Book XI.


other edi


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When it is required to draw a perpendicular from a point in one plane which is at right angles to another plane, unto this Jaft 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 11th prop. of a 17. 12 in the fame: But Euclid, Apollonius, and other geometers, when they have occafion for this problem, direct a perpendicuJar to be drawn from the point to the 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 mufl be perpendicular to the plane; but is expreffed in fewer words: Some editor, not perceiving this, thought it was neceffary to add this propofition, which can never be of any use to the 11th book, and its being near to the end among propofitions with which it has no connection, is a mark of its having been added to the text.

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In this it is fuppofed, that the ftraight lines which bifect the fides of the oppofite planes, are in one plane, which ought to have been demonftrated; as is now done.



Book XI. HE learned Mr Moor, profeffor of Greek in the Univer hty of Glasgow, obferved to me, that it plainly appears frem Archimedes's epiftle to Dofitheus, prefixed to his books of the Sphere and Cylinder, which epiftle he has reflored from antient manufcripts, that Eudoxus was the author of the chief propontions in this 12th book.

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At the beginning of this it is faid, "if it be not fo, the fquare "of BD fhall be to the fquare of FH, as the circle ABCD is "to fome space either lets than the circle EFGH, or greater "than it:" And the like is to be found near to the end of this propofition, as alfo in prop. 5. 11. 12. 18. of this book: Con


cerning which, it is to be obferved, that, in the demonstration Book XII. of theorems, it is fufficient, in this and the like cafes, that a thing made ufe of in the reasoning can poffibly exift, providing this be evident, though it cannot be exhibited or found by a geometrical construction: So, in this place, it is affumed, that there may be a fourth proportional to these three magnitudes, viz. the fquares of BD, FH, and the circle ABCD; because it is evident that there is fome fquare equal to the circle ABCD, though it cannot be found geometrically; and to the three rectilineal figures, viz. the fquares of BD, FH, and the fqure which is equal to the circle ABCD, there is a fourth square proportional; because to the three ftraight lines which are their fides, there is a fourth ftraight line proportional, and a 12. 6.. this fourth fquare, or a space equal 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 fhewn by Euclid, but left out by fome editor; for the lemma which some unskiltul hand has added to this propofition explains nothing of it.


In the Greek text and the tranflations, it is faid," and "because the two flraight 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: Becaufe 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 fimilar in a fhorter way by prop. 21. b. 6.

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

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In this, near to the end, are the words, as urgooder ideixen, " as was before shown," and the fame are found again in the end of prop. 18. of this book; but the demonftration referred to, except it be the useless lemma annexed to the 2d prop. is no where in these elements, and has been perhaps left out by fome editor who has forgot to cancel thofe words alfo.



Book XII.

a 20. 6.


A shorter demonftration is given of this; and that which is in the Greek text may be made fhorter by a step than it is: For the author of it makes ufe of the 22d 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.


The demonftration of this is imperfect, because it is not fhown, that the triangular pyramids into which those 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 demonstration of the corollary is as follows.

Upon the polygonal bafes 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 homologus fide FG.

Let the polygons be divided into the triangles ABE, EBC, ECD; FGL, LGH, LHK, which are fimilar each to each: b II. def. 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


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to EB, fo is FL to LG, because the triangles EAB, LFG are fimilar; therefore, ex aequali, as ME to EB, fo is NL to LG:


e b. II.

In like manner it may be fhewn that EB is to BM, as LG to Book XII. GN; therefore, again, ex aequali, as EM to MB, fo is LN to NG: Wherefore the triangles EMB, LNG having their fides proportionals are d equiangular, and fimilar to one another: d s. 6. Therefore the pyramids which have the triangles EAB, LFG for their bases, and the points M, N for their vertices, are fi milar to one another, for their folid angles are equal, and b 11. def. the folids themselves are contained by the fame number of finilar 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 because the pyramids EABM, LFGN are fimilar, and have triangular bafes, the pyramid EABM has f to LFGN the triplicate ratio of that which EB has to the f 8. 12. homologous fide LG. And, in the fame 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, fo 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, fo 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 pyra mid 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 thefe two propofitions, according to Euclid's manner, and is now restored: By which means, the first part of prop. 12. may be demonstrated in the fame words with the firft part of prop. 11.; on this account the demonstration of that first part is left out, and affumed from prop. 11.


In this propofition the common fection of a plane parallel to the bafes of a cylinder, with the cylinder itself, is fuppofed to be a circle, and it was thought proper briefly to demonstrate it; from whence it is fufficiently manifeft, that this plane divides the cylinder into two others: And the fame thing is underftood to be fupplied in prop. 14.

Z 4


Book XII.


"And complete the cylinders AX, EO," both the enunciation and expofition of the propofition reprefent the cylinders as well as the cones, as already defcribed: Wherefore the reading ought rather to be," and let the cones be ALC, ENG ; "and the cylinders AX, EO."

The first cafe in the fecond part of the demonftration is wanting; and fomething alfo in the fecond cafe of that part, before the repetition of the conftruction is mentioned; which are now added.


In the enunciation of this, propofition, the Greek words s την μείζονα σφαίραν στερεον πολυέδρων έγραψαι, μη ψανον τῆς ελασσονος σφαίρας κατά την επιφανειαν, are thus tranflated by Commandine and others," in majori folidum polyhedrum descri"bere quod minoris fphaerae fuperficiem non tangat," that is, "to defcribe in the greater fphere a folid polyhedron which "fhall not meet the fuperficies of the leffer sphere :" Whereby they refer the words κατα τὴν ἐπιφάνειαν to thefe next to them της ελάσσονες σφαιρας : But they ought by no means to be thus tranflated, for the folid polyhedron doth not only meet the fu perficies of the leffer fphere, but pervades the whole of that fphere Therefore the forefaid words are to be referred to TO σTIGION weλvedger, and ought thus to be tranflated, viz. to defcribe in the greater fphere a folid polyhedron whofe fuperficies fhall not meet the leffer fphere; as the meaning of the propofition neceffarily requires.

The demonftration of the propofition is spoiled and mutilated: For fome eafy 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 demonftration; and that the angle BZK is obtufe, in the second: Both which ought to have been demonftrated: Befides, in the first demonftration, it is faid, "draw Ko from the point K perpen"dicular to BD;" whereas it ought to have been faid, "join "KV," and it fhould have been demonftrated 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


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