Book XI. not equiangular. The Greek editor makes no mention of the first of these two last cases, but has inserted the demonstration of it as a part of that of the other: And therefore should have taken notiee 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, Book 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 solids described in the construction are parallelopipeds, which it is not to be thought that Euclid neglected: also the words, "of which the insisting straight lines are not in the same straight lines," have been added by some unskilful 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 necessary. Clavius has supplied this.

Also, in the construction, it is required to complete the solid of which the base is FH, and altitude the same with that of the solid CD: But this does not determine the solid to be completed, since there may be innumerable solids upon the same base, and of the same altitude: It ought therefore to be said, "complete the solid of which the base "is FH, and one of its insisting straight lines is FD:" The same correction must be made in the following Proposition 33.

PROP. D. B. XI.

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

PROP. XXXIV. B. XI.

In this the words, ὧν αἱ ἐφεστῶσαι ἐκ εἰσὶν ἐπὶ τῶν αὐτῶν Ebay, "of which the insisting 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 insisting "straight lines be, or be not, in the same straight lines:" For the other case is, without any reason, excluded; also the words wv rà ŋ, "of which the altitudes," are twice

put for w ai perTwoα, "of which the insisting straight Boox XI. "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 demonstrated to be right angles in a shorter way than in the Greek; and in the same way ACH, DFM, may be demonstrated to be right angles: Also the repetition of the same demonstration, which begins with "in the same manner," is left out, as it was probably added to the text by some editor: for the words "in like manner we may demonstrate," are not inserted except when the demonstration is not given, or when it is something different from the other if it be given, as in Prop. 26. of this Book. Campanus has not this repeti

tion.

We have given another demonstration of the corollary, besides the one in the original, by help of which the following 36th Prop. may be demonstrated without the 35th.

PROP. XXXVI. B. XI.

TACQUET in his Euclid demonstrates this proposition without the help of the 35th; but it is plain, that the solids mentioned in the Greek text in the enunciation of the proposition as equiangular, are such that their solid angles are contained by three plane angles equal to one another, each to each; as is evident from the construction. Now Tacquet does not demonstrate, but assumes, these solid angles to be equal to one another; for he supposes the solids to be already made, and does not give the construction by which they are made: But, by the second demonstration of the preceding corollary, his demonstration is rendered legitimate likewise in the case where the solids are constructed as in the text.

PROP. XXXVII. B. XI.

In this it is assumed, that the ratios which are triplicate of those ratios which are the same with one another, are likewise the same with one another: and that those ratios are the same with one another, of which the triplicate ratios are the same with one another: but this ought not to be granted without a demonstration; nor did Euclid assume the first and easiest of these two propositions, but demonstrated it in the case of duplicate ratios, in the 22d Prop. Book 6. On this account, another demonstration is

Boox XI. given of this Proposition, like to that which Euclid gives in Prop. 22, Book 6, as Clavius has done.

editions.

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 section 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 17. 12. case to do it by the 11th Prop. of the same: But Euclid, in other Apollonius, and other geometers, when they have occasion for this problem, direct a perpendicular to be drawn from the point to the plane, and conclude that it will fall upon the common section of the planes, because this is the very same thing as if they had made use of the construction above-mentioned, and then concluded that the straight line must be perpendicular to the plane; but is expressed in fewer words. Some editor, not perceiving this, thought it was necessary to add this proposition, which can never be of any use to the 11th book; and its being near to the end among propositions with which it has no connexion, is a mark of its having been added to the text.

PROP. XXXIX. B. XI.

In this it is supposed, that the straight lines which bisect the sides of the opposite planes, are in one plane, which ought to have been demonstrated; as is now done.

B. XII.

BOOK XII. THE learned Mr. Moore, Professor of Greek in the University of Glasgow, observed to me that it plainly appears from Archimedes's Epistle to Dositheus, prefixed to his books of the Sphere and Cylinder, which epistle he has restored from ancient manuscripts, that Eudoxus was the author of the chief propositions in this 12th book.

4

Ar the beginning of this it is said, "if it be not so, the square of BD shall be to the square of FH, as the circle "ABCD is to some space either less than the circle "EFGH, or greater than it," And the like is to be found near to the end of this proposition, as also in Prop. 5, 11, 12, 18 of this Book: Concerning which it is to be

observed, that in the demonstration of theorems, it is suffi- Book XII. cient, in this and the like cases, that a thing made use of in the reasoning can possibly exist, provided this be evident, though it cannot be exhibited or found by a geometrical construction: So, in this place, it is assumed, that there may be a fourth proportional to these three magnitudes, viz. the squares of BD, FH, and the circle ABCD; because it is evident that there is some square equal to the circle ABCD, though it cannot be found geometrically; and to the three rectilineal figures, viz. the squares of BD, FH, and the square which is equal to the circle ABCD, there is a fourth square proportional; because to the three straight lines which are their sides, there is a fourth straight line proportionala, and this fourth square, or a space equal ⚫ 12. 6. to it, is the space which in this proposition is denoted by the letter S: And the like is to be understood in the other places above cited: And it is probable that this has been shewn by Euclid, but left out by some editor; for the lemma which some unskilful hand has added to this proposition explains nothing of it.

PROP. III. B. XII.

In the Greek text and the translations, it is said, "and "because the two straight lines BA, AC, which meet one "another," &c. Here the angles BAC, KHL, are demonstrated to be equal to one another, by 10th Prop. B. 11, which had been done before: Because the triangle EAG was proved to be similar to the triangle KHL: This repetition is left out, and the triangles BAC, KHL, are proved to be similar in a shorter way by Prop. 21. B. 6.

PROP. IV. B. XII.

A FEW things in this are more fully explained, than in the Greek text.

66

PROP. V. B. XII.

IN this, near to the end, are the words, ws urgoσDEY εδείχθη, as was before shewn;" and the same are found again in the end of Prop. 18. of this Book; but the demonstration referred to, except it be in the useless lemma annexed to the 2d Prop. is no where in these Elements, and has been perhaps left out by some editor, who has forgot to cancel those words also.

BOOK XII.

PROP. VI. B. XII.

A SHORTER demonstration is given of this; and that which is in the Greek text may be made shorter by a step than it is: For the author of it makes use of the 22d Prop. of B. 5, twice: Whereas once would have served his purpose; because that proposition extends to any number of magnitudes which are proportionals taken two and two, as well as to three which are proportional to other three.

THE demonstration of this is imperfect, because it is not shown, that the triangular pyramids into which those upon multangular bases are divided, are similar to one another, as ought necessarily to have been done, and is done in the like case in Prop. 12. of this Book: The full demonstration of the corollary is as follows:

Upon the polygonal bases ABCDE, FGHKL, let there be similar and similarly situated pyramids which have the points M, N, for their vertices: The pyramid ABCDEM has to the pyramid FGHKLN the triplicate ratio of that which the side AB has to the homologous side FG.

Let the polygons be divided into the triangles ABE, a 20. 6. EBC, ECD; FGL, LGH, LHK, which are similara, each 11 Def.11. to each: And because the pyramids are similar, therefore b the triangle EAM is similar to the triangle LFN, and the 4. 6. triangle ABM to FGN: Wherefore ME is to EA, as NL to LF; and as AE to EB, so is FL to LG, because the

triangles EAB, LFG, are similar; therefore, ex æquali, as ME to EB, so is NL to LG: In like manner it may be shown, that EB is to BM, as LG to GN; therefore, again,