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Book XII. ficies of the less sphere, but pervades the whole of that sphere: Therefore the aforesaid words are to be referred to Torgiv Toλúdgor, and ought thus to be translated, viz. to describe in the greater sphere a solid polyhedron whose superficies shall not meet the less 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" Ko from the point K, per"pendicular 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 2, 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 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 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 less 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 these 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 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 definitions 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."

Book XII.




EUCLID'S DATA is the first in order of the books written by the ancient geometers to facilitate and promote the method of resolution or analysis. In the general, a thing is said to be given which is either actually exhibited, or can be found out, that is, which is either known by hypothesis, or can be demonstrated to be known; and the Propositions in the Book of Euclid's Data show what things can be found out or known from those that by hypothesis are already known; so that in the analysis or investigation of a problem, from the things that are laid down to be known or given, by the help of these propositions other things are demonstrated to be given, and from these, other things are again shown to be given, and so on, until that which was proposed to be found out in the problem is demonstrated to be given; and when this is done, the problem is solved, and its composition is made and derived from the compositions of the Data which were made use of in the analysis. And thus the Data of Euclid are of the most general and necessary use in the solution of problems of every kind.

EUCLID is reckoned to be author of the Book of the Data, both by the ancient and modern geometers; and there seems to be no doubt of his having written a book on this subject, but which, in the course of so many ages, has been much vitiated by unskilful editors in several places, both in the order of the propositions, and in the definitions and demonstrations themselves. To correct the errors which are now found in it, and bring it nearer to the accuracy with which it was, no doubt, at first written by Euclid, is the design of this edition, that so it may be rendered more useful to geometers, at least

to beginners who desire to learn the investigatory method of the ancients. And for their sakes, the compositions of most of the Data are subjoined to their demonstrations, that the compositions of problems solved by help of the Data may be the more easily made.

MARINUS, the philosopher's, preface, which, in the Greek edition, is prefixed to the Data, is here left out, as being of no use to understand them. At the end of it, he says, that Euclid has not used the synthetical, but the analytical method in delivering them; in which he is quite mistaken; for in the analysis of a theorem, the thing to be demonstrated is assumed in the analysis; but in the demonstrations of the Data, the thing to be demonstrated, which is, that something or other is given, is never once assumed in the demonstration, from which it is manifest, that every one of them is demonstrated synthetically; though indeed, if a proposition of the Data be turned into a problem, for example, the 84th or 85th in the former editions, which here are the 85th and 86th, the demonstration of the proposition becomes the analysis of the problem.

WHEREIN this edition differs from the Greek, and the reasons of the alterations from it, will be shown in the notes at the end of the Data.

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