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therefore the point H where they meet is the centre of the circle, and consequently H is given: And the point D is given, wherefore DH is given in magnitude: And because AD is parallel to FG, and GH equal to HA; DH is equal to HF, b 4. 6. and AD equal to GF: And DH is given, therefore 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. dat.
And because the straight line EFG is drawn from a given point F without or within the circle ABC given in position, therefore the rectangle EF, FG is given: And GF is equal d 95.or 96. to AD, wherefore the rectangle AD, EF is given.
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.
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 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 H 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 DC meet the circumference again in M, and let FH meet the same in E, and join MG, GF, GH.
Because the angles MDF, DFE are equal to one another, the circumferences MF, DE are equal; and adding or taking away the common part ME, the circumference DM is equal to EF; therefore the straight line DM is equal to the straight line EF, and the angle GMD to the angle GFE; and the angles GMC, GFH are equal to one another, because they are either the same with the angles GMD, GFE, or adjacent to them: And because D the angles KDL, LKD are together c 32. 1. equal to a right angle, that is, by the hypothesis, to the angles KDL, GCB; the angle GCB or GCH is equal to the 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 GMC is equal to GFH, and the d 26. 1. straight line GM to GF: therefored CG is equal to GH, and CM to HF: And because CG is equal to GH, the angle GCH is equal to GHC; but the angle GCH is given : Therefore GHC is given, and consequently the angle CGH is given; and CG is given in position, and the e 32. dat. point G; therefore GH is given in position; and CB is also given in position, wherefore the point H is given.
And because HF is equal to CM, the rectangle DC, FH f 95.or 96. is equal to DC, CM: But DC, CM is given f, because the point C is given, therefore the rectangle DC, FH is given.
a 26. 3.
b 8. 1.
THIS is made more explicit than in the Greek text, to prevent 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.
In the Greek text, def. 4. is thus: "Points, lines, spaces, "and angles are said to be given in position, which have al66 ways 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 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 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 less than the seventh part of the angle ABC; therefore, according to this definition, the straight line BD is given in position, as also the point D in which it meets the straight a 28. dat. 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 it is
b 44. dat. c 2. dat.
a 1. def.
b 2. def.
supposed that every thing in his Data may be found. This
The definition of an angle given by position is taken out of the fourth, 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 geometrically, but not in numbers; and the diameter is given but 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.
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.
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 magnitude 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 can be found ',' adds, "let it be found, and let it be the ratio of C to A." Now, from the second definition, nothing more follows han that some ratio, suppose the ratio of to Z, can be found, which is the same with the ratio of A to B; and when the author supposes that the ratio of C to 4, which is * See Dr Gregory's edition of the Data.
also the same with the ratio of A to B, can be found, he necessarily supposes, that to the three magnitudes E, Z, C, a fourth proportional ▲ 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 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, how can the angle ▲ be found to which C has the same ratio that E has to Z? Certainly no way until it be shown 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, though it be not explicitly mentioned.
THE order of the Propositions 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. 12. in the Greek is now made the 5th, and those which were the 22d and 23d are made the 11th and 12th, 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 24th in the Greek text is, for the same reason, made the 13th.
PROP. VI. VII.
THESE are universally true, though, in the Greek text, they are demonstrated by Prop. 2, which has a limitation; they are therefore now shown without it.