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PROP. LXXIX. LXXX. LXXXI. These are added to Euclid's Data, as Propositions which are often useful in the solution of Problems.
PROP. LXXXII. This which is Prop. 60. in the Greek text is placed before the 83. and 84. which in the Greek are the 58. and 59. because the Demonstration of these two in this Edition is deduced from that of Prop. 82. from which they naturally follow.
PROP. LXXXVIII. XC. Dr. Gregory in his preface to Euclid's Works which he published at Oxford in 1703, after having told that he had supplied the defects of the Greek text of the Data in innumerable places from several Manuscripts, and corrected Cl. Hardy's translation by Mr. Bernard's, adds, that the 86. Theorem “ or Propofition," seemed to be remarkably vitiated, but which could not be restored by help of the Manuscripts; then he gives three different translations of it in Latin, according to which he thinks it may be read; the two first have no distinct meaning, and the third which he says is the best, tho' it contains a true Proposition which is the go. in this Edition, has no connexion in the least with the Greek text. and it is strange that Dr. Gregory did not observe, that if Prop. 86. was changed into this, the Demonstration of the 86. must be cancelled, and another put in its place. but, the truth is, both the Enuntiation and the Demonstration of Prop. 86. are quite entire and right, only Prop. 87. which is more simple, ought to have been placed before it; and the deficiency which the Doctor justly observes to be in this part of Euclid's Data, and which no doubt is owing to the carelessness and ignorance of the Greek Editors should have been supplied, not by changing Prop. 86. which is both entire and necessary, but by adding the two Propofitions which are the 88. and go. in this Edition.
PROP. XCVIII. C. These were communicated to me by two excellent Geometers, the first of them by the Right Honourable the Earl Stanhope, and the other by Dr. Matthew Stewart; to which I have added the Demonstrations.
Tho' the order of the Propositions has been in many places changed from that in former Editions, yet this will be of little disadvantage, as the antient Geometers never cite the Data, and the Moderns very rarely.
S that part of the Composition of a Problem which is its
Construction may not be so readily deduced from the Anàlysis by beginners; for their fake the following Example is given in which the derivation of the several parts of the Construction from the Analysis is particularly shewn, that they may be allisted to do the like in other Problems.
PROBLEM. Having given the magnitude of a parallelogram, the angle of which ABC is given, and also the excess of the square of its side BC above the square of the side AB; To find its fides and describe it.
The Analysis of this is the same with the Demonstration of the 87. Prop. of the Data. and the Construction that is given of the Problem at the end of that Proposition, is thus derived from the Analysis.
Let EFG be equal to the given angle ABC, and because in the Analysis it is said that the ratio of the rectangle AB, BC to the parallelogram AC is given by the 62. 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
в B PDC
F G L O HN AC by what is shewn 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 excess of the square of BC above the square of BA, to which excess the rectangle CB, BD is equal, is exhibited by the rectangle HG, GL. then in the Analysis the rectangle AB, BC is said 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 consequently the ratio of the rectangle CB, BD, that is of the rectangle HG, GL, to AB, BC, that
is of the straight line DB to BA, is the same with the ratio (of the rectangle GL, GH to FE, GH, that is) of the straight line GL to FE, which ratio of DB to BA is the next thing said to be given in the Analysis. from this it is plain that the square of FE is to the square of GL, as the square of BA which is equal to the rectangle BC, CD is to the square of BD, the ratio of which spaces is the next thing faid to be given. and from this it follows that four times the square of FE is to the square of GL, as four times the rectangle BC, CD is to the square of BD; and, by Compofition, four times the square of FE together with the square of GL is to the square of GL, as four times the rectangle BC, CD together with the square of BD, is to the square of BD, that is [8. 6.] as the square of the straight lines BC, CD taken together is to the square of BD, which ratio is the next thing said to be given in the Analysis. and because four times the square of FE and the square of GL are to be added together, therefore in the perpendicular EG there is taken KG equal to FE, and MG equal to the double of it, because thereby the squares of MG, GL, that is, joining ML, the square of ML is equal to four times the square of FE and to the square of GL. and because the square of ML is to the square of GL, as the square of the straight 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 Composition, ML and LG together, that is, producing GL to N, so that ML be equal to LN, the straight 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 said to be given in the Analyfis. and from this it follows, that the rectangle HG, GO is to HG, GL, as the square of BC is to the rectangle CB, BD which is equal to the rectangle HG, GL, and therefore the square of BC is equal to the rectangle HG, GO, and BC is consequently found by taking a mean proportional betwixt HG and Go, as is faid in the Construction, and because it was shewn 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 likewise shewn that LG is to FE, or GK, as DB to BA, and the three first are now found, and theteby 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 rest of the Composition contains the Demonstration.
S the Propofitions from the 13. to the 28. may be thought
by beginners to be less useful than the rest, because they cannot so readily see how they are to be made use of in the solution of Problems; on this account the two following Problems are added, to thew that they are equally useful with the other Propofitions, and from which it may easily be judged that many other Problems depend upon these Propositions.
b. I. 6.
10 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 rectangle con. tained by the first and third is given.
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 2. 24. Dat. EF is given ; therefore a the excess of the first AB 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
H B AB above it has a given ratio to EF; and consequently the rectangle BA, AH has a
S G D given ratio to the rectangle AB, EF, which
last rectangle is given by the Hypothesis ; E F c. 2. Dat. therefore the rectangle BA, AH is given,
and BH the excess of its fides is given; K NML O d. 85. Dat. wherefore the fides AB, AH are givend. and
because the ratios of AB to CD, and of Al to EF are given; CD and EF are given.
The Composition. Let the given ratio of KL to KM be that which AB is required 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 by a square, e. 29. 6. and let BA, AH be its fides. then is AB the first of the straight lines required to be found. and by making as LK to KM, fo 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 aequali, as AH to EF, so is LK to KN. and as the rectangle BA, AH to the rectangle BA, EF, so is 8 the g. 1.6. rectangle LK, KO to the rectangle KN, KO. and, by the Construction, the rectangle BA, AH is equal to LK, KO, therefore h h. 14. 5. the rectangle AB, EF is equal to the given rectangle NK, 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.
O 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 remainder to BD is given ; therefore a the excess of the first AB above a given 2. 24. Dat. 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 BD be placed at right angles to EB, and join DE, then the triangle EBD is given in fpecies; wherefore the angle BED b. 44. Dat. is given. let AE which is given in magnitude be given also in pofition, and the straight line ED will be given in position. join AD, C. 32. Dat, and because the sum of the squares of AB, BD, that is d, the square d. 47. 6. of AD is given, therefore the straight line AD is given in magnitude; and it is also given in position, because from the given e. 34. Dat. point A it is drawn to the straight line ED given in position. therefore the point D in which the two straight lines AD, ED given in