be read thus, “ But if not, let the angle BAC be not less than Book XI. “ either of the other two, but greater than DAB.” At the end of this proposition it is said, “ in the same man“ ner it may be demonstrated,” though there is no need of any demonstration; because the angle BAC being not less than either of the other two, it is evident that BAC together with one of them is greater than the other. PROP. XXII. B. XI. And likewise in this, near the beginning, it is said, “ But if not, let the angles at B, E, H be unequal, and let the angle « at B be greater than either of those at E, H:” which words manifestly show this place to be vitiated, because the angle at B may be equal to one of the other two. They ought therefore to be read thus, “ But if not, let the angles at B, E, H be une qual, and let the angle at B be not less than either of the “ other two at E, H: therefore the straight line AC is not less “ than either of the two DF, GK.” PROP. XXIII. B. XI. The demonstration of this is made something shorter, by not repeating in the third case the things which were demonstrated in the first; and by making use of the construction which Campanus has given; but he does not demonstrate the second and third cases, the construction and demonstration of the third case are made a little more simple than in the Greek text. PROP. XXIV. B. XI. The word “ similar" is added to the enunciation of this proposition, because the planes containing the solids which are to be demonstrated to be equal to one another, in the 25th proposition, ought to be similar and equal, that the equality of the solids may be inferred from prop. C, of this book: and in the Oxford edition of Commandine's translation, a corollary is added to prop. 24, to show that the parallelograms mentioned in this proposition are similar, that the equality of the solids in prop. 25, may be deduced from the 10th def. of book 11. PROP. XXV. and XXVI. B. XI. In the 25th prop. solid figures, which are contained by the same number of similar and equal plane figures, are supposed Book XI. to be equal to one another. And it seems that Theon, or some other editor, that he might save himself the trouble of demonstrating the solid figures mentioned in this proposition to be equal to one another, has inserted the 10th def. of t.is book, to serve instead of a demonstration; which was very ignorantly done. Likewise in the 26th prop. two solid angles are supposed to be equal: if each of them be contained by three plane angles which are equal to one another, each to each. And it is strange enough, that none of the commentators on Euclid have, as far as I know, perceived that something is wanting in the demonstrations of these two propositions. Clavius, indeed, in a note upon the 11th def. of this book, afirms, that it is evident that those solid angles are even which are contained by the same number of plane angles, equal to one another, each to each, because they will coincide, if they be conceived to be placed within one another; but this is said without any proof, nor is it always true, except when the solid angles are contained by three plane angles only, which are equal to one another, each to each : 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, 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 Me nelaus, in the 4th prop. of his 1st book of spherics, explicitly demonstrates, that spherical triangles which are mutually equilateral, are also equiangular to one another; from which it is easy to show 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 shown in the notes on def. 10, 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 Book XI. solid angles to be such, “ 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 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 had supplied wiis 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 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: ? Y Book XI. 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 coroilary; 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 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. In this the worls, ων αι εφιστωσι εκ εισιν επι των αυτών ευθειαν, 5 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, ey Ta-ola of which si the altitudes," are twice put for we 'ai's®! Twoku, “ of which Book XI. “ the insisting straight lines ;” which is a plain mistake: for the altitude is alyays 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, DEM 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. Companus 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. |