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And by parity of reasoning, the like may be done in every other instance.

PROPOSITION VIII.

THEOREM.--If two spheres which touch one another externally, be

turned as one body about the two points which are the centres and which remain at rest, the point of contact shall remain unmored.

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Let the two spheres whose centres are A and B, touch one another externally in the point C; and in this situation let them be united as one body [by means

B of an inclosing body of hard matter, as be*Interc. 2. fore], and then be turned* about the two

points A and B which remain at rest. Their point of contact C shall remain unmoved.

Because the centre of each sphere remains at rest, each sphere + INTERC. 4. is turned about its own centre; wherefore each sphere willt be

without change of place. Hence their point of contact will at all times during the turning, coincide with the point in fixed space, in which the spheres touched one another before the turning began. For if it did not, the surfaces of the spheres would at one instant coincide in some point in fixed space and both of them pass through it, and at another instant they would not; which cannot be without one or both of the spheres having suffered change of place. And because the point of contact always coincides with one and the same point in fixed space, it remains unmoved.

And by parity of reasoning, the like may be proved of any other spheres. Wherefore, universally, if two spheres which touch one another externally, be turned as one body about the two points, which are the centres and which remain at rest, the point of contact shall remain unmoved. Which was to be demonstrated.

CoR. If along with the substantial spheres, the inclosing body of hard matter is turned about the two centres ; this inclosing body will also be without change of place, as respects the parts bounded by the hollow spherical surfaces.

For these hollow surfaces are always in contact with the surfaces of the substantial spheres, which are without change of place. Wherefore these hollow surfaces are also without change of place.

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PROBLEM.— From one of two assigned points to the other, to de

scribe a line, which being turned about its extreme points, every point in it shall be without change of place.

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Let the two assigned points be A and B.

It is required to describe a line from B to A, which being turned about its treme points B and A, every point in it shall be without

change of place. INTERC.3. About the centre A, with the central distance AB, describe* a Cor. 2.

sphere ; and about the same centre, with a central distance equal to the distance from A to any point C which is within the sphere last

described, describe another sphere AC; and about the centre B, + Interc. 7. describet a sphere touching the sphere AC externally, and let the Interc.5. single points in which it touches it, be F. If then the hard body INTERC. 2. in which are all the spheres, be turned* about the points A and tInterc. 8. B ; the point of contact F willt remain unmoved. And in like

manner if about the centre A, in the same hard body be described other spheres successively less than the sphere AC and than each other, as AD, AE ; and if about the centre B be described spheres respectively touching thesc, as in the points G, H; on the hard body in which are all the spheres being turned about A and B, the points of contact G, H, will also severally remain unmoved.

In this manner, therefore, between A and B may be determined any number of points that shall be desired, which on the hard body in which they are all situate being turned about A and B, will remain unmoved. Wherefore if one of the spheres that touch one another, as BF, be imagined to increase in magni. tude and the other to decrease, till the sphere BF meets the point A, and vice versa, (the spheres during such process remaining ever in contact); their point of contact will describe a line, every point in which, on the hard body in which the line is situate being

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turned about B and A, will remain unmoved, or be without change of place. Which was to be done.

And by parity of reasoning, the like may be done in every other instance.

NOMENCLATURE.--A line, which being turned about its extreme points, every point in it is without change of place, is called a straight line. A body or figure which is turned about two points in it that are also the extremities of a straight line, inasmuch as the whole of the straight line remains without change of place) is said to be turned round such straight line. When from any point to any other point, a straight line is described or made to pass; the two points are said to be joined. If to a straight line addition is made at either end, in such manner that the whole

continues to be a straight line, the original straight line is said to See Note. be prolonged, and the part added is called its prolongation. Any

straight line drawn from the centre of a sphere to the surface, is called a radius of the sphere. A straight line drawn through the centre and terminated both ways by the surface, is called a diameter of the sphere.

Cor. A straight line may be described or made to pass from
any one point to any other point.
For it may be done by Prop. IX. above.

PROPOSITION X.
THEOREM.— Between two points there cannot be more than one

straight line.
From the point A to the point B, let AB be

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A the straight line described as by Prop. IX. No

B other line between the points A and B, can be a straight line.

For, if this be disputed, let ACB be some other line.

Because AB is the straight line described as by Prop. IX, if the • INTERC.2. body in which it is situate be turned* about the extreme points A

and B, every point in the line AB will remain without change of Interc. 9. place. Wherefore the line ACB will be turned roundt the straight Nom.

line AB and moved into some other situation, as ADB. And because the line ACB on being turned about its extreme points A and B is

not without change of place, it is not a straight line ; for if it #Interc. 9. was af straight line, every point in it would be without change of

place. And in like manner may be shown that any other line from A to B which is not AB, is not a straight line.

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And by parity of reasoning, the like may be proved in every other instance. Wherefore, universally, between two points there cannot be more than one straight line. Which was to be demonstrated.

Cor. 1. Two straight lines cannot inclose a space.

For if they did, between two points there would be two different straight lines. Which (by the Proposition above) cannot be.

CoR. 2. Any portion of a straight line is also a straight line.

For inasmuch as when it is turned about its extreme points, every point in it is without change of place; every portion of it is at

the same time turned about its own extreme points without change *Interc. 9. of place, and consequently is a* straight line.

CoR. 3. The straight lines between equidistant points, are

equal. And the extremities of equal straight lines are equidistant. +1. Nom, 12. For if the points are equidistantt, they can be applied to one

another ; and if they are applied to one another, the straight lines

between them (by Cor. 1 above) will coincide ; and because they 1. Nom.14. coincide, they are equali.

And if two straight lines are equal, their extremities may be made to coincide. For if when one extremity of each are made to coincide, the other extremity of the one cannot be also made to coincide with the other extremity of the other; then the extremity of one of them may be made to coincide with a point in the other which is not the extremity, and one of the straight lines is greater than the other ; which cannot be, for they are equal. And because their extremities can be made to coincide, these extremities are equidistant.

CoR. 4. All the radii of the same or equal spheres are equal.

And spheres that have equal radii, are equal. * Constr.

For all the points in the surface are* equidistant from the centre ; wherefore (by Cor. 3 above) the straight lines from the centre to any points in the surface are equal.

And if two spheres have equal radii, the extremities of the radii

(by Cor. 3 above) are equidistant. Wherefore the spheres have + INTERC. 3. equal central distances; and consequentlyt are equal. Cor. 6.

CoR. 5. The sphere that has the greater radius, is the greater.

For if (See the Figure to Prop. IX) a straight line be described from the point B in the surface of a sphere, to the centre A, no point in that straight line, between B and A, as I, can also be in the surface of such sphere, but is necessarily within it; because

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every such point is (by the Construction in Prop. IX) situate in the surface of some other sphere concentric with the first and within it. Therefore the concentric sphere that cuts AB in any point that is between B and A (that is to say, which has a radius less than AB) is less than the sphere whose radius is AB; or the sphere whose radius is AB, is the greater.

Cor. 6. A sphere may be described about any centre, and

with a radius of any length that shall have been assigned. *INTERC.3.

For it has been shown* how it may be described with a central Cor. 2.

distance equal to the distance between any two points; therefore it may be described with a central distance equal to the distance between the extremities of the proposed radius.

CoR. 7. If two spheres touch one another externally, the straight line which joins their centres shall pass through the point of contact.

For it is one of the points through which the straight line +Interc.10. described as by Prop. IX, passes. And there can bet no other

straight line between the centres than this.

COR. 8. If the surfaces of two spheres pass through a point in the straight line which joins their centres; the spheres shall touch one another externally in that point, and no other point in the straight line shall be in the surface of either of the spheres.

For it is one of the points in which the various spheres by which the straight line is described as by Prop. IX, touch

another. And no other sphere can be described about either of INTERC.3. the same centres, whose surface shall pass through the same Cor. 4,

point; nor any other point in the straight line (as described by means of a succession of concentric spheres in Prop. IX) be in the surface of either of the touching spheres.

CoR. 9. A straight line from the centre of a sphere to a point

outside, coincides with the surface only in a point. *INTERC.7. For if about the point outside, another sphere be described*

touching the first; the straight line (by Cor. 7 above) will pass through the point of contact, and (by Cor. 8 above) no other point in the straight line will be in the surface of either of the spheres.

SCHOLIUM.—Henceforward, mention will never be made of distances, but See Notr. always of straight lines; nor of central distances, but always of radii. And

if one point be said to be farther or nearer than some other, it shall always be understood that the straight line drawn to it is greater or less than the straight line drawn to the other.

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