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 eighthe booke of Eu∣clides Elementes. (Book 8)

AFter that Euclide hath in the seuenth booke entreated of the proprieties of numbers in generall, and of certayne kindes thereof more specially, and of prime and composed numbers with others: now in this eight booke he prosecu∣teth farther, and findeth out and demonstrateth the pro∣perties and passions of certayne other kindes of numbers:* 1.1 as of the least numbers in proportion, and how such may be found out infinitely in whatsoeuer proportion: which thing is both delectable, and to great vse. Also here is en∣treated of playne numbers, and solide: and of theyr sides, and proportion of them. Likewise of the passions of num∣bers square and cube, and of the natures and conditions of their sides, and of the meane proportionall numbers of playne, solide, square, and cube numbers, with many other thinges very requisite and necessary to be knowne.

¶The first Theoreme. The first Proposition. If there be numbers in continuall proportion howmanysoeuer, and if their extremes be prime the one to the other: they are the least of all numbers that haue one and the same proportion with them.

SVppose that the numbers in continuall proportion be A, B, C, D. And let their extreames namely, A and D be prime the one to the other. Then I say that the numbers A, B, C, D, are the least of all numbers that haue

[illustration]

one and the same pro∣portion with thē. For if they be not, let E, F; G, H being lesse numbers then A, B, C, D, be in the selfe same proportion that ABCD are.* 1.2 And forasmuch as the numbers A, B, C, D, are in the selfe same proportion that the numbers E, F, G, H, are, & the mul••itude of these numbers E, F, G, H, is equall to the ••ult••••ude of these ••••mbers A, B, C, D, therefore of equalitie (by the 14. of the seuenth) as A is to D, so is E to H. But A and D are prime the one to the other, yea they are prime and the least that ha••e the same proportion with them: But the least numbers in a∣ny proportion measure the numbers that haue the same proportion with them equally, the antecedent the antecedent, and the consequēt the cons••••uen•• (by the 21. of the seuenth) wher¦fore A measureth E, the greater the lesse: which is impossible. Wherefore the numbers E, F, G, H, being lesse then A, B, C, D, are not in the same proportion that A, B, C, D, are, wher∣fore A, B, C, D, are the least of all numbers which ha••e one and the same proportion with

them•• which was required to be demonstrated.

¶ The 1. Probleme. The 2. Proposition. To finde out the least numbers in continuall proportion, as many as shall be required, in any proportion geuen.

SVppose that the proportion geuen in the lest numbers be A to B. It is required to finde out the lest numbers in continuall proportion, as many as shall be requi∣red, in the same proportion that A is to B.* 1.3 Let there be required foure. And let A multiplying him selfe produce C: and multiplying B let it produce D: and likewise let B multiplying him selfe produce E. And moreouer let A multiplying those num∣bers C, D, E,* 1.4 produce F, G, H: and let B multiplying E produce K. And forasmuch as A multiplying him selfe produced C, and multiplying B produced D, now then the number A multiplying two numbers A and B produced C & D. Wherefore (by the 17. of the seuenth) as A is to B, so is C to D. Againe, forasmuch as A multiplying B produced D, and B multi∣plying him selfe produced E, therefore ech of those numbers A and B multiplying B, bring∣eth forth these numbers D and E. VVherefore (by the 18. of the seuenth) as A is to B, so D to E. But as A is to B, so is C to D. Wherefore as C is to D, so is D to E. And forasmuch as A multiplying C and D produced F and G, therefore (by the 17. of the seuenth) as C is to D, so is F to G. But as C is to D, so is A to B. Wherefore as A is to B, so is F to G. Againe for∣asmuch as A multiplying D and E produced G and H, therefore (by the 17. of the seuenth) as D is to E, so is G to H. But as D is to E, so is A to B. Wherefore as A is to B, so is G to H.

[illustration]

And forasmuch as those numbers A and B multiplying E produced H and K, therefore (by the 18. of the seuenth) as A is to B, so is H to K. And it is proued, that as A is to B, so is •• to ••, and G to H: wherefore as F is to G, so is G to H, and H to K. Wherefore these numbers C, D, E, and F, G, H, K, are proportionall in the same proportion, that A is to B. Now I say, that they are also the lest. For forasmuch as A and B are the lest of all numbers that haue the same proportion with them: but the lest numbers that haue one & the same proportion with them are prime the one to the other (by the 24. of the seuenth): therefore A and B are prime the one to the other: and ech of these numbers A & B multiplying him selfe produced these numbers C and E, and likewise multiplying ech of these numbers C and E they produced F and K. Wherefore (by the 29. of the seuenth) C, E, and F, K, are prime the one to the other. But if there be n••mbers in continuall proportion how many soeuer, and if their extremes be prime the one to the other, they are the lest of all nūbers that haue the same proportion with them (by the first of the eight). Wherefore these numbers C, D, E, and F, G, H, K, are the lest

of ••ll n••mber ••hat ••a••e ••he same proportion with A and B•• And forasmuch 〈◊〉〈◊〉 (by the ••9. of the seuenth) that alwaies happeneth touching the extre•••••••• namely, that A and B ••••lti∣plying the numbers pr••duced 〈◊〉〈◊〉 shall produce ••ther prime numbers, namely, the ex∣tremes of fiue numbers in continuall proportion, therefore (by the first of this booke) all fiue are the lest of that proportion. And so infinitely: which was required to be done.

¶Corollary.

Hereby it is manifest, that if three numbers being in continuall proporti∣on, be the lest of all numbers that haue the same proportion with them, their extremes are squares: and if there be foure their extremes are cubes. For the extremes of three are produced of the multiplying of the nūbers A and B into them selues. And the extremes of foure are produced of the mul∣tiplying of the rootes A and B into the squares C and E, whereby are made the cubes F and K.

The 2. Theoreme. The 3. Proposition. If there be numbers in continuall proportion how many soeuer, and if they be the lest of all numbers that haue one and the same proportion with thē: their extremes shall be prime the one to the other.

SVppose that the numbers in continuall proportion being the least of all numbers that haue the same proportion with them be A, B, C, D.* 1.5 Then I say that their extremes A and D are prime the one to the other. Take (by the 2. of the eight, or by the 35. of the seuen••h) the two least numbers that are in the same proportion that A, B, C, D, are, and let the same be the numbers E, F. And after that take thre numbers

[illustration]

G, H, K, and so alwayes forward on more (by the former proposition) vntill the multitude ••aken be equall to the multitude of the numbers geuen A, B, C, D. And let those numbers be L, M, N, O. Wherfore (by the 29. of the seuenth) their extremes L, O,* 1.6 are prime the one to the other. For forasmuch as E and F are prime the one to the other, and eche of them mul∣tiplieng

himselfe produced G and K, & likewise ech of these, G & K multiplieng himself pro∣duced L & O, therfore (by the 29 of the seuenth) G & K are prime the 〈◊〉〈◊〉 to th•• other, & so likewise are L and O prime the one to the other. And forasmuch as A, B, C, D, are the least of

[illustration]

all numbers that haue the same proportion with them, and likewise L, M, N, O, are the least of all numbers that are in the same proportion that A, B, C, D, are, and the multitude of these numbers A, B, C, D, is equall to the multitude of these L, M, N, O: therfore euery one of these A, B, C, D, is equall vnto euery one of these L, M, N, O. Wherefore A is equall vnto L, and D is equall vnto O. And forasmuch as L and O are prime the one to the other, and L is e∣quall vnto A, and O is equall vnto D: therfore A and D are prime the one to the other: which was required to be proued.

The 2. Probleme. The 4. Proposition. Proportions in the least numbers how many soeuer beyng geuen, to finde out the least numbers in continuall proportion in the said proportions geuē.

SVppose that the proportions in the least numbers geuen, be A to B, C to D, and E to F. It is required to finde out the least numbers in continuall proportion, in the same proportion that A is to B, and that C is to D, and that E is to F. Take the least number whom B and C do measure, and let the same be G. And how often B measureth G, so many times let A measure H. And how oftē C measureth G, so many times let D measure K. Now E either measureth K or measureth it not.* 1.7 First let it measure it. And how often E measureth K, so many times let F measure L. And forasmuch as how often A measureth H, so many times doth B measure G: therfore by the 17. of the seuenth, as A is to B, so is H to G. And by the same reasō as C is to D•• so is •• to K, and moreouer as E is to F, so is K to L. Wherfore these numbers H, G, K, L, are in continuall proportion, and in the same proportion that A is to B, and that C is to D, and moreouer that E is to F. I say also that they are the least in those proportions••* 1.8 For if H, G, K, L, be not the least num∣bers in continuall proportion, and in the same proportions that A is to B, and C to D, and E to F, then are there some numbers less•• then H, G, K, L, in the same proportions that A is to B and C to D, and E to F, let those numbers be N, X, M, O. And forasmuch as A is to

B, so is N to X•• and A and B are the lest, but the least measure

[illustration]

those numbers that haue one and the same proportion with them equally, the greater, the greater, and the lesse the lesse, that is, the antecedent, the antecedent, & the consequent the consequent (by the 21. of the seuenth) therfore B measureth X. And by the same reason C also measureth X, wherfore B and C measure X. Wher∣fore the least number whom B and C measure, shall also by the 37 of the seuenth measure X. But the least number whome B and C measure is G. Wherefore G measureth X, the greater, the lesse, which is impossible. Wherfore there shall not be any lesse numbers then H, G, K, L, in continuall proportion, and in the same pro∣portions that A is to B, and C to D, and E to F.

But now suppose that E measure not K.* 1.9 And by the 36. of the seuenth, take the least number whome E and K measure, and let the same be M. And how often K measureth M, so often let either of these G and H measure either of these N and X.* 1.10 And how often E measureth M, so often let F measure O. And forasmuch as how often G measureth N, so often doth H measure X, therfore as •• is to G, so is X to N. But as H is to G, so is A to B Wherfore as A is to B, so is X to N. And by the same reason as C is to D, so is N to M. A∣gaine, forasmuch as how often E measureth M, so often F measureth O, therfore as E is to F so is M to O. Wherfore X, N, M, O, are in continuall proportion, and in the same proporti∣ons that A is to B, and C to D, and E to F. I say also that

[illustration]

they are the least in that proportion. For if X, N, M, O, be not the least in continuall proportion, and in the same pro∣portions that A is to B, and C to D, and E to F, then shall there be some numbers lesse then X, N, M, O, in continuall proportion, and in the same proportions that A is to B, and C to D, and E to F. Let the same be the numbers P, R, S, T. And for that as P is to R, so is A to B, and A and B are the least, but the least numbers measure those numbers that haue one & the same proportiō with them equally, the greater the greater, and the lesse the lesse, that is, the antece∣dent the antecedent, and the consequent the consequent by the 21. of the seuenth, therfore B measureth R. And by the same reason also C measureth R. Wherefore B and C mea∣sure R. Wherfore the least number wh••m B and C measure shall also measure R (by the 37. of the seuenth). But the lest number whom B and C measure is G, wherfore G measureth R. And as G is to R, so is K to S. Wherfore K measureth S. And E also measureth S. Wherefore E and K measure S. Wherfore the least number whom E and K measure, shall (by the selfe same) measure S. But the least number whom E and K measures M. Wherefore M measureth S the greater the lesse, which is impossible. Wherfore there are no numbers lesse then X, N, M, O, in continuall proportion, & in the same proportions that A is to B, and C to D, and E to F. Wherfore X, N, M, O are the least numbers in continuall proportion, and in the same proportions that A is to B, and C to D, and E to F: which was required to be done.

¶ The 3. Theoreme. The 5. Proposition. Playne or superficiall numbers are in that proportion the one to the other which is composed of the sides.

* 1.11SVppose that A and B be playne or superficiall numbers, and let the sides of A be the numbers C and D, and let the sides of B, be the numbers E and F. Then I say that A is to B in that proportion that is composed of the sides. Take (by the fourth of the

[illustration]

eight) the least numbers in continuall proportion, and in the same proportions that C is to E, and D to F.* 1.12 And let the same be the numbers G, H, K:* 1.13 so that as C is to E, so let G be to H, & as D is to F, so let H be to K. Wherefore those numbers G, H, K, haue the proporti∣ons of the sides: but the pro∣portion of G to K is con po∣sed of that which G hath to H and of that which H hath to K: wherefore G is vnto K in that proportion which is composed of the sides. Now I say that as A is to B, so is G to K. For let D multiplying E produce L. And forasmuch as D multiplying C produced A, and mul∣tiplying E produced L: therefore (by the 17. of the seuenth) as C is to E, so is A to L. But as C is to E, so is G to H, wherefore as G is to H, so is A to L. Agayne forasmuch as E mul∣tiplying D produced L, & multiplying F produced B: therefore (by the 17. of the seuenth) as D is to F, so is L to B. But as D is to F, so is H to K, wherefore as H is to K, so is L to B. And it is proued that as G is H, so is A to L. Wherefore of equalitie (by the 14. of the seuenth) as G is to K, so is A to B. But G is vnto K in that proportion which is composed of the sides, wherefore A is vnto B in that proportion which is composed of the sides: which was required to be demonstrated.

¶An other demonstration of the same after Campane.

Suppose that A and B be plaine numbers: and let the sides of A be the numbers C and D: and let the numbers E and F be the sides of the number B. And let D multiplying E produce the number G. Thē I say that the proportiō of

[illustration]

A to B is cōposed of the propor¦tiōs of C to E & D to F that is,* 1.14 of the sides of the superficial nū¦ber A to the sides of the super∣ficiall number B. For forasmuch as D multiplying E produced G, and multiplying C it produced A, therefore by (the 17. of the seuenth) A is to G as C is to E: agayne forasmuch as E multiplying D produced G and multiplying F it produ∣ceth B, therefore by the same G is to B as D is to F. Wherefore the proportions of the sides namely, of C to E and of D to F are one and the same with the proportions of A to G and G to B. But (by the fifth definition of the sixth) the proportion of the extremes A to B is composed of the proportions of the meanes, namely, of A to G and G to B, which are proued to be one and the same with the proportions of the sides C to E, and D to F. Wherefore the proportion of the superficiall numbers A to B is cōposed of the proportions of the sides C to E, and D to F. Wherefore pla••ne. &c. which was required to be proued.

¶The 4. Theoreme. The 6. Proposition. If there be numbers in continuall proportion how many soeuer, and if the first measure not the second, neither shall any one of the other measure any one of the other.

SVppose that there be numbers how many soeuer in continuall proportion, name∣ly, f••ue, A, B, C, D, E. And suppose that A measure not B. Then I say, that nei∣ther shall any other of the numbers A, B, C, D, E, measure any one of the other. That A, B, C, D, E, do not in continuall order measure one the other, it is mani∣fest: for A measureth not B. Now I say, that neither shall any other of them measure any o∣ther of them. I say that A shall not measure C. For how many in multitude A, B, C, are, take so many of the lest numbers that haue one and the same proportion with A, B, C, (by the 35. of the seuenth) and let the same be F, G, H.* 1.15 And forasmuch as F, G, H are in the selfe same

[illustration]

proportion that A, B, C, are: and the multitude of these numbers A, B, C, is equall to the mul∣titude of those numbers F, G, H, therefore of equalitie (by the 14. of the seuenth) as A is to C, so is F to H. And for that as A is to B, so is F to G, but A measureth not B, therefore nei∣ther doth F measure G. Wherefore F is not vnitie. For if F were vnitie, it should measure any number. But F and H are prime the one to the other (by the 3. of the eight). Wherefore F measureth not H: & as F is to H, so is A to C, wherefore neither doth A measure C. In like sort may we proue that neither shall any other of the numbers A, B, C, D, E, measure any other of the numbers A, B, C, D, E: which was required to be demonstrated.

¶The 5. Theoreme. The 7. Proposition. If there be numbers in continuall proportion how many soeuer, and if the first measure the last, it shall also measure the second.

SVppose that there be a multitude of numbers in continuall proportion, namely, A, B, C, D. And let A the first measure D the last.* 1.16

[illustration]

Then I say, that A the first measureth B the second. For if A do not measure B, neither shall any o∣ther measure any other (by the 7. of the eight): which (by supposition) is not true. For A is supposed to measure D. Now then A measuring D, shall also measure B: which was required to be proued.

¶ The 6. Theoreme. The 8. Proposition. If betwene two numbers there fall numbers in continuall proportion: how many numbers fall betwene them, so many also shall fall in continuall pro∣portion betwene other numbers which haue the selfe same proportion.

SVppose that betwene the two numbers A and B, do fall in continuall proporti∣on the numbers C and D. And as A is to B, so let E be to F. Then I say, that how many numbers in continuall proportion do fall betwene A and B, so many numbers also in continuall proportion shall there fall betwene E and F. How many A, B, C, D, are in multitude, take (by the 35. of the seuenth) so many of the least num∣bers that haue one and the same proportion with A, B, C, D, and let the same be G, H, K, L. Wherefore their extremes G and I. are prime the one to the other (by the 3. of the eight). And forasmuch as A and C, and D and B, are in the selfe same proportion that G & H, and K and L are,* 1.17 and the multitude of these numbers A, C, D, B, is equall to the multitude of these numbers G, H, K, L: therefore of equalitie (by the 14. of the seuenth) as A is to B, so is

[illustration]

G to L. But as A is to B, so is E to F. Wherefore as G is to L, so is E to F. But G and L are prime the one to the other: yea they are prime and the least. But the least numbers measure those numbers that haue the same proportion with them equally, the greater the greater, and the lesse the lesse (by the 21. of the seuenth) that is, the antecedent, the antecedent, & the con∣sequent, the consequent. Wherefore how many times G measureth E, so many times L mea∣sureth F•• How often G measureth E, so often let H measure M, and K measure N Where∣fore these numbers G, H, K, L, equally measure these numbers E, M, N, F. Wherefore (by the 18. of the seuēth) these nūbers G, H, K, L, are in the selfe same proportion that E, M, N, F, are. But G, H, K, L, are in the selfe same proportion that A, C, D, B, are: wherefore those numbers A, C, D, B, are in the selfe same proportion that E, M, N, F, are•• But A, C, D, B, are in continuall proportion: wherefore also E, M, N, F, are in continuall proportion. Wherfore how many numbers in continuall proportion fall betwene A and B, so many also in continu∣all proportion fall there betwene E and F: which was required to be demonstrated.

A Corollary added by Flussates.

* 1.18Betwene two numbers whose proportion is superparticular, or superbipartient, there falleth no mean•• proportio∣nall. For the least numbers of that proportion differ the one from the other onely by vnitie or by two.

But if betwene the greater numbers of that proportion there should fall a meane proportionall then should there fall also a meane proportionall betwene the least numbers which haue the same proporti∣on by this Proposition. But betwene numbers differing onely by vnitie or by two, there falleth no meane proportionall.

¶ The 7. Theoreme. The 9. Proposition. If two numbers be prime the one to the other, and if betwene them shall fall numbers in continuall proportion: how many numbers in continuall pro∣portion fall betwene them, so many also shall fall in continuall proportion betwene either of those numbers and vnitie.

SVppose that there be two numbers prime the one to the other A and B: and let there fall betwene them in continuall proportion these numbers C and D: and let E be vnitie. Then I say, that how many numbers in continuall proportion fall betwene A and B, so many also shall fall in continuall proportion betwene A and vnitie E:* 1.19 and likewise betwene B and vnitie E. Take (by the 35. of the seuenth) the two least numbers that are in the same proportion that A, C, D, B, are: and let the same be F and G: and then take three of the least nūbers that are in the same proportion that A, C, D, B, are: and let the same be H, K, L: and so alwaies in order one more, vntill the multitude of them be equall to the multitude of these numbers A, C, D, B: and those being so taken let them be M, N, X, O. Now it is manifest, that F multiplying him selfe produced H, and multiplying N produced M.* 1.20 And G multiplying him selfe produced L, and multiplying L produced O. And forasmuch as M, N, X, O, are (by supposition) the least of all numbers that haue the same proportion with G, F: and A, C, D, B, are (by the first of the eight) the least of all numbers that haue the same proportion with G, F: and the multitude of these numbers M, N, X, O, is equall to the multitude of these numbers A, ••, D, B: therefore euery one of these numbers M, N, X, O, is equall to euery one of these numbers A, C, D, B. Wherefore M is equall vnto

[illustration]

A, and O is eq••all vnto B. And forasmuch as F multiplying him selfe produced H: therfore •• measureth H by those vnities which are in F: and vnitie E measureth F by those vnities which are i•• F: wherfore (by the 15. of the seuenth) vnitie E, so many times measureth the number F, as F measureth H•• wherefore as vnitie E is to the number F, so is F to H. Againe forasmuch as F multiplying N produced M, therfore H measureth M by those vnities which are in F. And vnitie E measureth F by th••se vnities which are in F: wherefore (by the self same) vnitie E so many times measureth F, as H measureth M. Wherefore as vnitie E is to the numbers F, so is H to M. But it is proued, that as vnitie E is to the number F, so is F to H: wherefore as vnitie E is to the number F, so is F to H, and H to M. But M is equall vnto A•• wherefore as vnitie E is to the number F, so is F to H, & H to A. And by the same reason as vnitie E is to the number G, so is G to L and L to B. Wherefore how many num∣bers fall in continuall proportion betwene A and B: so many numbers also in continuall pro∣portion fall there betwene vnitie E and the number A, and likewise betwene vnitie E

and the number ••: which was required to be demonstrated.

¶ The 8. Theoreme. The 10. Proposition. If betwene two numbers and vnitie fall numbers in continuall proportion: how many numbers in continuall proportion fal betwene either of them & vnitie so many also shall there fall in continuall proportion betwene them.

SVppose that betwene the two numbers A, B, and vnitie C•• do fall these numbers in continuall proportion D, E, and F, G.* 1.21 Then I say that how many numbers in con∣tinuall proportion there are betwene either of these A, B, and vnitie C, so many

[illustration]

numbers also in continuall proportion shall there fall betwene A and B. Let D multiply∣ing F produce H, and let D multiplying H produce K, and like wise let F multiplying H produce L.* 1.22 And for that by supposition as vnitie C is to the number D, so is D to E, there∣fore how many times vnitie C measureth the number D,* 1.23 so many times doth D measure E. But vnitie C measureth D by those vnities which are in D•• wherefore D measureth E by those vnities which are in D. Wherefore D multiplying himselfe produceth E. Againe for that as vnitie C is to the number D, so is E to A, therefore how many times vnitie C mea∣sureth the number D, so many times E measureth A. But vnitie C measureth D, by those vnitie•• which are in D, therefore E measureth A by those vnities which are in D. Where∣fore D multilying E produced A. And by the same reason F multiplying himselfe produced G, and multiplying G produced B. And forasmuch as D multiplying himselfe produced E, and multiplying F produced H, therefore (by the 17. of the seuenth) as D is to F, so is E to H. And by the same reason as D is to F, so is H to G. Wherefore as E is to H, so is H to G. Agayne forasmuch as D multiplying E produced A, and multiplying H produced K, there∣fore (by the 17. of the seuēth) as E is to H, so is A to K. But as E is to H, so is D to F, there∣fore as D is to F, so is A to K. Againe forasmuch as D multiplying H produced K, and F multiplying H produced L, therefore (by the 17. of the seuenth) as D is to F, so is K to L. But as D is to F, so is A to K, wherfore as A is to K, so is K to L. Againe forasmuch as F multiplying H produced L and multiplying G produced B, therefore (by the 17. of the seuenth) as H is to G, so is L to B. But as H is to G so is D to F, wherefore as D is to F so i•• L to B. And it is proued that as D is to F, so is A to K, and K to L, and L to B. Wherfore the numbers A, K, L, B, are continuall proportion. Wherefore how many numbers in conti∣nuall proportion fall betwene either of these numbers A, B, & vnitie C, so many also in con∣tinuall proportion fall there betwene the numbers A and B: which was required to be proued.

¶ The 9. Theoreme. The 11. Proposition. Betwene two square numbers there is one meane proportional number. And

a square number to a square, is in double proportion of that which the side of the one is to the side of the other.

SVppose that there be two square numbers A and B, and let the side of A be C, & let the side of B be D. Then I say that betwene these square numbers A and B, there is one meane proportionall number, and also that A is vnto B in double proportion of that which C is to D. Let C multiplieng D produce E.* 1.24 And foras∣much as A is a square nūber, & the side thereof is C,

[illustration]

therfore C multiplieng himselfe produced A. And by the same reason D multiplieng himselfe produced B. Now forasmuch as C multiplieng C produced A, and multiplieng D produced E, therfore (by the 17. of the seuenth) as C is to D, so is A to E. Againe forasmuch as C multiplieng D produced E, and D multiplieng himselfe produced B, therefore these two numbers C and D multiplieng one number, namely, D, produce E and B. Wherfore (by the 18. of the seuenth) as C is to D, so is E to B. But as C is to D, so is A to E. Wherefore as A is to E, so is E to B. Wherefore betwene these square numbers A and B, there is one meane proportionall number, namely, E.* 1.25 Now also I say that A is vnto B in double proportion of that which C is to D. For forasmuch as there are three numbers in continuall proportion, A, E, B, therfore (by the 10. definition of the fift) A is vnto B in double proportiō of that which A is to E. But as A is to E, so is C to D. Wherefore A is vnto B in double proportion of that which the side C is vnto the side D: which was required to be proued.

¶ The 10. Theoreme. The 12. Proposition. Betwene two cube numbers there are two meane proportionall numbers. And the one cube is to the other cube in treble proportion of that which the side of the one is to the side of the other.

••Vppose that there be two cube numbers A and B, and let the side of A be C, and let the side of B be D. Then I say that betwene those cube numbers A and B, there are two meane proportionall numbers, and that A is vnto B in treble proportion of that which C is to D. Let C multiplieng himselfe produce E, and multiplieng D let it produce F,* 1.26 and let D multiplieng himselfe produce G. And let C multiplieng F produce H, and let D

[illustration]

multiplieng F produce K.* 1.27 And forasmuch as A is a cube number, and the side therof is C, & C multiplieng himselfe produceth E, therfore C multiplieng E produceth A. And by the same

reason for that D multiplieng himselfe, produced G, therfore D multiplieng G produceth B. And forasmuch as C multiplieng C and D produced E and F: therfore by the 17. of the fift, as C is to D, so is E to F. And by the same reason also, as C is to D, so is F to G. Againe foras∣much as C multiplieng E and F produced A and H, therfore as E is to F, so is A to H. But as E is to F, so is C to D. Wherfore as C is to D: so is A to H. Againe forasmuch as eche of these

[illustration]

numbers C and D multiplieng F produced H and K, therfore (by the 18. of the seuenth) as C is to D, so is H to K. Againe forasmuch as D multiplieng F and G produced K & B: ther∣fore (by the 17. of the seuenth) as F is to G, so is K to B. But as F is to G, so is C to D, where∣fore as C is to D, so is K to B. And it is proued that as C is to D, so is A to H, and H to K, and K to B: wherfore betwene these cube numbers A and B, there are two meane proportionall numbers, that is, H and K.

Now also I say, that A is vnto B in treble proportion of that which C is to D.* 1.28 For foras∣much as there are foure numbers proportionall A, H, K, B, therfore (by the 10. definition of the fift) A is vnto B in treble proportion of that which A is vnto H. But as A is vnto H, so is C to D, wherfore A is vnto B in treble proportion of that which C is to D: which was re∣quired to be proued.

¶ The 11. Theoreme. The 13. Proposition. If there be numbers in continuall proportion how many so euer, and ech multiplying himselfe produce certayne numbers, the numbers of them pro∣duced shall be proportinall. And if those numbers geuen at the beginning multiplying the numbers produced, produce other numbers, they also shalbe proportionall: and so shall it be continuing infinitely.

SVppose that there be a multitude of nūbers in cōtinuall proportiō, namely, A, B, C, as A is to B, so let B be to C. And let A, B, C, multiplying ech himself bring forth the nūbers D, E, F, & multiplying the nūbers D, E, F, let thē bring forth the nū∣bers G, H, K. Thē I say that D, E, F, are in cōtinuall proportiō, and also that. G, H, K, are in cōtinuall proportiō.* 1.29 For it is manifest that the nūbers D, E, F, are square numbers, & that the nūbers G, H, K, are cube nūbers. Let A multiplying B produce L. And let A & B mul∣tiplying L produce M and N. And againe let B multiplying C produce X: and let B and C multiplying X produce O and P.* 1.30 Now by the discourse of the proposition going before we

may proue that D, L, E, and also G, M,

[illustration]

N, H, are in continuall proportion and in the same proportion that A is to B and likewise that E, X, F, and also H, O, P, K, are in continuall proportion and in the same proportion that B is to C. But as A is to B, so is B to C. Wherefore D, L, E, are in one and the same proportion with E, X, F and moreouer G, M, N, H, are in one and the same proportion with H, O, P, K, and the multitude of these numbers D, L, E, is equall to the multitude of these numbers E, X, F, and likewise the multi∣tude of these numbers G, M, N, H, is e∣quall to the multitude of these numbers H, O, P, K, wherefore of equality (by the 14. of the seuenth) as D is to E, so is E to F. And as G is to H, so is H to K: which was required to be proued.

¶ The 12. Theoreme. The 14. Proposition. If a square number measure a square number, the side also of the one shall measure the side of the other. And if the side of the one measure the side of the other, the square number also shall measure the square number.

SVppose that there be two square numbers A and B, and let the sides of them be C and D•• and let A measure B. Wherefore C also shall measure D.* 1.31 Let C mul∣tiplying D produce E. Wherefore (by the 17. and 18. of the seuenth, and 13. of the eight) those numbers A, E, B, are in continuall proportion, and are in the same proportion that C is to D. And forasmuch as A, E, B, are in continuall proportion, and A measureth B; therefore (by the 7. of the eight) A measureth E. But as A is to E, so is C to D: wherefore C measureth D.

[illustration]

But now suppose that the side C do measure the side D.* 1.32 Then I say, that the square number A also measureth the square number B. For the same order of construction remayning, we may in like sort proue, that the numbers A, E, B, are in continuall proportion, & in the same proportion, that C is to D. And for that as C is to D, so is A to E, but C measureth D: ther∣fore A measureth E: and A, E, B, are in continuall proportion: wherefore A measureth B. If therefore a square number measure a square number, the side also of the one shall measure the side o•• the other. And if the side of the one measure the side of the other, the square num∣ber also shall measure the square number: which was required to be demonstrated.

¶ The 13. Theoreme. The 15. Proposition. If a cube number measure a cube number, the side also of the one shall mea∣sure

the side of the other. And if the side of the one measure the side of the other, the cube number also shall measure the cube number.

SVppose that the cube number A do measure the cube number B, and let the side of A be C, and the side of B be D. Then I say, that C measureth D. Let C mul∣tiplying him selfe produce E, & multiplying D let it produce F. And let D mul∣tiplying him selfe produce G. And moreouer let C and D multiplying F produce H and K.* 1.33 Now it is manifest (by the 17, and 18. of the seuenth, and 12. of the eight) that those numbers E, F, G, and also A, H, K, B, are in continuall proportion, & in the same pro∣portion that C is to D. And forasmuch as A, H, K, B, are in continuall proportion, and A measureth B, therefore (by the 7. of the eight) A also measureth H. But as A is to H, so is C to D. Wherefore C also measureth D.

[illustration]

* 1.34But now suppose that the side C do measure the side D. Then I say, that the cube number A also measureth the cube number B. For the same order of construction being kept, in like sort may we proue, that A, H, K, B, are in continuall proportion, and in the same proportion that C is to D. And forasmuch as C measureth D, but as C is to D, so is A to H, therefore A measureth H: wherefore A also measureth B. If therefore a cube number measure a cube number, the side also of the one shall measure the side of the other. And if the side of the one measure the side of the other, the cube number also shall measure the cube number: which was required to be proued.

¶ The 14. Theoreme. The 16. Proposition. If a square number measure not a square number, neither shall the side of the one measure the side of the other. And if the side of the one measure not the side of the other, neither shall the square number measure the square number.

* 1.35SVppose that A and B be two square numbers, and let

[illustration]

the side of A be C: and let the side of B be D. And be it that A measureth not B. Then I say, that neither shall C measure D. For if C do measure D, then (by the 14. of the eight) A also measureth B. But A by supposition measureth not B: wherefore neither doth C measure D.

* 1.36But now againe suppose that the side C measure not the side D. Then I say, that neither shall the square number A measure the square number B. For if A do measure B, then shall

C (by the 14. of the eight) measure D. But (by supposition) C measureth not D. Wherefore neither doth A measure B: which was required to be proued.

¶ The 15. Theoreme. The 17. Proposition. If a cube number measure not a cube number, neither shall the side of the one measure the side of the other. And if the side of the one measure not the side of the other, neither shall the cube nūber measure the cube number.

SVppose that the cube number A do not measure the cube number B:* 1.37 and let the side of A be C: and the side of B be D. Then I say, that C shall not measure D. For if C do measure D, then (by the 15. of the eight) A also shall measure B.* 1.38 But (by sup∣position) A measureth not B: wherefore nei∣ther

[illustration]

shall C measure D.

But now suppose that the side C measure not the side D.* 1.39 Then I say, that neither shall the cube number A measure the cube number B. For if A do measure B, then also (by the 15. of the eight) shall C measure D. But (by suppo∣sition) C measureth not D. Wherefore neither shall A measure B: which was required to be proued.

¶ The 16. Theoreme. The 18. Proposition. Betwene two like plaine or superficiall numbers there is one meane propor∣tionall number. And the one like plaine number is to the other like plaine number in double proportion of that which the side of like proportion, is to the side of like proportion.

SVppose that there be two like plaine or superficiall numbers A & B. And let the sides of A be the nūbers C, D: and the sides of B be the numbers E, F. And forasmuch as like plaine numbers are those which haue their sides proportionall (by the 22. defini∣tion of the seuenth) therefore as C is to D, so is E to F.* 1.40 Then I say that betwene A and B there is one meane proportionall number, and that

[illustration]

A is vnto B in double proportiō of that which C is vnto E, or of that which D is vnto F, that is, of that which side of like proportion is to side of like proportion. For for that as C is to D, so is E to ••, therefore alternately (by the 13. of the seuenth) as C is to E, so is D to F. And forasmuch as A is a plaine or superfi∣ciall number, and the sides thereof are C and D: therefore D multiplying C produced A. And by the same reason also E multiplying F produced B. Let D multiplying E produce G. And forasmuch as D multiplying C produced A, and multiplying E produced G, therefore (by the 17. of the seuenth) as C is to E, so is A to G. But as C is to E, so is D to F, wherefore as D is to F, so is A to G. Againe forasmuch as E multiplying D produced G, and multiplying •• produced B, therefore (by the 17. of the seuenth) as D is to F, so is G to B. But it is proued

that as D is to F, so is A to G: wherfore as A is to G, so is G to B. Wherefore these numbers A, G, B, are in continuall prorortion. Wherefore betwene A and B there is one meane proportio∣nall number.

* 1.41Now also I say that A is vnto B in dooble proportiō of that which side of like proporti∣on is to side of like proportion, that is, of that which C is vnto E, or of that which D is vnto F. For forasmuch as A, G, B, are in continuall proportion, therefore (by the 10. definition of the ••ift) A is vnto B in double proportion of that which A is vnto G. But as A is to G, so is C to E, and D to F: wherefore A is vnto B in double proportion of that which C is to E, or D to F: which was required to be demonstrated.

¶ The 17. Theoreme. Th 19. Proposition. Betwene two like solide numbers, there are two meane proportionall num∣bers. And the one like solide number, is to the other like solide number in treble proportion of that which side of like proportion is to side of lyke proportion.

SVppose that there be two like solide numbers A and B. And let the sides of the number A, be the numbers C, D, E. And let the sides of the number B, be the numbers F, G, H. And forasmuch as (by the 22. definition of the seuenth) lyke solide numbers haue their sides proportio∣nall,

[illustration]

therfore as C is to D, so is F to G, and as D is to E, so is G to H. Then I say that betwene A and B, there are two meane pro∣portionall numbers. And that A is vnto B in treble proportion of that which C is to F, or of that which D is to G, or moreouer of that which E is vnto H.

For let C multiplieng D produce K. And let F multiplieng G, produce L.* 1.42 And for∣asmuch as C, D, are in the self same propor∣tiō that F, G, are, & of C & D is produced K, and of F and G is produced L, therefore K and L are like plaine numbers. And ther∣fore betwene those numbers K and L, there is one meane proportionall number (by the 18. of the se••enth) Let the same be M. Wher¦••ore M is produced of D and F, as it is ma∣••••fest by the proposition goyng before. Wherfore as K is to M, so is M to L. And forasmuch as D multiplieng C produced K, and multiplieng F produced M: therfore (by the 17. of the seuenth) as C is to F, so is K to M, but as K is to M, so is M to L. Wherfore these numbers K M, L, are in continuall proportion and in the same proportion that C is to D. And for that as C is to D, so is F to G, therfore alternately (by the 13. of the seuenth) as C is to F, so is D to G. Againe, for that as D is to E, so is G to H, therfore alternately also as D is to G, so is E to H. Wherfore these numbers K, M, L, are in continuall proportion, and in the same propor∣tion that C is to F, and that D is to G, and moreouer that E is to H. Now let E and H multi∣plieng M produce N and X. And forasmuch as A is a solide number, and the sides thereof are C, D, E, therefore E multiplieng that which is produced of C and D, produceth A. But that which is produced of C and D is K. Wherfore E multiplieng K produceth A. And (by the same reason H multiplieng that which is produced of F and G, that is multiplieng L pro¦duceth

B. And forasmuch as E multiplieng K produced A and multiplieng M produced N, therfore (by the 17. of the seuēth), as K is to M, so is A to N. But as K is to M, so is C to F, & D to G. and moreouer E to H, therfore as C is to F, and D to G, and E to H, so is A to N. A∣gayne, forasmuch as E multiplieng M produced N, and H multiplieng M, produced X, ther∣fore (by the 18. of the seuenth) as E is to H, so is N to X. But as E is to H, so is C to F, and D to ••. Wherfore as C is to F, and D to G, and E to H, so is A to N, and N to X. Againe foras∣much as H multiplieng M, produced X, and multiplieng L produced B, therefore (by the 17. of the seuenth) as M is to L, so is X to B. But as M is to L, so is C to F, and D to G, and E to H therfore as C is to F, and D to G, and E to H•• so is not onely X to B, but also A to N, and N to X. Wherfore these numbers A, N, X, B, are in continuall proportion, and that in the proporti∣ons of the sides.* 1.43 I say moreouer that A is vnto B in treble proportion of that, which side of like proportion, is to side of like proportion, that is, of that which the number C hath to the num∣ber F, or of that which D hath to G, or moreouer of that which E hath to H. For forasmuch as there are foure numbers in continual proportion, that is, A, N, X, B, therfore (by the 10. de∣finition of the fift) A is vnto B in treble proportion of that which A is vnto N. But as A is to N, so is it proued that C is to F, and D to G, and moreouer E to H. Wherefore A is vnto B in treble proportion of that which side of like proportion is vnto side of like proportion, that is of that which the number C is to the number F, and of that which D is to G, and moreouer of that which E is ••o H: Which was required to be proued.

¶ The 18. Theoreme. The 20. Proposition. If betwene two numbers there be one meane proportionall number: those numbers are like plaine numbers.

SVppose that betwene the two numbers A and B there be one meane proportionall number, and let the same be C.* 1.44 Then I say, that those numbers A and B are like plaine numbers. Take (by the 35. of the seuenth) two of the least numbers that haue one & the same proportion with A, C, B: and let the same be the numbers D, E.* 1.45 Wherefore as D is to E, so is A to C, but as A is to C, so is C to B, wherefore as D is to E, so is C to B. Wherefore how many times D measureth A, so many times doth E measure C. How many times D measureth A, so many vnities let there be in F. Wherefore F multi∣plying D produceth A, and multiplying E it

[illustration]

produceth C: wherefore A is a plaine number: and the sides therof are D and F (by the 17. de∣finition of the seuenth). Againe forasmuch as D and E are the lest numbers that haue one & the same proportion with C, B, therefore (by the 21. of the seuenth) how many times D measu∣reth C, so many times doth E measure B. How often E measureth B, so many vnities let there be in G. Wherefore E measureth B by those v∣nities which are in G: wherefore G multiply∣ing E produceth B: wherefore B is a plaine number (by the 17. definition of the seuenth).* 1.46 And the sides thereof are E and G. Wherefore those two numbers A and B are two plaine numbers. I say moreouer that they are like. For forasmuch as F multiplying E produced C: and G multiplying E produced B: therefore (by the 17. of the seuenth) as F is to G, so is C to B, but as C is to B, so is D to E, wherefore as D is to E, so is F to G. Wherefore A and B are like plaine numbers, for their sides are proportionall: which was required to be proued.

¶The 19. Theoreme. The 21. Proposition. If betwene two numbers, there be two meane proportionall numbers, those numbers are like solide numbers.

* 1.47SVppose that betwene two numbers A and B, there be two meane proportionall numbers C, D. Then I say that A and B are like solide numbers. Take (by the 3•• of the seuenth, or 2. of the eight) three of the least numbers that haue one and the same proportion with A, C, D, B, and let the same be E, F, G. Wherefore (by the 3. of the eight) their extremes E, G are prime the one to the other. And forasmuch as betwene the numbers E and G there is one meane proportionall number:* 1.48 therfore (by the 20 of the eight) they are like plaine numbers. Suppose that the sides of E, be H and K. And let the sides of G, be L and M. Now it is manifest that these numbers E, F, G, are in continuall

[illustration]

proportion,* 1.49 and in the same proportion that H is to L, and that K is to M. And forasmuch a•• E, F, G are the least numbers that haue one and the same proportion with A, C, D, therefore of equalitie (by the 14. of the seuenth) as E is to G, so is A to D. But E, G, are (by the 3. of the eight) prime numbers, yea they are prime and the least, but the least numbers (by the 21. of the seuenth) measure those numbers that haue one & the same proportion with them equal∣ly, the greater the greater, and the lesse the lesse, that is, the antecedent the antecedent, & the consequent the consequent: therfore how many time•• E measureth A, so many times G mea∣sureth D. How many times E measureth A, so many vnities let there be in N. Wherefore N multiplieng E, produceth A. But E is produced of the numbers H, K. Wherfore N multipli∣eng that which is produced of H, K, produceth A. Wherefore A is a solide number, and the sides therof are H, K, N. Agayne, forasmuch as E, F, G, are the least numbers that haue one and the same proportion with C, D, B, therefore how many times E measureth C, so many times G measureth B. How oftētimes G measureth B, so many vnities let there be in X. Wher∣fore G measureth B by those vnities which are in X. Wherfore X multiplieng G produceth B. But G is produced of the numbers L, M. Wherefore X multiplieng that number which is pro∣duced of L and M, produceth B. Wherfore B is a solide number, and the sides therof are L, M X. Wherfore A, B are solide numbers. I say moreouer that they are like solide numbers. For forasmuch as N and X multiplieng E produced A and C: therfore by the 18. of the seuenth,

as N is to X, so i•• A to C, that is E ••o F. But as E is to F, so is H to L, and K to M: therefore as H is to L, so is K to M, and N to X. And H, K, N, are the sides of A, and likewise L, M, X, a•••• th•• sides of B•• wherfore A, B•• are like solide numbers: which was required to be proued.

¶ The 20. Theoreme. The 22. Proposition. If three numbers be in continuall proportion, and if the first be a square number, the third also shall be a square number.

SVppose that there be three numbers in continuall proportion A, B, C, and let the first be a square number.* 1.50 Then I say that

[illustration]

the third is also a square number. For for∣asmuch as betwene A and C there is one meane pro∣portionall number•• namely B, therefore (by the 20. of the eight) A and C are like playne numbers. But A is a square number. Wherefore C also is a square number: which was required to be proued.

¶The 21. Theoreme. The 23. Proposition. If foure numbers be in continuall proportion, and if the first be a cube nū∣ber, the fourth also shall be a cube number.

SVppose that there be foure numbers in

[illustration]

continuall proportion A, B, C, D.* 1.51 And let A be a cube number. Thē I say that D also is a cube number. For forasmuch as be∣twene A and D there are two meane proportio∣nall numbers B•• C. Therfore A, D are like solide numbers (by the 21. of this booke) But A is a cube number, wherfore D also is a cube number•• which was required to be demonstrated.

¶The 22. Theoreme. The 24. Proposition. If two numbers be in the same proportiō that a square number is to a square number, and if the first be a square number, the second also shall be a square number.

SVppose that two numbers A and B be in the same proportion, that the square number C is vnto the

[illustration]

squ•••••• nūber D. And let A be a square nū∣ber.* 1.52 Then I say that B also is a square number. For forasmuch as C and D are square numbers. Therfore G and D are like plaine numbers. Wherfore (by the 18. of the eight) betwene C and D there is one meane proportionall number. But as C is to D, so is A to B. Wherfore betwene A and B there is one meane proportionall number (by the 8. of the eight) But A is a square number. Wherfore (by the 22. of the eight) B also is a square number which was ••equired to be proued.

¶ The 23. Theoreme. The 25. Proposition. If two numbers be in the same proportion the one to the other, that a cube number is to a cube number, and if the first be a cube number, the second al∣so shall be a cube number.

SVppose that two numbers A and B be in the same proportiō the one to the other, that the cube nūber C is vnto the cube number D. And let A be a cube number. Then I say that B also is a cube nūber. For forasmuch as C, D, are cube nūbers, therfore C,* 1.53 D are like solide numbers, wherfore (by the 19. of the eight) betwene

[illustration]

C and D there are two proportionall numbers. But how many numbers fall in continual pro∣portion betwene C and D, so many (by the 8. of the eight) fal there betwene the numbers that haue the same proportion with them. Wherefore betwene A and B there are two meane pro∣portionall numbers which let be E and F. And forasmuch as there are foure numbers in con∣tinuall proportion, namely, A, E, F, B, and A is a cube number, therefore (by the 2••. of the eight) B also is a cube number: which was required to be demonstrated.

A Corollary added by Flussates.

* 1.54Betwene a square number and a number that is not a square number, falleth not the proportion of one square number to an other. For if the first be a square number, the second also should be a square number which is contrary to the supposition. Likewis•• betwene a cube number, and a number that is no cube number falleth not the propor∣tion of one cube number to an other. For if the first be a cube number, the second also should be a cube number, which is contrary to the supposition, & therfore impossible.

¶The 24. Theoreme. The 26. Proposition. Like playne numbers, are in the same proportion the one to the other, that a square number is to a square number.

SVppose that A and B be like plaine numbers. Then I say that A is vnto B in the same proportiō that a square number is to a square number. For forasmuch as A, B, are like plaine numbers,* 1.55 there∣fore

[illustration]

betwene A and B there falleth one meane proportional nūber (by the 18. of the eight). Let there fal such a number, and let the same be C. And (by the 35. of the seuenth) take the three least num∣bers that haue one and the same propor∣tion with A, C, B, and let the same be

D, E, F: wherefore (by the corollary of the 2•• of the eight) their 〈◊〉〈◊〉 that is D, F, are square numbers. And for that as D is to F, so is A to B, (by the 14. of the seuēth): and D, F, are square numbers. Therfore A is vnto B in that proportion, that a square nūber is vnto a square num••er: which was required to be proued.

The 25. Theoreme. The 27. Proposition. Like solide numbers are in the same proportion the one to the other, that a cube number is to a cube number.

SVppose that a A and B be like solide numbers. Then I say that A is vnto B, in the same proportion, that a cube numbe is to to a cube number. For forasmuch as A, B, are like solide numbers.* 1.56 Therefore (by the 19. of the eight) betwene A and B there

[illustration]

fall two meane proportionall numbers. Let there fall two such numbers, and let the same be C and D. And take (by the 35. of the seuenth) the least numbers that haue one and the same proportion with A, C, D, B,* 1.57 and equall also with thē in multitude, and let the same be E, F, G, H. Wherfore (by the corollary of the 2. of the eight) their extreames, that is, EH, are cube numbers. But as E is to H, so is A to B. Wherefore A is vnto B in the same proportion, that a cube number is to a cube number: which was required to be demonstrated.

¶A Corollary added by Flussates.

If two nnmbers be in the same proportion the one to the other that a square number is to a square number: those two numbers shall be like superficiall numbers.* 1.58 And if they be in the same propor∣tion the one to the other that a cube number is to a cube number, they shall be like solide nūbers.

First let the number A haue vnto the number B the same proportion, that the square number C hath to the square number D. Then I say, that A and B are like superficiall nūbers. For forasmuch as betwene the square numbers C and D there falleth a meane proportionall (by the 11. of this booke) there shall

[illustration]

also betwene A and B (which haue one and the same proportion with C and D) fall a meane proportio∣nall (by the 8. of this booke). Wherefore A and B are like superficiall nūbers (by the 20. of this booke).

But if A be vnto B, as the cube number C, is to the cube number D. Then are A & B like solide num∣bers. For forasmuch as C and D are cube numbers, there falleth betwene them ••wo meane proportio∣nall

[illustration]

numbers (by the 12. of this booke). And therefore (by the 8. of the same) betwene A and B (which are in the same proportion that C is to D) there falleth also two meane proportionall numbers. Wher∣fore (by the 21. of this booke) A and B are like solide numbers.

An other Corollary added also by Flussates.

If a number multiplying a square number, produce not a square number: the sayd number multiplying shall b•• no square number.* 1.59 For if it should be a square number, then should it and the number multiplied being like superficiall numbers (by reason they are square numbers) haue a meane proportionall (by the 18. of thys booke). And the number produced of the sayd meane should be equall to the number contayned vnder the extremes, which are square numbers (by the 20. of the seuenth). Wherefore the number produced of the extremes being equall to the square num∣ber produced of the meane, should be a square number. But the sayd number by supposition, is no square number. Wherefore neither is the number multiplying the square number, a square number.

The first part of the first Corollary is the conuerse of the 26. Proposition of this booke, and hath some vse in the tenth booke. The second part of the same also is the conuerse of the 27. Proposi∣tion of the same.

The end of the eighth booke of Euclides Elementes.

Notes

• * 1.1

  The Argu•••••••• of the eight books.
• * 1.2

  Demonstra∣tion leading to an absurd••••ie.
• * 1.3

  Construction.
• * 1.4

  Demonstra∣tion.
• * 1.5

  This proposi∣tion is the ••••∣uerse of the first.
• * 1.6

  Demonstra∣tion••
• * 1.7

  Two cases in this propositiō. The first case.
• * 1.8

  Demonstra∣tion leading to an absurditie.
• * 1.9

  The second case.
• * 1.10

  Demonstra∣tion.
• * 1.11

  This proposi∣tion in num∣bers answe∣reth to the of the sixth touching parel¦lelogrammes.
• * 1.12

  Construction.
• * 1.13

  Demonstra∣tion.
• * 1.14

  An other de∣monstratiō af∣ter Campane.
• * 1.15

  Demonstration.
• * 1.16

  Demonstra∣tion leading to an impossibi∣litie.
• * 1.17

  Demonstra∣tion.
• * 1.18

  A Corollary added by Flussates.
• * 1.19

  Construction.
• * 1.20

  Demonstra∣tion.
• * 1.21

  This proposi∣tion is the con∣uerse of the former.
• * 1.22

  Construction.
• * 1.23

  Demonstra∣tion.
• * 1.24

  The first part of this propo∣sition demon∣strated.
• * 1.25

  The second part demon∣strated.
• * 1.26

  Construction.
• * 1.27

  The first part of this pr••po∣sition de••••••∣strated.
• * 1.28

  The second part demon∣strated.
• * 1.29

  Construction.
• * 1.30

  Demonstra∣tion.
• * 1.31

  The first part of this propo∣sition.
• * 1.32

  The second part is the conuerse of the first.
• * 1.33

  The first part of this proposition.
• * 1.34

  The second part is the conuerse of the first.
• * 1.35

  A negat••ue proportion.

  The first part of this propo∣sition.
• * 1.36

  The second part is the cō∣uerse of the first.
• * 1.37

  A negatiue proposition.
• * 1.38

  The first part of this propo∣sition.
• * 1.39

  The second part is the cō∣uerse of the first.
• * 1.40

  Demonstra∣tion of the fi••st part of this proposition.
• * 1.41

  Demonstra∣tion of the second part.
• * 1.42

  Demonstra∣tion of the first part of this proposition.
• * 1.43

  The second part.
• * 1.44

  This proposi∣tion is the con∣uerse of the 18. proposition.
• * 1.45

  Construction.
• * 1.46

  Demonstra∣tion.
• * 1.47

  This proposi∣tion is the con∣uerse of the 19. proposition.
• * 1.48

  Construction.
• * 1.49

  Demonstra∣tion.
• * 1.50

  Demonstra∣tion.
• * 1.51

  Demonstra∣tion.
• * 1.52

  Demonstra∣tion.
• * 1.53

  Demonstra∣tion.
• * 1.54

  A Corollary added by Flussates.
• * 1.55

  Construction.
• * 1.56

  Construction.
• * 1.57

  Demonstra∣tion.
• * 1.58

  A Corollary added by Flussates.
• * 1.59

  Another Co∣rollary added by Flussates.