9. Similar folid figures are thofe which are contained under equal numbers of fimilar planes.

10. Equal and fimilar folid figures are fuch as are contained under fimilar planes equal in number and magnitude.

11. A folid angle is the inclination of more than two right lines, which touch one another and be not in the fame plane. Or thus, a folid angle is that which is contained under more than two plane-angles not lying in the fame plane, and ftanding at one point.

12. A pyramid is a folid figure contained under several planes ftanding upon a plane, and meeting in one point. 13. A prifm is a folid figure contained under planes, the two oppofite of which are equal, fimilar, and parallel, but the others are parallelograms.

14. A fphere is a folid figure generated by the entire revolution of a femicircle about its diameter, remaining at reft.

15. The axis of a sphere is that unmoveable right line about which the femicircle revolves.

16. The centre of a sphere is the fame as that of the femicircle generating it.

17. The diameter of a fphere is a right line drawn thro' the centre, and both ways terminating in the fuperficies of the fphere.

18. A cone is a folid figure generated by the entire revolution of a right angled triangle about one of the fides containing the right angle, which remains at reft during

b The former part of this definition, is not fo clear and diftinct as the latter.

A pyramid may perhaps be better defined thus: If a right line always paffing thro' a fixed point over a plane, moves from one angle of any right lined figure, in that plane, along every fide of that figure till it returns to the angle from whence it first moved, the folid contained under the fuperficies defcribed by the moving right line and the figure upon the plane is called a pyramid, and a prism may be defined much after the fame way.

d All definitions of figures from their generations are esteemed better than thofe derived from the properties of the figures;" and therefore this definition of a sphere is better than if it had been called a folid contained under one fuperficies, having a point within it fuch, that all right lines drawn from that point to the fuperficies are equal to one another.

the

Book XI. the motion of the triangle. And if the fixed fide be equal to the other fide containing the right angle, the cone is a right angled cone: If it be lefs, it is an obtufe angled cone; if greater, an acute angled cone.

19. The axis of the cone is that fixed fide of the triangle generating itf.

20. The bafe of a cone is the circle generated by the motion of one of the fides of the triangle generating the

cone.

21. A cylinder is a folid figure made by the entire revolution of a right angled parallelogram about one of its fides, which remains at reft during the revolution of the parallelogram 8.

22. The axis of the cylinder is that fide of the revolving parallelogram which remains at reft, while the cylin der is generating'

23. The bafes of a cylinder are the circles described by two of the oppofite fides of the parallelogram generating the cylinder.

24. Similar cones, and cylinders are those whose axes and the diameters of their bafes are proportional.

e This definition of a cone is too fcanty, it only taking in a right cone, viz. that whofe axis is at right angles to the plane of the bafe. Nor is the diftinction of a right cone, into right angled, oblique angled, and acute angled, of any use, at least in thefe Elements.

The full definition of a cone is this. If a right line always paffing thro' a fixed point over a plane, moves from one point of the circumference of a circle in that plane, quite round the circumference, returning again to the point from whence it first moved, the folid contained under the fuperficies generated by that moving right line and the circle in the plane, is called a cone.

f The bafe of a cone is that circle, and the axis, the right line drawn from the centre of the bafe to the fixed point

or vertex.

This definition of a cylinder is only particular, viz. only extending to a right cylinder whofe axis is at right angles to the plane of either of its bafes. The general definition is this. If a right line moves ronnd every part of the circumferences of two equal and parallel circles, the folid comprehended under the fuperficies generated by the moving right line, and the two parallel circles, is called a cylinder.

h The axis of a cylinder is rather the right line drawn thro' the centres of the parallel circles or bafes of the cylinder.

2

25. A cube

25. A cube is a folid figure contained under fix equal

fquares.

26. A tetrahedron is a folid figure contained under four equal equilateral triangles.

27. An octahedron is a folid figure contained under eight equal equilateral triangles.

28. A dodecahedron is a folid figure contained under twelve equal equilateral pentagons.

29. An icofahedron is a folid figure contained under twenty equal equilateral triangles.

i The 27th, 28th, and 29th definitions of an octahedron, dodecahedron, and icofahedron, and indeed the whole theory of thefe folids, laid down by Euclid in his 13th, 14th, and 15th books. feem to be more curious than useful; and I have often wondered why the five Platonic bodies should be held in fuch efteem by the ancient Platonic philofophers, as we find they were, and that even Euclid, who himself was a fectator of Plato, should have compiled the whole body of his Elements for the fake of this contemplation, as Proclus fays it is reported he did.

One part of a right line cannot be in a plane, and the other elevated above it.

For if poffible let the part A B of the right line A B C fie in a plane underneath, and the other part BC be ele

vated above it. Then will fome right line being the direct continuation of A B alfo be in that fame plane. Let this be B D. Wherefore two right lines A BC, ABD have both one common segment

C

D

B

A B, which is impoffible; for one right line cannot meet another in more points than one; otherwise the right lines will coincide.

Which

Therefore one part of a right line cannot lie in a plane, and the other part be elevated above it. was to be demonftrated.

PROP.

PROP. II. THEOR.

If two right lines mutually cut each other, they are both in the fame plane; alfo every triangle lies in one planek.

For let two right lines A B, CD cut one another in the point E: I fay the right lines A B, CD are both in the fame plane, alfo every triangle lies in one plane.

A

E

D

F G

For take any points F, G, in E B, E C. Join CB, FG, and draw F H, GK: First, I fay the triangle EBC lies in one plane: For if one part FHC or GBK of the triangle E B C lies in one plane, and the remaining part lies in another plane; then will one part of the lines EC, E B fall in one plane, and another part in another plane. And if the part CHKB FC BG of the triangle E Cв be in one plane, and the remaining part in another, one part of the right lines EC, EB, will be in one plane, and another part in another plane: which we have proved to be abfurd. Therefore the triangle E B C lies in one plane: But in that plane wherein the triangle B C E lies, both the right lines EC, EB do lie: And in that plane wherein are the right lines EC, E B, are [by 1.11.] the right lines A B, CD. Therefore the right lines A B, CD are both in the fame plane, and every triangle is in the fame plane. Which was to be demonstrated.

k From hence it appears why a ftool, table, &c. with only three legs, will always ftand firm, when thofe with four, or more, oftentimes will not.

If two planes cut one another, their common fection will be a right line.

For let two planes A B, BC cut one another, and let their common section be the line DB: I fay DB is a right line.

For if it be not, from the point D to B in the plane A B, draw the right line D E B, and in the plane B C, the right line DFB: Therefore the right lines DE B, D F B will

have the fame bounds, and include a space. Which [by 12. ax.] is impoffible: Wherefore DEB, DFB are not right lines. After the fame manner we demonftrate that no other line except BD, the common fection of the two planes A B, C D drawn from the point D to B, can be a right line.

Therefore if two planes cut one another, their common fection will be a right line. Which was to be demonftrated.

A

If a right line be at right angles to two right lines cutting one another in the common fection [of two planes]. it will be at right angles to the plane drawn thro' thofe lines.

For let any right line E F be at right angles to the right lines A B, CD. cutting one another in the point E: I fay alfo that E F is at right angles to the plane drawn through

A B, C D.

F

For take the equal right lines A E, E B, CE, ED; and through the point E draw the right line GEH any how. Join AD, CB; and from any point F draw FA, FG, FD, FC, FH, F B. Then because the two right lines A E, ED are equal to the two right lines c E, EB, and [by 15. 1.] they contain equal angles, the bafe AD [by 4. 1.] will be equal to the bafe CB, and the triangle A E D equal to the triangle C E B, and alfo the angle D A E equal to the angle EBC; and [by 15. 1.] the angle AEG is equal to the angle BEH: Therefore the two triangles AGE, EBH have two angles of the one equal to two angles of the other, each to each, and one fide AE of the one equal to one fide EB of the other, which lies between the equal angles: Wherefore [by 26. 1.] the remaining fides of the one are equal to the remaining fides of the other: Wherefore GE is equal to E H, and AG to BH. And because A E is equal to E B, and FE is common, and at right angles; the bafe FA [by 4. 1.] will be equal to the bafe F B.

E

B

By