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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 a HA is given ; and because as GI) a 2. dat. to AB, so is CG to HA, and so is b CD to HB; the ratio of Cub 12. 5. to HB is given : also because KD has a given ratio to LF, as KD to EF, so make CK to LE ; therefore

H the ratio of CK to LE is given : and CK' is

С L given, wherefore LE a is given : and because as KD to EF, so is CK to LE, and so G

A b is CD to LF; the ratio of CD to LF is

K given : but the ratio of CD to HB is given :

c9, dat. wherefore c the ratio of HB to LF is given: and from HB, LF the given magnitudes HA, B D F 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 d 19. dat. other d.

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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 E; 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 a 14. dat. ratio to AB; therefore a AB together with a given magnitude has a given ratio to C: let this given magni.

F tude be AF, wherefore FB has a given ratio to

G C: also because the excess of C above a given

A magnitude has a given ratio to DE; therefore a D together with a given magnitude has a given ratio to C : let this given magnitude be

B С E

b 9. data D';, wherefore GE has a given ratio to C: and FB has a given ratio to C, therefore b 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 c.

c 19. dat.

19.

PROP. XXVII.

If 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 abore 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

A
to CF is given ; and CF is given, wherefore a
GH is given ; and AG is given, wherefore
the whole AH is given : and because as GB с
to CD, so is GH to CF, and so is b the re-

F
mainder HB to the remainder FD ; the ratio H
of HB to FD is given : and the ratio of FD
to E is given, wherefore c the ratio of HB to
E is given : and AH is given ; therefore HB B
the excess of AB above a given magnitude AH has a given ra-
tio to E.

a 2. dat.

b 19. 5.

c 9. dat.

DE

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

A
magnitude has a given ratio to D: the excess
of AB above a given magnitude has a given

E
ratio to D.

Because EB has a given ratio to C, and the

excess of C above a given magnitude has a giv-F; d 24. dat. en ratio to D; therefore d the excess of EB

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 ra

B с D tio 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.

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IF two lines given in position cut one another, See N. the point or points in which they cut one another are given.

Let two lines AB, CD given in position cut one another in the point E ; the point E is gi С ven. Because the lines AB, CD

E are given in position, they have

A

B a 4. def. always the same situation a, and therefore the point, or points, in which they cut one another have always the same situation :

E

D and because the lines AB, CD can

A

B be found a, the point, or points, in which they cut one another,

D are likewise found; and therefore are given in position a.

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IF the extremities of a straight line be given in position; the straight line is given in position and magnitude.

Bulate

Because the extremities of the straight line are given, they can be founda; let these be the points A, B, between which a 4. def. a straight line AB can be drawnb;

b1 Postthis has an invariable position, be A cause 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.

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IF 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 a 1 def. to it can be found a ; let this be the straight line D: from the

greater straight line AC cut off AB
equal to the lesser D: therefore the A

B C other extremity B of the straight

-1
line AB is found: and the point B
has always the same situation ; be. D
cause any other point in AC, upon
the same side of Ā, cuts off between

it and the point A a greater or less straight line than AB, that h 4 def. is, than D ; therefore the point B is givenb: and it is plain

another such point can be found in AC produced upon the other side of the point A,

PROP. XXXI. 28.

IF 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. á 1. 1.

Through A drawa the straight line
DAE parallel to BC; the straight D

A E
line DAE has always the same posi-
tion, because no other straight line can
be drawn through A parallel to BC: B

C therefore the straight line DAE which b 4 def. has been found is given b in position.

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IF 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.

Bb 23. 1.

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

(

F E. with CB, is given in position. Because the angle is given, one

F a 1. def. equal to it can be founda: let this be the angle at D, at the given point C, in the given straight line A AB, make bthe angle ECB equal to the angle at D: therefore the straight line EC has always the same situation, because any other

D 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.

PROP. XXXIII.

IF 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 po- E

A

F sition. Through the point A, draw a the

a 31. 1. straight line EAF parallel to BC; and because through the given point A, the

B

D с straight line EAF is drawn parallel to BC, which is given in position, EAF is therefore given in positionb; and because the straight line AD meets the parallels b 31. dat.

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