right angles, the double of A and B together is less than four Book XI. 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 same 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 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 equal to the angles C, D, E, each to each And by Prop. 26, 11, at the point F, in the straight a 23. 11. 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 may be greater than the third, and also every two of the three C, D, M greater than the third: And, as in the preceding part, it may be demonstrated, that the three A, B, M are less than four right angles, as also that the three C, D, M, are less than four right angles. Make Book XI. therefore a solid angle at N, contained by the three plane angles ONP, PNQ, ONQ, a 23. 11. which are equal to A, B, M, each to each: And by Prop. N 26, 11, make at the point N, in the straight line ON, a R solid angle contained by three plane angles, of which one is Q P the angle ONQ equal to M, O equal to the angles C, D, each to each. Thus, at the point 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 quality 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. 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 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 to the words of this note, adds, "Now, can it possibly show any Book XI. "want of skill in an editor" (he means Euclid or Theon) "refer to an axiom which Euclid himself hath laid down, B. 1, "No. 14," (he means Barrow's Euclid, for it is the 10th in the Greek,)" and not to have demonstrated, what no man can de"monstrate?" 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 betwixt 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 having 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. 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 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 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: 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 Book XI. "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. THE words," and the triangle AED to the triangle BEC," are omitted, because the whole conclusion of the 4th Prop. B. 1, has been so often repeated in the preceding Books, it was needless to repeat it here. PROP. V. B. XI. In this, near to the end, id ought to be left out in the Greek text; and the word "plane" is rightly left out in the Oxford edition of Commandine's translation. PROP. VII. B. XI. THIS Proposition has been put into this Book by some unskilful editor, as is evident from this, that straight lines which are drawn from one point to another in a plane, are, in the preceding books, supposed to be in that plane: And if they were not, some demonstrations in which one straight line is supposed to meet another would not be conclusive, because these lines would not meet one another: For instance, in Prop. 30, B. 1, the straight line GK would not meet EF, if GK were not in the plane in which are the parallels AB, CD, and in which, by hypothesis, the straight line EF is: Besides, this 7th Proposition is demonstrated by the preceding 3d; in which the very thing which is proposed to be demonstrated in the 7th, is twice assumed, viz. that the straight line drawn from one point to another in a plane, is in that plane; and the same thing is assumed in the preceding 6th Prop., in which the straight line which joins the points B, D that are in the plane to which AB and CD are at right angles, is supposed to be in that plane: And the 7th, of which another demonstration is given, is kept in the book merely to preserve the number of the propositions; for it is evident from the 7th and 35th Definitions of the 1st Book, though it had not been Book XI. in the Elements. PROP. VIII. B. XI. IN the Greek, and in Commandine's and Dr Gregory's translations, near to the end of this Proposition, are the following words: "But DC is in the plane through BA, AD," instead of which, in the Oxford edition of Commandine's translation is rightly put, "but DC is in the plane through "BD, DA." But all the editions have the following words, viz. "because AB, BD are in the plane through BD, DA, "and DC is in the plane in which are AB, BD," which are manifestly corrupted, or have been added to the text, for there was not the least necessity to go so far about to show that DC is in the same plane in which are BD, DA, because it immediately follows, from Prop. 7. preceding, that BD, DA, are in the plane in which are the parallels AB, CD: Therefore, instead of these words, there ought only to be, "because all three are in the plane in which are the parallels "AB, CD." PROP. XV. B. XI. AFTER the words, "and because BA is parallel to GH," the following are added, "for each of them is parallel to DE, "and are not both in the same plane with it," as being manifestly forgotten to be put into the text. 66 PROP. XVI. B. XI. In this, near to the end, instead of the words, "but straight "lines which meet neither way," ought to be read, “but straight lines in the same plane which produced meet nei"ther way:" Because, though in citing this definition in Prop. 27, Book 1, it was not necessary to mention the words, "in the same plane," all the straight lines in the books preceding this being in the same plane; yet here it was quite necessary. PROP. XX. B. XI. In this, near the beginning, are the words, "But if not, "let BAC be the greater:" But the angle BAC may happen to be equal to one of the other two: Wherefore this place |