The Mathematicall Praeface to Elements of Geometrie of Euclid of Megara | Page 9

and for the second, I set 1{x}-3. And, now againe, I say, as 1. is
to 2. so is 4-1{x} to 1{x}-3. Wherfore, of these foure proportionall
numbers, the first and the fourth Multiplied, one by the other, do make
as much, as the second and the third Multiplied the one by the other.
Let these Multiplications be made accordingly. And of the first and the
fourth, we haue 1{x}-3. and of the second & the third, 8-2{x}. Wherfore,
our Aequation is betwene 1{x}-3: and 8-2{x}. Which may be reduced,
according to the Arte of Algiebar: as, here, adding 3. to eche part,
geueth the Aequation, thus, 1{x}=11-2{x}. And yet againe, contracting,
or Reducing it: Adde to eche part, 2{x}: Then haue you 3{x} aequall to
11: thus represented 3{x}=11. Wherefore, diuiding 11. by 3: the
Quotient is 3-2/3: the Valew of our 1{x}, Coss, or Thing, first supposed.

And that is the heigth, or Intension of the Forme resulting: which is,
Heate, in two thirdes of the fourth degree: And here I set the shew of
the worke in conclusion, thus. The proufe hereof is easie: by
subtracting 3. from 3-2/3, resteth 2/3. Subtracte the same heigth of the
Forme resulting, (which is 3-2/3) from 4: then resteth 1/3: You see, that
2/3 is double to 1/3: as 2.{P}. is double to 1.{P}. So should it be: by the
rule here geuen. Note. As you added to eche part of the Aequation, 3:
so if ye first added to eche part 2{x}, it would stand, 3{x}-3=8. And
now adding to eche part 3: you haue (as afore) 3{x}=11.
_______________________ | | | | {P}. 2. | Hote. 4. | 1/3 The forme | | |
3-2/3 resulting. | {P}. 1. | Hote. 3. | 2/3 |_________|___________|
And though I, here, speake onely of two thyngs Miscible: and most
commonly mo then three, foure, fiue or six, (&c.) are to be Mixed: (and
in one Compound to be reduced: & the Forme resultyng of the same, to
serue the turne) yet these Rules are sufficient: duely repeated and
iterated.
[Note.]
In procedyng first, with any two: and then, with the Forme Resulting,
and an other: & so forth: For, the last worke, concludeth the Forme
resultyng of them all: I nede nothing to speake, of the Mixture (here
supposed) what it is. Common Philosophie hath defined it, saying,
Mixtio est miscibilium, alteratorum, per minima coniunctorum, Vnio.
Euery word in the definition, is of great importance. I nede not also
spend any time, to shew, how, the other manner of distributing of
degrees, doth agree to these Rules. Neither nede I of the farder vse
belonging to the Crosse of Graduation (before described) in this place
declare, vnto such as are capable of that, which I haue all ready sayd.
Neither yet with examples specifie the Manifold varieties, by the
foresayd two generall Rules, to be ordered. The witty and Studious,
here, haue sufficient: And they which are not hable to atteine to this,
without liuely teaching, and more in particular: would haue larger
discoursing, then is mete in this place to be dealt withall: And other
(perchaunce) with a proude snuffe will disdaine this litle: and would be
vnthankefull for much more. I, therfore conclude: and wish such as

haue modest and earnest Philosophicall mindes, to laude God highly
for this: and to Meruayle, that the profoundest and subtilest point,
concerning Mixture of Formes and Qualities Naturall, is so Matcht and
maryed with the most simple, easie, and short way of the noble Rule of
Algiebar. Who can remaine, therfore vnpersuaded, to loue, alow, and
honor the excellent Science of Arithmetike? For, here, you may
perceiue that the litle finger of Arithmetike, is of more might and
contriuing, then a hunderd thousand mens wittes, of the middle sorte,
are hable to perfourme, or truely to conclude, with out helpe thereof.
Now will we farder, by the wise and valiant Capitaine, be certified,
what helpe he hath, by the Rules of Arithmetike: in one of the Artes to
him appertaining: And of the Grekes named
[#Taktike:#.]
#Taktike:#. "That is, the Skill of Ordring Souldiers in Battell ray after
the best maner to all purposes." This Art so much dependeth vppon
Numbers vse, and the Mathematicals, that Aelianus (the best writer
therof,) in his worke, to the Emperour Hadrianus, by his perfection, in
the Mathematicals, (beyng greater, then other before him had,) thinketh
his booke to passe all other the excellent workes, written of that Art,
vnto his dayes. For, of it, had written Aeneas: Cyneas of Thessaly:
Pyrrhus Epirota: and Alexander his sonne: Clearchus: Pausanias:
Euangelus: Polybius, familier frende to Scipio: Eupolemus: Iphicrates,
Possidonius: and very many other worthy Capitaines, Philosophers and
Princes of Immortall fame and memory: Whose fayrest floure of their