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and in this case the proposition is the same with this; that two spherical triangles that are equilateral to one another are also equiangular to one another, and can coincide: which ought not to be granted without a demonstration. Euclid does not assume this in the case of rectilineal triangles, but demonstrates it in Prop. 8. Book 1. that triangles which are equilateral to one another, are also equiangular to one another; and from this their total equality appears by Prop. 4. Book 1. And Menelaus, in the 4th Prop. of his first Book of Spherics, explicitly demonstrates, that spherical triangles which are mutually equilateral, are also equiangular to one another; from which it is easy to shew that they must coincide, providing they have their sides disposed in the same order and situation.
To supply these defects, it was necessary to add the three propositions marked A, B, C, to this Book. For the 25th, 26th, and 28th propositions of it, and consequently eight others, viz. the 27th, 31st, 32d, 33d, 34th, 36th, 37th, and 40th of the same, which depend upon them, have hitherto stood upon an infirm foundation; as also the 8th, 12th, Cor. of 17th and 18th of 12th Book, which depend upon the 9th Definition. For it has been shewn in the notes on Def. 10th 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 66 within one another do coincide": but this is 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 two other 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 in this edition, that is, of 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 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 parallelopipeds, which it is not to be thought that Euclid neglected: also the words, “of 66 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 66 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.
In this the words, ών αι έφεστωσαι ουκ εισίν επί των αυτών şübecāv, “ 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, “ whe“ther 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 wy Ta ütni, “ of which 66 the altitudes”, are twice put for åv ai ÉDEOTWO al, “ 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 way than in the Greek; and
in the same way ACH, DFM may be dernonstrated to be right angles: also the repetition of the same demonstration, which begins with in the same manner", is left oui, as it was probably added to the text by some editor: for the words “ in like manner we may des monstrate” 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.
PROP. XXXVIII. B. XI.
* 17. 12. in other edi. tions.
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 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 geometers, when they have occasion 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 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.
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.
The learned Mr. Moore, 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,