It is to be obferved that Tacquet, in his Euclid, defines equal Book XI. folid angles to be fuch, "as being put within one another do "coincide:" But this is an axiom, not a definition; for it is true of all magnitudes whatever. He made this ufelefs definition, that by it he might demonftrate the 36th prop. of this book, without the help of the 35th of the fame: Concerning which demonftration, fee the note upon prop. 36.

PROP. XXVIII. B. XI.

In this it ought to have been demonftrated, not affumed, that the diagonals are in one plane. Clavius has fupplied this defect.

PROP. XXIX. B. XI.

There are three cafes of this propofition; the first is, when the two parallelograms oppofite to the base AB have a fide common to both; the fecond is, when thefe parallelograms are feparated from one another; and the third, when there is a part of them common to both; and to this last only, the demonftration that has hitherto been in the elements does agree. The firft cafe is immediately deduced from the preceding 28th prop. which feems for this purpofe to have been premifed to this 29th, for it is neceffary to none, but to it, and to the 40th of this book, as we now have it, to which laft it would, without doubt, have been premifed, if Euclid had not made ufe of it in the 29th; but fome unfkilful editor has taken it away from the elements, and has mutilated Euclid's demonstration of the other two cafes, which is now reftored, and ferves for both at once.

PROP. XXX. B. XI.

In the demonstration of this, the oppofite planes of the folid CP, in the figure in this edition, that is, of the folid CO in Commandine's figure, are not proved to be parallel; which it is proper to do for the fake of learners.

PROP. XXXI. B. XI.

There are two cafes of this propofition; the firft is, when the infifting ftraight lines are at right angles to the bafes; the other, when they are not: The first cafe is divided again into two others, one of which is, when the bafes are equiangular parallelograms; the other, when they are not equiangular a

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 should have taken notice of it in a corollary; but we thought it better to give these two cafes feparately: The demonstration alfo is made fomething shorter 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 straight lines are not at right angles to the bafes, the editor does not prove that the folids defcribed in the construction are parallelepipeds, 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 unskilful hand; for they may be in the same straight lines.

66

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.

Alfo, 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 equiangular parallelograms in the 23d b. 6.

In this the words, ὧν αἱ ἐφεστῶσαι ἐκ εἰσὶν ἐπὶ τῶν αὐτῶν εὐθειῶν, "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, ar ra n, of which

the

"the altitudes," are twice put for av ai pestα, "of which Book XI. "the infifting straight lines;" which is a plain mittake: For the altitude is always at right angles to the bafe.

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: Alfo the repetition of the fame demonftration, which begins within 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. Companus has not this repetition.

We have given another demonftration 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 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 thofe 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 demonftration; nor did Euclid affume the firft and eatieft of these two propofitions, but demonftrated it in the cafe of duplicate ratios, in the 22d prop. book 6. On this account, another demontration is given of this propofition like to that which Euc.d gives prop. 22. book 6. as Clavius has done.

in

Book XI.

ditions.

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 plané, by Def. 4. of this book: And it would be foolish in this cafe to do it by the 11th prop. of a 1. 12. the fame: But Euclid, Apollonius, and other geometers. in other e- 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 ufe 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.

PROP. XXXIX. B. XI.

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

B. XII.

ས

T

HE learned Mr Moor, profeffor of Greek in the Univerfity of Glasgow, obferved to me, that it plainly appears from Archimedes's epiftle to Dofitheus, prefixed to his books of the Sphere and Cylinder; which epiftle he has restored 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 square of BD fhall be ro the fquare of FH, as the circle ABCD is "to tome 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 alfo in prop. 5. 11. 12. 18. of this book: Con

cerning

cerning which, it is to be obferved, 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, provi ding 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 some square 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 a 12. 6. their fides, there is a fourth straight line proportional a, and 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 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 an"other," &c. here the angles BAC, KHL are demonftrated to be equal to one another by 10th prop. b. 11. which had been done before: Becaufe 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 uzsorder iscíxon, 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 ufelefs 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 alfo.

23

PROP.