to the pyramid LGHN, and the pyramid ECDM to LHKN. and Book XII. because the pyramids EABM, LFGN are fimilar, and have triangular bafes, the pyramid EABM has f to LFGN the triplicate ratio f. 8. 12. of that which EB has to the homologous fide LG. and, in the fame 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, fo is the pyramid EBCM to the pyramid LGHN. in like manner, as the pyramid EBCM is to LGHN, fo is the pyramid ECDM to the pyramid LHKN. and as one of the antecedents is to one of the confequents, fo are all the antecedents to all the confequents. therefore as the pyramid EABM to the pyramid LFGN, fo 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 fide FG. Q. E. D. PRO P. XI. and XII. B. XII. The order of the letters of the Alphabet is not obferved in thefe two Propofitions, according to Euclid's manner, and is now reftored. by which means the firft part of Prop. 12. may be demonftrated in the fame words with the firft part of Prop. II. on this account the Demonftration of that first part is left out, and af fumed from Prop. 11. PROP. XIII. B. XII. In this Propofition the common fection of a plane parallel to the bafes of a cylinder, with the cylinder itfelf is fuppofed to be a circle, and it was thought proper briefly to demonftrate it; from whence it is fufficiently manifeft that this plane divides the cylinder into two others. and the fame thing is understood to be fupplied in Prop. 14. PROP. XV. B. XII. "And complete the cylinders AX, EO." both the Enuntiation and Expofition of the Propofition reprefent the cylinders as well as the cones as already defcribed. wherefore the reading ought rather Book XII. to be" and let the cones be ALC, ENG; and the cylinders AX, EO." The firft Cafe in the fecond part of the Demonftration is wanting; and fomething alfo in the fecond Cafe of that part, before the repetition of the conftruction is mentioned; which are now added. PROP. XVII. B. XII. In the Enuntiation of this Propofition the Greek words, c tài μείζονα σφαῖραν σεφεὸν πολύεδρον ἐγγράψαι, μὴ ψαῖον τῆς ἐλάσσονος σφαίρας κατὰ τὴν ἐπιφάνειαν, are thus tranflated by Commandine and others," in majori folidum polyhedrum defcribere quod minoris fphaerae fuperficiem non tangat;" that is, " to describe in the greater fphere a folid polyhedron which fhall not meet the fuper"ficies of the leffer fphere." whereby they refer the words xard τὴν ἐπιφάνειαν to thefe next to them τῆς ἐλάσσονος σφαίρας. but they ought by no means to be thus tranflated, for the folid polyhedron doth not only meet the fuperficies of the leffer sphere, but pervades the whole of that sphere. therefore the forefaid words are to be referred tὸ τὸ σερεὸν πολύεδρον, and ought thus to be tranflated. viz. to describe in the greater fphere a folid polyhedron whose fuperficies fhall not meet the leffer fphere; as the meaning of the Propofition neceffarily requires. The Demonftration of the Propofition is spoiled and mutilated. for fome eafy things are very explicitly demonftrated, while others not fo obvious are not fufficiently explained; for example, when it is affirmed that the fquare of KB is greater than the double of the fquare of BZ, in the firft Demonftration; and that the angle BZK is obtufe, in the fecond. both which ought to have been demonftrated. befides, in the firft Demonstration it is faid "draw K "from the point K perpendicular to BD;" whereas it ought to have been faid, " 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 Demonftration, that the Greek Editor did not perceive that the perpendicular drawn from the point K to the ftraight line BD must neceffarily fall upon the point V,for in the figure it is made to fall upon the point a different point from V, which is likewife fuppofed in the Demonftration. Commandine feems to have been aware of this; for in his figure he marks one and the fame point with the two let ters V, ; and before Commandine, the learned John Dee in the Book XII.' Commentary he annexes to this Propofition in Henry Billingsley's Tranflation of the Elements printed at London Ann. 1570, exprefsly takes notice of this error, and gives a Demonftration fuited to the Conftruction in the Greek Text, by which he fhews that the perpendicular drawn from the point K to BD, muft neceffarily fall upon the point V. Likewise it is not demonftrated that the quadrilateral figures SOPT, TPRY, and the triangle YRX do not meet the leffer fphere, as was necessary to have done. only Clavius, as far as I know, has obferved this, and demonftrated it by a Lemma, which is now premised to this Propofition, fomething altered and more briefly demonftrated. In the Corollary of this Propofition it is fuppofed that a folid po-' lyhedron is defcribed in the other sphere fimilar to that which is defcribed in the fphere BCDE. but as the Construction by which this may be done is not given, it was thought proper to give it, and to demonftrate that the pyramids in it are fimilar to thofe of the fame order in the folid polyhedron described in the sphere BCDE. From the preceding Notes it is fufficiently evident how much the elements of Euclid, who was a moft 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 lefs attentive and accurate in examining that Edition; whereby feveral errors, fome of them grofs enough, have escaped their notice from the age in which Theon lived to this time. Upon which account there is fome ground to hope that the pains we have taken in correcting thofe errors, and freeing the Elements as far as we could from blemishes, will not be un-' acceptable to good Judges who can difcern when Demonftrations are legitimate, and when they are not. The objections which, fince the first Edition, have been made against fome things in the Notes, efpecially against the doctrine of Proportionals, have either been fully anfwered in Dr. Barrow's Lect. Mathemat. and in these Notes; or are fuch, except one which has been taken notice of in the Note on Prop. 1. Book 1 1. as fhew that the perfon who made them has not fufficiently confidered the Z Book XII, things against which they are brought; fo that it is not necessary to make any further anfwer to these objections and others like them against Euclid's Definition of Proportionals, of which Definition Dr. Barrow juftly fays in page 297 of the above-named book, that "Nifi machinis impulfa validioribus aeternum perfiftet "inconcuffa." FINI S. |