The Elements of Euclid

ratio to the second, the same excess shall have a given ratio to the first; as is evident from thd 9th dat.

Ir there be three magnitudes, the excess of the first whereof above a given magnitude has a given ratio to the second; and the excess of the third above a given magnitude has a given ratio to the same second: the first shall either have a given ratio to the third, or the excess of one of them above a given magnitude shall have a given ratio to the other.

Let AB, C, DE be three magnitudes, and let the excesses of each of the two AB, DE above given magnitudes have given ratios to C; AB, DE either have a given ratio to one another, or the excess of one of them above a given magnitude has a given ratio to the other.

Let FB, the excess of AB above a given magnitude AF, have a given ratio to C; and let GE, the excess of DE above the given magnitude DG, have a given ratio to C; and because FB, GE have each of them a given ratio to C, they have a given ratio (9. dat.) to one another.

But to

FB, GE the given magnitudes AF, DG are added; therefore (18. dat.) the whole magnitudes AB, DE have either a given ratio to one another, or the excess of one of them above a given magnitude has a given ratio to the other.

PROP. XXVI.

If there be three magnitudes; the excesses of one of which above given magnitudes have given ratios to the other two magnitudes; these two shall either have a given ratio to one another, or the excess of one of them above a given magnitude shall have a given ratio to the other.

Let AB, CD, EF be three magnitudes, and let GD the excess of one of them CD above the given magnitude CG have a given ratio to AB; and also let KD the excess of the same CD above the given magnitude CK have a given ratio to EF: either AB has a given ratio to EF, or the excess of one of them above a given magnitude has a given ratio to the other.

Because GD has a given ratio to AB, as GD to AB, so make CG to HA; therefore the ratio of CG to HA is given and CG is given, wherefore (2. dat.) HA is given; and because as GD to AB, so is CG to HA, and so is (12. 5.) CD to HB; the ratio of CD to HB is given also because KD has a given ratio to EF, as KD to

EF, so make CK to LE: therefore the ratio H of CK to LE is given; and CK is given, wherefore LE (2. dat.) is given: and because as KD to EF, so is CK to LE, and so (12. 5.) is CD to LF; the ratio of CD to LF is given: but the ratio of CD to HB is given, wherefore (9. dat.) the ratio of HB to LF is given: and from HB, LF the given magnitudes HA, LE being taken, the remainders AB, EF shall either have a given ratio to one another, or the excess of one of them above a given magnitude has a given ratio to the other (19. dat.).

Another Demonstration.

Let AB, C, DE be three magnitudes, and let the excesses of one of them C above given magnitudes have given ratios to AB and DE: either AB, DE have a given ratio to one another, or the excess of one of them above a given magnitude has a given ratio to the other.

Because the excess of C above a given magnitude has a given ratio to AB; therefore (14. dat.) AB together with a given magnitude has a given ratio to C: let this given magnitude be AF, wherefore FB has a given ratio to C: also because the excess of C above a given magnitude has a given ratio to DE; therefore (14. dat.) DE together with a given magnitude has a given ratio to C: let this given magnitude be DG, wherefore GE has a given ratio to C: and FB has a given ratio to C, therefore (9. dat.) the ratio of FB to GE is given: and from FB, GE the given magnitudes AF, DG being taken, the remainders AB, DE either have a given ratio to one another, or the excess of one of them above a given magnitude has a given ratio to the other (19. dat.).

Ir there be three magnitudes, the excess of the first of which above a given magnitude has a given ratio to the second; and the excess of the second above a given magnitude has also a given ratio to the third; the excess of the first above a given magnitude shall have a given ratio to the third.

Let AB, CD, E be three magnitudes, the excess of the first of which AB above the given magnitude AG, viz. GB, has a given ratio to CD; and FD the excess of CD above the given magnitude CF, has a given ratio to E: the excess of AB above a given magnitude has a given ratio to E.

Because the ratio of GB to CD is given, as GB to CD, so make

GH to CF: therefore the ratio of GH to CF A is given; and CF is given, wherefore (2. dat.) GH is given and AG is given, wherefore the whole AH is given: and because as GB to CD, so is GH to CF, and so is (19. 5.) the remainder HB to the remainder FD; the ratio of HB to FD is given: and the ratio of FD to E is given, wherefore (9. dat.) the ratio of HB to E is given and AH is given; therefore HB, the excess of AB above a given magnitude AH, has a given ratio to E.

"Otherwise,

Let AB, C, D, be three magnitudes, the excess EB of the first of which AB above the given magnitude AE has a given ratio to C, and the excess of C above a given magnitude has a given ratio to D: the excess of AB above a given magnitude has a given ratio to D.

A |

Because EB has a given ratio to C, and the excess of C above a given magnitude has a given ratio to D; therefore (24. dat.) the excess of EB above a given magnitude has a given ratio to D: let this given magnitude be EF; therefore FB, the excess of EB above EF, has a given ratio to D and AF is given, because AE, EF are given therefore FB, the excess of AB above a given magnitude AF, has a given ratio to D."

Ir two lines given in position cut one another, the point or points in which they cut one another are given.*

Let two lines AB, GD given is position cut one another in the point E; the point E is given.

Because the lines AB, CD are given in position, they have always the same situation (4. def.), and therefore the point, or points, in which they cut one another, have always the same situation: and because the lines AB, CD can be found (4. def.), the point, or points, in which they cut one another, are likewise found; and therefore are given in position (4. def.)

PROP. XXIX.

Ir the extremities of a straight line be given in position; the

straight line is given in position and magnitude.

Because the extremities of the straight line are given, they can

* See Note.

be found (4. def.): let these be the points A, B, between which a straight line AB can be drawn (1. postulate); this has an invariable position,

-B

because between two given points there can be drawn but one straight line: and when the straight line AB is drawn, its magnitude is at the same time exhibited, or given therefore the straight line AB is given in position and magnitude.

Ir one of the extremities of a straight line given in position and magnitude be given: the other extremity shall also be given.

Let the point A be given, to wit, one of the extremities of a straight line given in magnitude, and which lies in the straight line AC given in position; the other extremity is also given.

Because the straight line is given in magnitude, one equal to it can be found (1. def.); let this be the straight line D; from the greater straight line AC cut off AB equal

to the lesser D: therefore the other extremity B of the straight line AB is found and the point B has always the D same situation; because any other point

B C

in AC, upon the same side of A, cuts off between it and the point A a greater or less straight line than AB, that is, than D; therefore the point B is given (4. def.): and it is plain another such point can be found in AC, produced upon the other side of the point A.

Ir a straight line be drawn through a given point parallel to a straight line given in position; that straight line is given in position.

Let A be a given point, and BC a straight line given in position; the straight line drawn through a parallel to BC is given in position.

Through A draw (31. 1.) the straight line DAE parallel to BC; the straight line DAE has always the same position, because no other straight line can be B drawn through A parallel to BC; there

A E

fore the straight line DAE, which has been found, is given (4. def.) in position.

Ir a straight line be drawn to a given point in a straight line given in position, and makes a given angle with it; that straight line is given in position.

Let AB be a straight line given in position, and C a given point

in it; the straight line drawn to C, which makes a given angle with CB, is given in position.

Because the angle is given, one equal to it can be found (1. def.); let this be the angle at D: at the given A point C, in the given straight line AB, make (23. 1.) the angle ECB equal to the angle at D: therefore the straight line EC has always the same situation, because any other straight line FC, drawn to the point

C, makes with CB a greater or less angle than the angle ECB, or the angle at D: therefore the straight line EC, which has been found, is given in position.

It is to be observed, that there are two straight lines EC, GC upon one side of AB that make equal angles with it, and which make equal angles with it when produced to the other side.

Ir a straight line be drawn from a given point to a straight line given in position, and makes a given angle with it; that straight line is given in position.

From the given point A, let the straight line AD be drawn to the straight line BC given in position, and make with it a given angle ADC; AD is given in position.

Through the point A, draw (31. 1.) the straight line EAF parallel to BC; and because through the given point A, the straight line EAF is drawn parallel to BC, which is given in position, EAF is therefore given in position (31. dat.): and because the straight line AD meets the parallels, BC, EF, the angle EAD is equal (29. 1.) to the angle ADC; and ADC is given, wherefore also the angle EAD is given therefore, because the straight line DA is drawn to the given point A in the straight line EF given in position, and makes with it a given angle EAD, AD is given (32. dat.) in position.

Ir from a given point to a straight line given in position, a straight line be drawn which is given in magnitude; the same is also given in position.*

Let A be a given point, and BC a straight line given in position; a straight line given in magnitude drawn from the point A to BC is given in position.

Because the straight line is given in magnitude, one equal to

* See Note.

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