لبهي Book XI. 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 laft three are less than four right angles, as has been demonstrated, wherefore also the angles C, D, E are together lefs 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 angles a. 23. 11. equal to the angles C, D, E, each to each 2. and by Prop. 26. 11. MA 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 folid angle conftituted 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 fuch, 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 ftrated that the three A, B, M, N are lefs than four right angles, as R are equal to A, B, M, each to each. and by Prop. 26. 11. make at the point N in the ftraight line ON a folid angle contained 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 Book XI. is a folid 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 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 confequently 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 purpofe with the angle D, and again from D or any one of thefe others, and the angles A, B, C, there may be found innumerable angles, fuch as E or M; it is plain that innumerable other folid angles may be conftituted which are each contained by the fame four plane angles, and all of them unequal to another. Q. E D. And from this it appears that Clavius and other Authors are mistaken who affert that those folid angles are equal which are contained by the fame 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 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 ftraight 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 affumed. 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 poffibly fhew 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 Barrow's Euclid, for it is the 10th in the Greek) " and not to have demonstrated, what no man can demonftrate?" But all that in this cafe can follow from that Axiom is, that if two ftraight lines could meet each other in two points, the parts of them betwixt these points must coincide, and fo they would have a fegment betwixt Book XI. these points common to both. Now, as it has not been shewn 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 fegment is deduced in the Greek Edition. but, on the contrary, because they cannot have a common segment, as is fhewn in Cor. of 11. Prop. B. 1. of 4to. Edition, it follows plainly that they cannot meet in two points, which the remarker fays no man can demonstrate. Mr. Simpson in the fame notes, p. 265. justly observes that in the Corollary of Prop. 11. Book 1. 4to. Edit. the straight lines AB, BD, BC, are fuppofed to be all in the fame plane, which cannot be affumed in 1. Prop. B. 11. this, foon after the 4to. Edition was published, I obferved 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 Curfus Vol. 1. and in place of it the Corollary of 11. Prop. Book 1. was added. PROP. II. B. XI. This Propofition 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 fecond part of the Enuntiation needs no Demonstration. befides a convex fuperficies may be terminated by three ftraight 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 Demonftration are changed into those now put into the Text. PROP. III. B. XI. In this Propofition 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 demonftrated that there can be no other "ftraight 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 ftraight 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 abfurd; 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 conclufion of the 4. Prop. B. 1. has been so often repeated in the preceding Books, it was needless to repeat it here. Book XI. PROP. V. B. XI. In this, near to the end, w, ought to be left out in the Greek text. and the word "plane" is rightly left out in the Oxford Edition of Commandine's Tranflation. PROP. VII. B. XI. This Propofition has been put into this Book by fome 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, fome Demonstrations in which one straight line is fuppofed to meet another would not be conclufive, because these lines would not meet one another. for instance, in Prop. 30. Book 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 Hypothefis, the ftraight line EF is. befides, this 7. Propofition is demonstrated by the preceding 3. in which the very thing which is propofed to be demonstrated in the 7. is twice affumed, viz. that the straight line drawn from one point to another in a plane, is in that plane; and the fame thing is affumed in the preceding 6. Prop. in which the ftraight line which joins the points B, D that are in the plane to which AB and CD are at right angles, is fuppofed to be in that plane. and the 7. of which another Demonstration is given, is kept in the Book merely to preserve the number of the Propofitions; for it is evident from the 7. and 35. Definitions of the 1. Book, tho' it had not been in the Elements. PROP. VIII. B. XI. In the Greek, and in Commandine's and Dr. Gregory's TranЛlations, near to the end of this Propofition, are the following words, "but DC is in the plane thro' BA, AD" instead of which in the Oxford edition of Commandine's tranflation, is rightly put "but "DC is in the plane thro' BD, DA." but all the Editions have Book XI. the following words, viz. "because AB, BD are in the plane thro' ❝ BD, DA, and DC is in the plane in which are AB, BD,” which are manifeftly corrupted, or have been added to the Text; for there was not the least neceffity to go so far about to fhew that DC is in the fame 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 fame plane with it," as being manifeftly 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 fame plane which produced meet neither way." because tho' in citing this Definition in Prop. 27. B. 1. it was not neceffary to mention the words, " in the fame plane" all the ftraight lines in the Books preceding this being in the fame 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 fhould be read thus, "but if not, let the angle BAC be not less than either of the other "two, but greater than DAB." At the end of this Proposition it is faid, " in the fame manner "it may be demonftrated," tho' there is no need of any Demon ftration; because the angle BAC being not lefs than either of the other two, it is evident that BAC together with one of them is greater than the other. PROP. XXII. B. XI. And likewise in this, near the beginning, it is said, "but if not, "let the angles at B, E, H be unequal, and let the angle at B be |