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Rook Xf. quired that B, C, D together be greater than A, from each of these mtaking away B, C, the remaining one D must be greater than the
excess of A above B and C. take therefore any magnitude D which is less than A, B, C together, but greater that the excess of A above B and C. then B, C, D together are greater than A; and because A is greater than either B or C, much more will A and D, together with either of the two B, C be greater than the other. and, by the construction, A, B, C are together greater than D.
Cor. If besides, it be required that A and B together shall not be less than C and D together; the excess of A and B together above C must not be less than D, that is D must not be greater than that excefs.
PROP. II. PROBLEM.
Four magnitudes A, B, C, D being given of which A and B together are not less than C and D together, and such that any three of them whatever are greater than the fourth; it is required to find a fifth magnitude E such, that any two of the three A, B, E shall be greater than the third, and also that any two of the three C, D, E Thall be greater than the third. Let A be not less than B, and C not less than D.
First, Let the excess of C above D be not less than the excess of A above B. it is plain that a magnitude E can be taken which is less than the sum of C and D, but greater than the excess of C abore D; let it be taken, then E is greater likewise than the excess of A above B; wherefore E and B together are greater than A; and A is not less than B, therefore A and E together are greater than B. and, by the Hypothesis, A and B together are not less than C and D together, and C and D together are greater than E; therefore likewise A and B are greater than E.
But let the excess of A above B be greater than the excess 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 preceeding case, it may be shewn 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 shewn that any two of the Book XI. 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 ano. ther, cach of which is contained by the fame four plane angles, placed in the same order.
Take three plane angles A, B, C, of which A is not less than eit ther 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 á 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 greater than the third, and also that any two of the angles C, D, E
Book X1. angles A, B, E, each to each, and because the angles C, D together
are not greater than the angles A, B together, therefore the angles C, D, E are not greater than the angles A, B, E. but these last three are less than four right angles, as has been demonstrated, wherefore also the angles C, D, E are together less than four right angles, and every two of them are greater than the third; therefore
a folid angle may be made which shall be contained by three plane a 23. 11. angles equal to the angles C, D, E, each to each and by Prop.
Ή 26. 11. at the point F in the straight line FG a solid angle may be made equal to that which is contained by the three plane angles that are equal to the angles C, D, E. let this be made, and let the angle GFK, which is equal to E, be one of the three; and let KFL, GFL be the other two which are equal to the angles C, D, each to each, thus, there is a solid angle constituted at the point F contained by the four plane angles GFH, HFK, KFL, GFL which are equal to the angles A, B, C, D, each to each.
Again, Find another angle M such, that every two of the three
to the angles C, D, each to each. thus at the point N there is a so- Book XI. lid angle contained by the four plane angles ONP, PNQ, ONR, o ONR which are equal to the angles A, B, C, D, each to each. and that the two folid angles at the points F, 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 confidering 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 three given plane angles A, B, C there can be found innumerable other angles that will serve the fame purpose with the angle D, and again from D or any one of these others, and the angles A, B, C, there may be found innumerable angles, such as E or M; it is plain that innumerable other solid angles may be constituted which are each contained by the fame 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 26. 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 affumed, and not demonstrated.
PROP. I. B. XI. The words at the end of this, “ for a straight line cannot meet “ a straight line in more than one point,” are left out, as an addition by fome unskilful hand; for this is to be demonstrated, not afa fumed.
Mr. Thomas Simpson in his notes at the end of the 2d Edition of his Elements of Geometry p. 262. after repeating the words of this note, adds “ Now can it possibly shew any want of skill in an editor” (he means Euclid or Theon) " to refer to an Axiom which * Euclid himself had laid down Book 1. No. 14." (he means Earrow's Euclid, for it is the roth in the Greek) “ and not to have “ demonstrated, what no man can demonstrate?” But all that is this case can follow from that Axiom is, that if two straighe lines could meet each other in two points, the parts of them betwixt thefe points must coincide, and so they would have a fegment
Book XI. betwixt these points common to both. Now, as it has not beeti
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 having a common segment is deduced in the Greek Edition. but, on the contrary, because they cannot have a common segment, as is sewn in Cor, of 11. Prop. B. 1. of 4to. Edition, it follows pl.inly that they cannot meet in two points, which the remarker fa; 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 fame plane, which cannot te allumed in 1. Prop. B.Il. this, foon after the 4to. Edition was published, I observed and corrected as it is now in this Edition. he is mistaken in thinking the roth Axiom he mentions here, to be Euclid's; it is none of Euclid's, but is the 1 oth in Dr. Barrow's Editio 1, who hal it from Herigon's Cursus Vol. 1. and in place of it the Corollary of 1. Prop. Book 1. was added.
PROP. II. B. XI. This Proposition seems to have been charged and vitiated by fo:ne Editor; for all the figures defined in the 1. Book of the Eleinants, and among them triangles, are, by the Hypothesis, plane figures; that is, such as are described in a plane; wherefore the fecond part of the Enuntiation needs no Demonstration. besides a Colex fuperficies may be terminated by three straight lines meeting one another, the thing that should have been demonstrated is, that two, or three struight lines, that meet one another, are in one plane, and as this is not sufficiently done, the Enuntiation and Dc mufratiun are changed into those now put into the Text.
PROP. III. B. XI.
Itraight 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 It aiglat line. and the force of the argument lies in this, viz. if the cominon fection of the planes be not a straight line, then two straight lines could include a space, which is absurd; therefore the common Sion is a straighi line.