Book XI. PROP. IV. B. XI. The words “ and the triangle AED to the triangle BEC" are omitted, because the whole conclusion of the 4. Prop. B. 1. has been fo often repeated in the preceding Books, it was needless to repeat it here. PRO P. V. B. XI. PRO P. 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, fome 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,ÇD, and in which, by Hypothesis, the straight line EF is. besides, this 7. Proposition is demonstrated by the preceding 3. in which the very thing which is proposed to be demonstrated in the 7, 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 6. 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 7. 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 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 Transla. tions, near to the end of this Proposition, are the following words, “ but DC is in the plane thro’ BA, AD” instead of which in the Oxford edition of Commandine's translation, 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 manifestly corrupted, or have been added to the Text; for there was not the least necessity to go so far about to thew 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 of whichi cre the parallels AB, CD.” PRO P. XV. B. XI. After the words, TM 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 manifestly forgotten to be put into the Text. PRO P. XVI. B. XI. In this, near to the end, instead of the words " but ļtraight 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 necessary to mention the words, “ in the fame 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 should 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 said, " in the same manner “ it may be demonstrated,” tho' there is no need of any Demon îtration ; because the angle BAC being not less 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 anglès at B, E, H be unequal, and let the angle at B be greater than either of those at E, H.” which words manifefly Book XI. fhew this place to be vitiated, because the angle at B may be equal to one of the other two. they ought therefore to be read thus,“ but “ if not, let the angles at B, E, H be unequal, and let the angle at B be not less than either of the other two at E, H. therefore “ the straight line AC is not less than either of the two DF, GK." PRO P. XXIII. B. XI. The Demonstration of this is made fomething shorter, by not repeating in the third case the things which were demonstrated in the first; and by making use of the construction which Campanus has given ; but he does not demonstrate the second and third cases. the construction and demonstration of the third cafe are made a little more simple than in the Greek Text. PROP. XXIV. B. XI. The word “ similar" is added to the Enuntiation of this Propo. sition, because the planes containing the folids which are to be de monstrated to be equal to one another in the 25. Proposition, cught to be fimilar and equal; that the equalicy of the folids may be inferred from Prop. C. of this Book. and in the Oxford Edi, tion of Commandine's Translation, a Corollary is added to Prop. 24. to Mew that the parallelograms mentioned in this Propofition are similar, that the equality of the solids in Prop. 25. may be deduced from the 10. Def. of B. II. PROP. XXV. and XXVI. B. XI, In the 25. Prop. folid figures which are contained by the fame number of fimilar and equal plane figures, are supposed to be equal to one another. and it seems that Theon, or some other Editor, that he might fave himself the trouble of demonstrating the folid figures mentioned in this Proposition to be equal to one another, has inserted the 10. Def. of this Book, to serve instead of a Dea monstration ; which was very ignorantly done. Likewise in the 26. Prop. two folid angles are supposed to be equal, if each of them bc contained by three plane angles which are equal to one another, each to each. and it is strange enough, that none of the Commentators on Euclid have, as far as I know, perceived that something is wanting in the demonstrations of thefe (wo Propositions. Clavius, indeed, in a note upon the 11. Defi Book XI, of this Book, affirms, that it is evident that those folid angles are equal which are contained by the same number of plane angles, equal to one another, each to each, because they will coincide, if they be conceived to be placed within one another; but this is said without any proof, nor is it always true, except when the solid angles are contained by three plane angles only, which are equal to one another, each to each. and in this case the proposition is the fame with this, that two spherical triangles that are equilateral to one another, are also equiangular to one another, and can coincide; which ought not to be granted without a demonstration. Euclid does not affume this in the case of rectilineal triangles, but demonstrates in Prop. 3. B. 1. that triangles which are equilateral to one another are also equiangular to one another; and from this their total equality appears by Prop. 4. B. 1. and Menelaus, in the 4. Prop. of his 1. Book of Spherics, explicitly demonstrates that spherical triangles which are mutually equilateral, are also equiangular to one another ; from which it is easy to new that they must coincide, providing they have their fides disposed in the fame order and situation. To supply these defects, it was necessary to add the three Propositions marked A, B, C to this Book. for the 25. 26. and 28. Propositions of it, and consequently eight others, viz. the 27. 31. 32. 33. 34. 36. 37. and 40. of the fame, which depend upon them, have hitherto stood upon an infirm foun. dation; as also, the 8. 12. Cor. of 17. and 18. of the 12. Book, which depend upon the 9. Definition. for it has been fewn in the Notes on Def. 10. of this Book, that solid figures which are contained by the same number of similar and equal plane figures, as also folid angles that are contained by the same number of equal plane angles are not always equal to one another. It is to be observed that Tacquet, in his Euclid, defines equal folid angles to be fach," as being put within one another do coin“ cide." but this is an Axiom, not a Definition, for it is true of all magnitudes whatever. he made this pseless Definition, that by it he might demonstrate the 36. Prop. of this book without the help of the 35. of the fame. concerning which Demonstra. țion, see the Note upon Prop. 36. PROP. XXVIII. B. XI. Book XI. PROP. XXIX. B. XI. There are three Cafes of this Proposition; the first is when the two parallelograms opposite to the base AB have a side common to both ; the second is, when these parallelograms are separated from one another; and the third, when there is a part of them common to both; and to this laft only the Demonstration that has hitherto been in the Elements does agree. The first Case is immediately deduced from the preceding 28. Propofition, which seems for this purpose to have been premised to this 29. for it is necesary to none but to it, and to the 40. of this Book, as we now have it, to which last it would, without doubt, have been premised, if Euclid had not made use of it in the 29. but some uníkilful Editor has taken it away from the Elements, and has mutilated Euclid's Demonstration of the other two Cafes, which is now restored, and ferves for both at once. PROP. XXX. B. XI. In the Demonstration of this, the opposite planes of the folid CP, in the figure in this Edition ; that is, of the folid CO in Commandine's figure, are not proved to be parallel ; which it is proper to do for the sake of learners. PROP. XXXI. B. XI. There are two Cases of this Proposition ; the first is when the insisting straight lines are at right angles to the bases; the other when they are not the first Case is divided again into two others, one of which is when the bases are equiangular parallelograms; the other when they are not equiangular. the Greek Editor makes no mention of the first of these two laft Cases, but has inserted the De. monstration of it as a part of that of the other. and therefore should have taken notice of it in a Corollary; but we thought it better to give these two cases separately. the Demonstration also is made something shorter by following the way Euclid has made use of in Prop. 14. B. 6. besides, in the Demonstration of the case in which the insisting straight lines are not at right angles to the bases, the Editor does not prove that the folids described in the construction are parallelepipeds, which it is not to be thought that Euclid neglected. also the words, “ of which the insisting straight lines are |