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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 cone

nes 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 thiat part, before the repetition of the construction is mentioned; which are now added.


In the enunciation of this proposition, the Greek words εις την μειζονα σφαιραν στερεον πολυεδρον εγραψαι μηψανον της ελασσονος σφαιρας κατατην επιφανειαν are thus translated by Commandine and others, “in majore solidum polyhedrum describere quod “ minoris sphæræ superficiem non tangat;" that is, “ to de“ scribe in the greater sphere a solid polyhedron which shall “ not meet the superficies of the lesser sphere:" whereby they refer the words rate TNV STi@avslay to these next to them 795 81cccovos apaigas. 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 foresaid words are to be referred to TO GTigcov Todvedgov, 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 nccessarily 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 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 Ka from the point K perpen« dicular to BD;" whereas it ought to have been said, “ join “ KV,” and it should have deen demonstrated that Ki is perpendicular to 1.D : 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 Book XII. 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 a different point from V,

a 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 the two letters V, ; and before Commandine, the learned John Dee, in the commentary he annexes to this proposition in Henry Billinsley's translation of the Elements, printed at London, anno 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.

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

Book XII. 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 validioribus “ æternum persistet inconcussa.”













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