“ 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, provided 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 proportional *, and this fourth square, or a space * 12. 6. 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 shewn by Euclid, but left out by some editor; for the lemma which some unskilful hand has added to this proposition explains nothing of it.


In the Greek text and the translations, it is said, 6 and because the two straight lines BA, AC, which “ meet one another," &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 similar to the triangle KHL: this repetition is left out, and the triangles BAC, KHL, are proved to be similar in a shorter way by Prop. 21. B. 6.


A few things in this are more fully explained, than in the Greek text.



In this, near to the end, are the words, ás fuerpoo Bev εδείχθη, , as was before shewn”; and the same found again in the end of Prop. 18. of this Book ; but the demonstration referred to, except it be in the useless lemma annexed to the 2d Prop. is no where in these Elements, and has been perhaps left out by some editor, who has forgot to cancel those words also.


A shorter demonstration 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 served his purpose ; because that proposition extends to any number of magnitudes which are proportionals taken two by two, as well as to three which are proportional to other three.


The demonstration of this is imperfect, because it is not shewn, that the triangular pyramids into which those upon multangular bases are divided, are similar to one another, as ought necessarily to have been done, and is done in the like case in Prop. 12. of this Book: the full demonstration of the corollary is as follows:

Upon the polygonal bases ABCDE, FGHKL, let there be similar and similarly situated pyramids which have the points M, N, for their vertices: the pyramid ABCDEM has to the pyramid FGHKLN the triplicate ratio of that which the side AB has to the homologous side FG.

Let the polygons be divided into the triangles ABE, EBC, ECD; FGL, LGH, LHK, which are similar*,

each to each: and because the pyramids are similar, *11Def.11. therefore * the triangle EAM is similar to the triangle

LFN, and the triangle ABM to FGN: wherefore * ME is to EA, as NL to LF; and as AE to EB, so is FL to LG, because the triangles EAB, LFG, are similar; therefore, ex æquali, as ME to EB, so is NL to LG: in like manner it may be shewn, that EB is to BM, as LG to GN; therefore, again, ex æquali, as EM to MB,

the 20. 6.

* 4.6.

so is LN to NG: wherefore the triangles EMB, LNG, having their sides proportionals, are * equiangular, and .

* 5. 6.

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similar to one another: therefore the pyramids which have the triangles EAB, LFG, for their bases, and the points M, N for their vertices, are similar* to one *11Def.11. another, for their solid angles are * equal, and the .B. 11. solids themselves are contained by the same number of similar planes: in the same manner the pyramid EBCM may be shewn to be similar to the pyramid LGHN, and the pyramid ECDM to LHKN: and because the pyramids EABM, LFGN, are similar, and have triangular bases, the pyramid EABM has * to * 8. 12. LFGN the triplicate ratio of that which EB has to the homologous side LG. And, in the same manner, the pyramid EBCM has to the pyramid LGHN the triplicate ratio of that which EB has to LG: therefore, as the pyramid EABM is to the pyramid LFGN, so is the pyramid EBCM to the pyramid LGHN: in like manner, as the pyramid EBCM is to LGHN, so is the pyramid ECDM to the pyramid LHKN: and as one of the antecedents is to one of the consequents, so are all the antecedents to all the consequents: therefore as the pyramid EABM to the pyramid LFGN, so is the whole pyramid ABCDEM to the whole pyramid FGHKLN: and the pyramid EABM has to the pyramid LFGN, the triplicate ratio of that which AB has to FG; therefore the whole pyramid has to the whole pyramid the triplicate ratio of that which AB has to the homologous side FG.

Q. E. D.

PROP. XI. and XII. B. XII.

The order of the letters of the alphabet is not observed in these two propositions, according to Euclid's manner, and is now restored: by which means, the first part of Prop. 12. may be demonstrated in the same words with the first part of Prop. 11.; on this account the demonstration of that first part is left out, and assumed from Prop. 11.


In this proposition, the common section of a plane parallel to the bases of a cylinder, with the cylinder itself, is supposed to be a circle, and it was thought proper briefly to demonstrate it; from which it is sufficiently manifest, that this plane divides the cylinder into two others: and the same thing is understood to be supplied in Prop. 14.


“ And complete the cylinders AX, EO.” Both the enunciation and exposition of the proposition represent the cylinders as well as the cones, as already described: wherefore the reading ought rather to be, “and let the ci cones be ALC, ENG; and the cylinders AX, EO.”

The first case in the second part of the demonstration is wanting; and something also in the second case of that part, before the repetition of the construction is mentioned; which are now added.

PROP. XVII. B. XII. In the enunciation of this proposition, the Greek words εις την μείζονα σφαίραν στερεών πολύεδρον έγγράψαι μη ψαύον της ελάσσονος σφαίρας κατά την επιφάνειαν are thus translated by Commandine and others, “in majori “ solidum polyhedrum describere quod minoris sphæræ "superficiem non tangat"; that is, to describe in the

greater sphere a solid polyhedron which shall not “ meet the superficies of the lesser sphere”: whereby they refer the words κατά την επιφάνειαν to these next to them της ελάσσονος σφαίρας : but they ought by no means to be thus translated : for the solid polyhedron doth not only meet the superficies of the lesser sphere,

but pervades the whole of that sphere: therefore the aforesaid words are to be referred to το στερεόν πολύεδρον, , and ought thus to be translated, viz. To describe in the greater sphere a solid polyhedron whose superficies shall not meet the lesser sphere; as the meaning of the proposition necessarily requires.

The demonstration of the proposition is spoiled and mutilated: for some easy things are very explicitly demonstrated, while others not so obvious are not sufficiently explained; for example, when it is affirmed, that the square of KB is greater than the double of the square of BZ, in the first demonstration; and that the angle BZK is obtuse, in the second: both which ought to have been demonstrated : besides, in the first demonstration, it is said, “ draw K12 from the point K, “ perpendicular to BD”; whereas it ought to have been said, “join KV”, and it should have been demonstrated, that KV is perpendicular to BD: for it is evident from the figure in Hervagius's and Gregory's editions, and from the words of the demonstration, that the Greek editor did not perceive that the perpendicular drawn from the point K to the straight line BD must necessarily fall upon the point V, for in the figure it is made to fall upon the point 2, a different point from V, which is likewise supposed in the demonstration. Commandine seems to have been aware of this; for in this figure he marks one and the same point with two letters V, 12; and before Commandine, the learned John Dee, in the commentary he annexes to this proposition in Henry Billingsley's translation of the Elements, printed at London, ann. 1570, expressly takes notice of this error, and gives a demonstration suited to the construction in the Greek text, by which he shews that the perpendicular drawn from the point K to BD, must necessarily fall upon the point V.

Likewise it is not demonstrated, that the quadrilateral figures SOPT, TPRY, and the triangle YRX, do not meet the lesser sphere, as was necessary to have been done; only Clavius, as far as I know, has observed this, and demonstrated it by a lemma, which is now premised to this proposition, something altered, and more briefly demonstrated.

In the corollary of this proposition, it is supposed that a solid polyhedron is described in the other sphere similar to that which is described in the sphere BCDE;

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