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Book XI. “ not in the fame straight lines," have been added by some unfall

ful hand; for they may be in the fame 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 neceffary. Clavius has supplied this.

Also, in the construction, it is required to complete the fold of which the base is FH, and altitude the fame with that of the folid CD; but this does not determine the solid to be completed, since there may be innumerable solids upon the same base, and of the fame altitude. it ought therefore to be said “complete the folid 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 equi. angular parallelograms in the 2 3. B. 6.

PROP. XXXIV. B. XI. In this the words, wx ai i pesway t'e cici éti Tŵr au tür Eulerov, “ of which the insisting straight lines are not in the fame 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 os

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 ta uun, “ of which the altitudes” are twice put for wai épswray," of which the insisting straight lines;" which is a plain mistake. for the altitude is always at right angles to the base,

PRO P. 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 faine Demonstration, which begins with " in the same manner,"is left out, as it was probably added to the Text by fome Editor ; for the words, " in like manner we may demonstrate” are not inserted

except when the Demonstration is not given, or when it is some- Book XI. thing different from the other, if it be given, as in Prop. 26. of this Book. Campanus has not this repetition.

We have given another Demonstration of the Corollary, besides the one in the Original, by help of which the following 36. Proposition may be demonstrated without tie 35.

PRO P. XXXVI. B. XI. Tacquet in his Euclid demonstrates this Proposition without the help of the 35. but it is plain that the solids mentioned in the Greck Text in the Enuntiation of the Proposition as equiangular, are such that their folid 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 folids 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 folids are constructed as in the Text.

PRO P. 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 ano. ther, 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 2 2. Prop.B. 6. on this account another Demonstration is given of this Proposition like to that which Euclid gives in Prop. 22. B. 6. as Clavius has done,

PRO P. 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 case to do it by the 11. Proposition of the fame. but Euclid", a. 17. 12. in Apollonius, and other Geometers, when they have occasion for thi Broblem, direct a perpendicular to be drawn from the point to the tions.

other Edi

Book XI. plane, and conclude that it will fall upon the common section of

the planes, because this is the very fame 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. fome Editor not perceiving this, thought it was necessary to add this Proposition, which can never be of any use, to the 11. Book. and its being near to the end among Propositions with which it has no congexion, is a mark of its having been added to the Text.

PRO P. 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.

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THE learned Mr. Moor, Professor of Greek in the University

of Glasgow, observed to me that it plainly appears from Archimedes Epistle to Dofitheus prefixed to his Books of the Sphere and Cylinder, which Epistle he has restored from antient Manuscripts, that Eudoxus was the Author of the chief Propositions in this twelfth Book.

PROP. II. B. XII. 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 ABCD is to fome “ 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, tho' it cannot be exhibited or found by a Geometrical construction. so in this place it is alsumed 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 fquare equal to the circle ABCD, tho' 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 Book XII, to the three straight lines which are their fides there is a fourth straight line proportional', and this fourth square, or a space equala. 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 fome Editor ; for the Lemma which some unskilful hand has added to this Proposition explains nothing of it.

PRO P. III. B. XII. 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. II. 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.

PRO P. IV. B. XII.
A few things in this are more fully explained than in the Greek
Text.

PROP. V. B. XII. In this, near to the end, are the words ws a poater ideix On, " 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 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.

PRO P. 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 2 2. 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 threc.

Book XII,

COR. PRO P. VIII. B. XII. The Demonstration of this is imperfect, because it is not shewn that the triangular pyramids into which those upon mukangular bales are divided, are similar to one another, as ought necessarily to have been done, and is done in the like case in Prop. 1 2. of this Book. the full Demonstration of the Corollary is as follows.

Upon the polygonal bafes ABCDE, FGHKL, let there be fimilar 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 fide FG.

Lee the polygons be divided into the triangles ABE, EBC, ECD;

FGL, LGH, LHK, which are similar * each to each. and because b. 15. Def. the pyramids are similar, therefore the triangle EAM is fimilar to

thie 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
M

N

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the triangles EAB, LFG are similar ; therefore, ex aequali, as ME
to EB, fo is NL to LG. in like manner it may be Thewn 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 sides proportionals are d equiangular, and similar to one ano-
ther. therefore the pyramids which have the triangles EAB, LFG
for their bases, and the points M, N for their vertices are similar
to one another, for their folid angles are e equal, and the folds
themselves are contained by the fame number of similar planes. in
the fame manner the pyramid EBCM may be fewn to be similar

c. B. 11.

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