Just a random but vaguely interesting point that occurred to me whilst writing this post.

We can imagine the amount of light at various times throughout the year as the function of a sine wave. From the vernal equinox to the autumnal equinox, the days are lighter than the nights, and so this portion of the year would represent that part of the sine wave which is positive, i.e. greater than zero. Between the autumnal equinox and the vernal equinox, the nights are longer than the days, so this portion of the year would represent the part of the sine wave which is negative, i.e. less than zero. For the avoidance of doubt, “less than zero” does not here mean “negative light,” but “less light than dark.”

The winter solstice – the shortest day – would represent the minimum point on that sine wave, whereas the summer solstice – the longest day – would represent the maximum point. Basic differential calculus tells us that minima and maxima are reached when the rate of change of a function passes through zero, i.e. changes from positive to negative in the case of a maximum, and from negative to positive in the case of a minimum.

Therefore the equinoxes represent those points where the function itself changes sign – the vernal equinox where it changes from negative to positive, and the autumnal equinox where it changes from positive to negative – and the solstices represent those points where the rate of change of the function or the first derivative of the function changes sign – the winter solstice where it changes from negative to positive, and the summer solstice where it changes from positive to negative.

The value of the function is positive above the x-axis, and negative below it. The rate of change of that function is negative when – following the line from left to right – it moves from a higher value to a lower value, i.e. between the summer solstice and the winter solstice. The rate of change is positive when it moves from a lower value to a higher value, i.e. between the winter solstice and the summer solstice.

Therefore each of the equinoxes and solstices represents a “zero point” or a “turning point” of one kind or another, but the turning points of the solstices are of a different order than the turning points of the equinoxes. Hence the comment in this post that “Note that the active signs of air and fire (air and fire are represented by upward pointing triangles) go with the equinoxes, and the passive signs of water and earth (water and earth are represented by downward pointing triangles) go with the solstices. That is, the active signs represent periods of change, from cold to hot in the case of fire, and from hot to cold in the case of air, whilst the passive signs represent the stable period of heat in the case of water, and the stable period of cold in the case of earth.” Technically this explanation should refer to “light and dark” rather than “cold and hot” but that is a mere technicality since most of the heat on the earth comes from the light of the Sun, even if the temperature maxima and minima lag behind those of the light. The “stable periods” in question refer to the fact that the solstices are in the middle of the positive and negative parts of the wave, respectively, and this explanation survives the fact that they too are “turning points” of a different order, relating to the rates of change of the qualities in question, rather than to the polarities of the qualities themselves.