Boox 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 B together is less than four right angles : But A and B together are greater than the angle E; wherefore the doubleof A, 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, 23. 11. D, E, each to eacha : And, by prop. 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 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 demonstrated, that 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 N IR P 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. Book XI. 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. Book XI. 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 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. 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 “neither 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 predeeling this being in the same plane; yet here it was drite neeestary. napf poi; PROP. XX. B. XI. In this, near the beginning, are the words, “ But if not, 6 let BAC be the greater:" but the angle BAC may happen to be equal to one of the other two: Wherefore this place |