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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 and two, as well as to three which are proportional to other three.


THE demonstration of this is imperfect, because it is not shown, 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, 20. 6. EBC, ECD; FGL, LGH, LHK, which are similara, each 11 Def. 11. to each: And because the pyramids are similar, therefore the triangle EAM is similar to the triangle LFN, and the 4. 6. 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

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triangles EAB, LFG, are similar; therefore, ex æquali, as ME to EB, so is NL to LG: In like manner it may be shown, that EB is to BM, as LG to GN; therefore, again,

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B. 11.

éx æquali, as EM to MB, so is LN to NG: Wherefore the Book XII. triangles EMB, LNG,, having their sides proportionals, 2 ared equiangular, and similar to one another: Therefore 5. 6. the pyramids which have the triangles EAB, LFG, for their bases, and the points M, N for their vertices, are similar to one another, for their solid angles are equal, and 11 Def. 11. the solids themselves are contained by the same number of similar planes: In the same manner the pyramid EBCM may be shown 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 LFGN the triplicate ra- f 3. 12. tio 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 FGH 'KLN: 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 whence 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.

Book XII.


"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 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.


In the enunciation of this proposition, the Greek words εις την μείζονα σφαιραν στερεον πολύεδρον εγγράψαι μη ψαύον της ελάσσονος σφαιρας κατα την επιφάνειαν are thus translated by Commandine and others, " in majori solidum polyhe"drum describere quod minoris sphæræ superficiem non "tangat ;" that is, "to describe in the greater sphere a so"lid polyhedron which shall not meet the superficies of "the lesser sphere :" Whereby they refer the words xara την επιφάνειαν to these next to them της ελασσονος 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 TO σTEGEOV TOXUεdov, 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 K 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

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BD must necessarily fall upon the point V, for in the fi- Book XII. gure 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, 2; 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 shows 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; but, as the construction by which this may be done is not given, it was thought proper to give it, and to demonstrate that the pyramids in it. are similar to those of the same order in the solid polyhedron described in the sphere BCDE.

FROM the preceding notes, it is sufficiently evident how much the Elements of Euclid, who was a most accurate geometer, have been vitiated and mutilated by ignorant editors. The opinion which the greatest part of learned men have entertained concerning the present Greek edition, viz. that it is very little or nothing different from the genuine work of Euclid, has without doubt deceived them, and made them less attentive and accurate in examining that edition; whereby several errors, some of them gross enough, have escaped their notice from the age in which Theon lived to this time. Upon which account there is some ground to hope that the pains we have taken in correcting those errors, and freeing the Elements as far as we could from blemishes, will not be unacceptable to good judges, who can discern when demonstrations are legitimate, and when they are not.

2 A

Book XII,

The objections which, since the first edition, have been made against some things in the notes, especially against the doctrine of proportionals, have either been fully answered in Dr. Barrow's Lect. Mathemat. and in these notes; or are such, except one which has been taken notice of in the note on Prop. 1. Book 11, as show that the person who made them has not sufficiently considered the things against which they are brought; so that it is not necessary to make any further answer to these objections and others like them against Euclid's definition of proportionals, of which definition Dr. Barrow justly says in page 297 of the above-named book, that "Nisi machinis impulsa vali"dioribus, æternùm persistet inconcussa."


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