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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) “ to “ refer to an axiom which Euclid himself hath laid down, B. l, « No. 14,” (he means Barrow's Euclid, for it is the 10th in the Greek,)“ and not to have demonstrated, what no man can de6 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, frizidw 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 bad not been Book XE. 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.
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, s in the same plane,” all the straight lines in the books preceding this being in the same plane; yet here it was quite necessary.
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
Book XI. should be read thus, “ But if not, let the angle BAC be not less
66 than either of the other two, but greater than DAB.”
At the end of this proposition it is said, “in the same man“ ner it may be demonstrated,” though there is no need of any demonstration : 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 angles at B, E, H be unequal, and let the
angle at B be greater than either of those at E, H:” Which words manifestly show 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.”
PROP. XXIII. B. XI.
The demonstration of this is made something 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 case are made a little more simple than in the Greek text.
PROP. XXIV. B. XI.
The word “ similar" is added to the enunciation of this proposition, because the planes containing the solids which are to be demonstrated to be equal to one another, in the 25th Proposition, ought to be similar and equal, that the equality of the solids may be inferred from Prop. C. of this book : And in the Oxford edition of Commandine's translation, a corollary is added to Prop. 24, to show that the parallelograms mentioned in this proposition are similar, that the equality of the solids in Prop. 25. may be deduced from the 10th Def. of Book XI.
PROP. XXV. and XXVI. B. XI.
In the 25th Prop. solid figures which are contained by the same number of similar and equal plane figures, are supposed to be equal to one another. And it seems that Theon, or some other editor, that he might save himself the trouble of demonstrating the solid figures mentioned in this proposition to be equal to one another, has inserted the 10th Def. of this book, to serve instead of a demonstration ; which was very ignorantly done.
Likewise in the 26th Prop. two solid angles are supposed to be equal: If each of them be 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 these two propositions. Clavius, indeed, in a note upon the 11th Def. of this Book, affirms that it is evident that those solid 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 same 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 assume this in the case of rectilineal triangles, but demonstrates in Prop. 8, Book 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, Book 1. And Menelaus, in the 4th Prop. of his first Book of Spherics, explicitly demonstrates, that spherical triangles which are mutually equilateral, are also equiangular to one another; from which it is easy to show that they must coincide, providing they have their sides disposed in the same order and situation.
To supply these defects, it was necessary to add the three Propositions marked A, B, C to this book. For the 25th, 26th, and 28th Propositions of it, and consequently eight others, viz. the 27th, 31st, 32d, 333, 34th, 36th, 37th, and 40th of the same which depend upon them, have hitherto stood upon an infirm foundation; as also, the 8th, 12th, Cor. of 17th and 18th of 12th Book, which depend upon the 9th definition. For it has been shown in the notes on Def. 10.