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 ninth booke of Eu∣clides Elementes. (Book 9)

IN THIS NINTH BOOKE Euclide continueth his purpose touching numbers: partly prosecuting thynges more fully,* 1.1 which were before somewhat spoken of, as of square and cube numbers: and partly setting out the na∣tures and proprieties of such kindes of number, as haue not yet bene entreated of: which yet are most necessary to be knowen. As are numbers euen, and odde: whose passions and conditions are in this booke largely taught, with their compositions, and subductions of the one from the other: with many other generall and speciall thinges to be noted, worthy the knowledge.

¶The 1. Theoreme. The 1. Proposition. If two like plaine numbers multiplying the one the other produce any num∣ber: the number of them produced shall be a square number.

SVppose that A and B be two like plaine numbers. And let A multiplying B produce the number C. Then I say, that C is a square number. For let A multiplying him selfe produce D. Wherefore D is a square number.* 1.2 And forasmuch as A multi∣plying him selfe produced D, and multiplying B produced C, therefore (by the 17. of the seuenth) as A is to B, so is D to C. And forasmuch as A, B, are like plaine numbers, therefore (by the 18. of the eight) betwene A and B there falleth a meane proportionall number. But if betwene two numbers fall num∣bers in continuall proportion, how ma∣ny

[illustration]

numbers fall betwene them, so many also (by the 8. of the eight) shall there fall betwene the numbers that haue the same proportion with them. Wherfore betwene C and D also there falleth a meane proportionall number. But D is a square number. Wherfore (by the 22. of the eight) C also is a square number: which was required to be proued.

¶The 2. Theoreme. The 2. Proposition. If two numbers multiplying the one the other produce a square number: those numbers are like plaine numbers.

* 1.3SVppose that two number••

[illustration]

A and B multiplying the one the other do produce C a square number. Then I say, that A and B are like plaine num∣bers. For let A multiplying him selfe produce D. Wherefore D is a square number. And forasmuch as A multi∣plying him selfe produced D,* 1.4 and multiplying B produced C, therefore (by the 17. of the se∣••enth) as A is to B, so is D to C. And forasmuch as D is a square number, and so likewise is C, therefore D and C are like plaine numbers. Wherefore betwene D and C there is (by the 18. of the eight) one meane proportionall number. But as D is to C, so is A to B. Wherefore (by the 8. of the eight) betwene A and B there is one meane proportionall number. But if be∣twene two numbers there be one meane proportionall number, those numbers are (by the 20. of the eight) like plaine numbers. Wherefore A and B are like plaine numbers: which was required to be proued.

A Corollary added by Campane.

H•••••••• it is manifest, th••t two squar•• numbers multiplyed the one into the, other do alwayes produce a squa•••• num••••r.* 1.5 For they are like superficiall numbers, and therefore the num∣ber produced of them, is (by the first of this booke) a square number. But a square num∣ber mul••••plye•• into a number not square, produceth a number not square. For if they should pro∣duce a square number, they should be like superficiall numbers (by this Proposition). But they are not. Wherefore they produce a number not square. But if a square num∣ber multiplyed into an other number produce a square number, that other number shall be a square number. For by this Proposition that other number is like vnto the square number which multiplyeth it, and therefore is a square number. But if a square number multiply∣ed into an other number produce a number not square, neither shall that other number also be a square number. For if it should be a square number, then being multiplyed into the square number it should produce a square number, by the first part of this Corollary.

The 3. Theoreme. The 3. Proposition. If a cube number multiplying himselfe produce a number, the number pro∣duced shall be a cube number.

SVppose that A being a cube number multiplieng himselfe, do produce the num∣ber B. Then I say that B is a cube number. Take the side of A, and let the same be the number C, and let C multiplieng himselfe produce the number D. Now it is manifest that C multiplieng D produceth A (by the 20. definition of the seuēth)* 1.6

[illustration]

And forasmuch as C multiplieng himselfe produced D, therfore C measureth D by those v∣nities

which are in C. But vnitie also measureth C by those vnities which are in C. Wherfore as vnitie is to C, so is C to D. Againe forasmuch as C multiplieng D produceth A: therefore D measureth A by those vnities which are in C. But vnitie measureth C by those vnities which are in C: wherefore as vnitie is to C, so is D to A. But as vnitie is to C, so is C to D, wherfore as vnitie is to C, so is C to D & D to A. Wherefore betwene vnitie & A there are two meane proportionall numbers, namely, C, D. Againe forasmuch as A multiplieng him∣selfe produced B, therefore A measureth B by those vnities which are in A. But vnitie also measureth A by those vnities which are in A. Wherfore as vnitie is to A, so is A to B. But be∣twene A and vnitie, there are two meane proportionall numbers. Wherfore betwene A and B also there are two meane proportionall numbers by the 8. of the eight. But if betwene two numbers, there be two meane proportionall numbers, and if the first be a cube number, the fourth also shall be a cube number by the 21. of the eight. But A is a cube number, wherefore B also is a cube number which was required to be proued.

¶The 4. Theoreme. The 4. Proposition. If a cube number multiplieng a cube number, produce any number, the number produced shall be a cube number.

SVppose that the cube number A multiplieng the cube number B, do produce the nū∣ber C. Then I say that C is a cube number. For let A multiplieng himselfe produce D. Wherefore D is a cube number (by the proposition going before).* 1.7 And forasmuch as A multiplieng

[illustration]

himselfe produced D, and mul∣tiplieng B, it produced C: ther∣fore (by the 17. of the seuenth) as A is to B, so is D to C. And forasmuch as A and B are cube numbers, therfore A and B are like solide numbers. Wherfore betwene A and B (by the 19. of the eight) there are two meane proportionall numbers. Wherefore also (by the 8. of the same) betwene D and C there are two meane proportionall numbers. But D is a cube number. Wher¦fore C also is a cube number (by the 23. of the eight) which was required to be demonstrated.

¶The 5. Theoreme. The 5. Proposition. If a cube number multiplying any number produce a cube nūber: the num∣ber multiplyed is a cube number.

SVppose that the cube number A, multiplying the number B, do produce a cube number, namely, C. Then I say, that B is a cube number. For let A multiplying him selfe produce D. Wherefore (by the 3. of the ninth) D is a cube nūber. And forasmuch as A multiplying him selfe produced D,* 1.8 and multiplying B, it produ∣ced C: therefore (by the 17. of the seuenth) as A is to B, so is D to C. And forasmuch as D

[illustration]

and C are cube numbers, they are also like solide nūbers. Wherefore (by the 19. of the eight) betwene D and C there are two meane proportionall numbers. But as D is to C, so is A to B.

Wherefore (by the 8. of the eight) betwene A and B there are two meane proportionall num∣bers. But A is a cube number. Wherefore B also is a cube number (by the 23. of the eight): which was required to be proued.

¶ A Corollary added by Campane.

Hereby it is manifest, that if a cube number multiply a number not cube, it shall produce a number not cube.* 1.9 For if it should produce a cube number, then the number multiplyed should also be a cube number (by this Proposition) which is contrary to the supposition. For it is supposed to be no cube number. And if a cube number multiplying a number produce a number not cube, the number multiplyed shall be no cube number. For if the number multi∣plied should be a cube number, the number produced should also be a cube number (by the 4. of this booke): which is contrary to the supposition, and impossible.

¶ The 6. Theoreme. The 6. Proposition. If a number multiplieng himselfe produce a cube number: then is that num∣ber also a cube number.

SVppose that the number A multipli∣eng

[illustration]

himself, do p••oduce B a cube nū∣ber. Then I say that A also is a cube number.* 1.10 For let A multiplieng B produce C. And forasmuch as A multiplieng himselfe produced B, & multiplieng B it produced C: therfore C is a cube number. And for that A multiplieng himselfe produced B, and multiplieng B it produced C, therfore (by the 17. of the seuenth) as A is to B, so is B to C. And for that B and C are cube numbers, they are also like solide numbers. Wherfore (by the 19. of the eight) betwene C and B there are two meane pro∣portional numbers. But as B is to C, so is A to B: wherfore (by the 8. of the eight) betwene A and B there are two meane proportional nūbers. But B is a cube number. Wherefore A also is a cube number by the 23. of the eight: which was required to be demonstrated.

¶ The 7. Theoreme. The 7. Proposition. If a composed number multiplieng any number, produce a number: the nū∣ber produced shall be a solide number.

SVppose that the composed number A multiplieng the nūber B, do produce the num∣ber C. Then I say that C is a solide number. For forasmuch as A is a composed nū∣ber, therfore some number measureth it (by the 14. definition). Let D measure it.* 1.11

[illustration]

And how o••ten D measureth A, so many vnities let there be in E. Wherefore E multiplieng D produceth A. And forasmuch as two numbers D and E, multiplieng themsel••es, produce A, which A againe multiplieng B produceth C: therfore C produced of three numbers mul∣tiplieng the one the other, namely, D, E, and B is (by the 18. definition of the seuenth) a so∣lide number. And the sides therof are the numbers D, E, B. If therefore a composed number &c, which was required to be proued.

¶ The 8. Theoreme. The 8. Proposition. If from vnitie there be numbers in continuall proportion how many soeuer: the third number from vnitie is a square number, and so are all forwarde leauing one betwene. And the fourth number is a cube number, and so are all forward leauing two betwene. And the seuenth is both a cube number

and also a square number, and so are all forward leauing fiue betwene.

SVppose that from vnitie there be these numbers in continuall proportion A, B, C, D, E, F. Then I say that the third number from vnitie, namely, B is a square number, and so are all forward leauing one betwene, namely, D and F.* 1.12 And that C the fourth number is a cube number, and so are all forwarde leauyng two be∣twene. And that F the seuenth number is both a cube number and also a square number, and so are all forward leauing fiue betwene. For for that as vnitie is to A, so is A to B. Therefore how many times vnitie measureth A, so many times A measureth B. But vnitie measureth A by those vnities which are in A, wherefore A measureth B by those vnities which are in A. And forasmuch as A measureth B by those vnities which are in A. Therfore A multiplieng himselfe produceth B. Wherfore B is a square number. And forasmuch as these numbers B, C, D, are in continuall proportion, and B is a square number, therfore by the 22. of the eight, D also is a square number. And by the same reason also F is a square number. And in like sort may we proue that leauing alwayes one betwene, all the rest forward are square num∣bers.

[illustration]

Now also I say that the fourth number from vnitie, that is, C, is a cube number, and so are all forward leauing two betwene. For for that as vnitie is to the number A, so is B to C,* 1.13 therefore how many times vnitie measureth the number A, so many times B measureth C. But vnitie measureth A by those vnities which are in A, wherfore B measureth C by those vnit••••s which are in A. Wherfore A multiplieng B produceth C. And forasmuch as A multi∣pli••ng himselfe produced B, and multiplieng B it produced C, therefore C is a cube number. And forasmuch as C, D, E, F, are in continuall proportion. But G is a cube number, therefore (by the 23. of the eight) F also is a cube number.

And it is proued, that F being the seuenth number from vnitie is also a square number.* 1.14 Wherfore F is both a cube number, and also a square number. In like sort may we proue, that lea••ing alwaies fiue betwene, all the rest forwarde, are numbers both cube and also square: which was required to be proued.

¶ The 9. Theoreme. The 9. Proposition. If from•• vnitie be numbers in continuall proportion how many soeuer: and if th•••• number which followeth next after vnitie be a square number, then all the rest following also be square numbers. And if that number which followeth next after vnitie be a cube number, then all the rest following shall be cube numbers.

SVppose that from vnitie there be these numbers in continuall proportion A, B, C, D, E, F. And let A which followeth next vnto vnitie be a square number. Then I say, that all the rest following also are square numbers.* 1.15 That the third number, namely, B, is a square number, & so all forward leauing one betwene, it is plaine by the Proposition next going before. I say also that all the rest are square numbers. For, forasmuch as A, B, C, are in continuall proportion, and A is a square number, therfore (by the 22. of the eight) C also is a square number. Againe forasmuch as B, C, D, are in con∣tinuall

proportion, and B is a square num∣ber,

[illustration]

therfore D also (by the 22. of the eight) is a square number. In like sort may we proue, that all the rest are square numbers.

* 1.16But now suppose that A be a cube num∣ber. Then I say, that all the rest following are cube numbers. That the fourth from v∣nitie, that is, C is a cube number, and so all forward leauing two betwene, it is plaine (by the Proposition going before). Now I say, that all the rest also are cube numbers. For, for that as vnitie is to A, so is A to B: therefore how many times vnitie measu∣reth A, so many times A measureth B. But vnitie measureth A by those vnities which are in A. Wherefore A also measureth B by those vnities which are in A. Wherefore A multiplying him selfe produceth B. But A is a cube number. But if a cube number mutiply∣ing him selfe produce any number, the number produced, is (by the 3. of the ninth) a cube number. Wherefore B is a cube number. And forasmuch as there are foure numbers in con∣tinuall proportion A, B, C, D, and A is a cube number, therefore D also (by the 23. of the eight) is a cube number. And by the same reason E also is a cube number, and in like sort are all the r••st following: which was required to be proued.

¶ The 10. Theoreme. The 10. Proposition. If from vnitie be numbers in continuall proportion how many soeuer, and if that number which followeth next after vnitie be not a square num∣ber, then is none of the rest following a square number, excepting the third from vnitie, and so all forward leauing one betwene. And if that number which ••olloweth next after vnitie be not a cube number, neither is any of the rest following a cube number, excepting the fourth from vnitie, and so all forward leauing two betwene.

SVppose that from vnitie be these numbers in continuall proportion A, B, C, D, E, F. And let A which followeth next after vnitie be no square number. Then I say,* 1.17 that neither is any of the rest a square number, excepting the third from vnitie, & so all forward leauing one betwene, namely, B, D, F, which are square numbers (by the 8. of this booke). For if it be possible, let C be a square number. But B also is a square number. Wher∣fore B is vnto C in that proportion that a square number is to a square number. But as •• is

[illustration]

••o C, so is A to B. Wherefore A is vnto B in th•••• proportion that a square number is to •• square number. But B is a square number. Wherefore A also is a square number (by the ••4. of the eigth)•• which is contrary to the supposition. Wherefore C is not a square number. And by the same reason none of all the other is a square number, excepting the third from vnitie, and so all forward leauing one betwene.

But now suppose that A be not a cube number•• Then I say,* 1.18 that none of all the rest is a cube number, excepting the fourth from vnitie, & so all forward leauing two betwene, name∣ly, C, and F, which (by the 8. of this booke) are cube numbers. For if i•• 〈◊〉〈◊〉 possible, l•••• D be a cube number. But C also is a cube number (by the 8. of the ninth). For it is the fourth from vnitie. But as C is to D, so is B to C. Wherefore B is vnto C, in that proportion tha•• a cube number is to a cube number. But C is a cube number. Wherefore B also is a cube number (by the 25. of the eight). And as vnitie is to A, so is A to B. But vnitie measureth A by those vnities which are in A. Wherefore A measureth B by those vnities which are in A. Wherfore A multiplying him••selfe produceth•• B a cube number. But if a number multiplying him selfe produce a cube number, then is that number also a cube number (by the 6. of the ninth)•• Wherefore A is a cube number: which is contrary to the supposition. Wherefore D is not a cube number. In like sort may we proue, that neither is any of the rest a cube number, excep∣ting the fourth from vnitie, and so all forward leauing two betwene: which was required to be proued.

¶ The 11. Theoreme. The 11. Proposition. If from vnitie be numbers in continuall proportion how many soeuer, the lesse measureth the greater by some one of them which are before in the said proportionall numbers.

SVppose that from vnitie A be these numbers in continuall proportion B, C, D, E. Then I say that of these numbers B, C, D, •••• E being the lesse, measureth E the grea¦ter by one of these numbers C or D. For for that as vnitie A is vnto the number B, so is D to E,* 1.19 therfore how many times vnitie A measureth

[illustration]

the number B, so many times D measureth E•• wherefore alternately (by the 15. of the seuenth) how many times vni∣ti•• A measureth the number D, so many times •• mea∣sureth E. But vnitie A measureth D by those vnities which are in D. Wherefore B also measureth E by those vnities which are in D. Wherefore •• the lesse, measu∣reth. E the greater by some one of the numbers which went before E in the proportionall numbers. And so likewise may we proue that E measureth D by some one of the numbers ••, C, D, namely, by C. And so of the rest. If therfore from vnitie &c. Which was required to be proued.

¶ The 12. Theoreme. The 12. Proposition. If from vnitie be numbers in continuall proportion how many soeuer, how many prime numbers measure the least•• so many also shal measure•• the num∣ber which followeth next after vnitie.

SVppose that from vnitie be these numbers in continuall proportion A, B, C, D. Th•• I say that how many prime nūbers measure D, so many also do measure A. Suppose that some prime number namely, E, do measure D. Thē I say that E also measureth A, which is next vnto vnitie. For if E do not measure A, and E is a prime number, but eue∣ry number is to euery number which it measureth not a prime number (by the 31. of the se∣uenth). Wherefore A and E are prime numbers the one to the other. And forasmuch as E measureth D, let it measure D by the number F.* 1.20 Wherefore E multiplieng F produceth D. Againe forasmuch as A measureth D by those vnities which are in C, therefore A multipli∣eng C produceth D. But E also multiplieng F produced D, wherfore that which is produced of the numbers A, C is equall to that which is produced of the numbers E, F. Wherfore as A is to E, so is F to C. But A, E, are prime numbers, yea they are prime and the least. But the lest numbers measure the numbers that haue one and the same proportion with them equally by the 21. of the seuenth, namely, the antecedent the antecedent, and the consequent the conse∣quent. Wherfore E measureth C. Let it measure it by G. Wherefore E multiplieng G produ∣ceth C. But A also multiplieng B produceth C. Wherfore that which is produced of the num∣bers

[illustration]

A, B, is equall to that which is produced of the numbers E, G. Wherfore as A is to E, so is G to B. But A, E are prime numbers, yea they are prime and the least. But the least num∣bers (by the 21. of the seuenth) measure the numbers that haue one and the same proportion with thē equally, namely, the antece••••s the antecedē••, & the cōsequēt the conseqēt. Wherfore E measureth B. Let it measure it by H. Wherefore B multiplieng H produceth B. But A also multiplieng himselfe produceth B, wherfore that which is produced of the numbers E, H, is equall to that which is produced of the number A. Wherfore as E is to A, so is A to H. But AE are prime nūbers, yea they are prime & the least, but the least numbers (by the 21. of the se∣uenth) measure the numbers that haue one and the same proportion with thē equally, name∣ly, the antecedēt the antecedent, and the cōsequent the consequent. Wherfore E measureth A and it also doth not measure it by ••••pp••sition, which is impossible. Wherfore A and E are not prime the one to the other, wherfore they are composed. But all composed numbers are measu∣red of some prime number, wherfore A and E are measured by some prime number. And for¦asmuch as E is supposed to be a prime number. But a prime number is not (by the definition) measured by any other number but of himselfe. Wherfore E measureth A and E, wherfore B measureth A, and it also measureth D. Wherfore E measureth these numbers A and D. And in like sort may we proue that how many prime numbers measure D, so many also shall mea∣sure A: which was required to be proued.

An other more briefe demonstration after Flussates.

* 1.21Suppose that from vnitie be nūbers in cō••••nuall proportion how many so euer, namely, A, B, C, D. And let some prime nūber, namely, •• measure the last nūber which is D. Thē I say that th•• same E mea∣sureth A which is the next number vnto vnitie. For if E doo not measure A, then are they prime the one to the other by the 31. of the seuenth. And forasmuch as A, B, C, D, are proportionall from vnitie,

therefore A multiplying himselfe produceth B. Wherfore B and E ar•• prim••.

[illustration]

numbers (by the 27. of the seuenth). And forasmuch as A multiplying B produceth C, therefore C is to E also a prime number by the 26. of the se∣uenth. And likewise infinitely A multiplying C produceth D: wherefore D and E are prime numbers the one to the other (b•• 〈◊〉〈◊〉 same ••6. of th•• s••∣uenth•• Wherefore E measureth not D as it was supposed, which is absurd, wherefore the prime number E measureth A, whiche is nexte vnto vnities which was required to be proued.

¶The 13. Theoreme. The 13. Proposition. If from vnitie be numbers in continuall proportion how many soeuer, and if that which followeth next after vnitie be a prime number: then shall no other number measure the greatest number, but those onely which are be∣fore in the sayd proportionall numbers.

SVppose that from vnitie be these numbers in continuall proportion A, B, C, D, and let that which followeth next after vnitie, that is, A, be a prime number. Then I say, that no other number besides these numbers A, B, C, measureth the greatest number of them which is D.* 1.22 For if it be possible, let E measure D. And let E be none of these numbers A, B, C, D. Now it is manifest that E is not a prime number. For if E be a prime number, & do also measure D, it shall likewise measure A being a prime number and not being one and the same with A, by the former Proposition: which is im∣possible. Wherefore E is not a prime number. Wherefore it is a composed number. But euery composed number (by the 33. of the seuenth) is measured by some prime number.

[illustration]

Now I say, that no other prime nūber besides A shall measure E. For if any other prime nūber do measure E, & E measureth D, therfore that number also shall measure D (by the 5. common sentence of the seuenth). Wherfore it shal also measure A (by the proposition next going before) being a prime number and not being one and the same with A: which is im∣possible. Wherefore euery the prime number A measureth E which measureth the greatest number D.

And forasmuch as E measureth D, let it measure it by F. Now I say, that F is none of these numbers A, B, C. For if F be one and ••he s••m•• with any of these numbers A, B, C, and is measureth D by ••, therefore one of these numbers A, B, C, measureth D by E. But one of these numbers A, B, C, measureth D by some one of these numbers A, B, C, therefore E is one and the same with one of these numbers A, B, C, which is cont••••ry 〈◊〉〈◊〉 th•• supposition. Wher∣fore F•••• no•• one and the same with any of these ••••••bers A, B, C.

In like sort may we pro••e, that onely the prime number A measureth F, pro••••••g first th••t

F is not a prime number. For if F be a prime number, and it measureth D, therefore it also measureth A being a prime number, and not being one and the same with A, by the former Proposition: which is impossible. Wherefore F is not a prime number: wherefore it is a com∣posed number, and therefore some prime number shall measure it. Now I say, that no other prime number besides A shall measure it. For if any other prime number do measure F, and F measureth D, therefore that number shall measure D (by the 5. common sentence of the se∣uenth). Wherefore it shall also measure A (by the former Proposition) being a prime num∣ber and not being one and the same with A: which is impossible. Wherefore onely the prime number A measureth F. And forasmuch as E measureth D by F, therefore E multiplying F produceth D, But A also multiplying C produceth D, therefore that which is produced

[illustration]

if A into C, is equall to that which is produced of E into F. Wherfore proportionally as A is to E, so is F to C. But A measureth E. Wherefore F measureth C. Let F measure C by G. And in like sort may we proue, that G is not one and the same with any of these numbers A, B, C, and that G is a composed number, and also that onely the prime number A measureth it. And forasmuch as F measureth C by G, therefore G multiplying F produced C. But A al∣so multiplying B produced C. Wherefore that which is produced of A into B, is equall to that which is produced of F into G. Wherefore proportionally as A is to F, so is G to B. But A measureth F. Wherefore G also measureth B. Let G measure B by H. Now in like sort as be∣fore may we proue, that H is not one and the same with A, and that H is a composed number, and measured onely of the prime number A. And forasmuch as G measureth B by th••se vni∣ties which are in H, therfore G multiplying H produced B. But A multiplying him selfe pro∣duced B. Wherfore that which is produced of H into G, is equall to the square number which is produced of A. Wherefore as H is to A, so is A to G. But A measureth G. Wherfore H mea∣sureth A being a prime number and not being one and the same with it: which is absurde. Wherfore no other number besides these numbers A, B, C, measureth the greatest number D: which was required to be demonstrated.

An other demonstration of the same after Campane.

* 1.23Suppose that E not being one and the same with the numbers A, B, C, D doo measure the num∣ber D. And let it measure it by the number P. And forasmuch as A being a prime number measureth the number D, which is produced of E into F: therefore by the 3••. of the fourth, A measureth either •• or ••. Let it measure E. Now forasmuch as D is produced of A into G, and also of E into F: therefore by the second part of the 19. of the seuenth, A is to E, as F is to C. But A measureth E: wherefore F measu∣reth C. Let it measure it by G. Wherefore by the 32. of the seuenth A shall measure either F or G. Let it measure F. Wherefore as before by the second part of the 1••. of the seuenth G shall measu••e B. Let it measure it by H. Now then as before it followeth by the 32. of the seuenth that A shall measure either G or H: suppose that it measure G. Wherefore by the second part of the 20. of the seuenth H shall mea∣sure •• (fo•• of A into himselfe is produced B, and of H into G also it produced B) I•• therefore H be not

equall vnto A, A shall be no prime number. Which is contrary to the supposition. ••••t if it be equall vn∣to A, then ••uery one of these numbers ••, F, E, shall be some one of the numbers A, B, C, D, by th•• 11.

[illustration]

proposition of the ninth repeted a•• of••en as neede requireth. Wherefore E is not a number diuerse from them, but is one and the same with some one of them: which is contrary to the supposi••ion; wher¦fore that is manifest which was required to be proued.

¶The 14. Theoreme. The 14. Proposition. If there be geuen the least number, whom certayne prime numbers geuen, do measure: no other prime number shall measure that nūber, besides those prime numbers geuen.

SVppose that the least number whom these prime numbers B, C, D, do measure, be A. Then I say that no other prime number besides B, C, D, measureth A. For if it be possible, let E being a prime number measure A, and let E be none of these numbers B, C, D.* 1.24 And forasmuch as E measu∣reth

[illustration]

A, let it measure it by F. Wherfore E multiplieng I produceth A: And these prime numbers B, C, D, measure A. ••ut if two numbers multiplieng the one the other produce any number. And if some prime number measure that which is produced, it shall also measure one of th••se numbers which were put at the beginning (by the 32. of the seuenth) Wherfore those numbers B, C, D, measure one of these numbers E or F. But they measure not E, for E is a prime number, and is not one and the same with any one of these numbers B, C, D. Wherfore they measure F being lesse then A which is impossible. For A is supposed to be the least whom B, C, D, measure. Wherefore no prime number besides B, C, D, measureth A: which was required to be demonstrated.

A proposition added by Campane.

If there be numbers how many soeuer in continuall proportion being the least in that proportion: a number measuring one of them, shall be a number not prime to one of the two least numbers in that proportion.

Suppose that there be numbers in continuall proportion how many soe••••r namely A, B, C, D, E which let be the least that haue the same proportion with them:* 1.25 and let the two least numbers in that proportion be F and G. And let some number as H measure some one of the numbers A, B, C, D, E, namely, C. Then I say that H is a number not prime either to F or G. Take (by the 2. of the eight) the

three least numbers in

[illustration]

the proportion of A to B: which let be P, Q, R. And afterward fower (by the same) which let be K, L, M, N: & so for∣ward till you come to the multitude of the numbers geuen A, B, C, D, E. Now it is manifest (by the demonstration of the second of the eight) that F multiplyed by P, Q, and R produceth K, L, M: and that F multiplyed by K, L, M, N produc••th A, B, C, D. And forasmuch as H measureth C: therfore H is either to F or to M not prime (by the corollary of the 32. of the seuenth added by Cāpane) If it be not prime vnto F: thē i•• that manifest which was required to be proued. But if H be not prime vnto M. Thē shal it not be prime either to F or to R (by the same corollary). If agayne it be not prime vnto F, then is that proued which was required. But if it be not prime vnto R, than (by the same corollary) shall it be a number not prime vnto G (which produceth R by the 2. of the eight)•• but G is one of the two least numbers F or G which are in the proportion of the numbers geuen at the beginning A, B, C, D, E. If therefore there be num∣bers how many soeuer. &c. which was required to be proued.

¶ The 15. Theoreme. The 15. Proposition. If three numbers in continuall proportion be the least of all numbers that haue one and the same proportion with them: euery two of them added to∣gether shall be prime to the third.

SVppose that there be three numbers in continuall proportion A, B, C, being the left of all numbers that haue one and the same proportion with them. Then I say, that euery two of these numbers A, B, C, added together,* 1.26 are prime to the third: name∣ly, that A, B, is prime to C, and B, C, to A, and A, C, to B. Take (by the 35. of the seuenth) two of the numbers that haue one and the same proportion with A, B, C, & let the same be the numbers DE, and EF.

Now it is manifest (by the sayd 35. Proposition) that DE multiplying him selfe produced A,* 1.27 and multiplying EF produced B, and moreouer EF multiplying him selfe produced C.

And forasmuch as DE and EF are the least in that proportion, they are also prime the one to the other (by the 24. of the seuenth). But

[illustration]

i•• two numbers be prime the one to the other, then both of thē added together, shall be prime to either of them (by the 30. of the seuenth). Wherefore the whole number DF is prime to either of these nūbers DE & EF. But DE also is prime vnto EF. Wherfore DF, & DE are prime vnto EF. Wherfore that which is produced of DF into DE, is (by the 26. of the se¦uēth) prime vnto EF. But if two nūbers be prime the one to the other, that which is produced of the one of thē into himselfe, is prime to the other (by the 27. of the seuēth). Wherfore that which is produced of DF into DE, is prime to that which is produced of EF into himsel••e. But that which is produced of FD into DE, is the square nūber which is produced of DE into himsel••e together with that which is produced of DE into EF (by the 3. of the second). Wherfore the square nūber which is produced of DE together with that which is produced of DE into EF, is prime to that which is produced of EF into himself. But that which is produ¦ced of DE into him selfe, is the number A, & that which is produced of DE into EF, is the number B: and that which is produced of EF into himselfe, is the number C. Wherefore the numbers A, B, added together are prime vnto C.

By the like demonstration also may we

[illustration]

proue, that the numbers B, C, are prime vnto the number A.

Now also I say, that the numbers A, C, are prime vnto the number B.

For forasmuch as DF is prime to either of

[illustration]

these DE and EF: therefore that which is produced of DF into him self, is prime to that which is produced of DE into EF. But that which is produced of DF into him selfe, is e∣quall to the square numbers which are produced of DE and EF together with that number which is produced of DE into EF, twise (by the 4. of the second). Wherefore the square nū∣bers which are produced of DE and EF together with that which is produced of DE into EF twise are prime to that which is produced of DE into EF. And by diuision also (by the 30. of the seuenth) the square numbers produced of DE and EF, together with that which is produced of DE into EF once are prime to that which is produced of DE into EF. A∣gaine (by the same 30. of the seuenth) the square nūbers produced of DE and EF, are prime to that which is produced of DE into EF. But that which is produced of DE into him selfe is A, and that which is produced of EF into him selfe is C, and that which is produced of DE into EF, is B. Wherefore the numbers A, C, added together are prime vnto the num∣ber B: which was required to be demonstrated.

This latter part of the demonstration, which proueth that the numbers A, & C are prime vnto B, is somewhat obscurely put of Theon. And therefore I will here make it playner.

Forasmuch as either of the numbers DE, and EF is prime to the whole DF:* 1.28 (as hath before b••ne proued) therefore that which is produced of DE into EF (which is the number B) is prime vnto DF, by the 26. of the seuenth. Wherefore by the 27. of the same that which is produced of DF into himselfe (which is the number composed of A and C and of the double of B by the 4. of the second) shall be prime vnto B. Wherefore it followeth that the number composed of A and C is prime vnto B. For if a number composed of two numbers, be prime to one of the said two numbers, as here the number com¦posed of A and C taken as one number and of the double of B, is prime vnto the double of B: then the two numbers whereof the number is composed, namely, the number composed of A and C, and the double of B shall be prime the one to the other (by the 30 of the seuenth). And therefore the number composed of A and C shall be prime to B taken once. For if any number should measure the two num∣bers, namely the number composed of A and C, and the number B, it should also measure the number composed of A and C, and the double of B (by the 5. common sentence of the seuenth): which is not possible, for that they are proued to be prime numbers.

Here haue I added an other demonstration of the former Proposition after Cam∣pane, which proueth that in nūbers how many soeuer, which is there proued onely tou∣ching three numbers: and the demonstration seement somwhat more perspicous then Theons demonstration. And thus he putteth the proposition.

If numbers how many soeuer being in continuall proportion be the least that haue one & the same proportion with them: euery one of them shalbe to the number composed of the rest prime.

Suppose that there be numbers in continuall proportion how many soeuer,* 1.29 and the least in their proportion: namely, A, B, C, D. Then I say that euery one of them, as for example 〈◊〉〈◊〉 D, is prime to the number composed of the rest, namely, of A, B, C. For if it be not, let some number, namely E mea¦sure D, and the number composed of A, B, C. Take the two least numbers in the same proportion that A, B, C, D are (by the 35. of the seuenth) which let

[illustration]

be F, G. And forasmuch as E measureth one of these number A, B, C, D, the same E shalbe a number not prime either to F or to G (by the proposition before added by Campane after the 14. proposition) wher∣fore some number shall measure E and one of these number•• For G: which let be H. ••nd forasmuch as H measureth But shall also measure D, which num∣ber D the number E also measureth (by the ••. com∣mon sentence of the seuenth). Moreouer forasmuch is H (by supposition) measureth one of these num∣bers

F or G, the same H shall measure all the meanes betwen•• A and D by the same cōmon sentēce. For e••••er of these numbers F or G produceth all the meanes by the next numbers in continuall proporti∣on and in the same proportion with them (as by L, I, K) by the second of the eight. Agayne forasmuch as H measureth E, which (by supposition) measureth the whole A, B, C: the same H shall also measure the whole, A, B, C (by the foresayd common sentence) and it measureth the part taken away, namely, the meanes B, C (as it hath bene proued) wherefore it also measureth the residue A (by the 4. common sentence o•• the seuenth) wherefore H measureth the extreames D and A, which are prime the one to the other (by the 3. of the eight) which were absurd. Wherefore D is a number prime to the number composed of the rest, namely, of A, B, C.

Secondly I say that this is so in euery one of them: namely that C is a prime number to the num∣ber composed of A, B, D. For if not, then as before let E measure C, and the number composed of A, B, D: which E shalbe a number not prime either to F or to G (by the former proposition added by Cam∣pane) wherefore let H measure them. And forasmuch as H measureth E, it shall also measure the whole A, B, C, D whom E measureth. And forasmuch as H measureth one of these numbers F or G, it shall measure one of the extreames A or D: which are produced of F or G (by the second of the eight) if they be multipl••ed into the meanes L or K. And moreouer the same H shall measure the meames, BC (by the 5. common sentence of the seuenth) when as by supposition it measureth either F or G. which mea∣sure B, C (by the second of the eight). But the same H measureth the whole A, B, C, D as we haue pro∣ued, for that it measureth E. Wherefore it shall also measure the residue, namely, the number composed of the extreames A and D (by the 4. common sentence of the seuenth). And it measureth one of these A or D (for it measureth one of these F or G which produce A and D) wherefore the same H shall measure one of these A or D and also the other of them (by the former common sentence) which num¦bers A and D are by the 3. of the eight prime the one to the other. Which were absurd. This may also be proued in euery one of these numbers A, B, C, D. Wherefore no number shall measure one of these numbers A, B, C, D and the numbers composed of the rest. Wherefore they are prime the one to the o∣ther: If therefore numbers how many soeuer. &c: which was required to be proued.

Here as I promised, I haue added Campanes demonstrations of those Propositions in numbers, which Eucl••de in the second booke demonstrated in lines. And that in thys place so much the rather, for that Theon as we see in the demonstration of the 15. Pro∣position seemeth to alledge the 3. & 4. Proposition of the second boke: which although they concerne lines onely, yet as we there declared and proued, are they true also in numbers.

¶ The first Proposition added by Campane.

That number which is produced of the multiplication of one number into numbers how many so∣euer: is equall to that number which is produced of the multiplication of the same number into the number composed of them.

This proueth that in numbers which the first of the second proued touching lines. Suppo•••• that the number A being multiplyed into the number B, and into the number C,* 1.30 and into the number D, doo produce the numbers E, F and G. Then I say that the number produced of A multiplyed into the number composed of B, C, and D is equall to the number composed of E, F, and G. For by the con∣uerse of the definition of a number multiplyed, what part vnitie is of A, the selfe same part is B of E, and C of F, and also D of G. Wherefore

[illustration]

by the 5. of the seuenth what part vnitie is of A, the selfe same part is the num∣ber composed of B, C, and D, of the num¦ber composed of E, F, and G. Wherfore by the definition that which is produ∣ced of A into the number composed of B, C, D, is equall to the number compo∣sed of E, F, G: which was required to be proued.

The second Proposition.

That number which is produced of the multiplication of numbers how many soeuer into one nū∣ber: is equall to that number which is produced of the multiplication of the number composed of them into the same number.

This is the conuerse of the former As if the ••••••bers •• and G and D multiplyed into the number A doo produce the numbers E and F and G.* 1.31

[illustration]

Then the number composed of B, C, D. mul∣tiplyed into the number A shall produce the number composed of the numbers E, F, G. Which thing is easly proued by the 16. of the seuenth and by the former proposition.

¶ The third Proposition.

That number which is produced of the multiplication of numbers how many soeuer into other numbers how many soeuer, is equall to that number which is produced of the multiplication of the number composed of those first numbers, into the number composed of these latter numbers.

As if the numbers A, B, C doo multiply the numbers D, E, F, ech one eche other, and if the num∣bers produced be added together. Then I say that the

[illustration]

number composed of the numbers produced is equall to the number produced of the number composed of the numbers A, B,* 1.32 C into the number composed of the num∣bers D, E, F. For by the former propositiō that which is produced of the number composed of A, B, C into D is equall to that which is produced of euery one of the sayd numbers into D: and by the same reason that which is produced of the number composed of A, B, C into E, is equal to that which is pro∣duced of euery one of the sayd numbers into E: and so likewise that which is produced of the number composed of A, B, C into F is equall to that which is produced of euery one of the sayd numbers into F. But by the first of these propositions th•••• which is produced of the number composed of these num¦bers A, B, C into euery one of these numbers D, E, F is equall to that which is produced of the number composed into the number composed: wherefore that is manifest which was required to be proued.

¶The fourth Proposition.

If a number be deu••••••d into partes how many soeuer: that nūber which is produced of the whole into him selfe, is equall to that number which is produced of the same number into all his partes.

This pro••eth in numbers that which the second of the second proued in lines. As if the number A, be deuided into the numbers B and C, and D.* 1.33 Then I

[illustration]

say, that that which is produced of A into himselfe, is e∣quall to that which is produced of A into all the sayd num¦bers B, C, and D. For putting the number •• equall to the number A, it is manifest by the first of these propositions that that which is produced of E into A,* 1.34 is equall to that which is produced of E into all the partes of A•• But by the common sentence that which is produced of E into A is equal to that which is produced of A into himselfe•• and that which is produced of E into the partes of A is equall to that which is pro∣duced of A into the selfe same partes. Wherefore that is manifest which was required to be proued.

¶The fift Proposition.

If a number be deuided into two partes that which is produced of the whole in••o one of the partes, as equall ••o that which is produced of the selfe same par•• into him selfe•• and into the other part.

This proueth in numbers that which in the 3. of the second was proued in lines.* 1.35 For let the num∣ber A be deuided into the numbers B and C. Then I say that that which is

[illustration]

produced of A into C, is equall to that which is produced of C into him∣selfe and into B. For by the 16. of the seuenth, that which is produced of A into C is equall to that which is produced of C into A. Now then put the number D equall to the number C.* 1.36 Wherefore that which is produced of A into C is equall to that which is produced of D into A. But by the first of these propositions that which is produced of D into A is equall to that which is produced of D into B and of D into C. Wher∣fore forasmu•••• •••• that which is produced of D into A and into B and into C is ••qu••ll to that which is produced of C into A•• and into B, and into himselfe, by reason of the equalitie of C and D: that is ma∣nifest which was required to be proued.

¶The sixt Proposition.

If a number be deuided into two partes: that which is produced of the multiplication of the whole ••nto him selfe, is equall to that which is produced of the multiplication of either of the partes into him selfe, and of the one into the other ••wise.

This proueth in numbers that which the fourth of the second proued touching lines. As if the number A be deuided into the numbers B and C. Then I say that

[illustration]

that which is produced of A into himself is equal to that which is produced of B into himselfe,* 1.37 and of C into himselfe, and of B into C twise. For by the 4 of these propositions, that which is produced of A into himselfe, is equall to that which is produced of A into B, and into C. But that which is produced of A into B, is equall to that which is produced of B into himselfe and into C (by the former proposition). And by the same that which is produced of A into C is equall to that which is produced of C into himselfe and into B.* 1.38 And forasmuch as that which is produced of C into B is equall to that which is produced of B into C by the 16. of the seuenth, it is manifest that that is true which was required to be proued.

¶The seuenth Proposition.

If a number be deuided into two equall partes, & into two vnequall partes: that which is produ∣ced of the greater of the vnequall partes into the lesse, together with the square nūber of the num∣ber set betwene, is equall to the square number produced of the halfe of the whole.

This proueth in numbers that which the 5. of the second proued in lines. As if the number AB be deuided into two equall numbers, which let be AC, and CB: and also in two vnequal numbers namely,* 1.39 AD and DB, of which let AD be the greater, and

[illustration]

DB the lesse. Then I say that that which is produced of the whole AD into DB together with the square number of CD, is equall to the square number of CB. For by the former propositiō the square of CB is equall to the square of CD and to the square of DB, and to that which is produced of BD into CD twise. But that which is produced of BD into himselfe and into CD is equall to that which is produced of BD into CB by the first of these propositions,* 1.40 and therefore vnto that which is produced of BD into AC. Wherefore that which is produced of BD into himselfe and into CD twise is equall so that which is produced of BD into AD. Wherefore by the same the square of CB exceedeth that which is produced of BD into AD by the square of CD: wher∣fore that is manifest which was required to be proued.

¶The 8. Proposition.

If a number be deuided into two equall part••s, and if vnto it be added an other number: that which is produced of the multiplication of the whole being composed into the number added, to∣gether with the square of the halfe, is equall to the square of the number composed of the halfe and the number added.

This proueth in numbers that which the 6. of the second proued touching lines. For suppose that the number AB be deuided into equall numbers, which let be AC and CB: and vnto it•• adde the number BD.* 1.41 Then I say, that that which is produced of the whole AD into DA together with the square of BC, is equall to the square of CD. For by the 6. of these

[illustration]

propositiōs the square of CD is equal to the square of DB, & to the square of BC, and to that which is produced of DB into BC twise. But by the ••. of these propositi∣ons,* 1.42 that which is produced of BD into himselfe and into BC twise is equall to that which is produ∣ced of BD into DA (for AC and CB are equall) wherefore the square of CD exceedeth that which is produced of BD into DA by the square of CB. Wherefore that is manifest which was required to be proued.

¶The 9. Proposition.

If a number be deuided into two partes: that which is produced of the whole number into him selfe together with that which is produced of one of the partes into him self, is equall to that which is produced of the whole into the sayd part twise together with that which is produced of the other part into him selfe.

This proueth in numbers that which the 7. of the second proued in lines. For let the number A be deuided into the numbers B and D.* 1.43 Then I say that the square of A together with the square of D is equall to that which is produced of A into D twise together with the

[illustration]

square of B. For it is manifest by the 6. of these propositions that the square of A is equall to the squares of B and D together with that which is produced of B into D twise. Wherefore the square of A together with the square of D, is equall to two squares of D••* 1.44 and to that which is produced of D into B twise together with the square of B. But by the first of these propositions two squares of D, and that which is produced of D into B twise is e∣quall to that which is produced of D into A twise. Wherfore that which is produced of D into A twise together with the square of B, is equall to the square of A together with the square of D: wherfore that is manifest which was required to be proued.

¶The 10. proposition.

If a number be deuided into two partes, and vnto it be added a number equall to one of the parts: the square of the whole number composed, is equall to the quadruple of that which is produced of the first number into the number added, together with the square of the other part.

This proueth in numbers, that which the 8. of the second proued in lines. Suppose that the num∣ber AB be deuided into the numbers AC and CB, vnto which

[illustration]

adde the number BD, which let be equall to the number CB.* 1.45 Then I say that the square of the whole number composed, namely, AD, is equall to that which is pro∣duced of AB into BD fower times together with the square of AC. For by the 6. of these propositions the square of AD, is equall to the square AB and to the square of BD together with that which is pro∣duced of AB into BD twise.* 1.46 And forasmuch as the square of BD is equal to the square CB: therfore the square of AD shall be equall to the square of AB and to the square of CB together with that which is produced of AB into BD twise. But by the former proposition the square of AB together with the square of CB, is equall to the square of AC together with that which is produced of AB into BC twise wherfore the square of AD is equal to that which is produced of AB into BD twise, and to that which is produced of AB into BC twise together with the square of AC. And for that that which is produced of AB into BC is equall to that which is produced of AB into BD, therefore is that manifest which was required to be proued.

The 11. proposition.

If a number be deuided into two equall partes, and into two vnequall partes: the squares of the two vnequall partes taken together, are double to the square of the halfe, and to the square of the excesse of the greater part aboue the lesse, the sayd two squares being added together.

This proueth in numbers that which the 9. of the second proued in lines. For suppose that the number AB be deuided into two equall partes: which let be AC and CB, and into two vnequall parts, which let be AD and DB.* 1.47 Then I say that the squares of the two numbers AD & DB, taken together, are double to the two squares of the two numbers AC and CD, taken together. For by the 6. of these propositions the square of AD is equall to the squares of

[illustration]

AC and CD, and to that which is produced of AC into CD twise. And forasmuch as the square of AC is equal to the square of CB, the square of AD shal be equall to the square of BC & to the square of CD together with that which is produced of BC into CD twise.* 1.48 Wherefore the square of AD together with the square of BD, is equall to the square of BC, and to the square of CD and to that which is produced of BC into CD twise together with the square of BD. But that which is produced of BC into CD twise together with the square of BD, is equall to the square of BC and to the square of CD by the 9. of these propositions. Wherfore the squares of the two numbers AD and DB are equall to the squares of the two numbers BC and CD, taken twise. And therefore the squares of the two numbers AD and DB are double to the squares of the two numbers BC and CD, that is AC and CD (for the numb••rs AC and BC are by supposition equall) wherfore that is manifest which was required to be proued.

The 12. proposition.

If a number be deuided into two equall partes, and vnto it be added an other number: the square of the whole number composed together with the square of the number added, is double to the square of the halfe, together with the square of the number composed of the halfe and the num∣ber added.

This proueth in numbers that which the 10. of the second proued in lines. Suppose that the num∣ber AB be deuided into two equall numbers AC and CB:* 1.49 and vnto it adde the number BD. Then I say that the square of AD together with the square of BD, is

[illustration]

double to the square of AC together with the square of CD. For forasmuch as the number CD is deuided into two partes, and vnto it is added the number AC which is equall to one of the partes (namely, to CB) therefore by the 10. of thes•• propositions,* 1.50 the square of AD is equall to that which is produced of CD into CA foure times to∣gether with the square of BD. And forasmuch as AC is equall to CB, therefore the square of AD, is equall to t••••t which is produced of DC into CB fower times together with the square of BD. Wh••r∣fore the square of AD together with the square of DB, is equall to that which is produced of DC into CB ••ower times together with two squares of BD. But by the 9. of these propositions, that which is produced of DC into CB fower times together with two squares of BD is double to the square of CD together with the square of CB (for the square of CD together with the square of CB is equal to that which is produced of DC into CB twise together with one square of CB). Wherefore forasmuch as the square of CB is equall to the square of AC, that is mani••est: which was required to be proued.

The 13. proposition.

It is impossible to deuide a number in such sorte: that that which is contayned vnder the whole and one o•• the partes, shall be equall to the square of the other part.

* 1.51That which the 11. of the second taught to be doone in lines is here proued to be impossible to be doone in numbers. ••or suppose that there be a number whatsoeuer namely AB. Then I say, that it is impossible to deuide it in such sort as is required in the proposition. For so should it be deuided ac∣cording to a proportion hauing a meane and two ex∣t••eames.

[illustration]

But i•• i•• be possible,* 1.52 let the number AB be so deuided in C. And as AB is to BC, so let BC be to CA. Wherefore AC shall be lesse then CB. Now then take away from CB a number equall to AC which let be CD. And forasmuch as the proportion of the whole AB to the whole BC, is as the proportion of the part taken away from AB, namely, BC to the part taken away from BC, namely, CD: therefore the proportion of the residue of AB, namely, AC, to the residue of BC, namely, to BD, is as the proportion of the whole AB to the whole BC (by the 11. of the seuenth). Wherefore BC is to CD, as CD is to DB. Wherfore CD is greater then DB. Wherefore sub••rahing DE out of CD, so that let DE be equall to DB: the proportion of BC to CD is as the proportiō of CD to DE. Wherfore the residue of CB, namely, DB, shal be to the residue of CD, namely, to CE, as the proportion of the whole BC to the whole CD. Wherfore CE may be subtra∣hed out of ED: wherfore there shalbe no end of this subraction: which is impossible.

¶The 16. Theoreme. The 16. Proposition. If two numbers be prime the one to the other, the second shall not be to any other number, as the first is to the second.

SVppose that these two numbers A and B be prime the one to the other. Then I say that B is not to any other nūber as A is to B.* 1.53 For if it be possible, as A is to B, so let B be to C. Now A and B are prime numbers, yea they are prime and the lest by the 23. of the seuenth. But (by the 21. of the seuenth) the least measure the numbers that haue one and the same proportion with them equally, the

[illustration]

antecedent the antecedent, and the consequent the consequent. Where∣fore the antecedent A, measureth the antecedent B, and it measureth also it selfe. Wherfore A measureth these numbers A & B being prime the one to the other, which is impossible. Wherfore as A is to B, so is not B to C: which was required to be proued.

¶The 17. Theoreme. The 17. Proposition. If there be numbers in continuall proportion how many soeuer, and if theyr

extremes be prime the one to the other, the lesse shall not be to any other number, as the first is to the second.

SVppose that there be these numbers in continuall proportion A, B, C, D, and let their extremes A and D be prime the one to the other. Then I say that D is not to any other number as A is to B. For if it be possible, as A is to B, so let D be to E. Wherfore alternately by the 13. of the seuenth, as A is to D, so is B to E. But A and D are prime,* 1.54 yea they are prime and

[illustration]

the least. But the least numbers (by the 21. of the seuenth) measure the numbers that haue one and the same proportiō with them equal∣ly, the antecedent the antecedent, and the con¦sequent. Wherefore the antecedent A measu∣reth the antecedent B: but as A is to B, so is B to C. Wherfore B also measureth C. Wherfore A also measureth C (by the 5. commō sentence of the seuenth) and forasmuch as B is to C, so is C to D, but B measureth C. Wher••ore C mea∣sureth D. But A measureth C. Wherfore A also measureth D by the same common sentence, and it also measureth it selfe. Wherefore A measureth these numbers A and D being prime the one to the other, which is impossible. Wherfore D is not to any other number as A is to B: which was required to be proued.

¶The 18. Theoreme. The 18. Proposition. Two numbers being geuen, to searche out if it be possible a third number in proportion with them.

SVppose that the two numbers geuen be A and B. It is required to searche out if it be possible a third number proportionall with them. Now A, B are either prime the one to the other or not prime.* 1.55 If they be prime, then (by the 16. of the ninth) it is mani∣nifest that it is impossible to finde out a third number proportional with them. But now sup∣pose that AB be not prime the one to the other. And let B multiplieng himselfe produce C. Now A either measureth C, or measureth it not. First,* 1.56 let it measure it and that by D. Wher∣fore A multiplieng D produceth

[illustration]

C.* 1.57 But B also multiplieng himself produced C. Wherfore that which is produced of A into D, is equall to that which is produced of B in∣to himselfe. Wherefore (by the se∣cond part of the 19. of the seuēth) as A is to B, so is B to D. Wherfore vnto these numbers A, B is found out a third number in proportion, namely, D.

But now suppose that A do not measure C••* 1.58 Then I say that it is impossible to ••inde out a third nū∣ber in proportion with these num∣bers A, B. For if it be possible, let there be found out such a number, and let the same be D.

Wherfore that which is produced of A into D, is equall to that which is produced of B into himselfe, but that which is produced of B into himselfe is C. Wherfore that which is produced of A into D is equall vnto C. Wherfore A multiplieng D produced C. Wherefore A measu∣reth G by D. But it is supposed also not to measure it, which is impossible. Wherefore it is not possible to finde out a third number in proportion with A & B, whensoeuer A measureth not C: which was required to be proued.

¶ The 19. Theoreme. The 19. Proposition. Three numbers beyng geuen, to search out if it be possible the fourth num∣ber proportionall with them.

* 1.59SVppose that the three numbers geuen be A, B, C. It is required to search out if it be possible a ••ourth number proportionall with them. Now A, B, C, are either in continuall proportion, and their extremes A, C are prime the one to the other or they are not in continuall proportion, and their extremes are yet prime the one to the other: or they are in continuall proportion, and their extremes are not prime the one to the other: or they are neither in continuall proportion, nor their extremes are prime the one to the other.

* 1.60If A, B, C, be in continuall proportion, and their extremes be prime the one to the other, it is manifest (by the 17. of the ninth) that it is impossible to finde out a fourth number pro∣portionall with them.

But now suppose that A, B, C, be not in continuall

[illustration]

proportion,* 1.61 and yet let their extremes be prime the one to the other. Then I say that so also it is impossible to finde out a fourth number proportional with thē. For if it be possible, let there be found such a number, and let the same be D. So that as A is to B, so let C be to D, and as B is to C, so let D bet•• E. And for that as A is to B, so is C to D, and as B is C, so is D to E, therfore of equallitie (by the 14. of the se∣uenth) as A is to C, so is C to E. But A and C are prime

[illustration]

the one to the other, yea they are prime, and the least: but the lest measure the numbers that haue one & the same proportion with them equally, the antecedent, the ante∣cedent, and the consequent the consequent (by the 21. of the seuenth). Wherfore A measureth C, namely, the an∣tecedent the antecedent, and it also measureth it selfe. Wherfore A measureth these numbers A and C being prime the one to the other, which is impossible. Wherfore it is not possible to finde out a fourth number proportionall with these numbers A, B, C.

But now againe suppose that A, B, C, be in

[illustration]

co••tinuall proportion,* 1.62 and let A and C not be prime the one to the other. Then I say that it is pos••ible to finde out a fourth number pro∣portionall with them. For let B multiplieng C produce D. Now A either measureth D, or measureth it not. First let it measure it, and that by E. Wherfore A multiplieng E produced D. But B also multiplieng C produced D. Wherfore that which is produced of AE is equal to that which is produced of BC: wherfore in what proportiō A is to B, in y^e same is C to E. Wher¦fore there is found out a fourth number, namely, E, proportionall with these nūbers A, B, C.

But now suppose that A do not measure D. Then I say that it is not possible to finde out a fourth number proportionall with these numbers A, B, C. For if it be possible, let there be found such a number, and let the same be E. Wherfore that which is produced of A into E is equall to that which is produced of B into C. But that which is produced of B into C is D. Wherfore that which is produced of A into E is equall vnto D. Wherefore A multiplieng E produced D, wherfore A measureth D, but it also measureth it not, which is impossible. Wher¦fore it is impossible to finde out a fourth number proportionall, with these numbers A, B, C, whensoeuer A measureth not D.

But now suppose

[illustration]

that A, B,* 1.63 C be nei∣ther in continuall proportiō, neither al¦so their extremes be prime the one to the other. And let B mul¦tiplieng C produce D. And in like sorte may we proue that if A do measure D, it is possible to finde out a fourth number proportionall with them. But if it do not measure D, thē is it vn∣possible: which was required to be proued.

¶ The 20. Theoreme. The 20. Proposition. Prime numbers being geuen how many soeuer, there may be geuen more prime numbers.

SVppose that the prime numbers geuen be A, B, C.* 1.64 Then I say, that there are yet more prime numbers besides A, B, C. Take (by the 38. of the seuenth) the lest number whom these numbers A, B, C do measure, and let the same be DE. And vnto DE adde vnitie DF. Now EF is either a prime number or not. First let it be a prime number,* 1.65 then are there found

[illustration]

these prime numbers A, B, C, and EF more in multi∣tude then the prime numbers ••irst geuen A, B, C.

But now suppose that EF be not prime.* 1.66 Wherefore some prime number measureth it (by the 24. of the se∣uenth). Let a prime number measure it, namely, G. Then I say, that G is none of these numbers A, B, C. For if G be one and the same with any of these A, B, C. But A, B, C, measure the nūber DE: wher∣fore G also measureth DE: and it also measureth the whole EF. Wherefore G being a num∣ber shall measure the residue DF being vnitie•• which is impossible. Wherefore G is not one and the same with any of these prime numbers A, B, C: and it is also supposed to be a prime number. Wherefore there are ••ound these prime numbers A, B, C, G, being more in multitude then the prime numbers geuen A, B, C: which was required to be demonstrated.

A Corollary.

By thys Proposition it is manifest, that the multitude of prime numbers is infinite.

¶ The 21. Theoreme. The 21. Proposition. If euen nūbers how many soeuer be added together: the whole shall be euē.

SVppose that these euen numbers AB, BC; CD, and DE, be added together. Then I say, that the whole number, namely, AE, is an euen number.* 1.67 For foras∣much as euery one of these numbers AB, BC, CD, and DE, is an euen num∣ber, therefore euery one of them hath an halfe. Wherefore the whole AE also hath an halfe. But an euen

[illustration]

number (by the definition) is that which may be deuided in∣to two equall partes. Where∣fore AE is an euen number: which was required to be proued.

¶ The 22. Theoreme. The 22. Proposition. If odde numbers how many soeuer be added together, & if their multitude be euen, the whole also shall be euen.

SVppose that these odde numbers AB, BC, CD, and DE, being euen in multitude, be added together. Then I say, that the whole AE is an euen number. For foras∣much as euery one of these numbers AB, BC, CD, and DE, is an odde number, is ye take away vnitie from e∣uery

[illustration]

one of them,* 1.68 that which remayneth o•• euery one of thē is an euen number. Wherefore they all added together, are (by the 21. of the ninth) an euen number: and the multitude of the vnities taken away is euen. Wherefore the whole AE is an euen number: which was required to be proued.

¶ The 23. Theoreme. The 23. Proposition. If odde numbers how many soeuer be added together, and if the multitude of them be odde, the whole also shall be odde.

SVppose that these odde numbers, AB, BC, and CD being odde in multitude be ad∣ded together. Then I say that the whole AD is an odde number. Take away from CD, vnitie DE, wherefore that which remaineth CE is an euen number. But AC also (by the 22. of the

[illustration]

ninth) is an euen num∣ber.* 1.69 Wherfore the whole AE is an euen number. But DE which is vnitie being added to the euen number AE, maketh the whole AD a•• odde number: which was required to be proued••

¶ The 24. Theoreme. The 24. Proposition. If from an euen number be takē away an euen number, that which remai∣neth shall be an euen number.

SVppose that AB be an euen number, and from

[illustration]

it take away an euen number CB.* 1.70 Then I say that that which remayneth, namely, AC is an euen number. For forasmuch as AB is an euen

euen number, it hath an halfe, and by the same reason also BC hath an halfe. Wherfore the residue CA hath an halfe. Wherfore AC is an euen number: which was required to be de∣monstrated.

¶ The 25. Theoreme. The 25. Proposition. If from an euen number be taken away an odde number, that which remai∣neth shall be an odde number.

SVppose that AB be an euen number, and

[illustration]

take away from it BC an odde number. Then I say that the residue CA is an odde number.* 1.71 Take away from BC vnitie CD. Wherfore DB is an euen number. And AB also is an euen number, wherefore the residue AD is an euen num∣ber (by the ••ormer proposition) But CD which is vnitie, being taken away from the euen nū∣ber AD maketh the residue AC an odde number: which was required to be proued.

¶ The 26. Theoreme. The 26. Proposition. If from an odde number be taken away an odde number, that which re∣mayneth shall be an euen number.

SVppose that AB be an odde number, and from it

[illustration]

take away an odde number BC. Thē I say that the residue CA is an euen number.* 1.72 For forasmuch as AB is an odde number, take away from it vnitie BD. Wherfore the residue AD is euen. And by the same reason CD is an euen number: wherfore the residue CA is an euen number (by the 24. of this booke)•• which was required to be proued.

¶ The 27. Theoreme. The 27. Proposition. If from an odde number be taken a way an euen number, the residue shall be an odde number.

SVppose that AB be an odde number, and from it

[illustration]

take away an euen number BC.* 1.73 Then I say that the residue CA is an odde number. Take away frō AB vnitie AD. Wherfore the residue DB is an euē number, & BC is (by supposition) euen. Wherfore the residue CD is an euen number. Wherefore DA which is vnitie, beyng added vnto CD which is an euen number maketh the whole AC an ••dde number: which was required to be proued.

¶ The 28. Theoreme. The 28. Proposition. If an odde number multiplieng an euen number produce any number, the number produced shall be an euen number.

SVppose that A being an odde number multiplieng B

[illustration]

being an euen number, do produce the number C. Then I say that C is an euen number.* 1.74 For forasmuch as A multiplieng B produced C, therfore C is composed of so many numbers equall vnto B as there be in vnities in A. But B is an euen nūber: wherfore C is composed of so many euen numbers, as there are vnities in A. But if euē numbers how ma∣ny soeu••r be added together, the whole (by the 21. of the ninth) is an euen number: wherfore C is an euen number: which was required to be demonstrated.

¶ The 29. Theoreme. The 29. Proposition. I•• an odde number multiplying an odde number produce any number, the number produced shalbe an odde number

SVppose that A being an odde number multiplying B being also an odde number, doo produce the number C. Then I say that C is an odde number. For forasmuch as A multiplying B produced C, therefore C is composed of so many numbers equall vnto B as there be vnities in A.* 1.75 But either of these num¦bers

[illustration]

A and B is an odde number. Wherefore C is com∣posed of odde numbers, whose multitude also is odde. Wherfore (by the 23. of the ninth) C is an odde nūber: which was required to be demonstrated.

A proposition added by Campane.

* 1.76If an odde number measure an euen number, it shall measure it by an euen number.

For if it should measure it by an odde number, then of an odde number multiplyed into an odde number should be produced an odde number, which by the former proposition is impossible.

An other proposition added by him.

* 1.77If an odde number measure an odde number, it shall measure it by an odde number.

For if it should measure it by an euen number, then of an odde number multiplyed into an euen number should be produced an odde number which by the 28. of this booke is impossible.

¶ The 30. Theoreme. The 30. Proposition. If an odde number measure an euen number, it shall also measure the halfe thereof.

SVppose that A being an odde number doo measure B being an euen number. Th•••• I say that it shall measure the halfe thereof. For forasmuch as A measureth B let i•• measure it by C.* 1.78 Thē I say that C is an euen number. For if not then, if it be possible le•• i•• be odde. And forasmuch as A measureth B by C: ther∣fore

[illustration]

A multiplying C produceth B. Wherfore B is composed of odde numbers whose multitude also is odde. Wherfore B is an odde number (by the 29. of this booke) which is ab∣surd•• for it is supposed to be euen: wherefore C is an euen num••er. Wherefore A measureth B by an euen number: and C measureth B by A. But either

of these numbers C and B hath an halfe part wherfore as C is to B, so is the halfe to the halfe. But C measureth B by A. Wherefore the halfe of C measureth the halfe of B by A: wherfore A multiplying the halfe of C produceth the halfe of B. Wherfore A measureth the halfe of B: and it measureth it by the halfe of C. Wherefore A measureth the halfe of the number B: which was required to be demonstrated.

¶ The 31. Theoreme. The 31. Proposition. If an odde number be prime to any number, it shal also be prime to the dou∣ble thereof.

SVppose that A being an odde number be prime vnto the number B: and let the dou∣ble of B be C.* 1.79 Then I say, that A is prime vnto C. For if A and C be not prime the one to the other, some one number measureth them both. Let there be such a num∣ber which measureth them both, and let the same be D.

[illustration]

But A is an odde number. Wherefore D also is an odde number. (For if D which measureth A should be an euen number, then should A also be an euen number (by the 21. of this booke): which is cōtrary to the suppositi∣on. For A is supposed to be an odde nūber: & therefore D also is an odde number). And forasmuch as D being an odde number measureth C, but C is an euē number (for that it hath an halfe, namely, B). Wherfore (by the Proposition next going before) D measureth the halfe of C. But the halfe of C is B. Wherefore D measureth B: and it also measureth A. Wherefore D measureth A and B being prime the one to the o∣ther: which is absurde. Wherefore no number measureth the numbers A & C. VVherfore A is a prime number vnto C. VVherefore these numbers A and C are prime the one to the other: which was required to be proued.

¶ The 32. Theoreme. The 32. Proposition. Euery nūber produced by the doubling of two vpward, is euenly euen onely.

SVppose that A be the number two: and from A vpward double numbers how many soeuer; as B, C, D. Then I say, that B, C, D, are numbers euenly euen onely. That e∣uery one of them is euenly euen, it is manifest:* 1.80 for euery one of them is produced by the doubling of two. I say also, that euery one of them is euenly

[illustration]

euen onely. Take vnitie E. And forasmuch as from vnitie are certaine numbers in continuall proportion, & A which follow∣eth next after vnitie is a prime number, therefore (by the 13. of the third) no number measureth D being the greatest number of these numbers A, B, C, D, besides the selfe same numbers in proportion. But euery one of these numbers A, B, C, is euenly euen. VVherefore D is euenly euen onely. In like sort may we proue, that euery one of these numbers A, B, C, is euenly euen onely: which was required to be proued.

¶The 33. Theoreme. The 33. Proposition. A number whose halfe part is odde, is euenly odde onely.

SVppose that A be a number whose halfe part is odde. Then I say that A is euenly od onely. That it is euenly odde it is manifest: for his halfe being odde measureth him by an euē number, namely, by 2.* 1.81 (by the defini∣tion).

[illustration]

I say also that it is euenly odde onely. For if A be euen∣ly euen, his halfe also is euen. For (by the definition) an euen number measureth him by an euen number. Wherefore that euen number which measureth him by an euen number shall also measure the halfe thereof being an odde number by the 4. common sentence of the seuenth which is absurd. Wherfore A is a number euenly odde onely: which was required to be proued.

An other demonstration to proue the same.

Suppose that the number A haue to his halfe an od nūber, namely, B. Thē I say that A is euēly od onely. That it is euenly odde needeth no profe: forasmuch as the number 2. an euen number measu∣reth it by the halfe thereof which is an odde number.* 1.82 Let C be the number 2. by which B measureth A

[illustration]

(for that A is supposed to be double vnto B). And let an euen number, namely, D measure A (which is possible for that A is an euen number by the definition) by F. And forasmuch as that which is produced of C into B is equall to that which is produced of D into F, therefore by the 19. of the seuenth, as C is to D, so is B to F. But C the number two measureth D being an euen number: wherfore F also measureth B which is the halfe of A. Wherfore F is an odde number. For if F were an euen number then should it in the B whome it measureth an odde number also by the 21. of this booke, which is contrary to the supposition. And in like maner may we proue that all the euē nūbers which measure the number A•• do measure it by odde numbers. Wherefore A is a number euenly odde onely: which was required to be proued.

¶ The 34. Theoreme. The 34. Proposition. If a number be neither doubled from two, nor hath to his half part an odde number, it shall be a number both euenly euen, and euenly odde.

SVppose that the nūber A be a nūber neither doubled frō the nūber two, neither also let it haue to his halfe part an odde nūber. Then I say that A is a nūber both euenly euen, and euenly odde.* 1.83 That A is euenly euen it is manifest, for the halfe therof is not odde, and is measured by the number 2. which is an euen number. Now I say that it is euenly odde also. For if we deuide A into two equall partes, and so conti∣nuing still, we shall at the length light vpon a certaine

[illustration]

odde number which shall measure A by an euen num∣ber. For if we should not light vpon such an odde nū∣ber, which measureth A by an euen number, we should at the length come vnto the number two, and so should A be one of those numbers which are doubled from two vpward, which is contrary to the supposition. Wherfore A is euenly odde. And it is proued that it is euenly euē: wherfore A is a number both euenly euen and euenly odde: whiche was required to be de∣monstrated.

This proposition and the two former manifestly declare that which we noted vp¦pon the tenth definition of the seuenth booke namely, that Campane and Flussates and diuers other interpreters of Euclide (onely Theon except) did not rightly vnderstand the 8. and 9. definitions of the same booke concerning a number euenly euen, and a num∣ber euenly odde. For in the one definition they adde vnto Euclides wordes extant in

the Greeke this word onely (as we there noted) and in the other this word all. So that af∣ter their definitions a number can not be euenly euen vnlesse it be measured onely by euen numbers: likewise a number can not be euenly odde vnlesse all the euen numbers which doo measure it, doo measure it by an odde number. The contrary whereof in this proposition we manifestly see. For here Euclide proueth that one number may be both euenly euen and euenly odde. And in the two former propositions he proued that some numbers are euenly euen onely, and some euenly odde onely: which word onely had bene in vaine of him added, if no number euenly euen could be measured by an odde number, or if all the numbers that measure a number euenly odde must needes measure it by an odde number. Although Campane and Flussates to auoyde this absurdi∣ty haue wreasted the 32. proposition of this booke frō the true sence of the Greeke and as it is interpreted of Theon. So also hath Flussates wreasted the 33. proposition. For wheras Euclide sayth, Euery nūber produced by the doubling of two vpward, is euēly euē only: they say, onely the numbers produced by the doubling of two, are euenly euen. Likewise whereas Euclide saith, A number whose hafle part is odde, is euenly odde onely, Flussates sayth, onely a number whose halfe part is od, Is euēly od. Which their interpretatiō is not true, neither can be applyed to the propositions as they are extāt in the Greeke. In dede the sayd 32. and 33. propositi∣ons as they put thē are true touching those numbers which are euenly euen onely, or euēly od onely. For no number is euenly euen onely, but those onely which are doubled from two vpward. Likewise no numbers are euenly odde onely, but those onely whose halfe is an odde number. But this letteth not, but that a number may be euenly euen although it be not doubled from two vpward & also that a number may be euēly odde although it haue not to his halfe an odde number. As in this 34. propositiō Euclide hath plainly proued. Which thing could by no meanes be true, if the foresayd 32. & 33. pro∣positons of this booke should haue that sence and meaning wherein they take it.

¶ The 35. Theoreme. The 35. Proposition. If there be numbers in continuall proportion how many soeuer, and if from the second and last be taken away numbers equall vnto the first, as the ex∣cesse of the second is to the first, so is the excesse of the last to all the nūbers going before the last.

SVppose that these numbers A, BC, D, and EF, be in continuall proportion be∣ginning at A the least. And from BC, which is the second, take away CG equall vnto the first, namely, to A, and likewise from EF the last take away FH e∣quall also vnto the first, namely, to A. Then I say, that as the excesse BG is to A the first, so is HE the excesse, to all the numbers D, BC, and A, which go before the last number, namely, EF.* 1.84 Forasmuch as EF is the greater (for the second is supposed greater then the first) put the number FL equall to the number D, and likewise the number FK equall to the number BC. And forasmuch as FK is equall vnto CB, of which FH is equall vnto GC, therefore the residue HK is equall vnto the residue GB. And for that as the whole F••, is to the whole FL, so is the part taken away FL, to the part taken away FK, therefore the residue LE is to

[illustration]

the residue KL, as the whole ••E is to the whole FL (by the 11. of the seuenth). So likewise for that FL is to FK, as FK is to FH, KL shall be to HK, as the whole FL is to the whole FK (by the same Proposition). But as FE is to FL, and as FL is to FK, and FK to FH, so were FE to D, and D to BC, and BC 〈◊〉〈◊〉 A. Wherefore as LE is to KL, and as KL is to HK, so is D to BC. Wherefore alter∣nately (by the 23. of the seuenth) as LE is to D, so is KL to be BC, and as KL is to BC, so is HK to A. Wherefore also as one of the antecedentes is to one of the consequentes; so are

all the antecedentes to all the consequentes. Wherefore as KH is to A, so are HK, KL, and LE, to D, BC, and A (by the 12. of the seuenth). But it is proued, that KH is e∣quall vnto BG. Wherefore as BG, which is the excesse of the second, is to A, so is EH the excesse of the last vnto the numbers going before D, BC, and A. Wherefore as the excesse of the second is vnto the first, so is the excesse of the last to all the numbers going before the last: which was required to be proued.

¶ The 36. Theoreme. The 36. Proposition. If from vnitie be taken numbers how many soeuer in double proportion continually, vntill the whole added together be a prime number, and if the whole multiplying the last produce any number, that which is produced is a perfecte number.

* 1.85SVppose that from vnitie be taken these numbers A, B, C, D, in double proportion continually, so that all those numbers A, B, C, D, & vnitie added together, make a prime number: and let E be the number composed of all those numbers A, B, C, D, & vnitie added together: and let E multiplying D, which is the last num∣ber, produce the number FG. Then I say, that FG is a perfect number.

[illustration]

* 1.86How many in multitude A, B, C, D, are, so many in continuall double proportion take be∣ginning at E, which let be the numbers E, HK, L, and M. VVherefore of equalitie (by the 13. of the seuenth) as A is to D, so is E to M. VVherefore that which is produced of E into D, is equall to that which is produced of A into M. But that which is produced of E in∣to D, is the number FG. VVherefore that which is produced of A into M, is equall vnto FG. VVherefore A multiplying M produceth FG. VVherefore M measureth FG by those vnities which are in A. But A is the number two. VVherefore FG is double to M. And the num∣bers M, L, HK, and E, are also in continuall double proportion. VVherefore all the num∣bers E, HK, L, M, and FG, are continually proportionall in double proportion. Take from the second number KH, and from the last FG a number equall vnto the first, namely, to E: and let those numbers taken be HN, & FX.* 1.87 VVherefore (by the Proposition going before) as the excesse of the second number is to the first number, so is the excesse of the last to all the

numbers going before it. VVherefore as NK is to E, so is XG to these numbers M, L, KH, and E. But NK is equall vnto E (for it is the halfe of HK, which is supposed to be double to E). VVherefore XG is equall vnto these numbers M, L, HK, and E. But XF is equall vnto E, and E is equall vnto these numbers A, B, C, D, and vnto vnitie. Wherfore the whole number FG is equall vnto these numbers E, HK, L, M,* 1.88 and also vnto these num∣bers A, B, C, D, and vnto vnitie. Moreouer I say, that vnitie and all the numbers A, B, C, D, E, HK, L, and M, do measure the number FG. That vnitie measureth it, it needeth no proufe. And forasmuch as FG is produced of D into E, therefore D and E do measure it. And forasmuch as the double from vnitie, namely, the nūbers A, B, C, do measure the num∣ber D (by the 13. of this booke) therefore they shall also measure the number FG (whom D measureth) by the ••. common sentence. By the same reason forasmuch as the nūbers E, HK,

[illustration]

L, and M, are vnto FG, as vnitie and the numbers A, ••, C, are vnto D (namely, in subdu∣ple proportion) and vnitie and the numbers A, B, C, do 〈◊〉〈◊〉 D, therefore also the num∣bers E, HK, L, and M, shall measure the number FG: No•• I say also, that no other num∣ber measureth FG besides these numbers A, B, C, D, E, HK, L, M, and vnitie. For if it be possible, let O measure FG. And let O not be any of these numbers A, B, C, D, E, HK, L, and M. And how often O measureth FG, 〈◊〉〈◊〉 vnities let there be in P. Wherefore O mul∣tiplying P produceth FG. But E also multiplying D produced FG. Wherefore (by the 19. of the seuenth) as E is to O, so is P to D. Wherefore alternately (by the 9. of the seuenth) as E is to P, so is O to D. And forasmuch as from vnitie are these numbers in continuall pro∣portion A, B, C, D, and the number A which is next after vnitie is a prime number, therfore (by the 13. of the ninth) no other number measureth D besides the numbers A, B, C. And it is supposed that O is not one and the same with any of these nūbers A, B, C. Wherefore O mea∣sureth not D. But as O is to D, so is E to P. Wherefore neither doth E measure P. And E is a prime number. But (by the 31. of the seuenth) euery prime number, is to euery number that it measureth not, a prime number. Wherefore E and P are prime the one to the other: yea they are prime and the least. But (by the 21. of the seuenth) the least measure the numbers that haue one and the same proportion with them equally, the antecedent the antecedent, and the consequent the consequent. And as E is to P, so is O to D. Wherefore how many times E measureth O, so many times P measureth D. But no other number measureth D besides the numbers A, B, C (by the 13. of this booke). Wherefore P is one and the same with one of these numbers A, B, C. Suppose that P be one and the same with B, & how many B, C, D, are

in multitude, so many take from E vpward, namely, E, HK, and L. But E, HK, and L, are in the same proportion that B, C, D, are. VVherefore of equalitie, as B is to D, so is E to L. VVherefore that which is produced of B into L, is equall to that which is produced of D into E. But that which is produced of D into E, is equall to that which is produced of P into O. VVherefore that which is produced of P into O, is equall to that which is produced of B into L. VVherefore as P is to B, so is L to O: and P is one & the same with B: wherefore L also is one and the same with O: which is impossible. For O is supposed not to be one and the same with any of the numbers geuen. VVherefore no num∣ber measureth FG besides these numbers A, B, C, D, E, HK, L, M, and vnitie. And it is proued, that FG is equall vnto these num∣bers A, B, C, D, E, HK, L, M, and vnitie, which are the partes therof (by the 39. of the seuenth). But a perfect nūber (by the definition) is that which is equall vnto all his partes. VVherfore FG is a perfect number•• which was re∣quired to be proued.

The end of the ninth booke of Euclides Elementes.

Notes

• * 1.1

  The Argumēt of the ni••th booke.
• * 1.2

  Demonstra∣tion.
• * 1.3

  This proposi∣tion is the con¦u••rse o•• t••e form••••.
• * 1.4

  Demonstra∣tion.
• * 1.5

  A Corollary a••ded by Campane.
• * 1.6

  Demonstra∣tion.
• * 1.7

  Demonstra∣tion.
• * 1.8

  Demonstra∣tion.
• * 1.9

  A Corollary added by Campane.
• * 1.10

  Demonstra∣tion.
• * 1.11

  Demonstra∣tion.
• * 1.12

  Demonstrati∣on of the first part.
• * 1.13

  The second part demon∣strated.
• * 1.14

  Demostratiō of the third part.
• * 1.15

  Demostra∣tion of the first part of this proposition.
• * 1.16

  The second p••rt demonstrated.
• * 1.17

  Demonstrati∣on of the first part lea∣uing to an absu••••itie.
• * 1.18

  Demonstra∣tion of the ••••∣cond p•••••• lea∣ding al••o to an absurditie.
• * 1.19

  Demonstra∣tion.
• * 1.20

  Demonstra∣tion leading to an absurditie.
• * 1.21

  An other de∣monstratiō a••∣ter Flussates.
• * 1.22

  Demonstra∣tion leading to an absurditie.
• * 1.23

  An other de∣monstratiō af∣ter Campane.
• * 1.24

  Demo••stration leading to an absurditie.
• * 1.25

  A propositio•• added by Campane.
• * 1.26

  Construc••ion.
• * 1.27

  Demonstra∣tion.
• * 1.28

  Demonstrati∣on to proue that the num∣bers A and C are prime to B.
• * 1.29

  Demonstra∣tion leading to an absurdi••ie.
• * 1.30

  Demonstra∣tiou.
• * 1.31

  This proposi∣tion is the cō∣uerse of the former.
• * 1.32

  Demonstra∣tion.
• * 1.33

  This answe∣reth to the 2. of the second.
• * 1.34

  Demonstra∣tion.
• * 1.35

  This answe∣reth to the 3. of the thirds.
• * 1.36

  Demonstra∣tion.
• * 1.37

  This answe∣r••th to th•• 4. of the second.
• * 1.38

  Demonstra∣tion.
• * 1.39

  This answe∣reth to the 5. of the second.
• * 1.40

  Demonstra∣tion.
• * 1.41

  This answe∣reth to the 6. of the second.
• * 1.42

  Demonstra∣tion.
• * 1.43

  This answe∣reth to the 7. of the second.
• * 1.44

  Demonstra∣tion.
• * 1.45

  This answe¦reth to the 8. of the second.
• * 1.46

  Demonstrati∣tion.
• * 1.47

  This answe∣reth to th•• 9. of the second.
• * 1.48

  Demonstra∣tion.
• * 1.49

  This answe∣reth to the 10. o•• the second.
• * 1.50

  Demonstra∣tion.
• * 1.51

  A negatiue propositi••n.
• * 1.52

  Demonstra∣tion lea••ing to an impossibi∣litie.
• * 1.53

  Demonstra∣tion leading to an absurditie.
• * 1.54

  Demonstra∣tion leading to an abjurditie.
• * 1.55

  Three cases in this proposi∣tion.
• * 1.56

  The first case.
• * 1.57

  The second case.
• * 1.58

  The third case.
• * 1.59

  Diuert cases ••n this proposi∣tion.
• * 1.60

  The first case.
• * 1.61

  The second case.
• * 1.62

  The third case.
• * 1.63

  The fourth case.
• * 1.64

  Two cases in this Proposi∣tion.
• * 1.65

  The first case.
• * 1.66

  The second case.
• * 1.67

  Demonstra∣tion.
• * 1.68

  Demonstra∣tion.
• * 1.69

  Demonstra∣tion
• * 1.70

  Demonstra∣tion.
• * 1.71

  Demonstra∣tion.
• * 1.72

  Demonstra∣tion.
• * 1.73

  Demonstra∣tion.
• * 1.74

  Demonstra∣tion.
• * 1.75

  Demonstra∣tion.
• * 1.76

  A proposition added by Campa••e.
• * 1.77

  An other ad∣ded by him.
• * 1.78

  Demonstra∣tion leading to an absurditie.
• * 1.79

  Demonstra∣tion.
• * 1.80

  Demonstra∣tion.
• * 1.81

  Demonstra∣tion leading to an absurditie.
• * 1.82

  An other de∣monstration.
• * 1.83

  Demonstra∣tion.
• * 1.84

  Demonstra∣tion.
• * 1.85

  This proposi∣tion teach••th how to finde out a perfect number.
• * 1.86

  Construction.
• * 1.87

  Demonstra∣tion.
• * 1.88

  Demonstra∣tion leading to an absurditie.