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about F and G have different centres, they cannot coincide ; be

cause if they coincided, they would make one sphere having two * INTERC.6.

centres, which cannot* be. And because the spheres described + INTERC, 3. Cor. 6. about F and G aret equal but do not coincide, and the point B Interc.11. is in the surface of both; they will* cut one another in a Cor.

self-rejoining line passing through B, or else touch in B only. But whichever of these they do, (inasmuch as the sphere whose centre is G must necessarily be exterior to the sphere whose centre is E, for it touches it externally), addition is made to the inclosing body last-mentioned in the parts which lie in one direction from B as for instance on the side of L, and subtraction at the same time made from it in the parts which lie in some other direction from B as for instance on the side of K. Wherefore the augmented parts NBL cannot upon turning occupy the place of the parts MBH as before, for they have been made greater; still more they cannot occupy the place of the parts MBK which have been made less. And if they cannot occupy the same place, the body

during the turning cannot be without change of place. The * I. Nom.26. assumption*, therefore, which involves the impossible consequence,

cannot be true; or ABD cannot also be a straight line. And in like manner may be shown that any other line, of which the segment AB is common to ABC and the remainder is not in one and the same line with ABC, cannot be a straight line.

But if no other line having the segment AB common as aforesaid, can be a straight line; two straight lines, which are not in one and the same line, cannot have the common segment AB.

And by parity of reasoning, the like may be proved in every other instance. Wherefore, universally, two straight lines, which are not in one and the same line, cannot have a common segment. Which was to be demonstrated.

CoR. 1. If any two points in one straight line are made to coincide with two points in another, the two straight lines shall coincide with one another to the extent of the length that is common to both, and be one straight line throughout.

For, First; If they fail to coincide between the points, two + Interc. 10. straight lines must inclose a space; which ist impossible. Cor. 1.

Secondly; If they afterwards fail to coincide beyond the points, to the extent of the length common to both; two straight lines, which are not in one and the same line, must have a common segment. Which (by Prop. XII above) is impossible.

А

B

D

B

line;

С

D

Thirdly; If there be two straight lines
as AB and CD, and their extremities B and
C are made to coincide each with a several
point between the extremities of the other А

the whole AD shall be a straight line.
For if the line ACBD be turned about the points C and B

which remain unmoved; because CB, which is a portion of CD, * Interc.9. is* a straight line, no point in it will change its place. And because Cor. 2.

CBD is a straight line, no point in BD will change its place ; for if it did, there would be two straight lines, not in one and the same line, having a common segment CB; which is impossible. And in like manner, because BCA is a straight line, no point in CA will change its place. Wherefore no point in all the line AD will change its place; and because no point in it changes its place, the extreme points A and D remain unmoved,

and the line is turned about its extreme points and every point in Interc. 10. it is without change of place. Therefore AD is at straight line. Nom.

CoR. 2. If two straight lines cut one another, they coincide only in a point.

For if they coincided in two points, (by Cor. 1 above) they must coincide throughout.

CoR. 3. Any straight line may be applied to any other, so that they shall coincide to the extent of the length common to both.

For two points in the one straight line may be made to coincide with two points in the other.

CoR. 4. Any straight line CD (see the last figure] may be united to any other AB, so that part of CD shall coincide with part of AB, and the remainder be in the same straight line.

For a point between C and D may be made to coincide with B, and at the same time C with some point between B and A.

Cor. 5. Any straight line CD may be added to any other straight line AB, so that their sum shall be one straight line. For any third straight line, as EF, may.

B (by Cor. 4 above) be united to AB, so that a part of it EB shall coincide with part of AB, and the remainder BF be in the same straight line. And afterwards the А E B F extremity C of CD may be made to coincide with the point B in AEBF, and some other point in CD with some other point in BF; whereupon (by Cor. 1 above) AD will be one straight line ; and it is the sum of AB and CD.

А

с

E

F

с

D

CoR. 6. A terminated straight line may be prolonged to any length in a straight line.

For (by Cor. 5 above) it may be multiplied [that is to say, straight lines equal to it may be added one to another] any

number of times, and the whole be one straight line. Wherefore INTERC.1. it may be thus multiplied, till* the straight line which is the result Cor. 17.

is greater than the straight line between any two points which shall have been specified.

NOMENCLATURE.—When a straight line is said to be of unlimited length, the meaning is, that no point is assigned at which it shall be held to be terminated, but on the contrary it shall without further notice be considered as prolonged to any extent a motive may ever arise for desiring.

Cor. 7. Any three points (which are not in the same straight line) being joined, there shall be formed a three-sided figure; and no point in any one side, shall coincide with any point in either of the other sides, except the points which were joined.

For if any points other than these, should coincide; the whole of the straight lines must coincide, and the three points be in the same straight line.

CoR. 8. Any straight line from a point within a sphere, being prolonged shall cut the surface.

For if BC be any straight line within the sphere whose centre is A, from any point outside as D let a straight line of

unlimited length be made to pass through + INTERC. 4. B.; and because the sphere may bet

turned about its centre without change of
place, D will during any such turning

continue to be outside the sphere, and a straight line may at any I INTERC. 9. time bef made to pass from D to B, which shall cut the surface of Cor.

the sphere. Let then the sphere be turned about its centre (the straight line of unlimited length DB, at the same time turning about the point D so as always to pass through B), till the point C be made to coincide with some point in DB as F, the point B at the same time occupying the situation E. After which let the sphere together with the straight line EFD be turned again about the centre A, till the points E and F are returned to their original situations B and C, and the straight line EFD is brought into the situation BCG. And because the sphere during all these turnings

B

G

•INTERC. 4. about the centre is* without change of place, the straight line BCG

which cuts the surface of the sphere after the last turning, cuts also the surface as it was before the first turning. That is to say, the straight line BC being prolonged before the first turning, cuts the surface.

PROPOSITION XIII.

PROBLEM. To describe a surface in which any two points

being taken, the straight line between them lies wholly in that surface.

+INTERC.10.

About any centre as A, and with any radius as AD, describet Cor. 6.

a sphere; and about any
other centre, and with a
radius equal to AD, de-

scribe another sphere, which INTERC.10. will be equal to the first ; Cor. 4.

and let these two equal

spheres be placed touching • INTERC. 5. one another in any point* +INTERC. 9. as C. Joint the centres Cor.

A and B, and the straight
Interc.10. line AB willf pass through
Cor. 7.

the point of contact C.
About A as a centre, with
the radius AB, describe a
sphere BFGH; and about
B as a centre, with the

radius BA, describe a sphere
INTERC.10.
Cor. 4.

AFIH, which will be equal
to the sphere BFGH. Be-
cause the spheres BFGH
and AFIH pass through
each other's centres, the
surface of each will be
partly within and partly
without the surface of the
other; therefore they cut
one another. And because the spheres are equal and cut one another,

G

E

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H

•N

*INTERC.11. their surfaces coincide* only in one self-rejoining line. Let them

coincide in FJHK; in which take any point as F, and join CF, TINTERC.12. and let CF bet prolonged to an unlimited length on the side Cor. 6.

of L. Let then the spheres BFGH and AFIH be united as one PINTERC. 9. body, and together with the straight line CL be turned round the Nom.

straight line AB which remains at rest. The straight line of un-
limited length CL, shall describe a surface as required.

Because the united spheres BFGH, AFIH are turned about *Interc. 4. their respective centres, each of them will* be without change

of place. Whence the intersection of their surfaces, which is
FJHK, will also be without change of place ; for if it was not,
the surfaces would at one instant intersect one another in some point
or points in fixed space and both of them pass through the same,
and at another instant they would not; which cannot be without
one or both of the spheres having suffered change of place. Also
because the intersection is without change of place, the point F,
which is a point in it before the turning begins, will at all times
during the turning be found in some part of FJHK; for if it
was not, the intersection would at one instant coincide

every where
with FJHK, and at another it would not; which cannot be, if
the intersection is without change of place. And if the body
formed by these united spheres be transposed, in such manner that
the points A and B shall change places; the straight line BA after
the transposition will coincide with and occupy the same place as AB

did before, for if not, between the same two points there would be +Interc.10. two different straight lines, which cannott be; and because the radii

Constr. of the two spheres are equal, the spheres are* equal, and each will
*INTERC.10.
Cor. 4.

coincide with and occupy the place formerly held by the other, and
consequently their point of contact C will occupy the same place as
before, and their intersection will coincide with their intersection
before the transposition, and, like it, will be without change of place
during any turning of the united spheres round the straight line
AB; also the point F in the straight line CL will at all times
during the turning be found in the same line in fixed space

FJHK
as before, and the united spheres may be turned round the straight
line BA till the point F occupies any place in the line in fixed
space FJHK which was occupied by it before the transposition.
Whence THE SURFACES DESCRIBED BY THE STRAIGHT LINE
CL BEFORE AND AFTER THE TRANSPOSITION, SHALL COINCIDE
THROUGHOUT ; for if there be any point N in the one surface, which

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