Book XI. The Greek editor makes no mention of the first of these two laft cafes, but has inferted the demonftration of it as a part of that of the other: And therefore fhould have taken notice of it in a corollary; but we thought it better to give these two cases feparately: The demonftration alfo is made fomething fhorter by following the way Euclid has made ufe of in prop. 14. book 6. Befides, in the demonftration of the cafe in which the infifting ftraight lines are not at right angles to the bafes, the editor does not prove that the folids defcribed in the conftruction are parallelopipeds, which it is not to be thought that Euclid neglected: Alfo the words, of which the infifting straight lines are not in the fame ftraight lines," have been added by fome unfkilful hand; for they may be in the fame ftraight lines.

4

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 fupplied this.

Allo, 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 folids 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 equiangular parallelograms in the 23d, B. 6.

PROP. XXXIV. B. XI.

In this the words, ὢν &ι εφεστώωσαι εκ εἰσιν ἐπὶ τῶν αὐτῶν ευθειών, "of which the infifting ftraight lines are not in the fame "ftraight 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, avra ifn, of which

"the

"the altitudes," are twice put for av å peas," of which Book XI. "the infifting ftraight lines;" which is a plain mistake: For the altitude is always at right angles to the base.

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 demonftrated to be right angles: Also 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 fomething 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 36th prop. may be demonftrated without the 35th.

PROP. XXXVI. B. XI.

Tacquet in his Euclid demonftrates this propofition without the help of the 35th; but it is plain, that the folids mentioned in the Greek text in the enunciation 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 these folid angles to be equal to one another; for he supposes 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 demonstration is rendered legitimate likewife in the cafe where the folids are conAructed 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 likewise 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 demonftrated it in the cafe of duplicate ratios, in the 22d prop. book 6. On this account, another demonftration is given of this propofition like to that which Euclid gives in prop. 22. book 6. as Clavius has done.

Z 2

PROP.

Book XI.

other edi

tions.

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 laft 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 a, Apollonius, and other geometers, when they have occafion for this problem, direct a perpendicular 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 fame thing as if they had made ufe of the conftruction above mentioned, and then concluded that the straight line must 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.

Book XH.

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.

B. XII.

HE learned Mr. Moor, profeffor of Greek in the Univer

Tfity of Glagow, obferved to me, that is plainly appears

from Archimedes's epiftle to Dofitheus, prefixed to his books of the Sphere and Cylinder, which epiftle he has reftored from ancient manuscripts, that Eudoxus was the author of the chief propofitions in this 12th book.

PROP. 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 lefs than the circle EFGH, or greater ❝ than it:" And the like is to be found near to the end of this propofition, as also in prop. 5. 11. 12. 18. of this book: Con

cerning

cerning which, it is to be observed, that, in the demonftration Book XII. of theorems, it is fufficient, in this and the like cafes, that a thing made use of in the reafoning can poffibly exift, providing this be evident, though it cannot be exhibited or found by a geometrical conftruction: 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 square which is equal to the circle ABCD, there is a fourth fquare proportional; because to the three ftraight lines which are their fides, there is a fourth ftraight line proportional a, 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 proba ble 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 ftraight 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: 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, sumpor edin, as was before fhown," 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 thefe Elements, and has been perhaps left out by fome editor who has forgot to cancel those words allo.

Book XII.

a 20. 6. bxI. def.

c 4.6.

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 22d 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. PROP. VIII. B. XII.

The demonftration of this is imperfect, because it is not shown, that the triangular pyramids into which those upon multangular bases 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 demonftration 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 theirv ertices: The pyramid ABCDEM has to the pyramid FGHKLN the triplicate ratio of that which the fide AB has to the homologous fide FG.

Let the polygons be divided into the triangles ABE, EBC, ECD; FGL, LGH, LHK, which are fimilar a 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 e ME is to EA, as NL to LF; and as AE

M

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