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

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 fupplied this.

Also, in the construction, it is required to complete the folid of which the base 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 ftraight lines is "FD." the fame correction must be made in the following Propofition 33.

PROP. D. B. XI,

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

PROP. XXXIV. B. XI.

In this the words, ὧν αἱ ἐφεσῶσαι ἐκ εἰσὶν ἐπὶ τῶν αὐτῶν εὐθειῶν, "of which the infifting ftraight lines are not in the fame straight "lines" are thrice repeated; but thefe 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, a ran," of which the altitudes" are twice put for av ai pesŵy, "of which the infifting 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 fhorter 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 Demonftration, 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 inferted

except when the Demonftration 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 Demonftration of the Corollary, befides the one in the Original, by help of which the following 36. Propofition may be demonftrated without the 35.

Tacquet in his Euclid demonftrates 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 conftruction by which they are made. but, by the fecond Demonftration of the preceding Corollary, his Demonftration is rendered legitimate likewife in the Cafe 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 thofe ratios which are the fame with one another, are likewife the fame with one another; and that those ratios are the fame with one ano ther, of which the triplicate ratios are the fame with one another; but this ought not to be granted without a Demonftration, nor did Euclid affume the first and easiest of these two Propofitions, but demonftrated it in the cafe of duplicate ratios, in the 2 2. Prop. B. 6. on this account another Demonstration 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 Apollonius, and other Geometers, when they have occafion for this other Edi Problem, direct a perpendicular to be drawn from the point to the tions.

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

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 Epistle he has reftored 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 faid, "if it be not fo, the fquare of "BD shall be to the fquare of FH, as the circle ABCD is to fome fpace 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 alfo in Prop. 5. 11. 12. 18.of this Book. concerning which it is to be obferved, that in the Demonstration 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, tho' it cannot be exhibited or found by a Geometrical conftruction. fo in this place it is af fumed that there may be a fourth proportional to thefe 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 the circle ABCD, there is a fourth fquare proportional; becaufe 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 equala. 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 Translations, it is faid, " and be"cause 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 w's Eμxpooder ideixon, "as was before shewn," and the fame are found again in the end of Prop. 18. of this Book; but the Demonftration referred to, except it be the ufelefs 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 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 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.

350

Book XII.

8. 20. 6.

COR. PRO P. VIII. B. XII.

The Demonftration of this is imperfect, because it is not fhewn 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 homologous fide FG.

a

Let the polygons be divided into the triangles ABE, EBC, ECD; FGL, LGH, LHK, which are fimilar each to each. and because b. 11. Def. 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

C. 4. 6.

d. 5. 6.

c. B. 11.

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 equiangular, and similar to one another. therefore the pyramids which have the triangles EAB, LFG for their bafes, and the points M, N for their vertices are fimilar 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 fewn to be fimilar