Book XI. quired that B, C, D together be greater than A, from each of these Ntaking away B, C, the remaining one D most be greater than the excess of A above B and C. take therefore any wagoitude Dwhich is less than A, B, C rogether, but greater than 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, to gether 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 tall 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 excess. PRO P. 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 shall be greater than the third. Let A be not less than B, and C not less than D. irst, 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 above 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 Hypothefis, 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 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 Book XI. 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. PRO P. III. THEORE M. There may be innumerable folid angles all unequal to one anos ther, each of which is contained by the fame four plane angles, placed in the fame 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 fuch, 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 be A E C F B II D H greater than the third. and because A and B together are less than two right angles, the double of A and B together is less than four right angles. but A and B together are greater than the angle E, wherefore the double of A and B together is greater than the three angles A, B, E together, which three are consequently less than four right angles ; and every two of the fame angles A, B, E are greater than the third ; therefore, by Prop. 23. 11. a solid angle may be made contained by three plane angles equal to the angles A, B, E, each to each. Let this be the angle F contained by the three plane angles GFH, HFK, GFK which are equal to the angles Y Book X1. 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 solid angle may be made which shall be contained by three plane angles 2. 23. 11. equal to the angles C, D, E, each to each'. and by Prop. 26.11. A E C F 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 angles A, B, M be greater than the third, and also every two of the three C, D, M be greater than the third. and, as in the preceding part, it may be demon N R 04 may be equal to the angles C, D, each to each. thus at the point N there Book XI. is a solid angle contained by the four plane angles ONP, PNQ, ONR, 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 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 folid 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 ferve the fame purpose with the angle D, and again from Dor 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 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 fufficiently demonstrated, because the equality of two folid angles, whereof each is contained by three plane angies which are equal to one another, each to each, is only assumed, and not demonstrated. PROP. I. B. XI. 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 assumed. Mr. Thomas Simpson, in his notes at the end of the ad Edition of his Elements of Geometry, p. 262. after repeating the words of this note, adds “ Now can it poffibly lew 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. N° 14." (he means Parrow's Euclid, for it is the icth in the Greek)" and not to have “ demonstrated, 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 betwixe Book XI these points common to both. Now, as it has not been shewa 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 mewn in Cor. of 11. Prop. B. 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. 1 1. Book 1. 4to. Edit. the straight lines AB, BD, BC, are suppofed to be all in the same plane, which cannot be assumed in 1. Prop. B. 11. 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 he mentions here, to be Euclid's; it is none of Euclid's, but is the ioth in Dr. Barsow's Edition, who had it from Herigon's Cursus Vol. 1. and in place of it the Corollary of 11. Prop. Book 1. was added. PROP. II. B. XI. This Proposition seems to have been changed and vitiated by fome Editor ; for all the figures defined in the 1. Book of the Elements, and among them triangles, are, by the Hypothefis, plane figures ; that is, such as are described in a plane ; wherefore the second part of the Enuntiation needs no Demonstration. besides a convex fuperficies may be terminated by three straight lines meeting one another. 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 fufficiently done, the Enuntiation and Demonstration are changed into those now put into the Text. PRO P. 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, |