Aristotle · a new plain-English translation from the original language
1| Why are those in hot places cowardly, while those in cold places are courageous? Is it because nature stands in the opposite condition to the places and the seasons, on account of the fact that things in a like condition would quickly wear away? Now those who are hot by nature are courageous, and those who are chilled are cowardly. It happens, then, that those who live in hot places become chilled (for since their body is porous, the heat escapes outward from them), while those in cold places have had their nature heated internally, because the flesh is condensed by the external cold, and as it condenses, the heat is compressed inward. Book 15. Problems Belonging to the Study of Mathematics. Why is it that, alone among the lines that bisect rectilinear figures, the line drawn from angle to angle is called a diameter? Is it because a diameter divides in half, as its name suggests, without destroying what is measured? Now the line that divides along the composite parts (I mean the angles) will be a diameter; for it does not destroy but divides, just as those who divide up a set of military equipment do. But the line that cuts through composite parts along the sides destroys; for the rectilinear figure is composed with respect to its angles. Why is it called a diameter? Is it because it alone divides in half, as if one were to say it is a 'half-measure'? And because it alone among the bisecting lines is called by this name? Or is it because it alone divides at the joints where the figure is bent, while the other lines divide along the sides?
2| Why do all human beings, both non-Greeks and Greeks, count up to ten, and not to some other number — say two, three, four, five — and then start doubling back, one-five, two-five, as in eleven, twelve? Nor again do they stop at some point further out than ten and then double back from there? For each of the numbers is indeed the one before it plus one or two, and so on for every further one, and yet they still count by setting the boundary at ten. For they plainly do not do this by chance, and they do it always; and what holds always and for everyone is not by chance but natural. Is it because ten is a perfect number? For it has all the species of number: even, odd, square, cube, length, plane, prime, composite. Or is it because the decad is a starting point? For one and two and three and four make ten. Or is it because the moving bodies are nine? Or is it because within ten ratios four cubic numbers are produced, out of which numbers the Pythagoreans say the universe is composed? Or is it because all human beings originally had ten fingers? Having these, as it were, as counters belonging to their own number, they count other things too by this quantity. Only one tribe among the Thracians counts to four, because, like children, they are unable to remember for long, and they have no use for anything in great quantity.
3| that the earth is the center; for the shapes that appear to us are always alike. This appears to be so if one does not observe from the midpoint, but the earth would sometimes seem triangular to us, sometimes trapezoidal, sometimes of other shapes, as our midpoint, if it were possible for us to observe from those other positions. For since the earth is spherical, its center and the center of the universe will be the same. But we live on top of the earth, so that we do not observe from that center itself, but such appearances present themselves to us at a distance of half the diameter away. What, then, prevents the illusion of the shapes from persisting even as the distance becomes greater?
4| Why, when the sun moves at a constant rate in equal times, is the increase and decrease of shadows not the same? Is it because the angles formed by the rays toward the object seen, under equal arcs, come out equal? And if these, when extended, also produce rays within the triangle contained by the first ray, the object seen, and the shadow — and if the angles are equal, the line to the object seen that is farther off must necessarily be greater than the one nearer; for this we know. Let the arc, then, be divided into equal parts, however many, and let Θ be the point seen. When, then, the sun at Α, having taken in the arc up to Θ, produces some shadow at ΘΑ, the ray must necessarily fall on Α. But when it comes to Β, the ray from Β will fall within ΘΑ, and likewise again when it passes on to Γ; otherwise a straight line would meet a straight line at two points. Since, then, ΑΒ is equal to ΒΓ, the angles subtended by it at Δ will also be equal, for they are at the center. And if equal to the one at Δ, then also in the triangle —
5| — for these are vertical to those. So that, since the angle is divided into equal parts, ΔΕ will be greater than ΕΖ by ΔΘ. And likewise for the other lines produced by the rays from the arc. At the same time it is also clear that the shadow must necessarily be shortest at noon, and that the increments are then least. For the sun stands most directly over us at noon, and stifling heat occurs both through the cause stated and because there is no wind; for when it separates out the air near the earth, wind arises. If, then, this happens at the same time in both hemispheres, it stands to reason that midnight and noon should be windless.
6| Why does the sun, when it comes through four-sided openings, not produce rectilinear shapes but circles, as, for instance, through wickerwork? Is it because the outward course of our sight is a cone, and the base of a cone is a circle, so that whatever the sun's rays strike appears circular in shape? For it is necessary that the figure made by the sun should also be bounded by straight lines, if indeed the rays are straight. For whenever straight lines fall against a straight line, they produce a rectilinear figure. And this is what happens with the rays: they fall against a straight line, namely the edge of the opening through which they shine, and this edge is straight, so that the outward course of the rays will be against a straight line. But because the rays that split off from our sight toward the extremities of the straight lines are weak, what is at the corners is not seen; rather, as much of the straight line as falls within the cone produces the figure, and the rest does not, but the rays of sight escape our notice as they fall on it. For we fail to see many of the things our sight reaches, such as things in darkness. Similar to this is also the fact that a square appears many-sided when close, but if one stands farther off, it appears a circle. For since the outward course of sight is a cone, when the figure withdraws into the distance, the rays of sight that split off toward the corners, being weak and few, fail to see, as the distance becomes greater, while those that fall on the middle, being massed together and strong, persist.
7| So then, when the figure is near, sight is able to see even what is at the corners, but when it becomes farther off, it is unable to. That is also why a curved line, when drawn out to a distance, appears straight. And the moon too seems to be bounded by straight lines on the eighth day, if the rays of sight fall not along its breadth but along the line that bounds it. For when the arc is near, sight can distinguish how much nearer one part of the arc is than another part; but when it becomes distant, sight no longer perceives this, but the arc seems to it to be equally distant throughout. That is why it also appears straight.
8| Why, though the moon is spherical, do we see a straight line when it is half-moon? Is it because our sight and the circular arc that the sun produces by striking the moon come to lie in the same plane? And when this happens, the sun appeared as a straight line. For since whatever casts its rays upon a sphere must see a circle, and the moon is spherical, and the sun looks upon it, the boundary produced by the sun would be a circle. So then, whenever this circle comes to be directly opposite us, it appears whole and seems to be a full moon; but when it shifts owing to the sun's passage, its arc comes to lie along our line of sight, so that it appears straight. And the other part is curved, because a hemisphere lies directly opposite our sight. Such a thing appeared as a half-circle. For the moon is always directly opposite our sight. But when the sun falls upon it we do not see that part, and it is filled in after the eighth day from the middle outward, because the sun, passing along further, makes the circle appear to us more slanted. And the circle, positioned thus relative to our sight, comes to resemble the section of a cone. It appears crescent-shaped, whenever the sun shifts position.
9| For whenever the sun's circle comes to be at the extreme points at which the moon appears half, the arc of the circle appears as an arc. For it is no longer in a straight line with our sight, but shifts aside. And when this happens, and the circle passes through the same points as before, it must appear crescent-shaped. For a certain part of the circle is straight relative to our sight, since the earlier part is directly opposite, so that a portion of the bright part is cut off; and then likewise the extremities remain in the same place, so that it must appear crescent-shaped. And it appears more or less so because of the sun's motion. For as the sun shifts, the circle it looks upon also turns, while remaining at the same points; for it admits of being tilted through infinitely many inclinations, since it is possible to draw infinitely many great circles through the same points. Why do the sun and moon, though spherical, appear flat? Is it because, of all things whose distance is not discernible, when they are farther off or nearer they appear equally distant? So that also in the case of a single thing having parts, if it does not differ in color, the parts must appear equally distant, and what is equally distant appears level and flat—
10| Why does the sun make shadows long when rising and setting, shorter as it climbs, and shortest at midday? Is it because, on rising, it will at first make the shadow parallel to the ground, and, since the ratio is unequal, project it as though infinite, and then long; and it is always shorter because the straight line from the higher point always falls inside the previous one? Let A B be the gnomon, and let the sun be at C, then at D; the ray from C, along which C Z lies, will fall outside C E. And the shadow B E is cast when the sun is higher, and B Z when it is lowest; and the shadow is shortest when the sun is highest, namely directly overhead. Why are the shadows cast by the moon greater than those cast by the sun, when they are on the same perpendicular? Is it because the sun is higher than the moon? It is then necessary that the ray from the higher point fall inside the other. Let the gnomon be A D, the moon B, the sun C. The ray from the moon is B Z, so that the shadow will be D Z; and the ray from the sun is C E, so that the shadow will necessarily be shorter; for it will be D E.
11| Why, during eclipses of the sun, if one observes through a sieve or through leaves—such as those of a plane tree or another broad-leaved tree—or by interlacing the fingers of one hand over the other, do crescent-shaped patches of light appear on the ground? Is it because, just as when light shines through a well-cornered opening it becomes round, a cone is formed? The cause is that two cones are formed, the one from the sun to the opening, and the one from there to the ground, and they share a common apex. So then, when things stand this way, and the light is cut off in a circle from above, there will be a crescent on the ground opposite the light. For the rays come from the arc of the crescent, and the openings in the fingers or the sieve act like small apertures; that is why it becomes more evident than through large openings. But from the moon such patches do not occur, whether it is eclipsing or waxing or waning, because the beams from its extremities are not sharply defined, but it shines from its middle; and the crescent has only a small middle part.
12| Why does a parhelion occur neither when the sun is at midheaven, nor above the sun, nor below the sun, but only to the sides? Is it because a parhelion occurs when our sight is bent back toward the sun, and this depends on the state of the air on which sight is reflected, which can be neither near the sun nor far from it? For if it is near, the sun will dissolve it, while if the reflecting surface is far, sight will not be reflected back; for sight, stretched out to a small mirror at a distance, becomes weak. That is also why halos do not occur there. So then, if the reflecting surface occurs directly opposite the sun and near it, the sun will dissolve it; and if it is far, a lesser amount of sight will strike it. But if it is to the side, the mirror can stand at such a distance that it neither is dissolved by the sun nor has the returning sight rise up massed together, because it travels beneath the earth. But below the sun it does not occur, because if it is near the earth it would be dissolved by the sun, while if it is high at midheaven, our sight would be torn apart. And in general it does not occur even to the side at midheaven, because if sight travels too far beneath the earth, little of it will reach the mirror, so that on being reflected back the whole of it will be weak.