The Mathematicall Praeface to Elements of Geometrie of Euclid of Megara | Page 8

the 4. principall elementall
Qualities. Hote and Colde, one against the other. And likewise Moyst
and Dry, one against the other. And in the Circle write Temperate.
Which Temperature hath a good Latitude: as appeareth by the
Complexion of man. And therefore we haue allowed vnto it, the
foresayd Circle: and not a point Mathematicall or Physicall.
[* Take some part of Lullus counsayle in his booke de Q. Essentia.]
Now, when you haue two thinges Miscible, whose degrees are * truely
knowen: Of necessitie, either they are of one Quantitie and waight, or
of diuerse. If they be of one Quantitie and waight: whether their formes,
be Contrary Qualities, or of one kinde (but of diuerse intentions and
degrees) or a Temperate, and a Contrary, The forme resulting of their
Mixture, is in the Middle betwene the degrees of the formes mixt. As for

example, let A, be Moist in the first degree: and B, Dry in the third
degree. Adde 1. and 3. that maketh 4: the halfe or middle of 4. is 2.
This 2. is the middle, equally distant from A and B
[* Note.]
(for the * Temperament is counted none. And for it, you must put a
Ciphre, if at any time, it be in mixture).
HOTE +C | | + | | + | | +E | MOIST A TEMPERATE B DRYE
+------+------+------+------+------+------+------+------+ |D | + | | + | | + | | +
COLD
Counting then from B, 2. degrees, toward A: you finde it to be Dry in
the first degree: So is the Forme resulting of the Mixture of A, and B, in
our example. I will geue you an other example. Suppose, you haue two
thinges, as C, and D: and of C, the Heate to be in the 4. degree: and of
D, the Colde, to be remisse, euen vnto the Temperament. Now, for C,
you take 4: and for D, you take a Ciphre: which, added vnto 4, yeldeth
onely 4. The middle, or halfe, whereof, is 2. Wherefore the Forme
resulting of C, and D, is Hote in the second degree: for, 2. degrees,
accounted from C, toward D, ende iuste in the 2. degree of heate. Of
the third maner, I will geue also an example: which let be this:
[Note.]
I haue a liquid Medicine whose Qualitie of heate is in the 4. degree
exalted: as was C, in the example foregoing: and an other liquid
Medicine I haue: whose Qualitie, is heate, in the first degree. Of eche
of these, I mixt a like quantitie: Subtract here, the lesse from the more:
and the residue diuide into two equall partes: whereof, the one part,
either added to the lesse, or subtracted from the higher degree, doth
produce the degree of the Forme resulting, by this mixture of C, and E.
As, if from 4. ye abate 1. there resteth 3. the halfe of 3. is 1-1/2: Adde
to 1. this 1-1/2: you haue 2-1/2. Or subtract from 4. this 1-1/2: you haue
likewise 2-1/2 remayning. Which declareth, the Forme resulting, to be
Heate, in the middle of the third degree.

[The Second Rule.]
"But if the Quantities of two thinges Commixt, be diuerse, and the
Intensions (of their Formes Miscible) be in diuerse degrees, and
heigthes. (Whether those Formes be of one kinde, or of Contrary kindes,
or of a Temperate and a Contrary, What proportion is of the lesse
quantitie to the greater, the same shall be of the difference, which is
betwene the degree of the Forme resulting, and the degree of the
greater quantitie of the thing miscible, to the difference, which is
betwene the same degree of the Forme resulting, and the degree of the
lesse quantitie. As for example. Let two pound of Liquor be geuen,
hote in the 4. degree: & one pound of Liquor be geuen, hote in the third
degree." I would gladly know the Forme resulting, in the Mixture of
these two Liquors. Set downe your numbers in order, thus.
__________________________ | | | | {P}. 2. | Hote. 4. | | | | | {P}. 1. |
Hote. 3. | |__________|____________|
Now by the rule of Algiebar, haue I deuised a very easie, briefe, and
generall maner of working in this case. Let vs first, suppose that
Middle Forme resulting, to be 1{x}: as that Rule teacheth. And because
(by our Rule, here geuen) as the waight of 1. is to 2: So is the difference
betwene 4. (the degree of the greater quantitie) and 1{x}: to the
difference betwene 1{x} and 3: (the degree of the thing, in lesse
quantitie. And with all, 1{x}, being alwayes in a certaine middell,
betwene the two heigthes or degrees). For the first difference, I set
4-1{x}: