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Book XI. plane, and conclude that it will fall upon the common fection of the planes, because this is the very fame thing as if they had made ufe of the conftruction above-mentioned, and then concluded that the straight line must be perpendicular to the plane; but is expreffed in fewer words. fome Editor not perceiving this, thought it was neceffary to add this Propofition, which can never be of any use, to the 11. Book. and its being near to the end among Propofitions 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 fides of the oppofite planes, are in one plane, which ought to have been demonstrated; as is now done.

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HE learned Mr. Moor, Profeffor of Greek in the University

Tof Glafgow, obferved to me that it plainly appears from

Archimedes Epiftle to Dofitheus prefixed to his Books of the Sphere and Cylinder, which Epistle he has restored from antient Manufcripts, that Eudoxus was the Author of the chief Propofitions in this twelfth Book.


At the beginning of this it is faid, "if it be not fo, the square of "BD fhall be to the fquare of FH, as the circle ABCD is to fome " space either lefs than the circle EFGH, or greater than it." and the like is to be found near to the end of this Propofition, as also in Prop. 5. 11. 12. 18. of this Book. concerning which it is to be obferved, that in the Demonstration of Theorems, it is sufficient, in this and the like cafes, that a thing made use of in the reasoning can poffibly exist, providing this be evident, tho' it cannot be exhibited or found by a Geometrical construction. fo in this place it is affumed 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 fome fquare equal to the circle ABCD, tho' it cannot be found geometrically; and to the three rectilineal figures, viz. the fquares of BD, FH, and the fquare which is equal

to the circle ABCD, there is a fourth fquare proportional; because Book XII. to the three straight lines which are their fides there is a fourth ftraight line proportional, and this fourth square, or a space equal a. 12. 6. to it, is the space which in this Propofition 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 fome Editor; for the Lemma which fome unfkilful hand has added to this Propofition 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 another" &c. here the angles BAC, KHL are demonstrated to be equal to one another by 10. Prop. B. 11. which had been done before. because the triangle EAG was proved to be fimilar to the triangle KHL. this repetition is left out, and the triangles BAC, KHL are proved to be similar in a fhorter 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 gode ideixen," as "was before fhewn," and the fame 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 2. Prop. is no where in these Elements, and has been perhaps left out by fome 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 fhorter by a ftep than it is. for the Author of it makes use of the 22. Prop. of B. 5. twice, whereas once would have ferved his purpofe; because that Propofition extends to any number of magnitudes which are proportionals taken two and two, as well as to three which are proportional to other three.

Book XII.

2. 20. 6.
b. 11. Def.


c. 4. 6.


The Demonftration of this is imperfect, becaufe it is not fhewn that the triangular pyramids into which thofe upon multangular bafes are divided, are fimilar to one another, as ought neceffarily to have been done, and is done in the like cafe in Prop. 12. of this Book. the full Demonftration of the Corollary is as follows.

Upon the polygonal bases ABCDE, FGHKL, let there be fimilar and fimilarly fituated 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 fide AB has to the homologous fide FG.

Let the polygons be divided into the triangles ABE, EBC, ECD; FGL, LGH, LHK, which are fimilar each to each. and because the pyramids are fimilar, therefore the triangle EAM is fimilar to the triangle LFN, and the triangle ABM to FGN. wherefore ME is to EA, as NL to LF; and as AE to EB, fo is FL to LG, because

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d. 5. 6.

e. B. II.

the triangles EAB, LFG are fimilar; therefore, ex aequali, as ME to EB, fo is NL to LG. in like manner it may be fhewn that EB is to BM, as LG to GN; therefore, again, ex aequali, as EM to MB, fo is LN to NG. wherefore the triangles EMB, LNG having their fides proportionals are a equiangular, and fimilar to one another. therefore the pyramids which have the triangles EAB, LFG for their bases, and the points M, N for their vertices are fimilar b to one another, for their folid angles are equal, and the folids themselves are contained by the fame number of fimilar planes. in the fame manner the pyramid EBCM may be fhewn to be fimilar


to the pyramid LGHN, and the pyramid ECDM to LHKN. and Book XII. because the pyramids EABM, LFGN are fimilar, and have triangular bases, the pyramid EABM has to LFGN the triplicate ratio f. 8. 12. of that which EB has to the homologous fide 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, fo is the pyramid ECDM to the pyramid LHKN. and as one of the antecedents is to one of the confequents, so are all the antecedents to all the confequents. therefore as the pyramid EABM to the pyramid LFGN, fo 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 has to the homologous fide FG. Q. E. D.

PROP. XI. and XII. B. XII.

The order of the letters of the Alphabet is not observed in these two Propofitions, according to Euclid's manner, and is now restored. by which means the first part of Prop. 12. may be demonftrated in the fame words with the firft part of Prop. II. on this account the Demonstration of that first is left out, and affumed from Prop. 11.



In this Propofition 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 demonftrate it; from whence it is fufficiently manifeft that this plane divides the cylinder into two others. and the fame thing is understood to be fupplied in Prop. 14.


"And complete the cylinders AX, EO." both the Enuntiation and Exposition of the Propofition represent the cylinders as well as the cones as already defcribed. wherefore the reading ought rather

Book XII. to be " and let the cones be ALC, ENG; and the cylinders AX, EO."

The first Cafe in the second part of the Demonstration is wanting; and fomething alfo in the fecond Cafe of that part, before the repetition of the construction is mentioned; which are now added.


In the Enuntiation of this Propofition the Greek words, es t μείζονα σφαῖραν σερεὸν πολύεδρον εγγράψαι, μή ψαύον τῆς ἐλάσονος σφαίρας κατὰ τὴν ἐπιφάνειαν, are thus tranflated by Commandine and others," in majori, folidum polyhedrum defcribere quod minoris "fphaerae fuperficiem non tangat ;" that is, " to defcribe in the "greater sphere a folid polyhedron which shall not meet the super❝ficies of the leffer sphere." whereby they refer the words xard τὴν ἐπιφάνειαν to thefe next to them τῆς ἐλάσονος σφαίρας. but they ought by no means to be thus tranflated, for the folid polyhedron doth not only meet the fuperficies of the leffer sphere, but pervades the whole of that fphere. therefore the forefaid words are to be referred to To spev Toλved pov, and ought thus to be translated, viz. to describe in the greater sphere a solid polyhedron whose superficies fhall not meet the leffer fphere; as the meaning of the Propofition neceffarily requires.

The Demonstration of the Propofition is spoiled and mutilated. for fome easy things are very explicitly demonftrated, while others not fo obvious are not fufficiently explained; for example, when it is affirmed that the fquare of KB is greater than the double of the fquare of BZ, in the first Demonstration; and that the angle BZK is obtufe, in the second. both which ought to have been demonftrated. befides, in the first Demonstration it is faid "draw K "from the point K perpendicular to BD;" whereas it ought to have been faid, "join KV," and it should have been demonstrated that KV is perpendicular to BD. for it is evident from the figure in Hervagius's and Gregory's Editions, and from the words of the Demonftration, that the Greek Editor did not perceive that the perpendicular drawn from the point K to the straight line BD must neceffarily fall upon the point V, for in the figure it is made to fall upon the point a different point from V, which is likewise fuppofed in the Demonstration. Commandine seems to have been aware of this; for in his figure he marks one and the fame point with the two let

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