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

¶The 4. Probleme. The 36. Proposition. Two numbers being geuen, to finde out the lest nūber which they measure.

SVppose that the ••••o numbers ge••e•• be A and B. It is required to finde 〈◊〉〈◊〉 the lest number which they measure•• No•• A and B are ei••h••r prime ••he one to the other, or not.* 1.1 Suppose first that A and •• be prime the one to the other: and let A multiplying B produce C: wherefore B, multiplying A produ•••••• also C (by the 16. of the seuenth.)* 1.2 Wherefore A and B measure C. Now also I say, that C is the lest nū∣ber which they measure••* 1.3 For if it be not, those numbers A and •• measure some number lesse the•• C: let them measur•• some number lesse then C, and let the same be D: and how often A measureth D, so many vnities let there be in E•• and how often •• measureth D, so many v∣nitie•• let there be in •••• Wherefore A multiplying E produce•••• D, and B multiplying F pro∣•••••••••••• also D. Wherefore that which is produced of A into ••,

[illustration]

is equall to that which is produced of B into ••: wherefore (by the 19. of the seuenth) as A is to B, so is F to E. But A and B are prime : yea they are prime and also the lest in that propor∣tion (by the 23. of the seuenth): but the lest numbers in any proportion measure those numbers that haue one and the same proportion with them equally: the greater the greater: and the lesse the lesse (by the 21 of the seuenth). Wherfore B measureth E, namely, the consequent, the consequent. And forasmuch as A multiplying B and E produced C and D: therefore (by the 17. of the seuenth) as •• is to E, so is C to D. But B measureth E. Wherefore C also measureth D, the greater, the lesse which is impossible. Wherefore if those numbers A and B be prime, they shall measure no number lesse then C. Wherefore C is the lest number which A and B measure.

But now suppose that A and B be not prime the one to the other,* 1.4 and take (by the 35. of the seuenth) the lest numbers that haue one and the same proportion with A and B, and let the same be F and E. Wherefore that which is produced of A into E, is equall to that which is produced of B into F (by the 19. of the seuenth). Let A multiplying E produce C•• wherfore B multiplying F produceth also C. Wherefore A and B measure C. Then I say, that C is the lest number that they measure. For if it be not, those numbers

[illustration]

A and B shall measure some number lesse then C: let them mea∣sure a number lesse then C, and let the same be D•• And how of∣ten A measureth D, so many vnities let there be in G, And how often •• measureth D•• so many vnities let there be in H.* 1.5 Now then A multiplying G produceth D. And B multiplying H pro∣duceth also D. Wherefore that which is produced of A into G is equall to that which is produced of B into H. Wherefore (by the 19. of the seuenth) as A is to B, so is H to G•• But as A is to B, so is F to E. Wherefore as F is to E, so is H to G: but the lest numbers in any proportion measure the numbers that haue the same proportion with them equally, the greater the grea∣ter, & the lesse the lesse (by the 21. of the seuēth). VVherfore E measureth G. And forasmuch as A multiplying G and E produced C and D•• therefore (by the 17. of the fift) as E is to G, so is C to D. But E measureth G. VVherefore C also measureth D, the greater the lesse: which is impossible. Wherefore those numbers. A and B do not measure any number lesse then ••. Wherefore •• is the lest number that is measured by A and B•• whi••h was required to be done.

Notes

• * 1.1

  Two cases in this propositiō.
• * 1.2

  The first case.
• * 1.3

  Demonstra∣tion leading to an absurditie.
• * 1.4

  The second case••
• * 1.5

  Demonstra∣tion leading to an absurditie.