04. 6. equal o to HF, and AD equal to GF: And DH is given, therefore HF is given in magnitude; and it is also given AS!! Gurd kor. * 80 Dat. in position, and the point H is given, therefore the point F is given. And because the straight line EFG is drawn from a given point F without or within the circle ABC given in position, • 95 or 96 therefored the rectangle EF, FG is given: And GF is Dat. equal to AD, wherefore the rectangle AD, EF is given. , PROP. C. If from a given point in a straight line given in position, 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 straight line which joins the given point and the centre 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. In Let the straight line CD be drawn from the giveri point C in the straight line AB given in position, to the circumference of the circle DEF given in position, of which Gis the centre; join CG, and 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 A is given; as also the rectangle CD, FH. Let CD, FH, meet one another in the point K, from which draw KL perpendicular to DF; and let D DC meet the circumference again in M, and let FH meet the same in E, and join MG, GF, GH. Al Because the angles MDF,DFE, are equal to one another, the circumferences MF, DE, are equala; A 3709 26.3. and adding or taking away the common part ME; the circumference DM is equal to EF; therefore the straight line DM is equal KVE to the straight line EF, and the angle GMD to the angle b GFE; b 8.1. and the angles GMC, GFH are equal to one another, because E they are either the same with the angles GMD, GFE, or adjacent to them: And because the angles | KX KDL, LKD, are together equal 32. 1. to a right angle, that is, by the hypothesis, to the angles KDL, GCB; the angle GCB or GCH is Hв equal to the angle (LKD, that is, 11 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 GMC is equal to GFH, and the straight line GM to GF; therefore d CG is equal to GH, and CM to HF: And be 26. la cause CG is equal to GH, the angle GCH is equal to GHC; but the angle ĠCH is given : Therefore GHC is given, and consequently the angle CGH is given ; and CG is given in position, and the point G; therefore e GH is given in posi- ' 32 Dat. tion; and CB is also given in position, wherefore the point H is given. And because HF is equal to CM, the rectangle DC, FH, is equal to DC, CM : But DC, CM is given f, because the 95 or 96 point C is given, therefore the rectangle DC, FH is given. Dat. LGL NOTES ON EUCLID'S DATA, DEFINITION II. This is made more explicit than in the Greek text, to pre. 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 shown to be equal, though 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, some propositions are demonstrated which in the Greek are not so well done by help of Prop. 2. DEF. 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 cases 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 B to the A opposite side AC, which shall cut off the angle DBC, which shall be the seventh part of the angle ABC; suppose this is done, therefore the B straight line BD is 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 lesser than the seventh part of the angle ABC; therefore, according to this definition, the straight line BD is given in position, as also a the point D in which it meets * 28 Dat. the 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 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 actually ex66 bibited, 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. DEF. XI. XII. XIII. XIV. XV. The 11th and 12th are omitted, because they cannot be given in English so as to have any tolerable sense : and therefore, wherever the terms defined occur, the words which express their meaning are made use of in their place. The 13th, 14th, 15th, 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 diameter is given geometri44 Dat. cally b, but not in numbers; and the diameter is given ; but 2 Dat. though 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 shown 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, from 7, 5. PROP. II. The limitation added at the end of this proposition between the inverted commas, is quite necessary, because without it the proposition cannot always be demonstrated : For the author having said*,“ because A is given, a magni: 1 Def." tude equal to it can be found a: let this be C; and because " the ratio of A to B is given, a ratio which is the same to it 2 Def. “ can be found b,” adds, “ let it be found, and let it be the “ ratio of Cto A.” Now, from the second definition, notbing * See Dr. Gregory's edition of the Data. |