dition, has no connection in the leaft with the Greek text. And it is ftrange that Dr Gregory did not obferve, that, if prop. 86. was changed into this, the demonftration of the 86th must be cancelled, and another put in its place: But the truth is, both the enunciation and the demonftration of prop. 86. are quite entire and right, only prop. 87. which is more fimple, ought to have been placed before it; and the deficiency which the Doctor juftly obferves to be in this part of Euclid's data, and which, no doubt, is owing to the careleffness and ignorance of the Greek editors, fhould have been fupplied, not by changing prop. 86. which is both entire and neceffary, but by adding the two propofitions, which are the 88th and goth in this edition. PRO P. XCVIII. C. Thefe were communicated to me by two excellent geometers, the first of them by the Right Honourable the Earl of Stanhope, and the other by Dr Matthew Stewart; to which I have added the demonftrations. Though the order of the propofitions has been in many places changed from that in former editions, yet this will be of little difadvantage, as the antient geometers never cite the data, and the moderns very rarely. A S that part of the compofition of a problem which is its conftruction may not be fo readily deduced from the analyfis by beginners: For their fake the following example is given, in which the derivation of the feveral parts of the conftruction from the analyfis is particularly fhown, that they may be affifted to do the like in other problems. PROBLEM. Having given the magnitude of a parallelogram, the angle of which ABC is given, and alfo the excefs of the square of its fide BC above the fquare of the fide AB; to find its fides, and defcribe it. The analysis of this is the fame with the demonftration of the 87th prop. of the data, and the conftruction that is given of the problem at the end of that propofition is thus derived from the analyfis. Let Let EFG be equal to the given angle ABC, and because in the analyfis it is faid that the ratio of the rectangle AB, BC to the parallelogram AC is given by the 62d prop. dat. therefore, from a point in FE, the perpendicular EG is drawn to FG, as the ratio of FE to EG is the ratio of the rectangle AB, BC to the parallelogram AC by what is shown at the end of prop. 62. Next, the magnitude of AC is exhibited by making the rectangle EG, GH equal to it; and the given excefs of the fquare of BC above the fquare of BA, to which excefs the rectangle CB, BD is equal, is exhibited by the rectangle HG, GL: Then in the analyfis, the rectangle AB BC is faid to be given, and this is equal to the rectangle FE, GH, because the rectangle AB, BC is to the parallelogram AC, as (FE to EG, that is, as the rectangle) FE, GH to EG, GH; and the parallelogram AC is equal to the rectangle EG, GH, therefore the rectangle AB, BC, is equal to FE, GH: And confequently the ratio of the rectangle CB, BD, that is, of the rectangle HG, GL, to AB, BC, that is, of the ftraight line. DB to BA, is the fame with the ratio (of the rectangle GL, GH to FE, GH, that is) of the ftraight line GL to FE, which ratio of DB to BA is the next thing faid to be given in the analyfis: From this it is plain that the fquare of FE is to the . fquare of GL, as the fquare of BA, which is equal to the rectangle BC, CD, is to the fquare of BD: The ratio of which fpaces is the next thing faid to be given: And from this it follows that four times the fquare of FE is to the fquare of GL, as four times the rectangle BC, CD is to the fquare of BD; and, by compofition, four times the fquare of FE together with the fquare of GL, is to the fquare of GL, as four times the rectangle BC, CD together with the fquare of BD, is to the fquare of BD, that is (8. 6.) as the fquare of the ftraight lines BC, CD taken together is to the fquare of BD, which ratio is the next thing faid to be given in the analyfis: And because four times the fquare of FE and the fquare of GL are to be added together; therefore in the perpendicular EG EG there is taken KG equal to FE, and MG equal to the double of it, because thereby the fquares of MG, GL, that is, joining ML, the fquare of ML is equal to four times the fquare of FE and to the fquare of GL: And because the fquare of ML is to the fquare of GL, as the square of the ftraight line made up of BC and CD is to the square of BD, therefore (22. 6.) ML is to LG, as BC together with CD is to BD; and, by compofition, ML and LG together, that is, producing GL to N, fo that ML be equal to LN, the ftraight line NG is to GL, as twice BC is to BD; and by taking GO equal to the half of NG, GO is to GL, as BC to BD, the ratio of which is faid to be given in the analyfis: And from this it follows, that the rectangle HG, GO is to HG, GL, as the fquare of BC is to the rectangle CB, BD which is equal to the rectangle HG, GL; and therefore the fquare of BC is equal to the rectangle HG, GO; and BC is confequently found by taking a mean proportional betwixt HG and GO, as is faid in the conftruction: And because it was shown that GO is to GL, as BC to BD, and that now the three first are found, the fourth BD is found by 12. 6. It was likewife fhown that LG is to FE, or GK, as DB to BA, and the three firft are now found, and thereby the fourth BA. Make the angle ABC equal to EFG, and complete the parallelogram of which the fides are AB, BC, and the construction is finished; the reft of the compofition contains the demonstration. A S the propofitions from the 13th to the 28th may be thought by beginners to be lefs ufeful than the reft, because they cannot fo readily fee how they are to be made use of in the folution of problems; on this account the two following problems are added, to fhow that they are equally ufeful with the other propofitions, and from which it may be eafily judged that many other problems depend upon thefe propofitions. T O find three ftraight lines fuch, that the ratio of the firft to the second is given; and if a given straight line be taken from the fecond, the ratio of the remainder to the third is given; alfo the rectangle contained by the first and third is given. Let Let AB be the firft ftraight line, CD the fecond, and EF the third: And because the ratio of AB to CD is given, and that. if a given ftraight line be taken from CD, the ratio of the remainder to EF is given; therefore the excefs of the first AB a 24. dat. above a given ftraight line has a given ratio to the third EF: Let BH be that given ftraight line; therefore AH, the excefs of AB above it, has a given ratio to EF; and A confequently the rectangle BA, AH, has a given ratio to the rectangle AB, EF, which HB laft rectangle is given by the hypothefis; C G D therefore the rectangle BA, AH is given, EF and BH the excefs of its fides is given; where b 1. 9. Ac 2. dat. fore the fides AB, AH are given: And be-K NML O ́d 85. dat. caufe the ratios of AB to CD, and of AH to с EF are given, CD and EF are given. The Compofition. Let the given ratio of KL to KM be that which AB is requi red to have to CD; and let DG be the given ftraight 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; alfo let the gi-. ven rectangle NK, KO be that to which the rectangle AB, EF is required to be equal: Find the given ftraight 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, fo GD to HB. To the given ftraight line BH apply a rectangle equal to LK, KO e 29. 6. exceeding by a fquare, and let BA, AH be its fides: Then is AB the firft of the ftraight lines required to be found, and by making as LK to KM, fo AB to DC, DC will be the fecond: And lastly, make as KM to KN, fo CG to EF, and EF is the third. e For as AB to CD, fo is HB to GD, each of thefe ratios being the fame with the ratio of LK to KM; therefore f AH is f 19. 5. to CG, as (AB to CD, that is, as) LK to KM; and as CG to EF, fo is KM to KN; wherefore, ex aequali, as AH to EF, fo is LK to KN: And as the rectangle BA, AH to the rectangle BA, EF, fo is the rectangle LK, KO to the rectangle KN, g 1. 6. KO: And, by the conftruction, the rectangle BA, AH is equal to LK, KO: Therefore the rectangle AB, EF is equal to the h 14. 5. given rectangle NK, KO: And AB has to CD the given ratio of KL to KM; and from CD the given ftraight line GD being taken, the remainder CG has to EF the given ratio of KM to KN. Q. E. D. PROB. T O find three ftraight lines fuch, that the ratio of the first to the second is given; and if a given straight line be taken from the fecond, the ratio of the remainder to the third is given; alfo the fum of the fquares of the firft and third is given. Let AB be the firft ftraight line, BC the fecond, and BD the third: And because the ratio of AB to BC is given, and that if a given ftraight line be taken from BC, the ratio of the remaina 24. dat. der to BD is given; therefore the excess of the firft AB above a given ftraight line, has a given ratio to the third BD: Let AE be that given ftraight line, therefore the remainder EB bas a given ratio to BD: Let BD be placed at right angles to EB, b 44. dat. and join DE; then the triangle EBD is given in fpecies; wherefore the angle BED is given: Let AE, which is given in magnitude, be given alfo in pofition, as alfo the point E, and e 32. dat. the ftraight line ED will be given in pofition: Join AD, and because the fum of the fquares of AB, BD, that is, the fquare of AD is given, therefore the ftraight line AD is given in mage 34. dat. nitude; and it is also given in pofition, because from the given point A it is drawn to the ftraight line ED given in pofition: Therefore the point D, in which the two ftraight lines AD, ED given in pofition cut one another, is given f: And the straight line DB which is at right angles to AB is given & in pofition, and AB is given in pofition, therefore the point B is given: And the points A, D are given, wherefore the ftraight lines AB, BD are given: And the ratio of AB to BC is given, and therefore BC is given. d 47. I. f 28. dat. g 33. dat. h 29. dat. i 2. dat. The Compofition. Let the given ratio of FG to GH be that which AB is required to have to BC, and let HK be the given ftraight line which is to be taken from BC, and let the ratio which the reD G H K AE BN M C F |
