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 fo often repeated in the preceding Books, it was needless to repeat it here.

PROP. V. B. XI.

In this, near to the end, 77id, 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 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 Demonftrations in which one ftraight line is fuppofed to meet another would not be conclufive, because these lines would not meet one another. for inftance, 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 Hypothefis, the straight line EF is. befides, this 7. Propofition is demonftrated by the preceding 3. in which the very thing which is propofed to be demonftrated 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 Demonftration 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 Transla tions, near to the end of this Propofition, are the following words, "but DC is in the plane thro' BA, AD" inftead 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.

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 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 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 "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.

1

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." becaufe tho' in citing this Definition in Prop. 27. B. 1. it was not neceffary to mention the words, " in the fame plane" all the straight 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 read thus, but if not, let the angle BAC be not lefs than either of the other "two, but greater than DAB.”

At the end of this Propofition it is faid, " in the fame manner "it may be demonftrated," tho' there is no need of any Demonstration; 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 likewife in this, near the beginning, it is faid, "but if not, flet 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 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 ftraight line AC is not lefs than either of the two DF, GK."

PROP. XXIII. B. XI.

The Demonftration of this is made fomething fhorter, by not repeating in the third cafe the things which were demonftrated in the firft; and by making ufe of the conftruction which Campanus has given; but he does not demonftrate the second and third cafes. the construction and demonftration of the third cafe are made little more fimple than in the Greek Text.

PROP. XXIV. B. XI.

2

The word "fimilar" is added to the Enuntiation of this Propo fition, because the planes containing the folids which are to be de monftrated to be equal to one another in the 25. 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 Edi tion 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 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 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 equal to one another, has inferted the 10. Def. of this Book, to ferve inftead of a Demonstration; which was very ignorantly done.

Likewife in the 26. 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 thefe two Propofitions. Clavius, indeed, in a note upon the 11, Def

Book XI, 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 spherical triangles that are equilateral to one another, are alfo equiangular to one another, and can coincide; which ought not to be granted without a demonstration. Euclid does not affume this in the cafe of rectilineal triangles, but demonftrates in Prop. 8. B. 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. B. 1. and Menelaus, in the 4. Prop. of his 1. Book of Spherics, 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 thefe defects, it was neceffary to add the three Propofitions marked A, B, C to this Book. for the 25. 26. 2nd 28. Propofitions of it, and confequently eight others, viz. the 27. 31. 32. 33. 34. 36. 37. and 40. of the fame, which depend upon them, have hitherto ftood upon an infirm foun dation; as alfo, the 8. 12. Cor. of 17. and 18. of the 12. Book, which depend upon the 9. 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.

It is to be obferved 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 pfelefs Definition, that by it he might demonftrate the 36. Prop. of this Book without the help of the 35. of the fame. concerning which Demonftra tion, fee the Note upon Prop. 36.

PROP. XXVIII. B. XI.

In this it ought to have been demonftrated, not affumed, that the diagonals are in one plane. Clavius has fupplied this defect.

There are three Cafes of this Propofition; the first is when the two parallelograms oppofite to the base AB have a fide common to both; the second is, when thefe 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 Demonftration that has hitherto been in the Elements does agree. The firft Cafe is immediately deduced from the preceding 28. Propofition, which feems for this purpose to have been premised to this 29. for it is neceffary to none but to it, and to the 40. of this Book, as we now have it, to which laft it would, without doubt, have been premifed, if Euclid had not made ufe of it in the 29. but fome unfkilful Editor has taken it away from the Elements, and has mutilated Euclid's Demonftration 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 oppofite 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 fake of learners.

PROP. XXXI. B. XI.

There are two Cafes of this Propofition; the first is when the infifting ftraight lines are at right angles to the bafes; the other when they are not. the first Cafe 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 last Cases, but has inferted the Demonstration 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 cafes feparately. the Demonftration alfo is made fomething shorter by following the way Euclid has made ufe of in Prop. 14. B. 6. befides, in the Demonftration of the cafe in which the infifting straight lines are not at right angles to the bafes, the Editor does not prove that the folids defcribed in the construction are parallelepipeds, which it is not to be thought that Euclid neglected. alfo the words, " of which the infifting straight lines are

Book XI.