PREFACE. UCLID'S DATA is the first in order of the books written EUC by the antient Geometers to facilitate and promote the method of Refolution or Analyfis. In the general, a thing is faid to be given which is either actually exhibited, or can be found out, that is, which is either known by Hypothefis, or that can be demonstrated to be known; and the Propofitions in the Book of Euclid's Data Thew what things can be found out or known from those that by Hypothefis are already known; fo that in the Analyfis or Investigation of a Problem, from the things that are laid down to be known or given, by the help of these Propofitions other things are demonstrated to be given, and from these other things are again shewn to be given, and fo 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 folved, and its Compofition is made and derived from the Compofitions of the Data which were made use of in the Analysis. And thus the Data of Euclid are of the moft general and neceffary ufe in the folution of Problems of every kind. Euclid is reckoned to be the Author of the Book of the Data both by the antient and modern Geometers; and there feems 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 feveral places, both in the order of the Propofitions, and in the Definitions and Demonftrations 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 defign of this Edition, that fo it may be rendered more ufeful to Geometers, at least to beginners who defire to learn the inveftigatory method of the Antients. And for their fakes the Compofition of most of the Data are fubjoined to their Demonftrations, that the Compofitions of Problems folved by help of the Data may be the more eafily made. Marinus the Philofopher's preface which in the Greek Edition is prefixed to the Data is here left out,, as being of no ufe to underftand them. at the end of it he fays that Euclid has not used the fynthetical, but the analytical method in delivering them; in which he is quite mistaken; for in the Analysis of a Theorem the thing to be demonftrated is affumed in the Analysis; but in the Demonftrations of the Data, the thing to be demonftrated, which is that fomething or other is given, is never once affumed in the Demonstration, from which it is manifeft that every one of them is demonftrated fynthetically; tho' indeed if a Propofition 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 Demonftration of the Propofition becomes the Analysis of the Problem. Wherein this Edition differs from the Greek, and the reasons of the alterations from it will be fhewn in the Notes at the end of the Data. EUCLID'S EUCLID'S DATA. SPA DEFINITION S. I. PACES, lines and angles are faid to be given in magnitude, when equals to them can be found. II. A ratio is faid to be given, when a ratio of a given magnitude to a given magnitude which is the fame ratio with it can be found. III. Rectilineal figures are faid to be given in fpecies, which have each of their angles given, and the ratios of their fides given. IV. Points, lines and spaces are faid to be given in pofition, which have always the fame fituation, and which are either actually exhibited, or can be found. A. An angle is faid to be given in pofition, which is contained by straight lines given in position. V.. A circle is faid to be given in magnitude, when a straight line from its center to the circumference is given in magnitude. VI. A circle is faid to be given in pofition and magnitude, the center of which is given in pofition, and a straight line from it to the circumference is given in magnitude. VII. Segments of circles are faid to be given in magnitude, when the angles in them, and their bases are given in magnitude. VIII. Segments of circles are faid to be given in pofition and magnitude, when the angles in them are given in magnitude, and their bafes are given both in pofition and magnitude. IX. A magnitude is faid to be greater than another by a given magnitude, when this given magnitude being taken from it, the remainder is equal to the other magnitude. See N. a. 1. Def. Dat. X. A magnitude is faid to be less than another by a given magnitude, when this given magnitude being added to it, the whole is equal to the other magnitude. TH PROPOSITION I. THE ratios of given magnitudes to one another is given. a Let A, B be two given magnitudes, the ratio of A to B is given. Because A is a given magnitude, there may be found one equal to it; let this be C. and becaufe B is given, one equal to it may be found; let it be D. and fince b. 7. 5. A is equal to C, and B to D; therefore b A is to See N. 2. B, as C to D; to B is given, and confequently the ratio of A because the ratio of the given magnitudes C, D which is the fame with it has been found. PRO P. II. A B C D 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 magni tude is given. Let the given magnitude A have a given ratio to the magnitude B; if a fourth proportional can be found to the three magnitudes above named, B is given in magnitude. Because A is given, a magnitude may be found a. 1. Def. equal to it; let this be C. and because the ratio EF of A to B is given, a ratio which is the fame with ABCD The figures in the margia fhew the number of the Propofitions in the other fo |