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does not coincide with some point in the other surface, then let the straight line CFL be brought into the situation in which it passes through the point N, and after transposition of the united spheres let them be turned round BA till the point F in CFL occupies the same place it occupied when CFL passed through N; and because the point N is not in the second surface, the straight line CFL after the transposition cannot pass through N, for if it did, the transposed surface would pass through N, and it does not ; and because in one situation of the straight line CFL the points C and F coincide with the points C and F in the other situation of it, but one of these straight lines passes through the point N. and the other does not, the two straight lines CFL coincide in
two points C and F, but do not coincide to the extent of the * Interc.12. length common to both, which is* impossible. And in the same Cor. 1.
manner may be shown that there is no other point in the one surface, either within or without the line FJHK, which does not after transposition of the united spheres coincide with some point in the other surface. Hert then, in the straight line of unlimited length CL ( See Fig. 2 in the neat page) take any point other than F, as M; and if about the centres A and B, with the radii AM and BM, spheres be described, THE POINT M, AT ALL TIMES DURING THE REVOLUTION OF CL, SHALL BE FOUND IN THE INTERSECTION OF THESE SPHERES. For, join AM, BM ; and afterwards let the united spheres be transposed, and turned round BA till M be in the same place as before. Because the extremities A and M occupy the places previously occupied by the extremities B and M, AM and BM are equal; wherefore, if about the centres A and B,
with the equal radii AM and BM, be described two spheres, these tInterc.10. willt be equal to one another; and because they are equal but have Cor. 4.
different centres A and B, and have the point M in their surfaces in INTERC.11.
1. common, they willț cut one another in one self-rejoining line pass
ing through M, or else touch externally in the point M only; but
they cannot touch externally in M, for then the straight line AB *INTERC.10. which joins their centres must* pass through M, and it does not ; Cor. 7.
wherefore they will cut one another in one self-rejoining line passing through M. Let this line be MQOP; and if afterwards the united spheres together with the straight line CL be turned round AB, M shall always be found in some part of the line MQOP and no where else; for because AM and BM are the radii of the two equal spheres that were described, the point M is always in the sur
face of both; whence, if it be not always in some part of MQOP,
the surfaces of these spheres must coincide elsewhere than in the *Interc11, one self-rejoining line MQOP, which cannot* be. And in the same
way of every other point in CL, or in the surface described by it. +INTERC.9. Whereto nert, in MQOP take any second point as Q, and joint
MQ; THE STRAIGHT LINE BETWEEN M AND Q, with its prolongation either way, SHALL LIE WHOLLY IN THE SURFACE DESCRIBED BY CL. For if MQ pass through the point C, it will without further question lie wholly in the surface described by CL, inasmuch as each half of it will coincide with part of CL as it was found at some instant of the revolution by which the
surface was described. But if MQ do not pass through C, join *INTERC.10. AM, AQ, BM, BQ, and these straight lines will be all equal one to Cor. 4.
another, because they are radii of the same or equal spheres; where
fore if about the centres M and Q, and with the radii MA and QA, were described two spheres, it might be shown as has been done in like circumstances before, that the surfaces of these would cut one another in one self-rejoining line in which is the point A, and that if the figure MBQ was turned round the straight line MQ, the point B would always be found in the self-rejoining line which passes through A, and consequently the turning might be continued till the point B coincided with A, every point in the straight line MQ remaining without change of place; whence, if in MQ had been taken any point as R, and RÅ and RB had been joined, when
the point B was made to coincide with the point A the straight "INTERC.12 line RB would at the same time have* coincided with the straight Cor. 1.
line RA, and therefore RA and RB are equal. About the centres A and B, with the radii AR and BR, describe two spheres Comitted in the figure]; which, it may be shown as has been done before, will cut one another in one self-rejoining line. Because AR and BR are equal, they are necessarily each greater than AC or BC (unless the point R coincides with C, in which event it is without further question in the surface described by CL); for if they were equal to AC and BC, they could meet in no point but C (inasmuch as the spheres AC and BC meet no where but in that point), and if they were less they could not meet at all. Wherefore the
sphere described about the centre A with the radius AR, will INTERC.10. bet greater than the sphere described about the same centre with Cor. 5.
the radius AC; and its surface will not coincide at all with
the surface of the other, but be exterior ; for if it coincided at INTERC. 3. all, the surfaces would coincide throughout and the spheres be Cor. 4.
equal; and if it did not coincide but was interior, the sphere would be the less, which cannot be, for it is the greater. And
because C is a point within the sphere described with the radius *Interc.12. AR, the straight line of unlimited length CL drawn from it will* cut Cor. 8.
the surface of such sphere and be cut by it. Let it be met in S; and See Note 1. join AS, BS. Because S is a point in CL, it may be shown as has
been done before, that AS and BS are equal; wherefore the point S shall be in the self-rejoining line which is the intersection of the spheres described with the radii AR and BR, for if not, then AS
would be equal to AR, but BS would not be equal to BR which is INTERC. 1. equal to AR; which ist impossible, for AS and BS are equal. And Cor. 3.
because, as has been shown before, the point S will at all times during the revolution of CL be found in some part of the inter
section of the spheres which pass through S, which intersection also passess through R; the point Sin CL, will by the revolution of CL be made to pass through R; wherefore R is in the surface described by the revolution of CL. And in like manner may be shown of every other point in the straight line MQ or in its prolongation either
way, that some point in the straight line of unlimited length CL passes through it on the united spheres being turned round AB; wherefore every point in MQ or in its prolongation either way, is in the surface described by CL. And if from M be drawn a straight line to any other point in MQOP, and prolonged to an unlimited length, in like manner may be shown that every point in this straight line or in its prolongation either way, is in the surface described by CL. Wherefore IF A STRAIGHT LINE OF UNLIMITED LENGTH AS MT, BE MADE TO TURN CONTINUOUSLY ABOUT M AND PASS ALWAYS THROUGH SOME POINT IN MQOP, this straight line (both the portion which may lie within MQOP, and its prolongation either way), will at all times lie wholly in the surface described by CL. And because MT may be turned till it passes through any point in the surface which is described by CL in or within the self-rejoining line MQOP; the straight line from M to any such point, as also its
prolongation either way may be shown to lie wholly in the surface See Note 2. described by CL. But every point in the surface described by
CL shall be in or within the self-rejoining line MQOP, which is not farther from the centres A and B than the point M. For if N be any point in the surface described by CL, it may
be shown as before, that N is in the intersection of the spheres that should be described with the equal radii AN and BN; wherefore if AN
and BN were respectively equal to AM and BM, N would be in *Interc10. MQOP, for the spheres respectively would* coincide, and conseCor. 4.
quently their intersections would coincide; and if AN and BN are
respectively less than AM and BM, the sphere with the radius AN +INTERC.10. willt be less than the sphere with the radius AM, and its surface Cor. 5. #Interc. 3. willf be interior to the surface of the other throughout, and in the Cor. 4.
same manner the surface of the sphere with the radius BN will be interior to the surface of the sphere with the radius BM; and because the surfaces of the two spheres are respectively interior to the surfaces of the two others, their intersection, in which is N, will not coincide with the intersection of the others but be within it. Wherefore the straight line from M TO ANY POINT IN THE SUR
FACE DESCRIBED BY CL WHICH IS NOT FARTHER FROM THE CENTRES A AND B THAN THE POINT M, as also its prolongation either way, may be shown to lie wholly in that surface. Lastly then, if in the surface described by the revolution of CL be taken any other two points whatsoever ; in the same way may be shown that from one of them (viz. either of them that is not nearer to the centres A and B than the other) A STRAIGHT LINE DRAWN TO THE OTHER, as also its prolongation either way, LIES WHOLLY IN THE SURFACE DESCRIBED BY THE REVOLUTION OF CL.
And by parity of reasoning, the like may be done by bringing into contact any other two equal spheres.
NOMENCLATURE.—A surface in which any two points being taken, the straight line between them lies wholly in that surface, is called a plane surface. The same when no particular boundaries to it are intended to be specified, is called a plane. If to a plane surface addition is made in any direction, in such manner that the whole continues to be a plane surface, the original plane surface is said to be prolonged, and the part added is called its prolongation. A figure which lies wholly in one plane is called a plane figure. The whole plane surface within the boundaries of a plane figure which is bounded on all sides, is called the area of the figure ; and its whole linear boundary, of whatever kind or composition, is called the perimeter. If a given straight line in a plane be turned in that plane about one of its extremities which remains at rest, till the straight line
B is returned to the situation from which it set out, the plane figure described by such straight line is called a circle, and its boundary the circumference. The point in which one extremity of the straight line remains at rest, is called the centre of the circle. Any straight line drawn from the centre of a circle to the circumference, is called a radius of the circle ; and any straight line drawn through the centre and terminated both ways by the circumference, is called a diameter of the circle.
When a circle is said to be described aboat the centre A with the radius
AB, the meaning is, that it is described by the revolution of the
given straight line AB about the extremity A. Cor. 1. A plane surface may be prolonged to any extent in a
plane. *INTERC.12. For the straight line CL, by the revolution of which it is Cor. 6. described, may be prolonged to any length.