The elements of geometrie of the most auncient philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, citizen of London. Whereunto are annexed certaine scholies, annotations, and inuentions, of the best mathematiciens, both of time past, and in this our age. With a very fruitfull præface made by M. I. Dee, specifying the chiefe mathematicall scie[n]ces, what they are, and wherunto commodious: where, also, are disclosed certaine new secrets mathematicall and mechanicall, vntill these our daies, greatly missed

* 1.1Those magnitudes are sayd to haue proportion the one to the other, which being multiplied may exceede the one the other.

Before he shewed and defined, what proportion was, now by this definition he declareth betwene what magnitudes proportion falleth, saying: That those quanti∣ties are said to haue proportion the one to the other, which being multiplyed, may excede the one the other.* 1.2 As for that the

[illustration]

line A being multiplied by what soeuer multiplication or nūber, as taken twise, thrise, or foure, fiue, or more times, or once and halfe, or once and a third, & so of any other part, or partes, may excede and become greater then the line B or contrariwise, then these two lines are said to haue proportion the one to the other. And so ye may see that betwene any two quātities of one kinde, there is a propor∣tion. For the one remayning vnmultiplied, & the other being certaine times mul∣tiplied, shall be greater then it. As 3. to 24. hath a proportion, for leauing 24. vnmul∣tiplied, and multiplying 3. by 9, ye shall produce 27: which is greater then 24, and excedeth it. Here is to be noted, that Euclide in defining what quantities haue pro∣portion,* 1.3 was compelled to vse multiplication, or els should not his definition be generall to either kinde of proportion: namely, to rationall and irrationall: to such proportion I say which may be expressed by number, and to such as cannot be ex∣pressed by any determinate number, but remaineth surd and innominable. In rati∣onall quantities which haue one common measure, the excesse of the one aboue the other is knowen, and by it is knowen the proportion, which may be expressed by some determinate number. But in irrationall quantities which haue no cōmon measure, it is not so. For in them the excesse of the one to the other is euer vn∣knowen,

& therefore is surd, and innominable. As betwene the side of a square and the diameter therof•• there is vndoubtedly a proportion, for that the side certaine times multiplied may excede the diameter. Likewise betwene the diameter of a circle and the circumference therof there is certainlie, by this definition, a propor∣tion, for that the diameter certaine times multiplied may excede the circumference of the circle: although neither of these proportions can be named & expressed by number. For this cause therefore vsed Euclide this maner of defining by multipli∣cation.

Notes

• * 1.1

  The fifth defi∣nition.
• * 1.2

  An example of this ••efini∣tion in mag∣nitudes.
• * 1.3

  Why Euclide in defining of Proportion vsed multi∣plication.