PRO P. CXV. Plate V. Fig. 10.

If a Space AB be contained under a refidual-line AC (CEAE) and a binomial CB, whose names CD, db, are commenfurable to the names CE, AE, of the refidualline, and in the fame proportion (CE: AE::CD : DB.) then the right-line F which containeth in power that Space AB, is rational.

Let G be . and make the rectangle CH-Gq; (4) then shall BH (HI-IB) be a refidual-line, and Hl (a) п CD (b) п CE. (a) and BI DB. (a) and HI: BI:: CD: DB (b): CE: EA. therefore by permutation HI: CE :: BI: EA. (c) therefore BH: AC:: HI : CE :: BI: EA. wherefore fince (d) HICE, (e) thence BH AC. (f) therefore the rectangle HC BA. But HC (Gq) (b) is pv. (g) therefor BA (Fq) is pr. and confequently F is §. Which was to be demonftrated.

Coroll.

a 113. 10.
b hyp.

c 19. 5.
d 12.10.

e 10. 10.

f 1. 6. and

10. 10.

g sch. 12.

10.

Hereby it appears that a rational fuperficies may be contained under two irrational right-lines.

PROP. CXVI. Fig. 11.

Of a medial-line AB are produced infinite irrational-lines BE, EF, &c. whereof none is of the fame kind with any of the precedent.

Let AC be propounded f. and AD a rectangle contained under AC, AB; (a) therefore AD is v. Take BE√ AD. (b) then BE is p, and the fame with none of the former. For no fquare of any of the former being applied to p, makes the breadth medial. Let the rectangle DE be finished, (a) then DE fhall be py, and (b) confequently EF (V DE) fhall be p, and not the fame with any of the former, for no fquare of the former being applied to , makes the latitude BE; therefore, &c.

a iem. 38.

10.

bil
b 11. 10.

2 47.1.

€ 9. 10.

PRO P. CXVII. Plate V. Fig. 12.

Let it be required to shew that in fquare figures BD, the diameter AC is incommenfurable in length to the fide AB.

For ACq: ABq (a) :: 2 : 1 (b) 1 (b): : not Q: Q (c) b cor. 24. 8. therefore AC AB. Which was to be demonftrated. This Theorem was of great note with the ancient Philofophers; fo that he who did not understand it was efteemed by Plato undeferving the name of a man, but rather to be reckoned among brutes.

The End of the Tenth Book.

THE

The ELEVENTH BOOK

O F

EUCLIDE's ELEMENT S.

I.

A

Definitions.

Solid is that which hath length, breadth and thickness.

II. The term, or extreme of a folid is a Superficies.

III. A right-line AB, Plate V. Fig. 13. is perpendicular to a Plane CD, when it makes right-angles ABD, ABE, ABF, with all the right-lines BD, BE, BF, that touch it, and are drawn in the faid Plane.

IV. A Plane AB, Fig. 14. is perpendicular to a Plane CD, when the right-lines FG, HK, drawn in one Plane AB to the line of common fection of the two Planes EB, and making right-angles therewith, do alfo make right-angles with the other Plane CD.

V. The inclination of a right-line AB, Fig. 15. to a Plane CD, is when a perpendicular AE is drawn from A the highest point of that line AB to the plane CD, and another line EB drawn from the point E, which the perpendicular AE makes in the plane CD, to the end B of the faid line AB which is in the fame plane, whereby the angle ABE which is contained under the infifting-line AB, and the line drawn in the plane EB

is acute.

VI. The inclination of a plane AB, Fig. 16. tô a plane C, is an acute angle FGH contained under the right-lines FH, GH which being drawn in either of the planes AB, CD to the fame point H of the common fection BE, make right-angles FHB, GHB, with the Common fection BE:

VII. Planes are faid to be inclined to other planss in the fame manner, when the faid angles of inclination are equal one to another.

VIII. Parallel planes are those which being prolonged never meet.

IX. Like folid figures are fuch as are contained under like planes equal in number.

X. Equal and like folid figures are fuch as are contained under like planes equal both in multitude and magnitude.

XI. A folid angle is the inclination of more than two right-lines which touch one another, and are not in the fame fuperficies.

Or thus

A folid angle is that which is contained under more than two plane angles not being in the fame fuperficies, but confifting all at one point.

XII. A Pyramid is a folid figure comprehended under divers planes fet upon one plane (which is the bafe of the pyramid,) and gathered together to one point.

XIII. A Prism is a folid figure contained under planes, whereof the two oppofite are equal, like, and parallel; but the others are parallelograms.

XIV. A Sphere is a folid figure made when the diameter of a femicircle abiding unmoved, the femicircle is turned round about, till it return to the fame place from whence it began to be moved.

Coroll.

Hence, all the rays drawn from the center to the fuperficies of a fphere, are equal amongst themselves. XV. The Axis of a fphere, is that fixed right-line, about which the femicircle is moved.

XVI. The Center of a sphere, is the fame point with the center of the femicircle.

XVII. The Diameter of a fphere, is a right-line drawn thro' the center, and terminated on either fide in the fuperficies of the fphere.

XVIII. A Cone is a figure made, when one fide of a rectangled triangle (viz. one of those that contain the right angle) remaining fixed, the triangle is turned round about till it return to the place from whence

it

it first moved. And if the fixed right-line be equal to the other which containeth the right-angle, then the Cone is a rectangled Cone: But if it be lefs, it is an obtufe-angled Cone; if greater, an acute-angled Cone. XIX. The Axis of a Cone is that fix'd line about which the triangle is moved.

XX. The Bafe of a Cone is the circle, which is defcribed by the right-line moved about.

XXI. A Cylinder is a figure made by the moving round of a right-angled parallelogram, one of the fides thereof, (namely, which contain the right-angle) abiding fix'd, till the parallelogram be turned about to the fame place, where it began to move.

XXII. The Axis of a Cylinder is that quiefcent right-line, about which the parallelogram is turned.

XXIII. And the Bases of a Cylinder are the circles which are defcribed by the two oppofite fides in their

motion.

XXIV. Like Cones and Cylinders, are those both whofe Axes and Diameters of their Bafes are proportional.

XXV. A Cube is a folid figure contained under fix equal fquares.

XXVI. A Tetraedron is a folid figure contained under four equal and equilateral triangles.

XXVII. An Octaedron is a folid figure contained under eight equal and equilateral triangles.

XXVIII. Â Dodecaedron is a folid figure contained under twelve equal, equilateral, and equiangular Pentagones.

XXIX. An Icofaedron is a folid figure contained under twenty equal and equilateral triangles.

XXX. A Parallelepipedon is a folid figure contained under fix quadrilateral figures, whereof those which are oppofite are parallel.

XXXI, A folid figure is faid to be infcribed in a folid figure, when all the angles of the figure infcribed are comprehended either within the angles, or in the fides, or in the planes of the figure wherein it is infcribed.

XXXII. Likewife a folid figure is then faid to be circumfcribed about a folid figure, when either the angles, or fides, or planes of the circumfcribed figure touch all the angles of the figure which it contains.