to the pyramid LGHN, and the pyramid ECDM to LHKN. and Book XII. because the pyramids EABM, LFGN are fimilar, and have triangu-n lar bases, the pyramid EABM has f to LFGN the triplicate ratio of f. 8. 12. 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. PROP. XI. and XII. B. XII. The order of the letters of the Alphabet is not obferved in these two Propofitions, according to Euclid's manner, and is now restored. by which means the first part of Prop. 12. may be demonstrated in the fame words with the firft part of Prop. 1 1. on this account the Demonstration of that firft part is left out, and affumed 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 itself is supposed 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 represent the cylinders as well as the cones as already described. wherefore the reading ought rather to Book XII. to be" and let the cones be ALC, ENG; and the cylinders AX, EO." The first Cafe in the second part of the Demonstration is wanting; and fomething alfo in the second Case of that part, before the repetition of the construction is mentioned; which are now added. In the Enuntiation of this Propofition the Greek words, &c Từ μείζονα σφαῖραν σερεὸν πολύεδρον ἐγγράψαι, μὴ ψανον 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 defcribe in the greater sphere a folid polyhedron which fhall not meet the fuper"ficies of the leffer sphere." whereby they refer the words xxτ2 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 fphere, but pervades the whole of that fphere. therefore the forefaid words are to be referred to rò sepeèv zoneffer, and ought thus to be tranflated, viz. to defcribe 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 Demonftration it is faid "draw KO "from the point K perpendicular to BD;" whereas it ought to have been faid," join KV," and it should have been demonftrated 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 ftraight line BD muft 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 Demonstration. Commandine feems to have been aware of this; for in his figure he marks one and the fame point with the two let ters ters V, ; and before Commandine, the learned John Dee in the Book XII. Commentary he annexes to this Propofition in Henry Billingsley's Translation of the Elements printed at London Ann. 1570, exprefly 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, must neceffarily fall upon the point V. Likewife 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 de#monftrated. In the Corollary of this Propofition it is fuppofed that a folid polyhedron 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 demonstrate that the pyramids in it are fimilar to those of the fame order in the folid polyhedron defcribed in the sphere BCDE. From the preceeding 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 prefent 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 unacceptable to good Judges who can difcern when Demonftrations are legitimate, and when they are not. The objections which, fince the firft Edition, have been made against some things in the Notes, especially against the doctrine of Proportionals, have either been fully anfwered in Dr. Earrow's Lect. Mathemat. and in thefe Notes; or are fuch, except one which has been taken notice of in the Note on Prop. 1. Book 11. as fhew that the perfon who made them has not fufficiently confidered the things Book XII. against which they are brought; fo that it is not neceffary 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. |