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Let AB be the first straight line, CD the second, and EF the third ; and because the ratio of AB to CD is given, and that if a given straight line be taken from CD, the ratio of the remainder to EF is given ; therefore a the excess of the first AB a 24. dat. above a given straight line has a given ratio to the third EF: let BH be that given straight line ; therefore AH, the excess of AB above it, has a given ratio to EF : and consequently b the rectangle BA, AH, has a A H B 61. 6. given ratio to the rectangle AB, EF, which last rectangle is given by the hypothesis ; therefore the rectangle BA, AH is given,
с G D
c 2. dat. and BH the excess of its sides is given ; where. fore the sides AB, AH are given d; and be
d 85. dat.
E F cause the ratios of AB to CD, and of AH to EF are given, CD and EF are è given.
K NM L O
-1-1-1 The Composition.
Let the giren ratio of KL to KM be that which AB is requir. ed to have to CD; and let DG be the given straight line which is to be taken from CD, and let the given ratio of KM to KN be that which the remainder must have to EF; also let the given rectangle NK, KO be that to which the rectangle AB, EF is required to be equal : find the given straight line BH which is to be taken from AB, which is done, as plainly appears from prop. 24, dat. by making as KM to KL, so GD to HB. To the given straight line BH apply e a rectangle equal to LK, KO exceeding e 29.6. by a square, and let BA, AH be its sides ; then is AB the first of the straight lines required to be found, and by making as LK to KM, so AB to DC, DC will be the second : and lastly, make as KM to KN, so CG to EF, and EF is the third.
For as AB to CD, so is HB to GD, each of these ratios being the same with the ratio of LK to KM; therefore f AH is to CG, f 19.5. as (AB to CD, that is, as) LK to KM; and as CG to EF, so is KM to KN; wherefore, ex æquali, aš AH to EF, so is LK 10 KN: and as the rectangle BA, AH to the rectangle BA, EF, so is & the rectangle LK, KO to the rectangle KN, KO: and by the g 1.6. construction, the rectangle BA, AH is equal to LK, KO: therefore h the rectangle AB, EF is equal to the given rectangle NK, h 14. 5. KO: and AB has to CD the given ratio of KL to KM; and from CD the given straight line GD being taken, the remainder CG has to EF the given ratio of KM to KN. Q. E. D.
To find three straight lines such, that the ratio of the first to the second is given; and if a given straight line be taken from the second, the ratio of the remainder to the third is given; also the sum of the squares of the first and third is given.
Let AB be the first straight line, BC the second, and BD the third : and because the ratio of AB to BC is given, and that if
a given straight line be taken from BC, the ratio of the remaina 24. 'dat. der to BD is given; therefore a the excess of the first AB above
a given straight line, has a given ratio to the third BD: let AE be that given straight line, therefore the remainder EB has a
given ratio to BD : let Bl) be placed at right angles to EB and b 44 dat. join DE; then the triangle EBD is b given in species; where
fore the angle BED is given: let AE, which is given in magni
tude, be given also in position, as also the point E, and the c 32 dat, straight line ED will be given c in position : join AD, and bed 47 1
cause the sum of the squares of AB, BD, that is d, the square
of aD is given, therefore the straight line AD is given in mage 34 dat, nitude ; and it is also given e in position, because from the given
point A it is drawn to the straight line ED given in position :
iherefore the point D, in which the two straight lines AD, ED f 28. dat. given in position cut one another, is given f; and the straight 833 dat. line DB which is at right angles to AB is given 8 in position, and
AB is given in position, therefore f the point B is given : and the h 29 dat. points A, D are given, wherefore h the straight lines AB, BD are i 2. dat. given : and the ratio of AB to BC is given, and therefore i BC
required to have to BD be the given ratio of HG to GL, and place GL at right angles to FII, and join LF, LH: next, as HG
is to GF, so make HK to AE; produce AE to N, so that AN be the straight line to the square of which the sum of the squares of AB, BD is required to be equal; and make the angle NED equal to the angle GFL; and from the centre A at the distance AN describe a circle, and let its circumference meet ED in D, and draw DB perpendicular to AN, and DM, making the angle BDM equal to the angle GLH. Lastly, produce BM to C, so that MC be equal to HK then is AB the first, BC the second, and BD the third of the straight lines that were to be found.
For the triangles EBD, FGL, as also DBM, LGH being eqiangular, as EB to BD, so is FG to GL; and as DB to BM, so is LG to GH; therefore, ex æquali, as EB to BM, so is (FG to GH, and so is) AE to HK or MC; whereforek, AB is to BC, k 12. 5. as AE to HK, that is, as FG to GH, that is, in the given ratio ; and from the straight line BC taking MC, which is equal to the given straight line HK, the remainder BM has to BD the given ratio of HG to GL; and the sum of the squares of AB, BD is equal d to the square of AD or AN, which is the given space. d 47. 1. Q. E. D.
I believe it would be in vain to try to deduce the preceding construction from an algebraical solution of the problem.