to the pyramid LGHN, and the pyramid ECDM to LHKN. and Book XII. because the pyramids EABM, LFGN are similar, and have triangular bases, the pyramid EALM hasf to LFGN the triplicate ratio f. 8.12. 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 LABM is to the pyramid LFGN, fo 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.

O. 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 rclored. 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 af fumed from Prop. 11.

PRO P. XIII. B. XII. In this Proposition the common section of a plane parallel to the bafes 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.

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“ And complete the cylinders AX, EO.” both the Enuntiation and Exposition of the Proposition represent the cylinders as well as the cones as already described. wherefore the reading ought rather

Book XII. to be " and let the cones bc ALC, ENG; and the cylinders AX,


The first Case in the second part of the Demonftration 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 Enuntiation of this Proposition the Greek words, eis Trà μείζονα σφαίραν σερεών πολύεδρον έγγράψω, μη ψωον της ελάσσονος o paipas nata Tivé 19dvetav, are thus translated by Commandine and others, “ in majori folidum polyhedrum defcribere quod minoris sphaerae superficiem non tangat;" that is, “ to describe in the

greater sphere a folid polyhedron which Mall not meet the fuper“ficies of the lesser sphere.” whereby they refer the words kata την επιφάνειας to thefe next to them της ελάσονος σφαίρας. but they ought by no means to be thus translated, for the folid polyhedro 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 το σερεών πολύεδρον, and ought thus to be tranlated. viz. to describe in the greater sphere a solid polyhedron whose fuperficies 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 1.2 “ 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 N a different point from V, which is likewise supposed in the Demonstration. Commandine seems to have been aware of this; for in his figure he marks one and the fame point with the two let

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ters V, 52; and before Commandine, the learned John Dee in the Book XII. 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 lie shews that the perpendicular drawn from the point K to BD, muft 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 leffer sphere, as was necessary to have done. only Clavius, as far as I know, has observed this, and demonstrated it by a Lemma, which is now premised to this Propofition, 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 defcribed in the sphere BCDE. but as the Conftruction 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.

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From the preceding Notes it is fufficiently evident how much the elements of Euclid, who was a most accurate Geometer, have 3

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 lefs attentive and accurate in examining that Edition; whereby several errors, fome 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 un-' acceptable to’good Judges who can discerni when" Demonstrations are legitimate, and when they are not.

The objections which, fince 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 Nore on Prop. 1. Book 1 1. as shew that the person who made them has not sufficiently considered the


Book XII.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 abote-named book, that “Nisi machinis impulsa validioribus aeternum perfiftet “ inconcussa.”

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Emeritus Professor of Mathematics in the University of Glasgow




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