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Book XI. of this Book, that solid figures which are contained by the same number of similar and equal plane figures, as also solid angles that are contained by the same number of equal plane angles, are not always equal to one another.
It is to be observed that Tacquet, in his Euclid, defines equal solid angles to be such," as being put within one an"other do coincide:" but this is an axiom, not a definition; for it is true of all magnitudes whatever. He made this useless definition, that by it he might demonstrate the 36th Prop. of this Book, without the help of the 35th of the same: Concerning which demonstration, see the note upon Prop. 36.
PROP. XXVIII. B. XI.
In this it ought to have been demonstrated, not assumed, that the diagonals are in one plane. Clavius has supplied this defect.
PROP. XXIX. B. XI.
THERE are three cases of this proposition; the first is, when the two parallelograms opposite to the base AB have a side common to both; the second is, when these parallelograms are separated from one another; and the third when there is a part of them common to both; and to this last only, the demonstration that has hitherto been in the Elements does agree. The first case is immediately deduced from the preceding 28th Prop., which seems for this purpose to have been premised to this 29th, for it is necessary to none but to it, and to the 40th of this Book, as we now have it; to which last it would, without doubt, have been premised, if Euclid had not made use of it in the 29th; but some unskilful editor has taken it away from the Elements, and has mutilated Euclid's demonstration of the other two cases, which is now restored, and serves for both at once.
PROP. XXX. B. XI.
In the demonstration of this, the opposite planes of the solid CP, in the figure of this edition, that is, in the solid CO in Commandine's figure, are not proved to be parallel; which it is proper to do for the sake of learners.
PROP. XXXI. B. XI.
THERE are two cases of this proposition; the first is, when the insisting straight lines are at right angles to the bases; the other, when they are not: the first case is divided again into two others, one of which is, when the bases are equiangular parallelograms; the other when they are not equiangular: The Greek editor makes no mention of the first of these two last cases, but has inserted the demonstration 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 cases separately: The demonstration also is made something shorter by following the way that Euclid has made use of in Prop. 14, Book 6. Besides, in the demonstration of the case in which the insisting straight lines are not at right angles to the bases, the editor does not prove that the solids described in the construction are parallelepipeds, which it is not to be thought that Euclid neglected: Also the words, "of which the "insisting straight lines are not in the same straight lines," have been added by some unskilful hand; for they may be in the same 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 necessary. Clavius has supplied this.
Also, in the construction, it is required to complete the solid of which the base is FH, and altitude the same with that of the solid CD: But this does not determine the solid to be completed, since there may be innumerable solids upon the same base, and of the same altitude: It ought therefore to be said, "complete the solid 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 equiangular parallelograms in the 23d, B. 6.
PROP. XXXIV. B. XI.
Is this the words, ὧν αἱ ἐφεστῶσαι οὐκ εἰσὶν ἐπὶ τῶν αὐτῶν εὐθειών, “ of "which the insisting 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 insisting straight lines be, or “be not, in the same straight lines:" For the other case is without any reason excluded; also the words avrà in, "of which "the altitudes," are twice put for av ai porca,“ of which "the insisting straight 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 demonstrated to be right angles in a shorter than in the Greek: And in the same way ACH, DFM may be demonstrated to be right angles: Also the repetition of the same demonstration, which begins with "in the same manner," is left out, as it was probably added to the text by some editor: for the words "in like manner "we may demonstrate," are not inserted except when the demonstration is not given, or when it is something 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 36th Prop. may be demonstrated without the 35th.
PROP. XXXVI. B. XI.
TACQUET in his Euclid demonstrates this proposition without the help of the 35th; but it is plain that the solids mentioned in the Greek text in the enunciation of the proposition as equiangular, are such that their solid 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 solids 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 solids are constructed as in the text.
PROP. 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 another, 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 22d Prop. Book 6. On this account, another demonstration is given of this proposition, like to that which Euclid gives in Prop. 22, Book 6, as Clavius has done.
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.
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, to 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 11th Prop. of the same: But Euclid, Apollonius, and other a 17. 12. geometers, when they have occasion for this problem, direct a in other 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 same 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: Some editor, not perceiving this, thought it was necessary to add this proposition, which can never be of any use to the 11th Book; and its being near to the end, among propositions with which it has no connexion, is a mark of its having been added to the text.
PROP. XXXIX. B. XI.
THE learned Mr Moor, professor of Greek in the University of Glasgow, observed to me, that it plainly appears from Archimedes's epistle to Dositheus, prefixed to his books of the Sphere and Cylinder, which epistle he has restored from ancient manuscripts, that Eudoxus was the author of the chief propositions in this 12th 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 some 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, though it cannot be exhibited or found by a geometrical construction: So in this place it is assumed, 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 square equal to the circle ABCD, though 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 to the three straight lines which are their sides, there is a fourth straight line propor
a 12. 6. tionala, and this fourth square, or a space equal 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 shown by Euclid, but left out by some editor; for the lemma which some unskilful hand has added to this proposition explains nothing of it.
PROP. 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 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