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s the altitudes," are twice put for wv 'di 'POTuomu, “ of which Book XI. “ the insisting straight lines ;" which is a plain mistake : for the altitude is always at right angles to the base.


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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, DEM 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. Companus has not this repetition.

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.


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.

Book XI.


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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 plant, 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 case to do it by the I Ith prop. of a 17. 12. in

the same : · but Euclid a, Apollonius, and other geometers, ofle edi- when they have occasion for this problem, direct a perpendicutions. lar 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 perpendicuiar 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. THE learned Mr. Moore, professor of Greek in the Univer

sity 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 minuscripts, 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 square “ of BD shall be to the square of FH, as the circle AFCD 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 Book XII. 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 D, FH, and the circle A CD ; because it is evident that there is some square equal to the circle ABCD though it cannot be found geometrically ; and to the three recti. lineal 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 a, and a 12. 6. this fourth square, or a space equal to it, is the space wi.ich 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 “ because the two straight lines BA, A which meet one ano

ther," &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.



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


In this, near to the end, are the words, ws in regoo.99 idugon, 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 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.


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 taking 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; b 11. def. and because the pyramids are similar, thereforeb the triangle 11. EAM is similar to the triangles LFN, and the triangle ABM C4. 6.

to FGN: wherefore c ME is to EA, as NL to LF; and as AE

a 20. 6.

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G 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 Book XII. GN; therefore again ex æquali, as EM is to MB, so is LN to GN: wherefore the triangles EMB, LNG having their sides proportionals are a equiangular, and similar to one another : a 5.6. therefore the pyramids which have the triangles EAB LFG for their bases, and the points M, N for their vertices, are similar b to one another, for their solid angles are c equal, and the b 11. def. solids themselves are contained by the same number of similar 11. planes : in the same manner, the pyramid EBCM may be c b. 11. 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 bases, the pyramid EABM has d to LFGN the triplicate ratio of that which EB has to the ho-d8. 12. mologous 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 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.

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