Consider these quotes from the recipe
"Know then, that the division that was made upon the Water, by the ancient philosophers, separates it into four substances; one into two, and three into one; the third part of which is colour, as it were – a coagulated moisture; but the second and third waters are the Weights of the Wise."
The philosopher's split the "water" into four portions, literally. As in the philosopher's bodies split the "water" when they digest it. The urine and semen/milk is thought to come back out as urine (two into one), and semen/milk, blood, and feces is thought to come back out as feces (three into one). The third part is a "coagulated moisture", which is to say soft stool. The second and third waters are the Weights of the Wise, which is to say the contents are what the philosophers would normally excrete on any other day.
"And know that the chief principle of the art, is the Crow which is the blackness of the night and clearness of the day, and flies without wings. From the bitterness existing in the throat, the tincture is taken; the red goes forth from his body, and from his back is taken a thin water."
Feces are "black", urine is "clear". They "fly without wings" during urination / defecation as they fall to the ground. "From his back is taken a thin water" - self explanatory.
"Know that this matter I call the Stone; but it is also named the feminine of magnesia, or the hen, or the white spittle, or the volatile milk, the incombustible oil, in order that it may be hidden from the inept and ignorant, who are deficient in goodness and self-control.
The disposition sought after by the philosophers, O son, is but one in our egg; but this in the hen's egg is much less to be found. But lest so much of the Divine Wisdom as is in a hen's egg should not be distinguished, our composition is, as that is, from the fPost too long. Click here to view the full text.