Book XI. " not in the fame ftraight lines," have been added by fome unskilful hand; for they may be in the fame straight lines.

The Editor has forgot to order the parallelogram FH to be ap plied in the angle FGH equal to the angle LCG, which is neceffary. Clavius has fupplied this.

Alfo, in the construction, it is required to complete the folid of which the bafe is FH, and altitude the fame with that of the folid CD; but this does not determine the folid to be completed, fince there may be innumerable folids upon the fame bafe, and of the fame altitude. it ought therefore to be faid "complete the folid of "which the bafe is FH, and one of its infifting straight lines is FD." the fame correction must be made in the following Propofition 33.

PRO P. D. B. XI.

It is very probable that Euclid gave this Propofition a place in the Elements, fince he gave the like Propofition concerning equiangular parallelograms in the 23. B. 6.

PROP. XXXIV. B. XI.

In this the words, ὧν αἱ ἐφες ῶσαι ἐκ εἰσὶν ἐπὶ τῶν αὐτῶν εὐθετῶν, “of which the infisting straight lines are not in the same 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 "whether the infifting ftraight lines be, or be not, in the fame "ftraight lines." for the other Cafe is without any reafon excluded. alfo the words, ar ran, "of which the altitudes" are twice put for ŵr ai épeswσay, " of which the infisting straight lines;" which is a plain mistake. for the altitude is always at right angles to the base.

ων

PROP. XXXV. B. XI.

The angles ABH, DEM are demonftrated to be right angles in a shorter way than in the Greek; and in the fame way ACH, DFM may be demonstrated to be right angles. alfo the repetition of the fame Demonstration, which begins with "in the fame manner," is left out, as it was probably added to the Text by fome Editor; for the words," in like manner we may demonftrate" are not inserted

except

except when the Demonstration is not given, or when it is fome- 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, befides the one in the Original, by help of which the following 36. Propofition may be demonstrated without the 35.

PRO P. XXXVI. B. XI.

Tacquet in his Euclid demonstrates this Propofition without the help of the 35. but it is plain that the folids mentioned in the Greek Text in the Enuntiation of the Propofition as equiangular, are fuch that their folid angles are contained by three plane angles equal to one another, each to each; as is evident from the conftruction. Now Tacquet does not demonftrate, but affumes thefe folid angles to be equal to one another; for he fuppofes the folids to be already made, and does not give the construction by which they are made. but, by the second Demonftration of the preceeding Corollary, his Demonstration is rendered legitimate likewise in the Case where the folids are conftructed as in the Text.

PROP. XXXVII. B. XI.

In this it is affumed that the ratios which are triplicate of those ratios which are the fame with one another, are likewife the fame with one another; and that those ratios are the fame with one another, of which the triplicate ratios are the fame with one another; but this ought not to be granted without a Demonstration, nor did Euclid affume the first and easiest of these two Propofitions, but demonstrated it in the cafe of duplicate ratios, in the 22. Prop. B. 6. on this account another Demonftration is given of this Propofition like to that which Euclid gives in Prop. 22. B. 6. as Clavius has done.

PROP. 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 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 11. Propofition of the fame. but Euclid, a. 17. 12. in other EdiApollonius, and other Geometers, when they have occafion for this Problem, direct a perpendicular to be drawn from the point to the

plane,

tions.

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 expreffed in fewer words. fome Editor not perceiving this, thought it was neceffary to add this Propofition, which can never be of any ufe, 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.

PROP. XXXIX. B. XI.

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

Book XII,

TH

B. XII.

HE learned Mr. Moor, Profeffor of Greek in the University of Glasgow, obferved to me that it plainly appears from Archimedes Epiftle to Dofitheus prefixed to his Books of the Sphere and Cylinder, which Epiftle he has reftored from antient Manufcripts, that Eudoxus was the Author of the chief Propofitions in this twelfth Book.

PRO P. II. B. XII.

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 less 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. concerning which it is to be obferved, that in the Demonftration of Theorems, it is fufficient, in this and the like cafes, that a thing made use of in the reasoning can poffibly exift, 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 fquares 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

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 fquare, 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 fhewn by Euclid, but left out by fome Editor; for the Lemma which fome unfkilful hand has added to this Propofition explains nothing of it.

PROP. III. B. XII.

In the Greek Text and the Tranflations, it is faid, "and because "the two straight lines BA, AC which meet one another" &c. here the angles BAC, KHL are demonftrated 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 fimilar in a fhorter way by Prop. 21. B. 6.

PROP. 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 as unpowder 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 thefe Elements, and has been perhaps left out by fome Editor who has forgot to cancel thofe words alfo.

PROP. VI. B. XII.

A fhorter Demonftration 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 ufe of the 22. Prop. of B. 5. twice, whereas once would have ferved his purpose; 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.

COR.

COR. PROP. VIII. B. XII.

The Demonstration of this is imperfect, because it is not shewn 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 homolo gous 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 b 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

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 d. 5. 6. their fides proportionals are equiangular, and fimilar to one another. therefore the pyramids which have the triangles EAB, LFG for their bafes, and the points M, N for their vertices are similar b e. B. 11. 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