved thing admits of being moved both faster and more slowly, then in every time the faster will also be able to move more slowly. These things being so, it is also necessary that time be continuous. I call continuous that which is always divisible into divisible parts, for once this is set down about the continuous, time must be continuous.” (Physics, VI.ii.232b21-26, Sachs).
- 5.
Recall how Aristotle discussed the object that is white prior to a given instant and not-white afterwards: up to, but not including that instant, the body is not done turning non-white. From that instant on, inclusive, it already is non-white. In the case of the rotating semi-infinite line AE, up to, but not including one quarter of a day after perpendicularity, it is not yet crossing BB; it only approaches crossing. From one quarter of a day, inclusive, it already crosses BB.
- 6.
A measure of ambiguity still remains, since the process of force decay irrespective of any locomotion it might engender, is itself “motion” in Aristotle’s general sense of any timed change. If the formal argument given in On the Heavens holds generally, then the force decay itself should be cut off and not continue indefinitely, and we now have a competition of thresholds. However, motion will stop when the first one is reached, and nowhere does Aristotle’s text help us figure out which will be the actual cause of the secession of motion. Of course, while treating the formal aspect of force decay as a type of change, remember that it must also have its own efficient aspect, and now we end up caught in a seemingly endless regression. For treating locomotion and its efficient aspect, we need only recall the following. (1) In the basic discussion of the threshold of motion in Physics Vii.v, a single person applies force to a ship, but fails to generate motion. (2) In Aristotle’s discussion of prolonged projectile motion in Physics VIII.x, once the force in the medium falls below the ability to generate more moving force and can generate motion only, the whole progression stops, and no further motion ensues. In studies of locomotion, therefore, we may ignore higher orders of thresholds.
- 7.
Strictly speaking, the properties of all matter, including the elements, are carried by the fundamental substrate of material properties—the hulé. The hulé that carries the properties of a clod of earth actually, also carries those of fire potentially. Since fire and earth are opposites, the hulé cannot carry both simultaneously in actual state, and the transition of potential fire into actual fire in the same bulk must be simultaneously accompanied by the transition of actual earth into potential earth.
- 8.
See Appendix A for objections to the suggestion that according to Aristotle, earth becomes less heavy as it recedes from the center of the cosmos, and heavier as it approaches it.
- 9.
Galileo refers to this as a “well known” argument against Aristotle. See I.E. Drabkin and S. Drake, (trans.), Galileo Galilei, On Motion and On Mechanics, (Madison, Wisconsin, University of Wisconsin Press, 1960), p. 97, or p. 326 in Favaro’s critical edition of Galileo’s works.
- 10.
Aristotle considers his argument against the possibility of un-disrupted reciprocating motion to apply generally, and not as confined to locomotion alone. He makes this explicitly clear in Physics, VIII.viii.264b2-8.
- 11.
For another example, consider the case of a bucket being gradually filled with water. Its weight grows gradually too, without interruption. Assume it starts out at weight A, ends up at weight C, and passes through weight B. In keeping with Aristotle’s analysis in Physics, VIII.viii. 263b10-23, the bucket is never in a state of weight B, it merely passes through this state at a cut in time. But now consider the bucket as tied to a rope wrapped round a pulley and connected on the other side to a load of constant weight B. Clearly, before it matches the weight of B, the bucket would move up. This motion should reverse (with the proper lag created by the motion prolonging effect of the air) once it exceeds the weight B. The change in the direction of moving force must be perfectly continuous, because the weight of the bucket grows continuously through B. Despite this, a period of rest will intervene between the resulting switch in the direction of motion, because the continuity of the change in force ensures that some of it must take place below the threshold of motion. Once again, then, the threshold of motion matches the dynamics to the purely formal requirement that rest must separate between the phases of reciprocating motion.
- 12.
This does not take into account the continuing decay of effective force in the air layers. The decay would be very small as long as the time interval (t 0, t 1) is sufficiently short, but in principle the force in the combined shell just after t 1 is less than or equal to the difference between the forces in the two air shells just prior to t 0.
- 13.
Wicksteed and Cornford (1929) omitted the bracketed lines in their translation, on suspicion of non-authenticity. Waterfield (1996) did the same.
- 14.
As Aristotle says of fire: “… yet the larger quantity {of fire} moves upwards more quickly than the small.” (On the Heavens, IV.ii.309b13.
- 15.
Strictly speaking, this is a “bad” ratio, to the extent that ratio is a relationship between magnitudes of the same kind. In the strict sense, two bodies should be placed on the scale of weight per unit matter by comparing the ratio of their respective heaviness to the ratio of their respective lightness. Namely, if per unit matter β 1:β 2 > κ 1:κ 2, then body 1 is heavier than body 2; if β 1:β 2::κ 1:κ 2, then they have the same weight; and if β 1:β 2 < κ 1:κ 2 then body 1 is lighter than body 2. To make the discussion less cumbersome, this more rigorous form is avoided. The same holds for the other mixed ratios used here.
- 16.
“In air, for instance, a talent of wood is heavier than a mina of lead, but in water it is lighter.” (On the Heavens, IV.iv.311b4, Guthrie).
- 17.
By “light” and “lighter” in a relative sense we mean that one of two bodies of the same size, each possessing {heaviness}, whose natural velocity in a downward direction is exceeded by that of the other.” (On the Heavens, IV.i.308a30-31, Guthrie).
- 18.
Beyond this short remark there is no further analysis of non-uniform motion in the chapter, so the observations on acceleration, deceleration, and climax serve merely as brief general reminders, not as analytical investigations in their own right. The entire chapter focuses on showing that the circular motion of the ethereal spheres is necessarily uniform. The introductory statement that sets out the subject for discussion notes parenthetically that this uniform motion is apparent only in the first sphere that provides the daily rotation common to all heavenly motions, and that below it the motion is already a combination of several components. This seems to reflect the Eudoxan homocentric structures, which should be kept in mind as background to the entire chapter.
Thus the idea proposed by Leo Elders, that the word άκμὴ does not denote “a state of highest intensity” but “the highest point of the trajectory” is unacceptable; Leo Elders, Aristotle’s Cosmology (Assen 1966), p. 210).
- 19.
The sphere around A need not be the size of the earth. It can be much larger, or much smaller. What must remain in all cases is that the circular arc subtended by points B and C must be a significant part of the sphere’s circumference.
- 20.
On the Heavens, tr. W. K. C. Guthrie, pp. 170–171, n. a.
- 21.
In the case of downward hurled heavy projectiles, for example, the dual action of the medium in Aristotle’s theory requires important distinctions. When hurled at a speed below the terminal speed of its natural fall, the peak speed will occur either at the end, or at the point of travel where it reaches terminal speed. When hurled at terminal speed, the projectile will travel downward at a constant speed throughout the duration of its journey. When hurled at a speed greater than its terminal speed, the projectile’s motion will invariably start at peak speed, decaying gradually into its terminal speed. Medium resistance determines the terminal speed of a given projectile. All the gradual speed transitions in the various cases of projectile flight are direct consequences of the motion prolonging effect of the medium in accordance with the discussion in Physics VIII.x.
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Yavetz, I. (2015). Some Refinements of the Basic Theory. In: Bodies and Media. SpringerBriefs in History of Science and Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-21263-0_3
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