whole E multiplying the last produce a number F, that which is produced Fhall be a perfect number.

Take as many numbers E, G, H, L, likewise in double proportion continually; then a of e-a 14. 7. quality A. D:: E. L. b therefore AL DE cb 19.7.

F

2

c hyp

F. d whence L= Wherefore E, G, H, L, F,d 7. ax. 7. are in double proportion. Let G-E be=M, and FEN; e then M. EN. E+ Gte 35.9. H+L. f But ME. g therefore N Ef3. ax. G+H+L. b therefore F+B+C+Dg 14. 5. E+G+H+LE+N. Moreover be- h 2. ax. 18 cause Dk measures DE (F) therefore every k 7.ax.7. one, 1, A, B, C, m meafuring D, as m alfo E,1 11. dx.7. G, H, L, does meafure F. And further,no otherm 11. 9. number measures the faid F. For if there do,let

it be P, which measures F by Q. n therefore PQ 0 19.7. FD. F. o therefore E.Q:: P.D. therefore feeing A a prime number measures D, and foP 13.9. no other P measures the fame, q confequently 9 20.def.7. E does not measure Q. Wherefore E being fup-r 31. 7. pofed a prime number, it fhall be prime to Q.1 23.7. wherefore E and Q are the leaft in their pro- t 21.7. portion; t and fo E measures P as many times. u 13.7. as Q does Du therefore Q is one of them A, B, C. Let it be B, feeing then of equality B.. DE. H x and fo BH DEF PQ. * and fo alfo Q. B:: H. P. y 14. 5. y therefore HP. therefore P is alfo one of them A, B, C, &c. against the Hypothefis. Wherefore no other be- z 22.def.7. fide the forefaid numbers meafures F, and z confequently F is a perfect number. Which was to be demonftrated.

The End of the ninth Book.

N

THE

x 19.7.

194

THE TENTH BOOK

O F

EUCLIDE's ELEMENTS.

Definitions.

YOmmenfurable magnitudes are thofe, which are measured by one and the fame measure.

C

The note of commenfurability is, as AB; that is, the line of 8 foot is commenfurable to the line B of 13 foot 3 becaufe D a line of one foot measures both A and B. Alfo√ 1850; because √2 measures both 18 and 50. For

I

•2 = √9 = 3. and √

√ 25= 5. wherefore 18. 50:: 3.5.

II. Incommenfurable magnitudes are fuch, of which no common measure can be found.

Incommenfurability is denoted by this mark; as √ 6 TL √ 15 (5;) that is, √6 is incommenfurable to the number 5, or to a magnitude defigned by that number; because there is no common meafure of them, as shall appear hereafter.

III. Right lines are commenfurable in power, when the fame space does measure their fquares.

1

The

AX

20

The mark of this commenfurability is ; as AB I CD. i. e. the line AB of 6 foot is in power commenfurable to the line CD, which Eis expreffed by √20, becaufe E the Space of one foot fquare does as well meafure ABq (36) as the rectangle Xr (20) to which the Square of the line CD (20) is equal. The fame note fometimes fignifies commenfurable in power only,"

IV. Lines incommenfurable in power are fuch; to whofe fquares no space can be found to be a common measure. This incommenfurability is denoted thus; s v/8 i. e. the numbers or lines 5, and v√8 are incommenfurable in power, because their Squares 25 and 8 are incommenfurable.

V. From which it is manifeft, that to any right line given right lines infinite in multitude are both commenfurable and incommenfurable; fome in length and power, others in power only. The right line given is called a Rational line. The note of which is p.

VI. And lines commenfurable to this line, whether in length and power, or in power only, are alfo called Rational, .

VII. But fuch as are incommensurable to it, are called Irrational.

And denoted thus p.

VIII. Alfo the fquare which is made of the faid given right line is called Rational, pv. IX. And likewife fuch figures as are commenfurable to it, are Rational, pa.

X. But such as are incommensurable, Irrational, pa

XI. And nhofe right lines alfo, which contain them in power, are Irrational p.

P

Schol.

B

That the last feven definitions may be rendred more clear by an example, let there be a circle ADBP, whofe femidiameter is CB, infcribe therein the fides of the ordinate figures, as of a Hexagone BP. of a triangle AP, of a fquare BD, of a Pentagone FD. Therefore, if according to the 5. def. the femidiameter CB be the Rational line given, expreffed by the number 2. to which the other lines BP, AP, BD, FD, are to be compared, then BP a BC= 2. a cor.15.4. wherefore BP is BC, according to the 6. def. b 47. I. Also AP b√12 (for ABg (16) - BPg (4) = 12) therefore AB BC. likewife according to the 6. def. and APq (12) is sy by the 9. def. Moreover BD b = √ DCq + BCq =√8 ; whence BD is

JJ BC; and BDq pr. Laftly, FDq = 10 −√ 10. (as fhall appear by the praxis to be delivered at the 10.13.) fhall be pv, according to the 10. def. and confequenily FD√: 10 √ 20 is §, according to the 11. def.

A Poftulate.

That any magnitude may be fo often multiplied, till it exceed any magnitude whatsoever of the fame kind.

Axiomes.

1. A magnitude meafuring how many magnitudes foever, does alfo meafure that which is compofed of them,

2. A magnitude measuring any magnitude whatsoever, does likewife measure every magnitude which that measures.

3. A magnitude measuring a whole magnitude and a part of it taken away, does alfo measure the refidue.

B

PROP. I.

Two unequal magnitudes AB, C, being given, if from the greater AB there be taken away more than half (AH) and from the refidue (HB) be again taken away more than half (HI) and this be done continuH ally, there ball at length be left a certain Fmagnitude IB, less than the less of the magnitudes first given C.

a Take C fo often, till its multiple a post. 10. ACD DE do fomewhat exceed AB, and there be DE FG-GE-C. Take from AB more than half HA,and from the remainder HB more than half HI, and fo continually, till the parts AH, HI, IB, be equal in multitude to the parts DF, FG, GE. Now it is plain, that FE, which is not lefs than DE, is greater than HB, which is less than AB DE. And in like manner GE, which is not lefs than FE, is greater than IB HB. therefore C, or GE

IB. Which was to be dem.

The fame may also be demonftrated, if from AB the half AH be taken away, and again from the refidue HB the half HI, and fo forward.