Ftwo like plane numbers A, B multiplying one another, produce a number AB, the number produced AB shall be a Square number.

For A.Ba Aq. AB; wherefore fince one mean proportionall bfalls between A and B, likewife one mean proportionall number fhall fall between Aq and AB therefore being the first Aq is a fquare number, 4 the third AB fhall be a fquare number too. W.W.to be Dem.

Or thus. Let a b, c d, be like plane numbers; namely a. b:: c.d. therefore a dbc. and fo like. wife a bed,or a dbca ad ad Q; ad, Qad,

PROP. II.

A, 6.
Aq, 36. AB, 324.

bers A,B are like plane

If two numbers A, B,multiplying one another, produce a fquare number AB,thofe numnumbers.

For A. Ba:: Aq.AB;wherefore being between Aq AB, b there falls one mean proportionall number, likewife one mean fhall fall between A and B. therefore A and B are like planes. W.W.to be Dem. '

PROP

1

PRO P. III.

A, 2. Ac, 3. Acc, 64.

If a cube number Ac multiplying it felf produce

number Acc, the number produced Acc fhall be a enbe number.

For I.A :: A.Aq; Aq.Ac. therefore between I and Ac fall two mean proportionalls. But 1. Ac b 17.71 a 15.def.7. :: Ac.Acc. therefore between Ac and Acc,fall also c 8. 8. two mean proportionalls and fo by confequence feeing Ac is a cube, Acc fhall be a cube also.which was to be Dem.

Orthus, aaa (Ac) multiplyed into it felf makes

aaaaa (Acc;)this is a cube, whofe fide is aa.

જપ

‚d 23. 8.

9 AC, 8X Bc,27,

Ifa cube number Acmui

Acc, 64 Ac, Be, z16. tiplying a cube number Bc

produce a number AcBc,

the produced number AcBc fhall be a cube. ́

b 12.80

c 8.8

For Ac.Bc :: Acc. AcBc. But between Ac and a 17.7. Bcb two mean proportionall numbers fall; therefore there fall as many between Acc and AcBc. So that whereas Acc is a cube number, fach alfo. w.w.to be Dem.

AcBc shall be

Or thus. AcBcaaabbb (ababab) C: ab.

If a cube number Ac multiplying a number В produce a cute

number AcB, the number multiplyed В fhall also be a cube.

d 23

2 17.7.

c 8.8.

For Acc. AcB :: Ac. B. But between Acc and ACB & fall two mean proportionalls; therefore alfo b12.8. as many hall fall between Ac and B. whence Ac & being a cube number, B (hall be a cube number too. w.w.to be DERE.

MTM

Γκάρ.

byp. b 19.def.7. €5.9°

a 13.def.7.

b 9.4x 7. € 17.def.7.

a byp.
b 20.7.
C 12. 8,

423.

PROP. VI.

A,8. Aq,64. Ac, 512. If a number A multiplying it felf produce cube Aq,that number A it felf is a cube.

For becaufe Aqa is a cube, and AqA (Ac) balfo a cube; therefore shall A be a cube.w.w.to be Dem.

PROP. VII. ge

A,6. B,11. AB, 66.

D, 2. E, 3.

If a compofed number "A

multiplying any number B, produce a number AB, the

2

number produced AB fhall be a folid number.
a fome other
Being A is a compofed number
number D measures it, conceive by E. therefore A
DE: whence DEBAB is a folid number.
w.w.to be Dem.

39

PROP. VIII. ds:

1. a, 3.a 2, 9. a 3, 27. a4, 81. af, 243:36,729.

If from a unite there be numbers continually propor tionall how many foever (1‚a,a ‚‚a 3,a4,&c.) the third number from a unite as is a fquare number; and fo are all forward, leaving one between (a4; a 6,a,&c.) But the fourth a 3 is a cube number; and fo are all forward, leaving two between (a 6,a 9, &c.) The seventh also a6 is both a cube number and a fquare; and fo are all forward,leaving five between (a 12,a 18, &c.)

For I.a a6aaaaaa

2. a

Q.a. and a 4

Q. aaa, &c.

aaaa — Q. aa. and

aaaC.a, and a6 aaaaaa C.aa. and aaaaaaaaa = C.aaa,&c.

3.aaaaaaa C.aa Q.aaa.therefore,&c, Or according to Euclide; Becaufe 1,a :: a. a bfhall aQ:a.therefore feeing a 2,a 3,a4,are ❤ ̧ the third a 4 fhall be a fquare number; and fo likewife a 6,a8, &c. Also because 1, a4 :: a2.a 3. therefore fhalla, baixa C; a. d therefore the fourth from a namely a6,fhall be likewise a cube,&c.and confequently a 6 is both a cube and afquare number, &c.

PROP.

7.

PROP. IX.

1.3, 4. 8, 16, a 3, 64. a4, 256, &c. 1. a,8. a1, 64. a 3, 5,12. a 4, 4096.

If from a unite there be numbers how

many foever continually proportionall (1,a,a,a 3,&c.) and the number following the unite (a) be a square;then all the rest, a2,a 3,a 4, &c. fhall be squares too. But if the number next the unite (a) be a cube, then all the following numbers a 2a 3,3 4,&c shall be cube numbers.

1.Hyp. For a,a4,a6, &c. are fquare numbers by the prec. prop. also being a is taken to be a fquare, therefore the third a 3 fhall be a fquare, and like- • 22, 8. wife a f,a7, &c, and fo all.

20.7.

d 3.9.

C23.8.

2.Hyp.a is taken to be a cube, b therefore a 4,a7, b 13.8. ao are cubes; but by the prec.a 3,a6, a 9, &c. are cubes: Iaftly because 1.a :: a. aa.c therefore fhall ai Q: a. but a cube multiplyed into it felf d produces a cube; therefore a 2 is a cube, e and confequently the fourth from it as, and in like manner a8, a 11,&c.are cubes, therefore all. W. W.to be Dem.

Peradventure more clearly thus. Let b be the fide of the square number a, and fo the feries a, a 2,a }, a 4,&c.will be otherwife expreffed, thus, bb, b 4,b, *b*, &c. It is evident that all these numbers are fquares,and may be thus expreffed, Q: b, Q: bb:Q. 、、bbb, Q: bbbb, &c.

In like manner, ifb be the fide of the cube a, the feries may be expreffed thus,b3, b6,b9, b ",&c. or C: b, C: b,C:b3, C; b4,&c.

PROP. X.

t

Į‚a, a 2, a 3, a 4, a 5, à 6. If from a unite there be 1, 2, 4, 8, 16, 32,64. numbers how many foever Continually proportionall(1,a,a,a,&c.) and the number next the unite (a) be not a square number; then is none of the reft following a fquare number, ex-. cepting as the third from the unite and fo all for ward, leaving one between (a4, a6, as, &c.) M 3

But

1

a hyp.

24.8.

But if that (a)which is next after the unite,be not a cube number,neither is any other of the following numbers a cube, faving as the fourth from the unite, and so all forward leaving two between,36,39,a", &c.

1.Hyp.For it it be poffible, let a , be a fquare nume ber; therefore because a. a*:: a 4.af sand by inver bfuppof and fion as a 4:: a.a; and also as and a4 b square numbers, and the firft a 2 a fquare, therefore a shall be likewife a fquare; contrary to the Hyp sola sv. Zasa 2.Hyp. Ifit may be, let a 4 be a cube, being of dis a6. aš equality a4. a6:: a. a,, and inversely a, a:: aaj and alfo being a6 and a4 are cubes, and the first as a o cube, e therefore a fhall be a cube alfo ;"against the p brud 25 Нур.

25.8.

5.x.y. 20 def.7. 14.7.

PROP. XI.

á, a, a, a, რá. 1,3,9, 27, 81, 243, 729.

If there be numbered how many foever in cons sy. tinuall proportion from a unite (1,a,a,a,&c.) the leffe mea fureth the greater by fome one of them that are amongst the proportionall

numbers.

Because 1, a:: a. aa. 4 therefore aaaaaa

Alfo becaufe 1.aab :: a.aaa.4therefore aaa≈ aa

aa.

a4as,&c.Laftly because ra36:: a. a 4.therefore

aa

24a3 =a6,&c.

Coroll.

H

Hence, If a number that measures any one of proportional numbers,be not one of the faid numbers, neither fhall the number by which it measures the pa faid proportionall numbers,be one of them.

PROF