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

Proportion deuided, or diuision of proportiō is,* 1.1 when the ex∣cesse wherein the antecedent excedeth the consequent, is com∣pared to the consequent.

Thys definition is the conuerse of the definition going next before:* 1.2 in it was vsed composition, and in thys is vsed diuision. As before so now suppose foure magnitudes in proportion AB the first, B the second, CD the third, and D the fourth: as AB to B: so CD to D:

[illustration]

AB,* 1.3 the antecedent of the first pro∣portion excedeth B the consequent of the first proportion by the magni∣tude A, wherfore A is the excesse of the antecedent AB aboue the consequent B: so likewise CD the antecedent of the second proportion, excedeth D the conse∣quent of the same proportion, by the quantitie C, wherefore C is the excesse of the antecedent CD aboue the consequent D. Now if ye compare A the excesse of AB the first antecedent, aboue the consequent B, as antecedent to B the con∣sequent, as to his consequent: also if ye compare D the excesse of the second an∣tecedent CD, aboue the consequent D, as antecedent to D the consequent, as to his consequent: then shall your magnitudes be in this order. As A to B, so is C to D: which is called diuision of proportion, or proportion deuided.* 1.4

And so in numbers, as 9. to 6, so 12. to 8, either proportion

[illustration]

is sesquialtera: the excesse of 9. the antecedent of the first proportion aboue 6. the consequent of the same is 3••: the ex∣cesse of 12. the antecedent of the second proportion aboue 8, the consequent of the same, is 4•• then if ye compare 3. the ex∣cesse of 9. the first antecedent aboue the consequent, as antecedent to 6, the conse∣quent, as to hys consequent: and also if ye compare .4 the excesse of 12. the second antecedent aboue the consequent, as antecedent, to 8. the consequent, as to hys consequent, ye shall haue your numbers after this maner by diuision of proporti∣on, as 3. to 6: so 4. to 8: for either proportion is subdupla.

Notes

• * 1.1

  The fi••t••ne definition.
• * 1.2

  This is the cō∣uerse of the former defini∣tion.
• * 1.3

  Example in magnitudes.
• * 1.4

  Example in numbers.