PREFACE. 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 that 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 the 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 accu. racy 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. Euclid's Data, DEFINITIONS. I. SPACES, lines, and angles, are said to be given in magnitude, when equals to them can be found. II. A ratio is said to be given, when a ratio of a given magnitude to a given magnitude which is the same ratio with it can be found. III. Rectilineal figures are said to be given in species, which have each of their angles given, and the ratios of their sides given. IV. Points, lines, and spaces, are said to be given in position, which have always the same situation, and which are either actually exhibited, or can be found. A. An angle is said to be given in position which is contained by straight lines given in position. V. A circle is said to be given in magnitude, when a straight line from its centre to the circumference is given in magnitude. VI. A circle is said to be given in position and magnitude, the cen tre of which is given in position, and a straight line from it to the circumference is given in magnitude. VII. Segments of circles are said to be given in magnitude, when the angles in them, and their bases, are given in magnitude. VIII. Segments of circles are said to be given in position and magni tude, when the angles in them are given in magnitude, and their bases are given both in position and magnitude. IX. A magnitude is said to be greater than another by a given mag nitude, when this given magnitude being taken from it, the remainder is equal to the other magnitude. X. tude, when this given magnitude being added to it, the whole 1. • PROPOSITION I. See N. a 1. def. dat. THE ratios of given magnitudes to one another is given. Let A, B be two given magnitudes, the ratio of A to B is given. Because A is a given magnitude, there may a be found one equal to it; let this be C : and because B is given, one equal to it may be found ; let it be D; and since A is equal to C, and B to D; therefore b A is to B, as C to D; and consequently the ratio of A to B is given, because the ratio of the given magnitudes C, D which is the same with it, has been A B C D found. b 7. 5. 2. PROP. II. See N. IF a given magnitude has a given ratio to another magnitude, " and if unto the two magnitudes by “ which the given ratio is exhibited, and the given magnitude, a fourth proportional can be found ;" the other magnitude is given. Let the given magnitude A have a given ratio to the mag. nitude B; if a fourth proportional can be found to the thret magnitudes above named, B is given in magnitude. Because A is given, a magnitude may be found equal to it a; let this be C: and because the ratio of A to B is given, a ratio which is the same with it may be found, let this be the ratio of the given magnitude E A B C D to the given magnitude F : unto the magni E F tudes E, F, C find a fourth proportional D, which, by the hypothesis, can be done. Wherefore, because A is to B, as E to F; and as E to F, so is C to D; A is b to B, as C to * The figures in the margin show the number of the propositions in the other editions, a 1. def. b 11. 5. D. But A is equal to C; therefore c B is equal to D. The c 14. 5. magnitude B is therefore given a because a magnitude D equal a 1. def. to it has been found. The limitation within the inverted commas is not in the Greek text, but is now necessarily added; and the same must be understood in all the propositions of the book which depend upon this second proposition, where it is not expressly mentioned. See the note upon it. IF any given magnitudes be added together, their sum shall be given. Let any given magnitudes AB, BC be added together, their sum AC is given. Because AB is given, a magnitude equal to it may be founda; a 1. def. let this be DE : and because BC is gi- A B C ven, one equal to it may be found ; let this be EF : wherefore, because AB is equal to DE, and BC equal to EF; the D E F whole AC is equal to the whole DF: 1 AC is therefore given, because DF has been found which is equal to it. PROP. IV. IF a given magnitude be taken from a given magnitude; the remaining magnitude shall be given. From the given magnitude AB, let the given magnitude AC be taken ; the remaining magnitude CB is given. Because AB is given, a magnitude equal to it may a be a 1. def. found ; let this be DE: and because A CB AC is given, one equal to it may be found ; let this be DF: wherefore because AB is equal to UE, and AC to D F E DF; the remainder CB is cqual to the remainder FE. CB is therefore given a, because FE which is equal to it has been found. |