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



WHEN it is required to draw a perpendicular from a point in one plane which is at right angles to another plane, to 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 case to do it by the 11th Prop. of the same: But Euclid, Apollonius, and other a 17. 12. geometers, when they have occasion for this problem, direct a in other 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.


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.

Book XII.

a 12. 6.


THE learned Mr Moor, 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.


At the beginning of this it is said, "if it be not so, the 66 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 sufficient, in this and the like cases, that a thing made use of in the reasoning can possibly exist, providing 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 proportional, and this fourth square, or a space equal 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 shown by Euclid, but left out by some editor; for the lemma which some unskilful hand has added to this proposition explains nothing of it.


IN the Greek text and the translations, it is said, “and be"cause the two straight lines BA, AC which meet one an"other," &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, Book XII. and the triangles BAC, KHL are proved to be similar in a shorter way by Prop. 21, B. 6.


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


IN this, near to the end, are the words, as gooder ideíxen, "as was before shown ;" and the same are found again in the end of Prop. 18. of this Book; but the demonstration referred to, except it be the useless lemma annexed to the second Prop., is nowhere in these Elements, and has been perhaps left out by some editor, who has forgot to cancel those words also.


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 one 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, EBC, ECD; FGL, LGH, LHK, which are similar a, each to each; a 20. 6. And because the pyramids are similar, therefore the triangle b11.def.11. EAM is similar to the triangle LFN, and the triangle ABM

c 4. 6.

Book XII. 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, ex æquali, as EM to MB, so is LN to NG: Wherefore the triangles EMB, LNG having their sides proportionals, ared equiangular, and similar to one another: Therefore the pyramids which have the triangles EAB, LFG for their bases, and the points M, N for their b 11. def. vertices, are similar to one another, for their solid angles are equal, and the solids themselves are contained by the

d 5. 6.


c B. 11.

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same number of similar planes: In the same manner, the pyramid EBCM may be shown to be similar to the pyramid LGHN, and the pyramid ECDM to LHKN: And because the pyramids EABM, LFGN are similar, and have triangular f 8. 12. bases, the pyramid EABM has to LFGN the triplicate ratio of that which EB has to the homologous side LG. And in the same manner, the pyramid EBCM has to the pyramid LGHN the triplicate ratio of that which EB has to LG: Therefore as the pyramid EABM is to the pyramid LFGN, so is the pyramid EBCM to the pyramid LGHN: In like manner, as the pyramid EBCM is to LGHN, so is the pyramid ECDM to the pyramid LHKN: And as one of the antecedents is to one of the consequents, so are all the antecedents to all the consequents: Therefore as the pyramid EABM to the pyramid LFGN, so is the whole pyramid ABCDEM to the whole pyramid FGHKLN: And the pyramid EABM has to the pyramid LFGN the triplicate ratio of that which AB has to FG; therefore the whole pyramid

has to the whole pyramid the triplicate ratio of that which AB Book XII. has to the homologous side FG. Q. E. D.

PROP. XI. and XII. B. XII.

THE order of the letters of the alphabet is not observed in these two propositions, according to Euclid's manner, and is now restored: By which means the first part of Prop. 12. may be demonstrated in the same words with the first part of Prop. 11.; on this account the demonstration of that first part is left out, and assumed from Prop. 11.


In this proposition, the common section of a plane parallel to the bases of a cylinder, with the cylinder itself, is supposed to be a circle, and it was thought proper briefly to demonstrate it; from whence it is sufficiently manifest, that this plane divides the cylinder into two others: And the same thing is understood to be supplied in Prop. 14.


"AND complete the cylinders AX, EO," both the enunciation and exposition of the proposition represent the cylinders as well as the cones, as already described: Wherefore the reading ought rather to be, "and let the cones be ALC, "ENG; and the cylinders AX, EO."

The first case in the second part of the demonstration is wanting; and something also in the second case of that part, before the repetition of the construction is mentioned, which are now added.


In the enunciation of this proposition, the Greek words, εἰς τὴν μείζονα σφαῖραν στερεὸν πολύεδρον ἐγγράψαι μὴ ψαῦον τῆς ἐλάσσονος σφαίρας κατὰ τὴν ἐπιφάνειαν, are thus translated by Commandine and others," in majori solidum polyhedrum describere quod "minoris sphæræ superficiem non tangat;" that is, "to de"scribe in the greater sphere a solid polyhedron which shall "not meet the superficies of the less sphere:" Whereby they refer the words κατὰ τὴν ἐπιφάνειαν to these next to them τῆς ἐλάσε Covos palgas: But they ought by no means to be thus translated; for the solid polyhedron doth not only meet the super

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