Book XI.

PROP. I. B. XI.

The words at the end of this, "for a ftraight line cannot "meet a ftraight line in more than one point," are left out, as an addition by fome unfkilful hand; for this is to be demonftrated, not affumed.

Mr Thomas Simpfon, 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 "fkill in an editor (he means Euclid or Theon) to refer to an "axiom which Euclid himself hath laid down, book 1. No 14." he means Barrow's Euclid, for it is the roth in the Greek," and "not to have demonftrated, what no man can demonftrate?" But all that in this cafe can follow from that axiom is, that, if two straight lines could meet each other in two points, the parts of them betwixt thefe points muft coincide, and fo they would have a fegment betwixt these points common to both. Now, as it has not been fhewn in Euclid, that they cannot have a com mon fegment, 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 fegment, as is fhewn in Cor. of 11th prop. book 1. of 4to edition, it follows plainly that they cannot meet in two points, which the remarker fays no man can demonftrate.

Mr Simpson, in the fame notes, p. 265. justly observes, that in the corollary of prop. 11. book 1. 4to edit. the ftraight lines AB, BD, BC, are fuppofed to be all in the fame plane, which cannot be affumed in ift prop. book 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 10th prop. book 1. was added.

PRO P. II. B. XI.

This propofition feems to have been changed and vitiated by fome editor; for all the figures defined in the 1ft book of the elements, and among them triangles, are, by the hypothefis, plane figures; that is, fuch as are defcribed in a plane; where fore the fecond part of the enunciation needs no demonstration. Befides, a convex fuperficies may be terminated by three ftraight

lines meeting one another: The thing that fhould have been Book XI. demonftrated is, that two, or three ftraight lines, that meet one another, are in one plane. And as this is not fufficiently done, the enunciation and demonstration are changed into those now put into the text.

PRO P. 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 demonstrated that there can be "no other straight line between the points D, B:" Because from this, that two lines include a fpace, 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 fection of the planes be not a ftraight line, then two ftraight lines could include a space, which is abfurd; therefore the common fection is a ftraight line,

PROP. IV. B. XI.

The words and the triangle AED to the triangle BEC" are omitted, because the whole conclufion of the 4th prop. b. 1. has been fo often repeated in the preceding books, it was needless to repeat it here.

PROP. V. B. XI.

In this, near to the end, iw, 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 unfkilful editor, as is evident from this, that ftraight lines which are drawn from one point to another in a plane, are, in the preceding books, fuppofed to be in that plane: And if they were not, fome demonstrations in which one ftraight line is fuppofed to meet another would not be conclufive, because thefe lines would not meet one another: For instance, in prop. 30. b. 1. the ftraight 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: Betides, this 7th propofition is demonftrated by the preceding 3. in which the very thing which is propofed to be demonftrated in the 7th, is twice affumed, viz. that the ftraight line drawn from one point to another in a plane, is in that plane; and the fame thing is affumed in the preceding 6th prop. in which the ftraight line

which

Book XI. 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 7th, of which another demonftration is given, is kept in the book merely to preferve the number of the propofitions; for it is evident from the 7th and 35th definitions of the ift book, though it had not been in the elements.

PROP. VIII. B. XI.

In the Greek, and in Commandine's and Dr Gregory's tranf lations, near to the end of this propofition, are the following words: "But DC is in the plane through BA, AD" instead of which, in the Oxford edition of Commandine's tranflation, 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 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 leaft neceffity to go fo far about to fhew that DC is in the fame plane in which are BD, DA, because it immediately follows from prop7. preceding, that BD, DA are in the plane in which are the parallels AB, CD: Therefore, inftead of these words, there ought only to be "becaufe all three are in the plane in which "are the parallels AB, CD."

PROP. XV. B. XI.

After the words," and becaufe 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, inftead of the words, "but ftraight "lines which meet neither way" ought to be read, "but straight lines in the fame plane which produced meet neither way:" Becaufe, though in citing this definition in prop. 27. book 1. it was not neceflary 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 neceffary.

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

be read thus," But if not, let the angle BAC be not less than Book XI. either of other two, but greater than DAB."

At the end of this propofition it is faid, " in the fame man"ner it may be demonftrated," though there is no need of any demonftration; 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 likewife in this, near the beginning, it is faid, "But "if not, let the angles at B, E, H be unequal, and let the angle at B be greater than either of thofe at E, H:" Which words manifeftly 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 lefs than either "of the other two at E, H: Therefore the ftraight line AC " is not less than either of the two DF, GK.”

PROP. XXIII. B. XI.

The demonftration of this is made fomething shorter, by not repeating in the third cafe the things which were demonftrated in the firft; and by making use of the conftruction which Campanus has given; but he does not demonftrate the fecond and third cafes: The conftruction and demonftration of the third cafe are made a little more fimple than in the Greek text.

PRO P. XXIV. B. XI.

The word "fimilar" is added to the enunciation of this propofition, because the planes containing the folids which are to be demonftrated to be equal to one another, in the 25th propofition, ought to be fimilar and equal, that the equality of the folids may be inferred from prop. C. of this book: And, in the Oxford edition of Commandine's tranflation, a corollary is added to prop. 24. to fhew that the parallelograms mentioned in this propofition are fimilar, that the equality of the folids in prop. 25. may be deduced from the 10th def. of book 1I.

PRO P. XXV. and XXVI. B. XI.

In the 25th prop. folid figures which are contained by the fame number of fimilar and equal plane figures, are fuppofed to be equal to one another. And it feems that Theon, or fome other editor, that he might fave himself the trouble of demonftrating the folid figures mentioned in this propofition to

be

Book XI. be equal to one another, has inferted the 10th def. of this book, to ferve instead of a demonstration; which was very ignorantly done.

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Likewife in the 26th prop. two folid angles are fuppofed 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 ftrange enough, that none of the commentators on Euclid have, as far as I know, perceived that fomething is wanting in the demonftrations of these two propofitions. Clavius, indeed, in a note upon the 11th def. of this book, affirms, that it is evident that thofe folid angles are equal which are contained by the fame 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 faid without any proof, nor is it always true, except when the folid angles are contained by three plane angles only, which are equal to one another, each to each: And in this cafe the propofition is the fame with this, that two fpherical triangles that are equilateral to one another, are alfo equiangular to one another, and can coincide; which ought not to be granted without a demonftration. Euclid does not affume this in the case of rectilineal tri. angles, but demonftrates in prop. 8. book 1. that triangles which are equilateral to one another are alfo equiangular to one another; and from this their total equality appears by prop. 4. book 1. And Menelaus, in the 4th prop. of his ift book of fpherics, explicitly demonftrates that spherical triangles which are mutually equilateral, are alfo equiangular to one another; from which it is eafy to fhew that they muft coincide, providing they have their fides difpofed in the fame order and fituation.

To fupply these defects, it was neceffary to add the three propofitions marked A, B, C to this book. For the 25th, 26th, and 28th propofitions of it, and confequently eight others, viz. the 27th, 31ft, 32d, 33d, 34th, 36th, 37th, and 40th of the fame, which depend upon them, have. hitherto ftood upon an infirm foundation; as alfo, the 8th, 12th, Cor. of 17th and 18th of 12th book, which depend upon the 9th definition. For it has been fhewn in the notes on def. 10. of this book, that folid figures which are contained by the fame number of fimilar and equal plane figures, as alfo folid angles that are contained by the fame number of equal plane angles, are not always equal to one another.

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