The Mathematicall Praeface to Elements of Geometrie of Euclid of Megara | Page 6

most part, of their execution, differre from the fiue operations of like
generall property and name, in our Whole numbers practisable, So
often, (for a more distinct doctrine) we, vulgarly account and name it,
an other kynde of Arithmetike." And by this reason:
[1.]
the Consideration, doctrine, and working, in whole numbers onely:
where, of an Vnit, is no lesse part to be allowed: is named (as it were)
an Arithmetike by it selfe. And so of the Arithmetike of Fractions.
[2.]
In lyke sorte, the necessary, wonderfull and Secret doctrine of
Proportion, and proportionalytie hath purchased vnto it selfe a peculier
maner of handlyng and workyng: and so may seme an other forme of
Arithmetike.
[3.]
Moreouer, the Astronomers, for spede and more commodious
calculation, haue deuised a peculier maner of orderyng numbers, about
theyr circular motions, by Sexagenes, and Sexagesmes. By Signes,
Degrees and Minutes &c. which commonly is called the Arithmetike of
Astronomical or Phisicall Fractions. That, haue I briefly noted, by the
name of Arithmetike Circular. Bycause it is also vsed in circles, not
Astronomicall. &c.
[4.]
Practise hath led Numbers farder, and hath framed them, to take vpon
them, the shew of Magnitudes propertie: Which is Incommensurabilitie

and Irrationalitie. (For in pure Arithmetike, an Vnit, is the common
Measure of all Numbers.) And, here, Numbers are become, as Lynes,
Playnes and Solides: some tymes Rationall, some tymes Irrationall.
And haue propre and peculier characters, (as [2rt]. [3rt]. and so of other.
Which is to signifie Rote Square, Rote Cubik: and so forth:) & propre
and peculier fashions in the fiue principall partes: Wherfore the
practiser, estemeth this, a diuerse Arithmetike from the other. Practise
bryngeth in, here, diuerse compoundyng of Numbers: as some tyme,
two, three, foure (or more) Radicall numbers, diuersly knit, by signes,
of More & Lesse: as thus [2rt]12 + [3rt]15. Or thus [4rt]19 + [3rt]12 -
[2rt]2. &c. And some tyme with whole numbers, or fractions of whole
Number, among them: as 20 + [2rt]24. [3rt]16 + 33 - [2rt]10. [4rt]44 +
12-1/4 + [3rt]9. And so, infinitely, may hap the varietie. After this:
Both the one and the other hath fractions incident: and so is this
Arithmetike greately enlarged, by diuerse exhibityng and vse of
Compositions and mixtynges. Consider how, I (beyng desirous to
deliuer the student from error and Cauillation) do giue to this Practise,
the name of the Arithmetike of Radicall numbers: Not, of Irrationall or
Surd Numbers: which other while, are Rationall: though they haue the
Signe of a Rote before them, which, Arithmetike of whole Numbers
most vsuall, would say they had no such Roote: and so account them
Surd Numbers: which, generally spoken, is vntrue: as Euclides tenth
booke may teach you. Therfore to call them, generally, Radicall
Numbers, (by reason of the signe [rt]. prefixed,) is a sure way: and a
sufficient generall distinction from all other ordryng and vsing of
Numbers: And yet (beside all this) Consider: the infinite desire of
knowledge, and incredible power of mans Search and Capacitye: how,
they, ioyntly haue waded farder (by mixtyng of speculation and practise)
and haue found out, and atteyned to the very chief perfection (almost)
of Numbers Practicall vse. Which thing, is well to be perceiued in that
great Arithmeticall Arte of Aequation: commonly called the Rule of
Coss. or Algebra. The Latines termed it, Regulam Rei & Census, that is,
the +Rule of the thyng and his value+. With an apt name:
comprehendyng the first and last pointes of the worke. And the vulgar
names, both in Italian, Frenche and Spanish, depend (in namyng it,)
vpon the signification of the Latin word, Res: +A thing+: vnleast they
vse the name of Algebra. And therin (commonly) is a dubble error. The

one, of them, which thinke it to be of Geber his inuentyng: the other of
such as call it Algebra. For, first, though Geber for his great skill in
Numbers, Geometry, Astronomy, and other maruailous Artes, mought
haue semed hable to haue first deuised the sayd Rule: and also the
name carryeth with it a very nere likenes of Geber his name: yet true it
is, that a Greke Philosopher and Mathematicien, named Diophantus,
before Geber his tyme, wrote 13. bookes therof (of which, six are yet
extant: and I had them to *vse,
[* Anno. 1550.]
of the famous Mathematicien, and my great frende, Petrus Montaureus:)
And secondly, the very name, is Algiebar, and not Algebra: as by the
Arabien Auicen, may be proued: who hath these precise wordes in
Latine, by Andreas Alpagus (most perfect in the Arabik tung) so
translated. Scientia faciendi Algiebar & Almachabel. i. Scientia
inueniendi numerum ignotum, per additionem Numeri, & diuisionem &
aequationem. Which is to say: +The Science of workyng Algiebar and
Almachabel+, that is, the +Science