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

* 1.1Magnitudes which are in one and the selfe same proportion, are called Proportionall.

As if the lyne A, haue the same proportion to the line B, that the lyne C hath to the lyne D, then are the

[illustration]

said foure magnitudes A, B, C, D, called proportionall. Also in numbers for that 9. to 3. hath that same proportiō that 12 hath to 4:* 1.2 therefore these foure nūbers 9.3.12.4.* 1.3 are said to be proportionall. Here is to be noted that this likenes or idemptitie of proportiō which is called, as before was said proportionalitie, is of two sortes: the one is continuall, the other is discontinuall.* 1.4 Continuall proportionalitie is, when the magnitudes set in lyke proportion, are so ioyned together, that the second which is consequent to the first, is antecedent to the third, and the fourth which is consequent to the third, is antecedent to the fift, and so continually forth. So eue∣ry quantitie or terme in this proportionalitie, is both antecedent and consequent (consequent in respect of tha which went before,* 1.5 & antecedent in respect of that which followeth) except the first, which is onely antecedent to that which follow∣eth, and the last which is onely consequent to that which went before. Take an ex∣ample in these numbers, 16.8.4.2.1.* 1.6 In what proportion 16. is to 8, in the same is 8. to 4, in the same also is 4. to 2, and likewise 2. to 1. For they all are in duple pro∣portion: 16. the first is antecedent to 8, and 8. is consequent vnto it: and the selfe same 8. is antecedent to 4: which 4 beyng consequent to 8. is antecedent to 2, which 2 likewise is consequent to 4. and antecedent to 1: which because he is the last, is onely consequent, and antecedent to none, as 16. because it was the first, was antecedent onely, and consequent to none. Also in this proportionalitie all the magnitudes must of necessitie be of one kynde,* 1.7 by reason of the continuation of the proportions in this proportionalitie, because there is no proportion betwene quantities of diuers kyndes. Discontinuall proportionalitie is,* 1.8 when the magni∣tudes which are set in lyke proportion, are not continually set, as before they were, hauyng one terme referred both to that which went before, and to that which fo∣loweth, but haue their termes distinct and seuered asonder: as the first is antece∣dent to the second, so is the third antecedent to the fourth. Example in numbers, as 8 is to 4.* 1.9 so is 6. to 3. for either proportion is duple. Where ye see, how ech pro∣portion hath hys owne antecedent and consequent distinct from the antecedent and consequent of the other, and no one number is antecedent and consequent in diuers respectes. And by reason of the discontinuaunce of the proportions in this proportionalitie,* 1.10 the quantities compared, may be of diuers kyndes, because the consequent in the first proportion is not the antecedent in the second proportion. So that ye may compare superficies to superficies, or body to body in the selfe same proportion that ye do lyne to lyne.

Notes

• * 1.1

  The seuenth definition
• * 1.2

  9 12
• * 1.3

  3 4
• * 1.4

  Proportiona∣lity of two sortes, conti∣••uall and dis∣continuall.
• * 1.5

  An example of continuall proportionali∣tie in num∣bers.
• * 1.6

  16.8.4.2.1.
• * 1.7

  In coutinnall proportionali∣tie the quan∣tities cannot be of one kinde.
• * 1.8

  Discontinuall prop••rtiona∣litie.
• * 1.9

  Example of discontinual proportionali∣ty in numbers.
• * 1.10

  In discōtinual proportionali∣tie the propor∣tions may be of diuers kindes.