Book XII.

PROP. XV. B. 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.

PROP. XVII. B. XII. In the enunciation of this proposition, the Greek words εις την μειζονα σφαιραν στερεον πολυεδρον εγγραψαι μη ψαυον ons enco roves o paigas nata TTYY ETIQavelay 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 xata TYV, ETI Pavelar to these next to them the endocovos codipes: · 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 OTEPEOY Torvedpov, 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 K1 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 fi- Boox XII. gure it is made to fall upon the point n, 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 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 edi. tion, 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 legiti. mate, and when they are not.

Boox 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 vali66 dioribus, æternùm persistet inconcussa.”







By ROBERT SIMSON, M. D. Emeritus Professor of Mathematics in the University of


LONDON: Printed for F. WINGRAve, and the rest of the Proprietors.

i 1814.

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