"greater than either of those 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 straight 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 demonstrated in the firft; and by making use of the construction which Campanus has given; but he does not demonstrate the second and third cafes. the conftruction and demonstration of the third cafe are made a little more fimple than in the Greek Text.

PROP. XXIV. B. XI.

The word "fimilar" is added to the Enuntiation of this Propofition, because the planes containing the solids which are to be demonstrated 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 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 ro. Def. of B. 11.

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 fome other Editor, that he might fave himself the trouble of demonstrating the solid 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.

Likewise in the 26. Prop. two folid 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 fomething is wanting in the demonftrations of these two Propofitions. Clavius, indeed, in a note upon the 11. Def.

Book XI. pf this Book, affirms, that it is evident that those 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 proposition 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 demonstrates 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 easy to fhew that they must coincide, providing they have their fides disposed 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. and 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 foundation; as also, 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 fuch, " 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 ufelefs 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, see the Note upon Prop. 36.

PROP. XXVIII. B. XI.

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

PROP. XXIX. B. XI.

There are three Cafes of this Propofition; the first is when the two parallelograms oppofite to the bafe 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 last only the Demonstration that has hitherto been in the Elements does agree. The first Case 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 last it would, without doubt, have been premised, if Euclid had not made use of it in the 29. but fome unskilful Editor has taken it away from the Elements, and has mutilated Euclid's Demonftration of the other two Cafes, which is now reftored, 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 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 bafes 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 something shorter by following the way Euclid has made use of in Prop. 14. B. 6. befides, in the Demonstration of the cafe in which the infifting 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 infifting straight lines are

Book XI.

Book XI. " not in the same straight lines," have been added by fome unskil

ful hand; for they may be in the same straight lines.

PROP. XXXII. B. XI.

The Editor has forgot to order the parallelogram FH to be applied in the angle FGH equal to the angle LCG, which is neceffary. Clavius has supplied this.

Alfo, in the conftruction, it is required to complete the folid of which the bafe is FH, and altitude the fame with that of the folid CD; but this does not determine the solid to be completed, fince there may be innumerable folids upon the fame base, and of the fame altitude. it ought therefore to be faid "complete the folid of "which the base is FH, and one of its infisting straight lines is "FD." the fame correction must be made in the following Propofition 33.

PROP. D. B. XI.

It is very probable that Euclid gave this Propofition a place in the Elements, fince he gave the like Propofition concerning equiangular parallelograms in the 23. B. 6.

PROP. XXXIV. B. XI.

In this the words, ὧν αἱ ἐφεςῶσαι ἐκ εἰσὶν ἐπὶ τῶν αὐτῶν εὐθειῶν, "of which the infisting straight lines are not in the same straight "lines" are thrice repeated; but these words ought either to be left out, as they are by Clavius, or in place of them ought to be put "whether the infifting straight lines be, or be not, in the fame "straight lines." for the other Cafe is without any reafon excluded. also the words, v Tan, " of which the altitudes" are twice put for av ai spesãoa, “ of which the infisting straight lines;" which is a plain mistake. for the altitude is always at right angles to the base.

PROP. XXXV. B. XI.

The angles ABH, DEM are demonftrated to be right angles in a fhorter way than in the Greek; and in the fame way ACH, DFM may be demonstrated to be right angles. also the repetition of the fame Demonstration, which begins with " in the fame manner," is left out, as it was probably added to the Text by fome Editor; for the words, "in like manner we may demonftrate" are not inferted

except when the Demonstration is not given, or when it is fome- Book XI. thing different from the other, if it be given, as in Prop. 26. of this Book. Campanus has not this repetition.

We have given another Demonstration of the Corollary, besides the one in the Original, by help of which the following 36. Propofition may be demonftrated without the 35.

PROP. XXXVI. B. XI.

Tacquet in his Euclid demonstrates this Propofition without the help of the 35. but it is plain that the folids mentioned in the Greek Text in the Enuntiation of the Propofition as equiangular, are fuch that their folid angles are contained by three plane angles equal to one another, each to each; as is evident from the conftruction. Now Tacquet does not demonftrate, but affumes these folid angles to be equal to one another; for he supposes the folids to be already made, and does not give the conftruction by which they are made. but, by the fecond Demonstration of the preceding Corollary, his Demonftration is rendered legitimate likewise in the Cafe where the folids are constructed as in the Text.

PROP. XXXVII. B. XI.

In this it is affumed that the ratios which are triplicate of those ratios which are the fame with one another, are likewise the fame with one another; and that those ratios are the fame with one another, of which the triplicate ratios are the fame with one another; but this ought not to be granted without a Demonstration, nor did Euclid affume the firft and eafieft of thefe two Propofitions, but demonstrated it in the cafe of duplicate ratios, in the 22. Prop. B. 6. on this account another Demonftration is given of this Propofition like to that which Euclid gives in Prop. 22. B. 6. as Clavius has done,

PROP. XXXVIII. B. XI.

When it is required to draw a perpendicular from a point in one plane which is at right angles to another plane, unto this last plane, it is done by drawing a perpendicular from the point to the common fection of the planes; for this perpendicular will be perpendicular to the plane, by Def. 4, of this Book. and it would be foolish in this cafe to do it by the 11. Propofition of the fame. but Euclid a, a. 17. 12. in Apollonius, and other Geometers, when they have occafion for this Problem, direct a perpendicular to be drawn from the point to the

other Edi

tions,