tude. and because AD is parallel to FG, and GH equal to HA; DH is equal to HF, and AD equal to GF. and DH is given, there. b. 4. 6. fore HF is given in magnitude; and it is also given in position, and the point H is given, therefore the point F is given. C. 30. Dati And because the straight line EFG is drawn from a given point F without or within the circle ABC given in position, therefored d 95. or the rectangle EF, FG is given. and GF is equal to AD, wherefore 96. Dat. the rectangle AD, EF is given. PRO P. C. a straight line be drawn to any point in the circumference of a circle given in position; and from this point a straight line be drawn making with the first an angle equal to the difference of a right angle and the angle contained by the straight line given in position, and the traight line which joins the given point and the center of the circle; and from the point in which the second line meets the circumference again, a third straight line be drawn making with the second an angle equal to that which the first makes with the second. the point in which this third line meets the straight line given in position is given; as also the rectangle contained by the first straight line and the segment of the third betwixt the circumference and the straight line given in position, is given. Let the straight line CD be drawn from the given point C in the straight line AB given in position, to the circumference of the circle DEF given in position of which G is the center; join CG, and Ff 2 from B В from the point D let DF be drawn making the angle CDF equal to the difference of a right angle and the angle BCG, and from the point F let FE be drawn making the angle DFE equal to CDF, meeting AB in H. the point H is given; as also the rectangle CD, FH. Let CD, FH meet one another in the point K from which draw KL perpendi D cular to DF; and let DC meet the cir G cumference again in M, and let FH meet F the fame in E, and join MG, GF, GH. A ABecause the angles MDF, DFE are e MY taking away the common part ME, the D E because the angles KDL, LKD are togeC. 32. 1. ther equal to a right angle, that is, by the hypothesis, to the angles KDL, GCB; H 3 angle (LKD, that is to the angle) LKF or GKH. therefore the points C, K, H, G are in the circumference of a circle ; and the angle GCK is therefore equal to the angle GHF; and the angle d. 26. 1. GMC is equal to GFH, and the straight line GM to GF; therefore! CG is equal to GH, and CM to HF. and because CG is equal to therefore GHC is given ; and consequently the angle CGH is given. €. 32. Dat and CG is given in position, and the point G; therefore o GH is given in position; and CB is also given in position, wherefore the point H is given. And becaufe HF is equal to CM, the rectangle DC, FH is equal 8.95.or 96 to DC, CM. but DC, CM is given f, because the point C is given; Dat, therefore the rectangle DC, FH is given. F HIS is made more explicit than in the Greek text, to pie vent a mistake which the Author of the second Demonstration of the 24th Proposition in the Greek Edition has fallen into, of thinking that a ratio is given to which another ratio is shewn to be equal, tho' this other be not exhibited in given magnitudes. See the Notes on that Proposition which is the 13th in this Edition. besides by this Definition, as it is now given, fome Propositions are demonstrated, which in the Greek are not so well done by help of Prop. 2. D E F. IV. In the Greek text Def. 4. is thus “ Points, lines, spaces and angles are said to be given in position which have always the same “ situation.” but this is imperfect and useless, because there are innumerable cafes in which things may be given according to this Definition, and yet their position cannot be found. for instance, let the triangle ABC be given in position, and let it be proposed to draw a straight line BD from the angle at A B to the opposite side AC which shall cut off the angle DBC which shall be the feventh part of the angle ABC. fuppose this is done, therefore the straight line BD is B C invariable in its position, that is, has always the same situation; for any other straight line drawn from the point B on either side of BD cuts off an angle greater or leflcr than the seventh part of the angle ABC; therefore, according to this Definition, the straight line BD is given in position, as also the poiat D in which it meets the a. 28. Dat. straight line AC which is given in position. but from the things here given, neither the straight line BD nor the point D can be found Ff 3 found by the help of Euclid's Elements only, by which every thing in his Data is supposed may be found. this Definition is therefore of no use. we have amended it by adding" and which are either ac“ tually exhibited or can be found;" for nothing is to be reckoned given, which cannot be found, or is not actually exhibited. The Definition of an angle given by position is taken out of the 4th, and given more distinctly by itself in the Definition marked A. DE F. XI. XII. XIII. XIV. XV. The 11th and 12th are omitted because they cannot be given in English so as to have any tolerable fense, and therefore wherever the terms defined occur, the words which express their meaning are made use of in their place. The 13. 14. 15. are omitted as being of no use. It is to be observed in general of the Data in this book, that they are to be understood to be given Geometrically, not always Arithmetically, that is, they cannot always be exhibited in numbers; for instance, if the side of a square be given, the ratio of it to its b. 44. Dat. diameter is given 6 geometrically, but not in numbers; and the diac. 2. Dat. meter is given, but tho' the number of any equal parts in the side be given, for example 10, the number of them in the diameter cannot be given. and the like holds in many other cases. PROPOSITION I. In this it is shewn that A is to B, as C to D, from this that A is to C, as B to D, and then by permutation; but it follows directly, without these two steps, froin 7.5. PRO P. II. The limitation added at the end of this Proposition between the inverted commnas is quite necessary, because without it the Propofi tion cannot always be demonstrated. for the Author having said * 1. 1. Dèf. “ because A is given, a magnitude equal to it can be found, let “ this be C; and because the ratio of A to B is given, a ratio b. Def. which is the same to it can be found b” adds, “ let it be found, “ and let it be the ratio of C to 4.” Now from the second Definition nothing more follows than that some ratio, suppose the ratio of E to Z, can be found, which is the same with the ratio of A to B; and when the Author fupposes that the ratio of C to 4, which is See Dr. Gregory's Edition of the Data. also 2. also the same with the ratio of A to B, can be found, he necessarily fupposes that to the three magnitudes E, Z, C a fourth proportional A may be found; but this cannot always be done by the Elements of Euclid; from which it is plain Euclid must have understood the Proposition under the limitation which is now added to his text. An Example will make this clear ; let A be a given angle, and B A ВА another angle to which A has a given ratio, for instance, the ratio of the given straight line E to the given one Z, then, having found an angle C equal to A, C how can the angle o be found to which C has the same ratio that E has to Z? Z certainly no way, until it be shewn how to find an angle to which a given angle has a given ratio, which cannot be done by Euclid's Elements, nor probably by any Geometry known in his time. Therefore in all the Propositions of this book which depend upon this second, the above mentioned limitation must be understood, tho' it be not explicitly mentioned. PRO P. V. The order of the Propofitions in the Greek text between Prop.4. and Prop. 25. is now changed into another which is more natural, by placing those which are more simple before those which are more complex ; and by placing together those which are of the same kind, some of which were mixed among others of a different kind. thus Prop. 1 2. in the Greek is now made the 5. and those which were the 22. and 23. are made the 11. and 12. as they are more simple than the Propositions concerning magnitudes the excess of one of which above a given magnitude has a given ratio to the other, after which these two were placed ; and the 24. in the Greek text is, for the same reason, made the 13. PRO P. VI. VII. These are universally true, tho' in the Greek text they are demonstrated by Prop. 2. which has a limitation; they are therefore now shewn without it. PRO P. Ff4 |