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But let the excess of A above B be greater than the ex- Book XI. cess of C above D: And because, by the hypothesis, the three B, C, D, are together greater than the fourth A; C and D together are greater than the excess of A above B : Therefore a magnitude may be taken which is less than C and D together, but greater than the excess of A above B. Let this magnitude be E: and because E is greater than the excess of A above B, B together with E is greater than A: And as, in the preceding case, it may be shown that A together with E is greater than B, and that A together with B is greater than E: Therefore, in each of the cases, it has been shown, that any two of the three A, B, E, are greater than the third.
And because in each of the cases E is greater than the excess of C above D, E together with D is greater than C; and by the hypothesis, C is not less than D; therefore E together with C is greater than D; and, by the construction, C and D together are greater than E: Therefore any two of the three C, D, E, are greater than the third.
PROP. III. THEOREM. THERE may be innumerable solid angles, all unequal to one another, each of which is contained by the same four plane angles placed in the same order.
Take three plane angles, A, B, C, of which A is not less than either of the other two, and such, that A and B together are less than two right angles; and, by Problem 1, and its corollary, find a fourth angle D such, that any three whatever of the angles A, B, C, D, be greater than the remaining angle, and such, that A and B together be not less than C and D together: And, by Problem 2, find a fifth angle E such, that any two of the angles A, B, E, be
H Η greater than the third, and also that any two of the angles
Book XI. C, D, E, be greater than the third: And because A and B
together are less than two right angles, the double of A and
be contained by three plane angles equal to the angles C, 23. 11. D, E, each to each a : And, by prop. 26, 11, at the point
F, in the straight line FG, a solid angle may be made equal
Again: Find another angle M such, that every two of the
the three A, B, M, are less than four right angles, as also Book XI. that the three C, D, M, are less than four right angles. Make therefore a a solid angle
2 23. 11. at N contained by the three plane angles OŃP, PNQ, OND, which are equal to A, B, M, each to each : And by prop. 26, ll, make at the point n, in the straight line ON, a solid angle contained Oe. by three plane angles, of which one is the angle ONQ equal to M, and the other two are the angles QNR, ONR, which are equal to the angles C, D, each to each. Thus, at the point N, there is a solid angle contained by the four plane angles ONP, 'PNQ, QNR, ONR, which are equal to the angles A, B, C, D, each to each. 'And that the two solid angles at the points P, N, each of which is contained by the above-named four plane angles, are not equal to one another, or that they cannot coincide, will be plain by considering that the angles GFK, ONQ, that is, the angles E, M, are unequal by the construction; and therefore the straight lines GF, FK, cannot coincide with ON, NQ, nor consequently can the solid angles, which therefore are unequal.
And because from the four plane angles A,B,C,D, there can be found innumerable other angles that will serve the same purpose with the angles E and M: it is plain that innumerable other solid angles may be constituted which are each contained by the same four plane angles, and all of them unequal to one another. Q.E.D.
And from this it appears, that Clavius and other authors are mistaken, who assert that those solid angles are equal which are contained by the same number of plane angles that are equal to one another, each to each. Also it is plain, that the 26th prop. of book 11, is by no means sufficiently demonstrated, because the equality of two solid angles, whereof each is contained by three plane angles which are equal to one another, each to each, is only assumed, and not demonstrated.
PROP. I. B. XI.
not meet a straight line in more than one point,” are left out, as an addition by some unskilful hand; for this is to be demonstrated, not assumed.
Mr. Thomas Simpson, in his notes at the end of the second edition of his Elements of Geometry, p. 262, after repeating the words of this note, adds, “Now, can it possibly 16 show any want of skill in an editor (he means Euclid or “ Theon) to refer to an axiom which Euclid himself hath “ laid down, book ), No. 14,” he means Barrow's Euclid, for it is the 10th in the Greek, " and not to have de
monstrated what no man can demonstrate ?" But all that in this case can follow from that axiom is, that, if two straight lines could meet each other in two points, the parts of them betwixt these points must coincide, and so they would have a segment betwist these points common to both. Now, as it has not been shown in Euclid, that they cannot have a common segment, this does not prove that they cannot meet in two points, from which their not have ing a common segment is deduced in the Greek edition : But, on the contrary, because they cannot have a common segment, as is shown in Cor. of 11th Prop. Book 1, of 4to edition, it follows plainly, that they cannot meet in two points, which the remarker says no man can demonstrate.
Mr. Simpson, in the same notes, p. 265, justly observes, that in the corollary of Prop. 11, Book 1, 4to edit. the straight lines AB, BD, BC, are supposed to be all in the same plane, which cannot be assumed in 1st Prop. Book 1). This, soon after the 4to edition was published, I observed and corrected as it is now in this edition : He is mistaken in thinking the 10th axiom hie mentions here to be Eu. clid's; it is none of Euclid's, but is the 10th in Dr. Barrow's edition, who had'it from Herigon's Cursus, vol. I. and in place of it the corollary of 11th Prop. Book 1, was added.
PROP. II. B. XI. This proposition seems to have been changed and vitiated by some editor; for all the figures defined in the 1st Book of the Elements, and among them triangles, are, by the hypothesis, plane figures; that is, such as are described in a plane: wherefore the second part of the enunciation needs no demonstration. Besides, a convex superficies may be terminated by three straight lines meeting one another: Book XT, The thing that should have been demonstrated is, that two or three straight lines, that meet one another, are in one plane. And as this is not sufficiently done, the enunciation and demonstration are changed into those now put into the text.
PROP. III. B. XI. In this proposition the following words near to the end of it are left out, viz. " therefore DEB, DFB, are not straight “ lines; in the like manner, it may be demonstrated, that “ there can be no other straight line between the points D, «B:" Because from this, that two lines include a space, it only follows that one of them is not a straight line : And the force of the argument lies in this, viz. if the common section of the planes be not a straight line, then two straight lines could include a space, which is absurd ; therefore the common section is a straight line.
PROP. IV. B. XI.
PROP. V. B. XI.
PROP. VII. B. XI.